Reconstruction stabilizer and active vision

FIELD AND BACKGROUND OF THE INVENTION

0001The present invention relates to radioactive-emission measurements of a volume. More particularly, the present invention relates to the accurate reconstruction of the volume, based on measurements from non-uniform views of a volume, and on the dynamic selection of views during the data acquisition process. Of particular interest is view selection for medical imaging and/or in conjunction with medical instruments, such as guided minimally invasive surgical instruments.

0002Radionuclide imaging is one of the most important applications of radioactivity in medicine. Its purpose is to obtain a distribution image of a radioactively labeled substance, e.g., a radiopharmaceutical, within the body following administration thereof to a patient. Radioactive-emission imaging relies on the fact that in general, pathologies, such as malignant tumors, malfunctioning organs, and inflammations, display a level of activity different from that of healthy tissue. Thus, radiopharmaceuticals, which circulate in the blood stream, are picked up by the active pathologies to a different extent than by the surrounding healthy tissue; in consequence, the pathologies are operative as radioactive-emission sources and may be detected by radioactive-emission imaging. It will be appreciated that the pathology may appear as a concentrated source of high radiation, a hot region, as may be associated with a tumor, or as a region of low-level radiation, which is nonetheless above the background level, as may be associated with carcinoma.

0003A reversed situation is similarly possible. Dead tissue has practically no pick up of radiopharmaceuticals, and is thus operative as a cold region.

0004Thus radiopharmaceuticals may be used for identifying active pathologies as well as dead tissue.

0005In the discussion that follows, the term body structure is intended to include organs, portions of organs, a part of the body, and a whole body. The term organ target is intended to include pathological features within organs. These pathological features may be expressed, by radioactive-emission imaging, as any one of the following: <ol id="ol0001" compact="compact"><li>i. hot regions, of a radioactive emission intensity higher than the background level;</li><li>ii. regions of low-level radioactive emission intensity, which is nonetheless above the background level; and</li><li>iii. cold regions, of a radioactive emission intensity, lower than the background level.</li></ol>

0006Examples of radiopharmaceuticals include monoclonal antibodies or other agents, e.g., fibrinogen or fluorodeoxyglucose, tagged with a radioactive isotope, e.g., ^99Mtechnetium, ⁶⁷gallium, ²⁰¹thallium, ¹¹¹indium, ¹²³iodine, ¹²⁵iodine and ¹⁸fluorine, which may be administered orally or intravenously. The radiopharmaceuticals are designed to concentrate in the area of a tumor, and the uptake of such radiopharmaceuticals in the active part of a tumor, or other pathologies such as an inflammation, is higher and more rapid than in the tissue that neighbors the tumor. Thereafter, a radiation-emission-measuring-probe, which may be configured for extracorporeal or intracorporeal use, is employed for locating the position of the active area. Another application is the detection of blood clots with radiopharmaceuticals such as ACUTECT from Nycomed Amersham for the detection of newly formed thrombosis in veins, or clots in arteries of the heart or brain, in an emergency or operating room. Yet other applications include radioimaging of myocardial infarct using agents such as radioactive anti-myosin antibodies, radioimaging specific cell types using radioactively tagged molecules (also known as molecular imaging), etc.

0007The usual preferred emission for such applications is that of gamma rays, which emission is in the energy range of approximately 11-511 KeV. Beta radiation and positrons may also be detected.

0008Radioactive-emission imaging is performed with a radioactive-emission-measuring detector, such as a room temperature, solid-state CdZnTe (CZT) detector, which is among the more promising that are currently available. It may be configured as a single-pixel or a multi-pixel detector. Alternatively, another solid-state detector such as CdTe, HgI, Si, Ge, or the like, or a combination of a scintillation detector (such as NaI(Tl), LSO, GSO, CsI, CaF, or the like) and a photomultiplier, or another detector as known, may be used.

00091a - 1i schematically illustrate detecting units 102 and detecting blocks 101 of various geometries and constructions, and radioactive-emission-measuring probes associated with them.

0010<figref idref="f0001">Fig. 1a</figref> schematically illustrates a detecting unit 102, formed as a single-pixel detector 104, for example, a room-temperature solid-state CdZnTe (CZT) detector, having a diameter D and a thickness τ_d. Both the detector diameter D, or a diameter equivalent in the case of a non-circular detector, and the detector thickness τ_d affect the detecting efficiency. The detector diameter determines the surface area on which radioactive emission impinges; the greater the surface area, the greater the efficiency. The detector thickness affects the stopping power of the detector. High energy gamma rays may go through a thin detector, and the probability of their detection increases with detector thickness. By itself, a single-pixel detector cannot generate an image; rather, all counts are distributed over the surface area of the detector.

0011<figref idref="f0001">Fig. 1b</figref> schematically illustrates the detecting unit 102 with a collimator 108, formed as a single cell of a diameter D, a length L, and a septa thickness τ, attached to the detector 104. The collimator 108 may be, for example, of lead, tungsten or another material which substantially blocks gamma and beta rays.

0012The collimator's geometry, and specifically, the ratio of D/L, provides the detecting unit 102 with a collection angle δ analogous to a viewing angle of an optical camera. The collection angle δ limits the radioactive-emission detection to substantially only that radioactive emission, which impinges on the detector 104 after passing through a "corridor" of the collimator 108 (although in practice, some high-energy gamma rays may penetrate the collimator's walls).

0013<figref idref="f0001">Fig. 1c</figref> schematically illustrates a block 101 of the detecting units 102, with the collimator 108, formed as a multi-cell collimator, of a cell diameter D. The collection angle δ is defined for each of the detecting units 102 in the block, and each of the detecting units 102 forms a pixel in the block 101.

0014<figref idref="f0001">Fig. 1d</figref> schematically illustrates a radioactive-emission-measuring probe 100 which comprises several detecting units 102, of different geometries and different collection angles δ, within a housing 107.

0015<figref idref="f0002 f0003">Figs. 1e - 1i</figref> schematically illustrate the block 101, formed as a combination of a scintillation detector (such as NaI(Tl), LSO, GSO, CsI, CaF, or the like), a collimator grid, and photomultipliers.

0016As seen in <figref idref="f0002">Fig. 1e</figref>, the block 101, having proximal and distal ends 109 and 111, respectively, vis a vis an operator (not shown), is formed of the scintillation detector 104, of a single pixel, and the collimators 108, to create the detecting units 102. A plurality of photomultipliers 103 is associated with the single pixel scintillation detector 104, and with proper algorithms, as known, their output can provide a two dimensional image of the scintillations in the single pixel scintillation detector 104. In essence, this is an Anger camera, as known.

0017The distal view 111 of the collimator grid is seen in Fig. If.

0018Two optional proximal views 109 of the photomultipliers 103 are seen in <figref idref="f0002">Figs. 1g and 1h</figref>, as a square grid arrangement, and as an arrangement of tubes.

0019An Anger camera 117, of the block 101 in the housing 107 is seen in <figref idref="f0003">Fig. 1i</figref>.

0020In each of the cases of <figref idref="f0001 f0002 f0003">Figs. 1a - 1i</figref>, the geometry of the collimator 108 determines the collection angle δ, wherein with no collimator, the collection angle δ, is essentially a solid angle of 4π steradians. Thus, the collimator's geometry affects both the detection efficiency and the image resolution, which are defined as follows: <ol id="ol0002" compact="compact"><li>i. The detection efficiency is the ratio of measured radiation to emitted radiation; and</li><li>ii. The image resolution is the capability of making distinguishable closely adjacent radioactive-emission organ targets, or the capability to accurately determine the size and shape of individual radioactive-emission organ targets.</li></ol>

0021Naturally, it is desired to optimize both the detection efficiency and the image resolution. Yet, they are inversely related to each other. The detection efficiency increases with increasing collimator's collection angle, and the image resolution decreases with increasing collimator's collection angle. For example, when the ratio of D/L is 1/2, the collection angle δ is substantially 2.5 steradians, so the cell views incident radiation within the confinement of about a 2.5-steradian sector. However, when the ratio of D/L is 1/12, the collection angle δ is substantially 0.31 steradians, so the cell views incident radiation within the confinement of about a 0.31-steradian sector.

0022Once the emission data is obtained, the data is processed to reconstruct the intensity distribution within the measured volume. The reconstruction process is generally complex, due to the large quantity of data which must be processed in order to obtain an accurate reconstruction. The following statistical model may be used to perform reconstruction.

0023We assume an intensity distribution, <i>I</i>, defined over an input space <i>U</i>, where <i>U</i> comprises a set of basic elements (<i>e.g</i>., pixels in two dimensional spaces, voxels in three dimensional spaces), and <i>I</i>(<i>u</i>) is the intensity of a given basic element <i>u</i> ∈ <i>U.</i> A detecting unit positioned on a radiation-emission-measuring-probe takes a series of measurements <maths id="math0001"><math display="inline"><mi mathvariant="bold">y</mi><mo>=</mo><msubsup><mfenced><msub><mi>y</mi><mi>i</mi></msub></mfenced><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>T</mi></msubsup></math><img file="EP1908011B1_D0001.tif" /></maths> from different positions and orientations around the volume <i>U</i>. The geometrical and physical properties of the detecting unit, together with its position and orientation in a given measurement <i>i</i>, determine the detection probability <i>φ_i</i>(<i>u</i>) of a photon emitted from location <i>u</i>. Thus the effective intensity of location <i>u</i> as viewed by the detecting unit during measurement <i>i</i> is <i>φ_i(u)I(u)</i>.

0024The random count <i>X_i</i>(<i>u</i>) of photons that are emitted from location <i>u</i> and detected in measurement <i>i</i> is modeled by a Poisson process with mean φ<i>_i</i>(<i>u</i>)<i>I</i>(<i>u</i>). The total count of photons detected in measurement <i>i</i> is thus: <maths id="math0002" num="(1a)"><math display="block"><msub><mi>y</mi><mi>i</mi></msub><mo>∼</mo><mi>Poisson</mi><mfenced><msub><mi mathvariant="normal">Σ</mi><mrow><mi>u</mi><mo>∈</mo><mi>U</mi></mrow></msub><mo>⁢</mo><msub><mi mathvariant="italic">φ</mi><mi mathvariant="italic">i</mi></msub><mfenced><mi mathvariant="italic">u</mi></mfenced><mo>⁢</mo><mi mathvariant="italic">I</mi><mfenced><mi mathvariant="italic">u</mi></mfenced></mfenced></math><img file="EP1908011B1_D0002.tif" /></maths> or in matrix notation: <maths id="math0003" num="(1b)"><math display="block"><mi mathvariant="bold">y</mi><mo>=</mo><mi>Poisson</mi><mfenced><mi mathvariant="bold">ΦI</mi></mfenced></math><img file="EP1908011B1_D0003.tif" /></maths> where <b>y</b> is the vector of measurements <i>y_i</i>, Φ is a matrix of detection probabilities over measurements <i>i</i> and voxels <i>u</i>, and <b>I</b> is a vector of intensity per voxel <i>u</i>. The reconstruction problem is to reconstruct the intensities <b>I</b> from the measurements <b>y</b>.

0025The 2-D Radon transform is a mathematical relationship which may be used to reconstruct the emission intensities of volume <i>U</i> when the set of measurements <maths id="math0004"><math display="inline"><msubsup><mfenced><msub><mi>y</mi><mi>t</mi></msub></mfenced><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow><mi>T</mi></msubsup></math><img file="EP1908011B1_D0004.tif" /></maths> is unconstrained. The Radon transform is not statistical and does not take into account the Poissonian nature of the counts. In addition, it models the views as line projections. The Radon transform maps the spatial domain (x,y) to the Radon domain (p,φ). For a fixed projection angle, the Radon transform is simply a projection of the object. A technique known in the art as filtered back-projection (FBP) uses a back-projection operator and the inverse of the Radon transform to reconstruct the intensity distribution in volume <i>U</i> from measurements <maths id="math0005"><math display="inline"><msubsup><mfenced><msub><mi>y</mi><mi>t</mi></msub></mfenced><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow><mi>T</mi></msubsup><mn>.</mn></math><img file="EP1908011B1_D0005.tif" /></maths>

0026The basic, idealized problem solved by the FBP approach is to reconstruct an image from its Radon transform. The Radon transform, when properly defined, has a well-defined inverse. However, in order to invert the transform one needs measured data spanning 180°. In many medical imaging situations, the positioning of the detecting unit relative to the emitting object is constrained, so that complete measured data is not available. Reconstruction based on filtered back-projection is therefore of limited use for medical imaging. Maximum likelihood (ML) and Maximum A Posteriori (MAP) estimation methods, which address the statistical nature of the counts, have been found to provide better image reconstructions than FBP.

0027Limited-angle tomography is a reconstruction technique in the related art which reconstructs an image from projections acquired over a limited range of angular directions. The success of the reconstruction process depends upon the extent of the angular range acquired compared with the angular range of the missing projections. Any reconstruction from a limited range of projections potentially results in spatial distortions (artifacts) in the image. Limited angle techniques can be applied for both the Radon transform and the statistical models, but better results are generally achieved within the statistical framework. While it is known that the severity of the artifacts increases with the increasing angular range of the missing projections, limited-angle tomography does not provide information on which projections should be used in order to most effectively reconstruct the image.

0028Maximum likelihood (ML) estimation is a widely used method in the related art for reconstructing an image from a constrained set of measurements. A parameterization of the generative model described above is obtained by assigning an intensity <i>I</i>(u) to every voxel in <i>U</i>. The likelihood of the observed data y = (<i>y_t</i>)<i>_t</i>, given the set of parameters <b>I</b> = {<i>I</i> (<i>u</i>) : <i>u</i> ∈ <i>U</i>} is: <maths id="math0006" num="(2)"><math display="block"><mtable columnalign="left"><mtr><mtd><mi>L</mi><mfenced><mi>y</mi><mrow><mo>|</mo></mrow><mi mathvariant="bold">I</mi></mfenced><mo>=</mo><mi>lu</mi><mo>⁢</mo><mi>P</mi><mfenced><mi>y</mi><mrow><mo>|</mo></mrow><mi mathvariant="bold">I</mi></mfenced></mtd><mtd><mo>=</mo><mi>lu</mi><mstyle displaystyle="true"><munder><mo>∏</mo><mi>t</mi></munder></mstyle><mi>P</mi><mfenced><msub><mi>y</mi><mi>t</mi></msub><mrow><mo>|</mo></mrow><mi mathvariant="bold">I</mi></mfenced><mo>=</mo><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mi>t</mi></munder></mstyle><mi>lu</mi><mo>⁢</mo><mi>P</mi></mstyle><mfenced><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mi>u</mi></munder></mstyle><msub><mi>x</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced><mrow><mo>|</mo></mrow><mi mathvariant="bold">I</mi></mstyle></mfenced></mtd></mtr><mtr><mtd><mspace width="1em" /></mtd><mtd><mo>=</mo><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mi>t</mi></munder></mstyle><mi>ln Poission</mi></mstyle><mspace width="1em" /><mfenced><msub><mi>y</mi><mi>t</mi></msub><mrow><mo>|</mo></mrow><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mi>u</mi></munder></mstyle><msub><mi>φ</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced><mrow><mo>|</mo></mrow><mi mathvariant="italic">I</mi><mfenced><mi>u</mi></mfenced></mstyle></mfenced></mtd></mtr><mtr><mtd><mspace width="1em" /></mtd><mtd><mo>=</mo><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mi>t</mi></munder></mstyle></mstyle><mspace width="1em" /><mfenced open="{" close="}"><mo>-</mo><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mi>u</mi></munder></mstyle><msub><mi>φ</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced><mo>⁢</mo><mi mathvariant="italic">I</mi><mfenced><mi>u</mi></mfenced></mstyle><mo>+</mo><msub><mi>y</mi><mi>t</mi></msub><mspace width="1em" /><mi>ln</mi><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mi>u</mi></munder></mstyle><msub><mi>φ</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced><mo>⁢</mo><mi mathvariant="italic">I</mi><mfenced><mi>u</mi></mfenced></mstyle><mo>-</mo><mi>ln</mi><mfenced><msub><mi>y</mi><mi>t</mi></msub><mo>!</mo></mfenced></mfenced></mtd></mtr></mtable></math><img file="EP1908011B1_D0006.tif" /></maths> Note that the lower and upper bound of an indexing variable (such as voxels u and time index t) are omitted in the following description, when they are clear from the context.

0029There is currently no analytic way to solve Eqn. 2 for the maximum of the likelihood function. However, optimization methods that find local maxima of the likelihood are known. One such method is the Expectation-Maximization (EM) process. In EM estimation, there is no guarantee that the sequence converges to a maximum likelihood estimator. For multimodal distributions, this means that an EM algorithm will converge to a local maximum (or saddle point) of the observed data likelihood function, depending on starting values.

0030Since the data generated by the model is only partially observable by our measurements, a basic ingredient of the Expectation-Maximization formalism is to define a set of random variables that completely define the data generated by the model. In the current case, since <i>Y_t</i> = ∑_uX<i>_t</i>(<i>u</i>), the set of variables {X_u(<i>t</i>) : <i>u</i> ∈ <i>U</i>; <i>t</i> = <i>1</i>, ..., <i>T</i>} is such a set; the generated data is <b>x</b> = (<b>x</b><i>_t</i>)<i>_t</i>, where <b>x</b><i>_t</i> = (x_t(u))<i>_u</i>, and the observed data <b>y</b> is completely determined by <b>x</b>. The main tool in the EM formalism is the complete data likelihood: <maths id="math0007" num="(3)"><math display="block"><mtable columnalign="left"><mtr><mtd><mi>ln</mi><mo>⁢</mo><mi>P</mi><mfenced><mi mathvariant="normal">x</mi><mrow><mo>|</mo></mrow><mi>I</mi></mfenced></mtd><mtd><mo>=</mo><mi>ln</mi><mstyle displaystyle="true"><munder><mo>∏</mo><mi>t</mi></munder></mstyle><mi>P</mi><mfenced><msub><mi>x</mi><mi>t</mi></msub><mrow><mo>|</mo></mrow><mi>I</mi></mfenced><mo>=</mo><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mi>t</mi></munder></mstyle><mi>ln</mi><mstyle displaystyle="true"><munder><mo>∏</mo><mi>u</mi></munder></mstyle><mi>Poisson</mi><mfenced><msub><mi>x</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced><mrow><mo>|</mo></mrow><msub><mi>φ</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced><mo>⁢</mo><mi>I</mi><mfenced><mi>u</mi></mfenced></mfenced></mstyle></mtd></mtr><mtr><mtd><mspace width="1em" /></mtd><mtd><mo>=</mo><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mi>t</mi></munder></mstyle><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mi>u</mi></munder></mstyle><mfenced open="{" close="}"><mo>-</mo><msub><mi>φ</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced><mo>⁢</mo><mi>I</mi><mfenced><mi>u</mi></mfenced><mo>+</mo><msub><mi>x</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced><mo>⁢</mo><mi>ln</mi><mfenced><msub><mi>φ</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced><mo>⁢</mo><mi>I</mi><mfenced><mi>u</mi></mfenced></mfenced><mo>+</mo><mi>ln</mi><mfenced><msub><mi>x</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced><mo>!</mo></mfenced></mfenced></mstyle></mstyle></mtd></mtr></mtable></math><img file="EP1908011B1_D0007.tif" /></maths>

0031Since the likelihood depends on the complete data, which is only partially observable, we take its expectation with respect to the space of the unobserved data, given the current set of hypothesized parameters (i.e. the current estimator). The result is a function <i>Q</i>(<b>I</b>|<b>I</b>') which assigns likelihood to sets <b>I</b> of model parameters, given the current set <b>I</b>', and given the observed data <b>y</b>: <maths id="math0008" num="(4)"><math display="block"><mtable columnalign="left"><mtr><mtd><mi>Q</mi><mfenced><mi mathvariant="bold">I</mi><mrow><mo>|</mo></mrow><mi mathvariant="bold">Iʹ</mi></mfenced></mtd><mtd><mo>=</mo><mi>E</mi><mfenced open="[" close="]"><mi>ln</mi><mo>⁢</mo><mi>P</mi><mfenced><mi mathvariant="normal">x</mi><mrow><mo>|</mo></mrow><mi mathvariant="bold">I</mi></mfenced><mrow><mo>|</mo></mrow><mi mathvariant="normal">y</mi><mo>;</mo><mi mathvariant="bold">Iʹ</mi></mfenced></mtd></mtr><mtr><mtd><mspace width="1em" /></mtd><mtd><mo>=</mo><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mi>t</mi></munder></mstyle><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mi>u</mi></munder></mstyle><mfenced open="{" close="}"><mo>-</mo><msub><mi>φ</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced><mo>⁢</mo><mi>I</mi><mfenced><mi>u</mi></mfenced><mo>+</mo><mi>E</mi><mfenced open="[" close="]"><msub><mi>x</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced><mrow><mo>|</mo></mrow><msub><mi>y</mi><mi>t</mi></msub><mo>;</mo><mi mathvariant="bold">Iʹ</mi></mfenced><mo>⁢</mo><mi>ln</mi><mfenced><msub><mi>φ</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced><mo>⁢</mo><mi>I</mi><mfenced><mi>u</mi></mfenced></mfenced><mo>+</mo><mi>C</mi></mfenced></mstyle></mstyle></mtd></mtr></mtable></math><img file="EP1908011B1_D0008.tif" /></maths> where <i>C</i> is a term which is independent of the intensities <b>I</b>. The function <i>Q(I</i>|<i>I')</i> is maximized by the following new estimates: <maths id="math0009" num="(5)"><math display="block"><mi>I</mi><mfenced><mi>u</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mstyle displaystyle="false"><mstyle displaystyle="false"><munder><mo>∑</mo><mi>t</mi></munder></mstyle><msub><mi>φ</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced></mstyle></mfrac><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mi>t</mi></munder></mstyle><mi>E</mi><mfenced open="[" close="]"><msub><mi>x</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced><mrow><mo>|</mo></mrow><msub><mi>y</mi><mi>t</mi></msub><mo>;</mo><mi mathvariant="bold">Iʹ</mi></mfenced></mstyle><mo>;</mo><mo>∀</mo><mi>u</mi><mo>∈</mo><mi>U</mi><mn>.</mn></math><img file="EP1908011B1_D0009.tif" /></maths>

0032The expectation in Eqn. 5 is obtained as follows: <maths id="math0010" num="(6)"><math display="block"><mtable columnalign="left"><mtr><mtd><msub><mi>P</mi><mrow><msub><mi>X</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced></mrow></msub><mfenced><msub><mi>x</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced><mrow><mo>|</mo></mrow><msub><mi>y</mi><mi>t</mi></msub><mo>;</mo><mi mathvariant="bold">Iʹ</mi></mfenced><mo>=</mo><mfrac><mrow><msub><mi>P</mi><msub><mi>Y</mi><mi>t</mi></msub></msub><mfenced><msub><mi>y</mi><mi>t</mi></msub><mrow><mo>|</mo></mrow><msub><mi>x</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced><mo>;</mo><mi mathvariant="bold">Iʹ</mi></mfenced><mo>⁢</mo><msub><mi>P</mi><mrow><msub><mi>X</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced></mrow></msub><mfenced><msub><mi>x</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced><mrow><mo>|</mo></mrow><mi mathvariant="bold">Iʹ</mi></mfenced></mrow><mrow><msub><mi>P</mi><msub><mi>Y</mi><mi>t</mi></msub></msub><mfenced><msub><mi>y</mi><mi>t</mi></msub><mrow><mo>|</mo></mrow><mi mathvariant="bold">Iʹ</mi></mfenced></mrow></mfrac></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mrow><mi>Poisson</mi><mo>⁢</mo><mfenced><msub><mi>y</mi><mi>t</mi></msub><mo>-</mo><msub><mi>x</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced><mrow><mo>|</mo></mrow><mstyle displaystyle="false"><munder><mo>∑</mo><mrow><mi>v</mi><mo>≠</mo><mi>u</mi></mrow></munder><mo>⁢</mo><msub><mi>φ</mi><mi>t</mi></msub><mfenced><mi>v</mi></mfenced><mo>⁢</mo><mi mathvariant="italic">Iʹ</mi><mfenced><mi>v</mi></mfenced></mstyle></mfenced><mo>⁢</mo><mi>Poisson</mi><mfenced><msub><mi>x</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced><mrow><mo>|</mo></mrow><msub><mi>φ</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced><mo>⁢</mo><mi mathvariant="italic">Iʹ</mi><mfenced><mi>u</mi></mfenced></mfenced></mrow><mrow><mi>Poisson</mi><mfenced><msub><mi>y</mi><mi>t</mi></msub><mrow><mo>|</mo></mrow><mstyle displaystyle="false"><munder><mo>∑</mo><mi>v</mi></munder><mo>⁢</mo><msub><mi>φ</mi><mi>t</mi></msub><mfenced><mi>v</mi></mfenced><mo>⁢</mo><mi mathvariant="italic">I</mi><mfenced><mi>v</mi></mfenced></mstyle></mfenced></mrow></mfrac></mtd></mtr><mtr><mtd><mo>=</mo><mi>Binomial</mi><mfenced><msub><mi>x</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced><mrow><mo>|</mo></mrow><mfrac><mrow><msub><mi>φ</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced><mo>⁢</mo><mi mathvariant="italic">Iʹ</mi><mfenced><mi>u</mi></mfenced></mrow><mstyle displaystyle="false"><munder><mo>∑</mo><mi>v</mi></munder><msub><mi>φ</mi><mi>t</mi></msub><mfenced><mi>v</mi></mfenced><mi mathvariant="italic">Iʹ</mi><mfenced><mi>v</mi></mfenced></mstyle></mfrac><mo>;</mo><msub><mi>y</mi><mi>t</mi></msub></mfenced></mtd></mtr></mtable></math><img file="EP1908011B1_D0010.tif" /></maths>

0033It follows that <maths id="math0011"><math display="inline"><mstyle mathvariant="normal"><mi>E</mi></mstyle><mfenced open="[" close="]"><msub><mi mathvariant="italic">x</mi><mi mathvariant="italic">t</mi></msub><mfenced><mi mathvariant="italic">u</mi></mfenced><mrow><mo>|</mo></mrow><msub><mi>y</mi><mi>t</mi></msub><mo>;</mo><mi mathvariant="italic">Iʹ</mi></mfenced><mo>=</mo><msub><mi>y</mi><mi>t</mi></msub><mo>⁢</mo><mfrac><mrow><msub><mi>φ</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced><mo>⁢</mo><mi mathvariant="italic">Iʹ</mi><mfenced><mi>u</mi></mfenced></mrow><mrow><msub><mi mathvariant="normal">Σ</mi><mi>v</mi></msub><mo>⁢</mo><msub><mi>φ</mi><mi>t</mi></msub><mfenced><mi>v</mi></mfenced><mo>⁢</mo><mi mathvariant="italic">Iʹ</mi><mfenced><mi>v</mi></mfenced></mrow></mfrac><mo>,</mo></math><img file="EP1908011B1_D0011.tif" /></maths> and hence the EM iteration is: <maths id="math0012" num="(7)"><math display="block"><mi>I</mi><mfenced><mi>u</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mstyle displaystyle="false"><mstyle displaystyle="false"><munder><mo>∑</mo><mi>t</mi></munder></mstyle><msub><mi>φ</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced></mstyle></mfrac><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mi>t</mi></munder></mstyle><msub><mi>y</mi><mi>t</mi></msub></mstyle><mfrac><mrow><msub><mi>φ</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced><mo>⁢</mo><mi mathvariant="italic">Iʹ</mi><mfenced><mi>u</mi></mfenced></mrow><mstyle displaystyle="false"><mstyle displaystyle="false"><munder><mo>∑</mo><mi>v</mi></munder></mstyle><msub><mi>φ</mi><mi>t</mi></msub><mfenced><mi>v</mi></mfenced><mi mathvariant="italic">Iʹ</mi><mfenced><mi>v</mi></mfenced></mstyle></mfrac></math><img file="EP1908011B1_D0012.tif" /></maths>

0034It is provable that each EM iteration improves the likelihood. Thus, given a random starting estimator, the EM algorithm iterates the above improvement step until it converges to a local maximum of the likelihood. Several random starts increase the chance of finding a globally good estimator.

0035It is usually desired to maximize the expected posterior probability (given a proper prior) rather than the expected likelihood. In that case we assume a prior probability on the intensities <i>P</i>(<b>I</b>) = Π<i>_uP</i>(<i>I</i>(<i>u</i>)). A proper conjugate prior for the Poisson distribution is the Gamma distribution: <maths id="math0013" num="(8)"><math display="block"><mi>P</mi><mfenced><mi>I</mi><mfenced><mi>u</mi></mfenced></mfenced><mo>=</mo><mi>Gamma</mi><mfenced><mi>I</mi><mfenced><mi>u</mi></mfenced><mrow><mo>|</mo></mrow><msub><mi>α</mi><mi>u</mi></msub><mo>;</mo><msub><mi>β</mi><mi>u</mi></msub></mfenced><mo>=</mo><mfrac><msubsup><mi>β</mi><mi>u</mi><mrow><msub><mi>α</mi><mi>u</mi></msub><mo>+</mo><mn>1</mn></mrow></msubsup><mrow><mi mathvariant="normal">Γ</mi><mo>⁢</mo><mfenced><msub><mi>α</mi><mi>u</mi></msub><mo>+</mo><mn>1</mn></mfenced></mrow></mfrac><mo>⁢</mo><mi>I</mi><mo>⁢</mo><msup><mfenced><mi>u</mi></mfenced><msub><mi>α</mi><mi>u</mi></msub></msup><mo>⁢</mo><msup><mi>e</mi><mrow><mo>-</mo><msub><mi>β</mi><mi>u</mi></msub><mo>⁢</mo><mi>I</mi><mfenced><mi>u</mi></mfenced></mrow></msup></math><img file="EP1908011B1_D0013.tif" /></maths>

0036Now the maximization is done on <i>Q</i>(<b>I</b>|<b>I</b>'<i>) = E</i>[<i>lnP(</i><b>x</b>|<i>I)p(I)</i> |<b>y</b>; <i>I</i>']. Plugging the Gamma prior into <i>Q</i>, and solving for <i>I(u),</i> we get the following EM iteration for the maximum posterior estimation: <maths id="math0014" num="(9)"><math display="block"><mi>I</mi><mfenced><mi>u</mi></mfenced><mo>=</mo><mfrac><mrow><msub><mi>α</mi><mi>u</mi></msub><mo>+</mo><mstyle displaystyle="false"><munder><mo>∑</mo><mi>t</mi></munder><mo>⁢</mo><mi>E</mi><mfenced open="[" close="]"><msub><mi>x</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced><mrow><mo>|</mo></mrow><msub><mi>y</mi><mi>t</mi></msub><mo>;</mo><mi mathvariant="bold">Iʹ</mi></mfenced></mstyle></mrow><mrow><msub><mi>β</mi><mi>u</mi></msub><mo>+</mo><mstyle displaystyle="false"><munder><mo>∑</mo><mspace width="1em" /></munder><mo>⁢</mo><msub><mi>φ</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced></mstyle></mrow></mfrac></math><img file="EP1908011B1_D0014.tif" /></maths><maths id="math0015" num="(9)"><math display="block"><mo>=</mo><mfrac><mn>1</mn><mrow><msub><mi>β</mi><mi>u</mi></msub><mo>+</mo><mstyle displaystyle="false"><mstyle displaystyle="false"><munder><mo>∑</mo><mi>t</mi></munder></mstyle><msub><mi>φ</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced></mstyle></mrow></mfrac><mo>⁢</mo><mfenced open="[" close="]"><msub><mi>α</mi><mi>u</mi></msub><mo>+</mo><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mi>t</mi></munder></mstyle><msub><mi>y</mi><mi>t</mi></msub><mo>⁢</mo><mfrac><mrow><msub><mi>φ</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced><mo>⁢</mo><mi mathvariant="italic">Iʹ</mi><mfenced><mi>u</mi></mfenced></mrow><mstyle displaystyle="false"><mstyle displaystyle="false"><munder><mo>∑</mo><mi>v</mi></munder></mstyle><msub><mi>φ</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced><mi mathvariant="italic">Iʹ</mi><mfenced><mi>u</mi></mfenced></mstyle></mfrac></mstyle></mfenced></math><img file="EP1908011B1_D0015.tif" /></maths>

0037The EM update step can be formulated in matrix notation as follows. Let Φ be the matrix of the projections [<i>φ_t</i>(<i>u</i>)]<i>_t,u,</i> and let <b>I</b>, <b>I'</b>,<b>y</b>,α and β be represented as column vectors. Eqn. 10 can be written in vector and matrix notations as: <maths id="math0016" num="(11)"><math display="block"><mi mathvariant="normal">I</mi><mo>=</mo><mfrac><mrow><mi>α</mi><mo>+</mo><mi>Iʹ</mi><mo>⋅</mo><mfenced><msup><mi mathvariant="normal">Φ</mi><mi>T</mi></msup><mo>⁢</mo><mfrac><mi>v</mi><mi mathvariant="normal">ΦIʹ</mi></mfrac></mfenced></mrow><mrow><mi>β</mi><mo>+</mo><msup><mi mathvariant="normal">Φ</mi><mi>T</mi></msup><mo>⁢</mo><mn>1</mn></mrow></mfrac></math><img file="EP1908011B1_D0016.tif" /></maths> where the explicit multiplication and division denote element-wise operations, and where 1 is a vector (of the appropriate length) consisting solely of 1's.

0038Limited computational resources (<i>i.e</i>., when the entire projection matrix Φ cannot be kept in memory) may require breaking the update computation according to a partition of Φ into a set of sub-matrices (Φ<i>_i</i>). In that case the intensities can be updated gradually (using only one sub-matrix at each step) according to the following computation: <maths id="math0017" num="(12)"><math display="block"><mi mathvariant="normal">I</mi><mo>=</mo><mfrac><mrow><mi>α</mi><mo>+</mo><mi>Iʹ</mi><mo>⋅</mo><mstyle displaystyle="false"><munder><mo>∑</mo><mi>i</mi></munder><mo>⁢</mo><msubsup><mi mathvariant="normal">Φ</mi><mi>i</mi><mi>T</mi></msubsup><mo>⁢</mo><mfrac><msub><mi>y</mi><mi>i</mi></msub><mrow><msub><mi mathvariant="normal">Φ</mi><mi>i</mi></msub><mo>⁢</mo><mi mathvariant="normal">Iʹ</mi></mrow></mfrac></mstyle></mrow><mrow><mi>β</mi><mo>+</mo><munder><mo>∑</mo><mi>i</mi></munder><mo>⁢</mo><msubsup><mi mathvariant="normal">Φ</mi><mi>i</mi><mi>T</mi></msubsup><mo>⁢</mo><mn>1</mn></mrow></mfrac></math><img file="EP1908011B1_D0017.tif" /></maths> where <b>y</b><i>_i</i> is the vector of observations that are obtained using the views of Φ<i>_i</i>.

0039In the context of image reconstruction from a constrained set of views, the utility of the reconstruction algorithms described above is limited. Many estimation algorithms, including EM and the Radon transform, require measurements from a complete set of views surrounding the imaged object. Although algorithms for EM with missing views have been developed, these algorithms are based on equally spaced views surrounding the imaged object, of which a number of views are not available. These algorithms do not provide a generalized solution for an unconstrained set of views, which may not be equally spaced or available for all directions surrounding the object.

0040Singular value decomposition (SVD) is a known technique for factorizing a rectangular real or complex matrix, with applications in signal processing and statistics. SVD may be considered a generalization of Eigenvalue decomposition to m*n matrices, whereas Eigenvalue decomposition is applicable only to square matrices.

0041SVD states that given the m-by-n matrix M whose entries are either from the field of real numbers or the field of complex numbers, there exists a factorization of the form: <maths id="math0018" num="(13)"><math display="block"><mi mathvariant="bold">M</mi><mo mathvariant="bold">=</mo><msup><mi mathvariant="bold">UDV</mi><mi mathvariant="bold">T</mi></msup></math><img file="EP1908011B1_D0018.tif" /></maths> where <b>U</b> is an m-by-m unitary matrix, <b>D</b> is m-by-n with nonnegative numbers on the diagonal and zeros off the diagonal, and <b>V^T</b>, the conjugate transpose of <b>V</b>, is an n-by-n unitary matrix. The elements along the diagonal of <b>D</b> are denoted the singular values. Such a factorization is called a singular-value decomposition of <b>M</b>. A common convention is to order the singular values D_i,i in non-increasing fashion, so that the diagonal matrix <b>D</b> is uniquely determined by <b>M.</b>

0042The condition number of a matrix is defined as the ratio of the matrix's largest singular value to its smallest singular value. In numerical analysis, the condition number associated with a problem is a measure of that problem's amenability to digital computation. A problem with a low condition number is said to be well-conditioned, while a problem with a high condition number is said to be ill-conditioned. For example, the condition number associated with the linear equation <b>x</b> = <b>My</b> gives a bound on how accurate the solution y will be after approximate solution.

0043SVD may be employed for the solution of linear inverse problems. The inverse problem for a set of linear equations, <b>x</b> = <b>My,</b> is to calculate vector <b>y</b> from a known <b>x</b> and <b>M.</b> Using SVD, with <b>M=UDV*,</b> the problem may be restated as <b>x</b>= <b>UDV*</b>y. If <b>D</b> is an invertible matrix, y is obtained as: <maths id="math0019" num="(14)"><math display="block"><mi mathvariant="bold">y</mi><mo mathvariant="bold">=</mo><msup><mi mathvariant="bold">VD</mi><mrow><mo mathvariant="bold">-</mo><mn mathvariant="bold">1</mn></mrow></msup><mo>⁢</mo><msup><mi mathvariant="bold">U</mi><mi mathvariant="bold">T</mi></msup><mo>⁢</mo><mi mathvariant="bold">x</mi></math><img file="EP1908011B1_D0019.tif" /></maths> where <b>U</b> and <b>V^t</b> are unitary matrices, and <b>D</b> is a diagonal matrix containing the singular values of <b>M.</b> Since <b>U</b> and <b>V^t</b> are easily transposable, a solution to Eqn. 13 is obtainable only if <b>D</b> is invertible. As <b>D</b> is a diagonal matrix of singular values e_l to e_n (where e_k denotes the k-th singular value D_k,k), <b>D^-1</b> is a diagonal matrix of the reciprocals of the singular values. That is: <maths id="math0020" num="(15)"><math display="block"><msup><mi mathvariant="normal">D</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msubsup><mi>e</mi><mn>1</mn><mrow><mo>-</mo><mn>1</mn></mrow></msubsup></mtd><mtd><mspace width="1em" /></mtd><mtd><mspace width="1em" /></mtd><mtd><mspace width="1em" /></mtd></mtr><mtr><mtd><mspace width="1em" /></mtd><mtd><msubsup><mi>e</mi><mn>2</mn><mrow><mo>-</mo><mn>1</mn></mrow></msubsup></mtd><mtd><mspace width="1em" /></mtd><mtd><mspace width="1em" /></mtd></mtr><mtr><mtd><mspace width="1em" /></mtd><mtd><mspace width="1em" /></mtd><mtd><mo>⋯</mo></mtd><mtd><mspace width="1em" /></mtd></mtr><mtr><mtd><mspace width="1em" /></mtd><mtd><mspace width="1em" /></mtd><mtd><mspace width="1em" /></mtd><mtd><msubsup><mi>e</mi><mi>n</mi><mrow><mo>-</mo><mn>1</mn></mrow></msubsup></mtd></mtr></mtable></mfenced></math><img file="EP1908011B1_D0020.tif" /></maths>

0044If the dimension of <b>y</b> is larger than the dimension of <b>x</b>, <b>y</b> cannot be determined directly but rather can only be estimated, which is often performed by an iterative procedure. If the condition number of <b>D</b> is very large, the multiplicative factors e_i^-1 will vary greatly, thus multiplying any measurement or calculation errors based on individual or linear combinations of elements of <b>y</b>, and decreasing the likelihood of convergence of the iterative estimation procedure.

0045Truncated SVD is a known technique for reducing sensitivity to inaccuracies or noise when solving a set of linear equations. In truncated SVD, the constraints associated with the smaller singular values are eliminated from the estimation. In terms of the inverse problem of Eqn. 14, this is accomplished for the i-th singular value by setting e_i^-1 equal to zero in matrix <b>D</b>^-1. Thus the lower valued singular values no longer affect the estimation process.

0046In order to achieve a reconstructed image which is adequate for medical diagnostic and treatment purposes, a reliable, high-resolution image of the tested object (i. e. body structure) must be obtained. Currently, reliable reconstruction algorithms are available only for complete data sets, which provide coverage of the entire volume. Such data is generally not available during medical imaging. Additionally, when high-resolution detecting units are used, their efficiency is relatively low, and the detecting units must remain at each position for a relatively long time in order to achieve a high probability of detection. Since during medical testing, measurements are generally performed at many locations as the detecting unit is moved relative to the observed body structure, the testing procedure generally requires a long time and is physically and emotionally difficult for the patient.

0047Additionally, reconstruction is based upon a large quantity of data, and is a lengthy and computationally complex process. A method and an apparatus according to the prior art is known from <patcit id="pcit0001" dnum="EP887661A"><text>EP 887 661</text></patcit>.

0048There is thus a widely recognized need for, and it would be highly advantageous to have, an apparatus, system and method devoid of the above limitations.

SUMMARY OF THE INVENTION

0049According to an aspect of the present invention there is provided a method for stabilizing the reconstruction of an imaged volume as defined in appended claim 1.

0050According to an aspect of the present invention there is provided a reconstruction stabilizer, for improving the reliability ofreconstruction of an imaged volume, as defined in appended claim 18.

BRIER DESCRIPTION OF THE DRAWINGS

0051The invention is herein described, by way of example only, with reference to the accompanying drawings. With specific reference now to the drawings in detail, it is stressed that the particulars shown are by way of example and for purposes of illustrative discussion of the preferred embodiments of the present invention only, and are presented in the cause of providing what is believed to be the most useful and readily understood description of the principles and conceptual aspects of the invention. In this regard, no attempt is made to show structural details of the invention in more detail than is necessary for a fundamental understanding of the invention, the description taken with the drawings making apparent to those skilled in the art how the several forms of the invention may be embodied in practice.

0052In the drawings: <ul id="ul0001" list-style="none" compact="compact"><li><figref idref="f0001 f0002 f0003">Figs. 1a-1i</figref> show detecting units and blocks of various geometries and constructions and radioactive-emission-measuring probes, associated with them.</li><li><figref idref="f0004">Figs. 1j and 1k</figref> pictorially illustrate a view and viewing parameters associated with it, in accordance with definitions of the present invention.</li><li><figref idref="f0005">Fig. 2a</figref> is a simplified flowchart of a method for stabilizing the reconstruction of an imaged volume, according to a preferred embodiment of the present invention.</li><li><figref idref="f0006">Fig. 2b</figref> is a simplified flowchart of a method for analyzing a detection probability matrix, according to a preferred embodiment of the present invention.</li><li><figref idref="f0007">Fig. 2c</figref> is a simplified block diagram of a reconstruction stabilizer, according to a preferred embodiment of the present invention.</li><li><figref idref="f0008">Fig. 2d</figref> is a simplified flowchart of a method of performing radioactive-emission measurements of a body structure, according to a preferred embodiment of the present invention.</li><li><figref idref="f0009">Fig. 3</figref> shows an object shaped as a cylinder with a front protrusion, and having a high-emittance portion (hotspot).</li><li><figref idref="f0010">Fig. 4a</figref> illustrates an object having two high-emission regions of interest.</li><li><figref idref="f0010">Fig. 4b</figref> illustrates the added information provided by each of views V_A to V_F.</li><li><figref idref="f0011">Figs. 5a</figref> and <figref idref="f0012">5b</figref> are simplified flowcharts of iterative methods of performing radioactive-emission measurements of a body structure, according to a first and a second preferred embodiment of the present invention.</li><li><figref idref="f0013 f0014">Figs. 6a and b</figref> are simplified flowcharts of methods for dynamically defining further views, according to a first and a second preferred embodiment of the present invention.</li><li><figref idref="f0015">Figs. 7a and 7b</figref> present the principles of modeling, for obtaining an optimal set of views, in accordance with the present invention;</li><li><figref idref="f0016">Fig. 8</figref> shows four models of a prostate emittance model.</li><li><figref idref="f0017 f0018 f0019">Figs. 9a-9f</figref> show emittance models of a given volume to illustrate view selection using the separability criterion.</li><li><figref idref="f0020 f0021 f0022">Figs. 9g-9i</figref> show emittance models of a given volume to illustrate view selection using a weighted-combination criterion.</li><li><figref idref="f0023">Fig. 10</figref> is a simplified flowchart of an iterative method for selecting further views, according to a preferred embodiment of the present invention.</li><li><figref idref="f0024">Fig. 11</figref> is a simplified flowchart of a single iteration of a view selection method, according to a preferred embodiment of the present invention.</li><li><figref idref="f0025">Fig. 12</figref> is a simplified flowchart of a method for dynamically defining further views, according to a third preferred embodiment of the present invention.</li><li><figref idref="f0026">Fig. 13</figref> is a simplified block diagram of measurement unit for performing radioactive-emission measurements of a body structure, according to a preferred embodiment of the present invention.</li><li><figref idref="f0027">Fig. 14</figref> is a simplified flowchart of a method for selecting an optimal initial predetermined set of views of a volume to be imaged, according to a preferred embodiment of the present invention.</li><li><figref idref="f0028">Fig. 15</figref> shows an example of a volume with two views.</li><li><figref idref="f0029">Fig. 16</figref> illustrates the concept of uniform coverage of a volume.</li><li><figref idref="f0030">Fig. 17</figref> is a simplified flowchart of method for stabilizing the reconstruction of an imaged volume with dynamic view definition, according to a preferred embodiment of the present invention.</li></ul>

DESCRIPTION OF THE PREFERRED EMBODIMENTS

0053The present embodiments teach providing modifications of the reconstruction and/or imaging processes in order to obtain a reliable reconstruction of the imaged volume. Specifically, the methods teach analyzing the reliability of the reconstruction obtainable from collected emission data over a set of views to determine modifications which are expected to improve reliability. The present embodiments further teach using radioactive-emission measurements to define views for further radioactive-emission measurements of a body structure, to be performed during the current measurement process.

0054With non-uniform scanning, the amount of information available for different voxels is not uniform, thereby constraining the ability to obtain an accurate reconstruction of the volume. In addition, there may be a need to focus on a region or regions of interest, and to control reconstruction resolution and accuracy where and when necessary. In such cases it is important to provide features and algorithmic components which handle or compensate for the lack of uniformity of information, and are capable of focusing on features or regions of interest, while maintaining short and efficient data acquisition.

0055In the context of medical imaging the imaged volume corresponds to a body structure, which may include a whole body, portion of a body, target organ and so forth. Those non-limiting embodiments presented below which are directed at imaging a body structure are to be understood as applying to any imaged volume.

0056Before explaining at least one embodiment of the invention in detail, it is to be understood that the invention is not limited in its application to the details of construction and the arrangement of the components set forth in the following description or illustrated in the drawings. The invention is capable of other embodiments or of being practiced or carried out in various ways. Also, it is to be understood that the phraseology and terminology employed herein is for the purpose of description and should not be regarded as limiting.

0057Referring now to the drawings, <figref idref="f0004">Figs. 1j and 1k</figref> pictorially illustrate a view and viewing parameters associated with it, in accordance with definitions of the present invention.

0058Seen in <figref idref="f0004">Fig. 1j</figref> is a volume U, subdivided into voxels u. The volume U is defined in a six-degree coordinate system (x;y;z;ω;θ;σ) and has a point of origin P0(x0; y0; z0; ω0; θ0; σ0). A detecting unit 102 is positioned at a location and orientation P1(x1; y1; z1; ω1; θ1; σ1). The detecting unit 102 has a detector 104 of a specific detector material of a thickness τ_d, and a collimator 108 of a diameter D and a length L, so as to define a collection angle δ. The location, orientation, and collection angle parameters determine a three-dimensional sector S, which is the portion of the volume U that is within the detector unit's field of view. The probability of detection of a given voxel outside of sector S is negligible, since a photon emitted from the given voxel outside of sector S will have a very low probability of reaching detector 104 through collimator 108.

0059<figref idref="f0004">Fig. 1k</figref> schematically illustrates the emission rate of the volume U, as a function of time, given that a radioactive material of a specific half-life has been administered at a time T0.

0060A view may thus be defined as a group of nonzero probabilities of detecting a radioactive emission associated with all the voxels that form sector S (<figref idref="f0004">Fig. 1j</figref>).

0061A view is sometimes referred to as a projection, and the two terms are synonymous. Furthermore, a view defined over the sector S can be naturally extended to be defined over the group of all voxels in the volume U, by simply associating a zero probability with every vovel outside the sector S. This enables applying mathematical operations over the entire volume U.

0062The viewing parameters, which are the factors affecting the detection of radioactive emissions, are as follows:

i. <u>location and orientation parameters:</u>

0063a location and an orientation in a six-dimensional space, P1(x1; y1; z1; ω1; θ1; σ1), with respect to the origin P0(x0; y0; z0; ω0; θ0, σ0) of the volume U, in which the detecting unit 102 is positioned;

ii. <u>detector-unit parameters:</u>

0064<ul id="ul0002" list-style="bullet" compact="compact"><li>the detector unit geometry (e.g. length L, diameter D, and/or collection angle δ), which together with the location and orientation parameters, P1(x1; y1; z1; ω1; θ1; σ1) with respect to the origin P0(x0; y0; z0; ω0; θ0; σ0) define the sector S;</li><li>the septa thickness τ, which affects the probability that a photon that enters the collimator will reach the detector as well as crosstalk effects (which occur when a photon which entered a neighboring cell penetrates the collimator and reaches the detector), hence, the detector efficiency</li><li>the detector material, which affects the detector efficiency; and</li><li>the detector thickness τ_d, which affects the detector's stopping power, hence, its efficiency;</li></ul>

iii. <u>attenuation parameters:</u>

0065attenuation properties of all the voxels within the sector S, as they affect the probabilities that radioactive emissions from a specific voxel within the sector S will reach the detector, wherein different voxels within the sector S may have different attenuation properties, since several types of tissue may be involved;

iv. <u>time parameters:</u>

0066since the radiopharmaceutical decays with a specific half-life, the time t₁ since administration, and the duration of the measurement Δt₁, affect the number of emissions that occur during the radioactive-emission measurement.

v. <u>radiopharmaceutical parameters:</u>

0067The half-life t_1/2, of the radiopharmaceutical, the types of radioactive emission, whether gamma or beta, and the energies of the radioactive emission affect the probability of detection.

0068Some of these viewing parameters are fixed for a particular situation. Specifically, the tissue attenuation parameters are given. Additionally, the time t₁ since administration of the radiopharmaceutical is generally governed by the blood pool radioactivity, since it is generally necessary to wait until the blood pool radioactivity dies out for low-level detection to be possible. For the remaining viewing parameters, optimization may be carried out.

0069To recapitulate the problem described above, an intensity distribution I, in terms of radioactive emissions per seconds, is defined over the volume U, forming our input space. Volume U comprises a set of basic elements u (e.g., pixels in two dimensional spaces, voxels in three dimensional spaces), and I(u) is the intensity in a given basic element u ∈ U. A view (also denoted a projection) ϕ∈Φ is defined by the set of probabilities {ϕ(u):u∈U}, where ϕ(u) is the probability of detecting a radioactive emission from a voxel u, as defined by viewing parameters, such as the physical and geometrical properties of the detecting unit, as well as the attenuation parameters of the viewed volume U, and the time parameters. A measurement is obtained by choosing a view ϕ∈Φ, and then sampling according to the viewing parameters.

0070As shown in Eqn. 1b, the relationship between the measurements, y, and the emission intensities over the body structure may be represented as <b>y</b> = Poisson(Φ<b>I</b>). The reconstruction problem we are faced with is to calculate the intensity vector <b>I</b> from the measurement vector <b>y</b>, given a known probability matrix Φ.

0071Applying SVD analysis to the probability matrix Φ, we obtain: <maths id="math0021" num="(16)"><math display="block"><mi mathvariant="normal">Φ</mi><mo mathvariant="normal">=</mo><mi>UDVt</mi><mo mathvariant="normal">=</mo><mo mathvariant="normal">></mo><mspace width="1em" /><mi mathvariant="normal">y</mi><mo mathvariant="normal">=</mo><mi>Poisson</mi><mfenced><mi>UDVtI</mi></mfenced></math><img file="EP1908011B1_D0021.tif" /></maths> where <b>U</b> and <b>V</b><i>^t</i> are unitary matrices, and <b>D</b> is a diagonal matrix containing the singular values of Φ. Any destabilizing singular values in D are likewise destabilizing to the Poisson process. Specifically, those elements of <b>I</b> (or linear combinations of elements) for which insufficient data is available are equivalent, regardless of whether or not the photon emissions are modeled as Poissonian.

0072The following description is directed at a non-limiting preferred embodiment wherein the relationship between the measurements <b>y</b> and intensities <b>I</b> are modeled as a set of linear equations, <b>y</b> = Φ<b>I</b>, for the purposes of reconstruction. This model is suitable for least squares optimization when photon emissions have a Gaussian distribution, and thus may be reasonably applied when the measured counts (i.e. y) are relatively high. However the embodiments are extendable to modeling the emissions as Poissonian. In a preferred embodiment, when a given voxel or voxels are not within Gaussian range, the y values of several low-intensity views are combined and set equal to the sum of all the related linear equations, in order to bring the total count into Gaussian range.

0073Assuming intensities with a Gaussian distribution, then by applying SVD to the present inverse problem, <b>I</b> is obtained as: <maths id="math0022" num="(17)"><math display="block"><mi mathvariant="bold">I</mi><mo mathvariant="bold">=</mo><msup><mi mathvariant="bold">VD</mi><mrow><mo mathvariant="bold">-</mo><mn mathvariant="bold">1</mn></mrow></msup><mo>⁢</mo><msup><mi mathvariant="bold">U</mi><mi mathvariant="bold">t</mi></msup><mo>⁢</mo><mi mathvariant="bold">y</mi></math><img file="EP1908011B1_D0022.tif" /></maths> As discussed above, the condition number of D may serve as an indicator of the stability of the reconstruction, particularly during an iterative reconstruction process. Furthermore, known techniques such truncated SVD may be used for optimization and reconstruction purposes.

0074The present embodiments are of a method, apparatus, and system which analyze the reliability of the reconstruction possible by a given imaging constellation, and define modifications in order to improve the quality of reconstruction. The modifications may be to the reconstruction and/or data collection aspects of the imaging process. As described below, the modifications may use active view selection during imaging and/or to guide non-uniform scanning of the imaged volume.

0075Reference is now made to <figref idref="f0005">Fig. 2a</figref>, which is a simplified flowchart of a method for stabilizing the reconstruction of an imaged volume, according to a preferred embodiment of the present invention. In step 115, an analysis of the reliability of reconstruction of a radioactive-emission density distribution of the volume is performed. The analysis is made to determined the reliability of the data which may be collected by radioactive-emission measurements taken over a specified set of views. In step 120, modifications to improve reliability of reconstruction are defined in accordance with the analysis. The modifications may involve a change or adaptation of the reconstruction process and/or a change or adaptation of the data collection process. Both the analysis and modification aspects of the present embodiment are described in detail below.

0076The present embodiment tailors the reconstruction and/or data collections processes in order to obtain a more accurate reconstruction of the imaged volume. Low reliability data may cause large errors in the reconstruction process. As a result the reconstruction process may be noisy and unstable, and fail to converge properly. The reconstructed image may contain artifacts, unsupported frequencies and other errors.

0077The present embodiments are applicable to all stages of the imaging and/or reconstruction processes. The analysis may be performed prior to data collection, in order to define a scanning procedure which provides high-reliability data. During data collection, the scan pattern may be adapted or views may be added (e.g. active vision/adaptive scanning), utilizing the new information provided by the collected data. After data collection, modifications may be made to the reconstruction process in order to counteract the effects of unreliable data and non-uniformities, and to improve reconstruction accuracy and stability.

0078In a first preferred embodiment, the analysis of the reliability of reconstruction is based upon on analysis of the singular values of the probability matrix Φ. A non-limiting preferred embodiment of such an analysis is presented below.

0079Reference is now made to <figref idref="f0006">Fig. 2b</figref>, which is a simplified flowchart of a method for analyzing a detection probability matrix, according to a preferred embodiment of the present invention. In step 125, a detection probability matrix is provided. As described above, the detection probability matrix gives the detection probability for each of the views over the voxels of the volume. More explicitly, in the preferred embodiment each view has an associated row in the detection probability matrix. The values in a given row are derived from the probability of detection of a photon emerging from each of the voxels of the volume, by a detector at the associated view. In step 130, the singular values of the detection probability matrix are calculated. In the preferred embodiment, the singular values are calculated by employing SVD on the detection probability matrix. Performing SVD on the detection probability matrix enables easy implementation of stabilization methods such as truncated SVD for stabilizing the reconstruction process. In step 135, those singular values which may have a destabilizing effect on the reconstruction process are identified as destabilizing singular values.

0080In the preferred embodiment, a measure of the reconstruction reliability (denoted herein the reliability measure) is determined. The reliability measure is used for determining if any modification of the imaging and/or reconstruction processes is required, or if a stable reconstruction is possible without such modification.

0081Preferably, the condition number of the probability matrix is utilized as a reliability measure. If the condition number is satisfactory the reconstruction may be considered stable, with no need for modification. Preferably, the identifying of destabilizing singular values is performed only if the condition number is below a specified magnitude.

0082In an alternate preferred embodiment, the information-theoretic Fisher information is calculated as a reliability measure from intensity distributions provided of the imaged volume. An intensity distribution may be obtained either as an emittance model created prior to the data measurement process, or as an intermediate result of the reconstruction process. The Fisher information is described in detail below, in the context of the reliability criterion for active vision.

0083In the preferred embodiment, the identification of destabilizing singular values is performed as follows. The ratio of the largest singular value to each of the remaining singular values is calculated. A singular value for which the ratio is above a specified threshold is considered destabilizing. One may consider that if the effects of the destabilizing singular values upon the reconstruction are eliminated, the condition number of the detection probability matrix is effectively below the threshold.

0084Preferably, unreliable voxels or combinations of voxels associated with a given destabilizing singular value are identified. The intensity levels of such unreliable voxels or combination of voxels are unsupported in the collected emission data, due to the destabilizing effects of the associated singular value. Therefore the accurate reconstruction of such voxels and/or combinations is unlikely, and may destabilize the reconstruction process. The destabilization of the reconstruction by unsupported voxels may lead to artifacts and unsupported frequencies in the reconstructed intensities. These effects are not necessarily limited to the unsupported regions, but may extend into other portions of the volume and create artifacts there as well.

0085The voxels and linear combinations of voxels associated with a singular may be determined via SVD decomposition of the intensity distribution matrix Φ. The matrix V obtained by SVD decomposition of Φ indicates voxels or linear combinations of voxels whose intensity after reconstruction is most highly affected by each of the singular values in <b>D</b>. Each row of V indicates a weighting associated with a given singular value for each voxel in I. Consider the following example of a volume with four voxels i1-i4: <maths id="math0023" num="(18)"><math display="block"><mi mathvariant="bold">y</mi><mo mathvariant="bold">=</mo><msup><mi mathvariant="bold">UDV</mi><mi mathvariant="bold">T</mi></msup><mo>⁢</mo><mi mathvariant="bold">I</mi><mo mathvariant="bold">=</mo><mi mathvariant="bold">U</mi><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>e</mi><mn>1</mn></msub></mtd><mtd><mspace width="1em" /></mtd><mtd><mspace width="1em" /></mtd><mtd><mspace width="1em" /></mtd></mtr><mtr><mtd><mspace width="1em" /></mtd><mtd><msub><mi>e</mi><mn>2</mn></msub></mtd><mtd><mspace width="1em" /></mtd><mtd><mspace width="1em" /></mtd></mtr><mtr><mtd><mspace width="1em" /></mtd><mtd><mspace width="1em" /></mtd><mtd><msub><mi>e</mi><mn>3</mn></msub></mtd><mtd><mspace width="1em" /></mtd></mtr><mtr><mtd><mspace width="1em" /></mtd><mtd><mspace width="1em" /></mtd><mtd><mspace width="1em" /></mtd><mtd><msub><mi>e</mi><mn>4</mn></msub></mtd></mtr></mtable></mfenced><mo>⁢</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>v</mi><mn>11</mn></msub></mtd><mtd><msub><mi>v</mi><mn>12</mn></msub></mtd><mtd><msub><mi>v</mi><mn>13</mn></msub></mtd><mtd><msub><mi>v</mi><mn>14</mn></msub></mtd></mtr><mtr><mtd><mn>..</mn></mtd><mtd><mspace width="1em" /></mtd><mtd><mspace width="1em" /></mtd><mtd><mn>..</mn></mtd></mtr><mtr><mtd><mn>..</mn></mtd><mtd><mspace width="1em" /></mtd><mtd><mspace width="1em" /></mtd><mtd><mn>..</mn></mtd></mtr><mtr><mtd><msub><mi>v</mi><mn>41</mn></msub></mtd><mtd><msub><mi>v</mi><mn>42</mn></msub></mtd><mtd><msub><mi>v</mi><mn>43</mn></msub></mtd><mtd><msub><mi>v</mi><mn>44</mn></msub></mtd></mtr></mtable></mfenced><mo>⁢</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mi>i</mi><mo>⁢</mo><mn>1</mn></mtd></mtr><mtr><mtd><mi>i</mi><mo>⁢</mo><mn>2</mn></mtd></mtr><mtr><mtd><mi>i</mi><mo>⁢</mo><mn>3</mn></mtd></mtr><mtr><mtd><mi>i</mi><mo>⁢</mo><mn>4</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mi mathvariant="normal">U</mi><mo>⁢</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mo>⋯</mo></mtd></mtr><mtr><mtd><mo>⋯</mo></mtd></mtr><mtr><mtd><mo>⋯</mo></mtd></mtr><mtr><mtd><msub><mi>e</mi><mn>4</mn></msub><mo>⁢</mo><mfenced><msub><mi>v</mi><mn>41</mn></msub><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>42</mn></msub><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mn>4</mn><mo>+</mo><msub><mi>v</mi><mn>43</mn></msub><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mn>3</mn><mo>+</mo><msub><mi>v</mi><mn>44</mn></msub><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mn>4</mn></mfenced></mtd></mtr></mtable></mfenced></math><img file="EP1908011B1_D0023.tif" /></maths>

0086In a first example, assume that e4 is a destabilizing singular value, and that the fourth row of V<b>, v</b>_<b>4</b>, equals [0.001 0.01 0.97 0.015]. Examining <b>v₄</b> it is seen that the coefficient for the single voxel i3 is significantly larger the coefficients of the other voxels. It is therefore quite apparent that the reliability of data regarding voxel i3 is particularly low, and it is unlikely that i3 can be accurately reconstructed. In this example, the preferred embodiment is to perform smoothing of i3 relative to the neighboring voxels. Smoothing may be a practical approach where an inaccuracy occurs in spatially localized regions of the volume. An exemplary criterion to determine which voxels should be smoothed, is to select voxels whose corresponding coefficient of <b>v_i</b> is above a certain threshold.

0087In a second example, assume that e3 is a destabilizing singular value, and that the third row of V, <b>v</b>₃, equals [0.5 - 0.5 0.5 - 0.5]. In the present example, the unreliability of the data relates to a combination of voxels which may be distributed over the volume, rather than to an independent, spatially located voxel or group of voxels. In the present example, the reliability of data regarding the linear combination 0.5i1-0.5i2+0.5i3-0.5i4, as reflected by measurements y, is low, which may lead to instability and inaccuracies during reconstruction. Since there are no dominant voxels it may be less effective to perform smoothing. Instead, the preferred embodiment is to constrain intensities of the combination of voxel intensities between iterations.

0088A proposed approach for applying such a constraint is as follows. Given a reconstructed intensity vector <b>I</b>', modify the intensities prior to performing the next iteration as follows: <maths id="math0024" num="(18)"><math display="block"><mi mathvariant="bold">Iʹ</mi><mo mathvariant="bold">=</mo><mo mathvariant="bold">></mo><mi mathvariant="bold">Iʹ</mi><mo mathvariant="bold">-</mo><msub><mi mathvariant="bold">v</mi><mi mathvariant="bold">k</mi></msub><mo>⁢</mo><mi mathvariant="bold">α</mi><mfenced><msup><msub><mi mathvariant="bold">v</mi><mi mathvariant="bold">k</mi></msub><mi mathvariant="bold">T</mi></msup><mo>⁢</mo><mi mathvariant="bold">I</mi></mfenced></math><img file="EP1908011B1_D0024.tif" /></maths> where <b>v</b>_k is the row of <b>V</b> associated with a destabilizing singular number, <b>e</b>_k, and α is a proportionality factor which may be related to the magnitude of <b>e</b>_k. The effect of the voxel combination on the following iteration is therefore reduced as desired, where the reduction may be correlated with the dominance of the associated singular value.

0089Preferably, constraints are added during the reconstruction process to reduce or eliminate the effects of the unreliable data associated with destabilizing singular values.

0090In the preferred embodiment, the modifications include defining constraints on the reconstruction process. The reconstruction may be considered unreliable if one or more destabilizing singular values have been found, and the modifications may be made in order to reduce the effect of the destabilizing singular values upon the reconstruction. Examples of such constraints may include: <ol id="ol0003" compact="compact"><li>1) Smoothing or piecewise smoothing for all or portions of the volume</li><li>2) Uniting voxels</li><li>3) Subdividing voxels</li><li>4) Adjusting the levels of one or more voxels after a reconstruction iteration</li><li>5) Constraining the input data</li><li>6) Performing reconstruction with differing resolutions for different portions of the volume</li></ol> The above exemplary constraints are now discussed in turn.

0091Smoothing a reconstructed image is performed by calculating the intensity level of a voxel based on the intensities of surrounding voxels, in order to control the magnitude and rate of the fluctuations in intensities between voxels. For example, the intensity of a given voxel may be corrected to better reflect its value as an average of the surrounding voxels. Smoothing may be performed over the entire volume, or only on portions of the volume deemed unreliable.

0092When a non-uniform scan is performed, it is possible that not all directions surrounding a voxel will have equal support. It may therefore be easier to distinguish voxel from some neighbors than from others. In the preferred embodiment, smoothing is performed directionally. In directional smoothing, the smoothing is weighted more strongly in certain directions than in others.

0093In a first example, directional smoothing is used to overcome the limitations of unreliable data by smoothing the voxel in the weaker directions. In a second example, a higher resolution is available for some portions of the volume than is available in other portions. If it is important to obtain a reconstruction having a uniform resolution, directional smoothing may be applied reduce the resolution in highly reliable portions of the volume to the resolution obtainable elsewhere.

0094A directional smoothing policy may be based on the Fisher information measure, which provides directionality information. The Fisher information matrix indicates directionality and cross-relationships between voxels, but may be difficult to calculate. In the preferred embodiment, a scalar Fisher information is calculated for groups of views, where each group is localized in a given direction. The scalar Fisher information measures may then be analyzed together, in order to determine the relationships between the different directions.

0095Low reliability voxels may be united or combined with surrounding voxels in order to increase their reliability. Voxels may combined by assigning the intensity level of a neighboring high-reliability voxel to the low-reliability volel. Voxel merging may be carried out more extensively, by repeatedly merging the lowest reliability voxel with one of the neighboring voxels to form an aggregate of voxels, until no voxel is left with a reliability below a threshold. It will be appreciated that the aggregate voxels have a lower resolution locally, but all other regions with good coverage remain of high resolution, according to their coverage.

0096An opposite approach for high-reliability portions of the volume is to subdivide high-reliability voxels into smaller voxels in order to improve resolution while still achieving a required reliability. For example, if there are 1000 different views independently covering 1 cubic cm and almost not affected by the surrounding voxels, then theoretically that volume can be divided up to about 1000 voxels, if the views create a linearly independent set of equations with a good condition number.

0097A further possible modification of the reconstruction process may be to adjust the effect of certain unreliable voxels or linear combinations of voxels after each of one or more iterations of the reconstruction process. For example, the magnitude of unreliable voxels or combinations of voxels associated with a destabilizing singular value may be reduced towards zero or towards the value of neighboring voxels by a specified factor, possibly in proportion to their respective weightings in the associated row of matrix V and/or to the associated singular value.

0098An additional possible modification is to constrain the value of the input. For example, a realistic range of counts may be defined overall or per view based on view parameters, typical spatial structures and the like. Examples include adding a prior constraint such as a gamma distribution of intensities, a constant or piecewise-constant progression of intensities, a linear or piecewise-linear progression of intensities, smoothing or piecewise smoothing constraints, an intensity distribution based on the shape of or magnitude of the object being scanned (possibly determined during a pre-scan), or a maximal or known range of expected intensities,

0099The reconstruction process may be performed in a manner that obtains varying resolutions over the volume. In one approach, the entire volume is defined as a single huge voxel, and split over the reconstruction iterations to form smaller voxels. The process is performed repeatedly over all or some of the voxels of the volume, as long as the result of the split maintains a reconstruction reliability high enough for stable results.

0100In a further preferred embodiment, the information-theoretic Fisher information is used as a reliability measure. Modifications to the reconstruction and/or data collection processes may be performed when the Fisher information is deemed to be outside a specified range. The calculation of the Fisher information is described in detail below, in the context of the reliability criterion for active vision. The Fisher information may be calculated from the results of a previous reconstruction iteration, or, initially, from an emittance model provided of the imaged volume.

0101The following addresses a preferred embodiment in which the modification is implemented by defining views for imaging the volume. The defined views serve to guide the data collection process in order to obtain measurements which enable performing a stable and accurate reconstruction of the intensity distribution of the volume. In this manner, detecting resources may be invested effectively in order to improve reconstruction reliability. Scanning resources include, for example, detector dwell time, number of detectors, angular and translational increments, and the like-features that increase the amount of data collection.

0102Such modification may be performed when a region is interesting but is still left with coarse resolution, so as to allocate more scanning resources, such as dwell time, number of detectors, angular and translational increments, to cover that region and to form more independent views such as to increase the reliability of the reconstruction of that area. Preferably, the views are defined so as to obtained a desired resolution over the region of interest.

0103The defined views may yield a non-uniform scanning procedure. A scanning density may be specified by the angular and translational increment size, or steps, the smaller the increment, the higher the density. Additionally, the scanning density may be defined by the acquisition time at each position - the longer the time, the higher the density.

0104Non-uniform scanning may be defined by specifying varying scan densities for imaging the volume. For example, the modification may entail adjusting a local scanning density to scan a region of interest with high density, and to scan other regions with low density.

0105The non-uniform scans may relate to non-uniform angular steps of the detector along a sweep, non-uniform detector translational steps, or different steps by different detectors. Some detectors may employ dense steps and others may employ sparse steps, for example, based on active vision as taught hereinbelow.

0106The scan density may be adapted to the distance to the object of interest. Since resolution decreases with distance, the higher density may compensate for increased distance.

0107Additionally or alternatively, the angular steps may increase in density when scanning a region of interest, and decrease in density, when scanning other portions of the volume.

0108Furthermore, more than one region of interest may be scanned with dense steps, simultaneously. The two regions of interest may be, for example, a tissue region and a blood pool region. This has applicability, for example, to dynamic studies of blood perfusion, by providing even scanning resources both to the blood and to the tissue.

0109Additionally, convex scans may be employed.

0110Variable scans, where a same region is scanned first with a first density and then with another density, may be employed. Alternatively, a same region may be scanned by a first group of detectors with a first density and then by a second group of detectors with another density, concurrently, or at different times. Thus the same region is scanned with at some density by a given detector and at a different density by another detector.

0111In a further preferred embodiment, view definition is performed dynamically during radioactive-emission measurements of the volume. The definition of further views for measurement during the data collection process is denoted active vision herein and is described in detail below. Active vision may be performed independently or in conjunction with the present preferred embodiment of stabilizing the reconstruction of an imaged volume.

0112In the preferred embodiment, the method includes the further step of iteratively reconstructing the radioactive-emission density distribution of the volume, preferably by EM estimation. Reconstruction reliability is preferably evaluated after every iteration.

0113Reference is now made to <figref idref="f0007">Fig. 2c</figref>, which is a simplified block diagram of a reconstruction stabilizer, according to a preferred embodiment of the present invention. Reconstruction stabilizer 140 includes reliability analyzer 145 which analyses the reconstruction reliability of a radioactive-emission density distribution of a volume. The reconstruction reliability may be determined from measures such as the condition number of the probability density matrix or the Fisher information, as described above. Modifier 150 defines modifications to improve the reliability of the reconstruction, based on the analysis performed by reliability analyzer 145. Modifier 150 preferably applies at least one constraint to the reconstruction process and/or data collection. The constraints may include applying smoothing, uniting voxels, adjusting the levels of voxels, defining new views or non-uniform scanning, and other constraints described above. The views may be defined before or during emission data collection.

0114In the preferred embodiment, reconstruction stabilizer 140 is included in a system for generating a three-dimensional image of volume. In the preferred embodiment, the system further includes radiological imaging camera 155 and reconstructor 160. Camera 155 includes detectors capable of independent movement during data acquisition, and which are capable of detecting radiation emitted from the volume thereby to provide radiation data. In accordance with embodiments of the present invention, each block of the imaging camera construction or each detecting unit, where single-pixel detecting units are used, may be provided with at least one, and preferably, two, three, or as many as six degrees of motion such as, for example, rotational motion around the x, y, and z, axis, oscillatory motion about these axes, or translational motion along these axes. Camera 155 thus enables flexible view definition by modifier 150, increasing the likelihood that a reliable reconstruction will be achieved. Reconstructor 160 analyzes the radiation data provided by camera 155 so as to reconstruct a three-dimensional image of the volume.

0115Reconstructor stabilizer 140 guides camera 155 and reconstructor 160, by providing one or both with the modifications determined to improve reconstruction reliability.

0116In practice, it is generally necessary to reconstruct volumes at a high resolution (i.e. a large number of voxels) from many measurements. Implementing the techniques described above then requires the manipulation of extremely large matrices. The large size of these matrices poses difficulties both for performing the calculations and for memory management. These difficulties are particularly problematic when active view selection is performed.

0117In the preferred embodiment, calculations are performed in a localized manner. That is, calculations are performed for different sections of the volume in turn, progressing through the volume until it is covered in its entirety. When calculations are being made of a particular section of the volume, the variables relating to the other sections are "frozen". Thus first a selected section of the volume may be stabilized, so that the reconstruction of following sections is based on data with improved reliability. Localized calculations, such as SVD decomposition, are performed on smaller sub-matrices rather than on a single large matrix, alleviating the difficulties of manipulating very large matrices.

0118The division of the volume into sections may be devised in any practical way, for example as sequential slices or by another spatial connection. The division may be based on the scan pattern, so that well-supported sections may be stabilized first, followed by less supported section. The division may also or alternately be based on knowledge of the body being imaged, such as its shape or composition. When imaging a body structure, the sections may be based on the known axes or structures of the organ

0119The following embodiments are of a method for determining further views for the imaging of a body structure, denoted active vision herein. Active vision addresses the problem of ensuring that the quality of data gathered during the measurement process is adequate to provide a high quality image. The collected data and/or the image reconstructed from the collected data is analyzed the while the measurement process is taking place. Based on the analysis, further views are defined. Since each view is associated with known values of the viewing parameter(s), selecting a view effectively specifies known viewing parameter values. The defined further views thus define a set of viewing parameter values, which are used during the current measurement process in order to collect data which yields a high-quality reconstruction of the body structure.

0120Active vision may be performed independently, as described in the embodiments presented below. Additionally or alternatively, active vision may be performed in conjunction with the above-described method for stabilizing reconstruction, by modifying data collection by dynamically providing additional views to improve reconstruction reliability (see <figref idref="f0030">Fig. 17</figref>).

0121The following embodiments are not confined to a specific reconstruction algorithm. Further views are preferably defined based on one or more of the following: <ol id="ol0004" compact="compact"><li>1) Detector photon count</li><li>2) Geometric properties of the reconstructed body structure</li><li>3) Information theoretic measures that quantify the quality of the data fed to the reconstruction algorithm</li></ol>

Each of these criteria is discussed in detail below.

0122Reference is now made to <figref idref="f0008">Fig. 2d</figref>, which is a simplified flowchart of a method of performing radioactive-emission measurements of a body structure, according to a preferred embodiment of the present invention. In step 200, radioactive-emission measurements of the body structure are performed at predetermined views, preferably in vivo. Preferably the measurements are performed for diagnostic purposes. These predetermined views are selected prior to the measurement process, based on a model of the body structure being imaged. In the model more and less informative viewing directions have been identified. The predetermined views of step 200 preferably include those views expected to be informative, based on an analysis of the model.

0123Preferably the body structure is all or a portion of: a prostate, a heart, a brain, a breast, a uterus, an ovary, a liver, a kidney, a stomach, a colon, a small intestine, an oral cavity, a throat, a gland, a lymph node, the skin, another body organ, a limb, a bone, another part of the body, and a whole body.

0124In step 210 the radioactive-emission measurements are analyzed. Preferably the analysis includes one or more of: <ol id="ol0005" compact="compact"><li>1) Analyzing detector photon count(s)</li><li>2) Analyzing detector photon count rate(s)</li><li>3) Identifying detector saturation</li><li>4) Reconstructing a body structure image from emission measurements</li><li>5) Identifying geometric properties of the reconstructed image</li><li>6) Applying information-theoretic measures to the reconstructed image</li></ol>

0125In step 220, further views for measurements are dynamically defined, based on the analysis performed in step 210. Preferably, each of the views is associated with viewing parameters selected from the group consisting of: detector unit location, detector unit orientation, collection angle, and measurement duration. Defining a view consists of providing a value for each of the parameters associated with the given view. The analysis (step 210) and/or dynamic view definition (step 220) may take into account external parameters including: measurement duration, time elapsed from the administration of the pharmaceutical to the measurement, radiopharmaceutical half life, radioactive emission type, and radioactive emission energy.

0126Each of these analysis techniques, and their application to view definition, is now discussed in turn. While each of the analysis/view determination techniques is discussed as a separate embodiment, multiple techniques may be used together to obtain the desired image quality.

0127In a first preferred embodiment, a photon count analysis ensures that the photon count at a given view yields an acceptable measurement error. As discussed above, the radiative emissions of the body structure being imaged is a Poisson process. In a Poisson process the Poisson noise grows inversely to the square root of the number of photons detected. In other words, if N photons are collected from a given view, the resulting signal to noise ratio (SNR) equals: <maths id="math0025"><img file="EP1908011B1_D0025.tif" /></maths>

0128The unprocessed detector photon count at a given view thus provides significant information regarding the quality of the information obtained at a given view. If the photon count is too low, it may be desired to continue to collect photons at the current location/orientation in order to obtain a satisfactory SNR. Alternatively, it may be determined that enough photons have already been collected, and to terminate the current view and move on to the next view.

0129The analysis is preferably performed by defining a global or local required measurement error, and comparing the square root of the obtained photon count to the required measurement error. Photon count analysis can be applied to the current and/or previous views. When a photon count of a current view is found to be too low, the duration of the current view is preferably extended in order to obtain the required error value. When a photon count of a past view is found to be too low, an emission measurement at substantially the same location and orientation but having a longer duration than previously is preferably performed. Alternately or additionally, the collection angle at the given location/orientation is preferably increased.

0130In an additional preferred embodiment, a detector photon count is analyzed to identify detector saturation at a given view. Preferably, when a detector is determined to have saturated, a new view or views are selected to reinforce those views that have saturated. In an alternate preferred embodiment, further views are defined to avoid highly-radiating portions of the body structure.

0131In a second preferred embodiment, a photon collection rate at a given view is analyzed to determine if it is within a specified range. In the preferred embodiment, the photon count rate is used to identify regions of high or low interest. In prostate imaging, for example, a region of high interest may be identified by a high photon rate, indicative of a tumor. In a second example, a region of high interest may be identified in heart imaging by a low photon rate, indicative of non-functional tissues. After one or more areas of high and/or low interest are found, further views are preferably defined by selecting views to concentrate on regions of high interest and/or to avoid regions of low interest. It is thus possible to zoom in on a suspected pathology without repeating the emission measurement process.

0132In a further preferred embodiment, the analyzing of step 210 includes reconstructing a radioactive-emission density distribution of the body structure. Reconstruction may be performed according to any applicable technique known in the art. The reconstruction is then used as the basis for further analysis.

0133Reconstruction based on the data collected from the predetermined views provides information regarding the quality of information obtained from the preceding measurements, and which further views are likely to be most informative. Selecting new views based on reconstruction is intended to bring us into viewing from the more informative views or combinations of views.

0134Reference is now made to <figref idref="f0009">Figs. 3</figref> and <figref idref="f0010">4a-4b</figref>, which pictorially illustrate how different views provide differing types and quality of information. <figref idref="f0009">Fig. 3</figref> shows an object 300 shaped as a cylinder with a front protrusion, and having a high-emittance portion (hotspot) 310. Four views of object 300 are shown, which can be seen to provide different levels of information. Front views, such as V₁, provide little information regarding the shape of object 300 and have relatively little attenuation between the detector and hotspot 310. Side views, such as V₂, provide edge information regarding the object shape or profile, and when correlated with front views help locate hotspot 310 spatially within object 300. Top views, such as V₃, provide information, regarding the cylinder edge region 320. Finally, rear views, such as V₄, are uninformative about the shape of object 300 and have high attenuation relative to hot region 310.

0135<figref idref="f0010">Figs. 4a and 4b</figref> demonstrate how the proper selection of views may improve the quality of information obtained for the body structure, for example in distinguishing between two regions of interest within a given volume.

0136<figref idref="f0010">Fig. 4a</figref> illustrates an object 400 having two high-emission regions of interest (ROI), 410 and 420. For clarity the views V_A to V_F are shown as lines in <figref idref="f0010">Fig. 4a</figref>, however in practice they will each have a finite collection angle δ. The position of ROIs 410 and 420 are assumed to have been estimated based on a model of object 400 and/or a previously performed prescan. The goal of the present invention is to select an additional new view or views which increase the information we have regarding the separation of ROIs 410 and 420 within object 400.

0137In simple terms, consider the object as having three regions: ROI 410 with intensity I₁, ROI 420 with intensity I₂, and a low-emission region 430 between the two ROIs with intensity I₃. The detected intensity at a given detector is proportional to <maths id="math0026"><math display="inline"><mmultiscripts><msub><mo mathvariant="normal">/</mo><msubsup><mi mathvariant="normal">r</mi><mi>ni</mi><mn mathvariant="normal">2</mn></msubsup></msub><mprescripts /><none /><msub><mi mathvariant="normal">I</mi><mi mathvariant="normal">n</mi></msub></mmultiscripts><mo mathvariant="normal">,</mo></math><img file="EP1908011B1_D0026.tif" /></maths> where I_n is the emission intensity of region n and r_i is the distance of region n from detector V_i.

0138<figref idref="f0010">Fig. 4b</figref> illustrates the added information provided by each of the shown views, V_A to V_F. Views V_B and V_C collect emissions from all three regions, and are therefore least informative. Views V_D and V_E collect emissions from only low emittance region 430, and therefore provide most information regarding the location of each ROI within the volume and the separation between ROIs 410 and 420. Views V_A and V_F pass only through a single ROI, and therefore provide an intermediate level of information. It is a goal of the present invention to determine, while the emission measurements of the body structure are taking place, that views in the vicinity of V_D and V_E are highly informative, and to add these further views to the measurement process.

0139A body structure reconstruction can be utilized in several ways to define further views. A first way is to identify interesting portions of the contour and structure of the reconstruction. For example, it is seen in <figref idref="f0009">Fig. 3</figref> that top views are infonnative about edge region 320. Further top view measurements will therefore be informative re edge region 320, and may enable defining the edge more accurately.

0140In a preferred embodiment, the reconstruction is analyzed to identify textural edges within the reconstruction, and view definition preferably includes selecting views at an angle to the textural edges. In the preferred embodiment, the angle is a substantially sharp angle in order to provide information regarding the edge.

0141In another preferred embodiment, the reconstruction is analyzed to identify volumetric boundaries within the reconstruction, and view definition preferably includes selecting views at an angle to the volumetric boundaries. It is expected that the defined views will provide information regarding the boundary and differences in surrounding tissues on either side of the boundary. In the preferred embodiment, the angle is a substantially sharp angle.

0142Another way to utilize the reconstruction to define further views is to identify suspected organ targets within the reconstruction, and to select further view(s) in close proximity to the suspected organ targets. A suspected organ target is typically detected by identifying portions of the reconstruction whose emission intensity distribution and spatial characteristics are typical of a suspect region.

0143In a first preferred embodiment, a suspected organ target is defined as a high-emittance portion of the reconstruction. In prostate imaging, for example, a suspected organ target may be a high-emittance portion of the structure, indicating a tumor. A high-emittance portion is characterized by an intensity that is greater than the background intensity by a factor of at least (1+α), where α is a parameter specified by the user. In practice, a hotspot is usually detectable only if the radiation levels within the hotspot are higher than the background level by a factor of 1.5-2. α is therefore typically defined between 0.5-1. However, the detectability of a hotspot rises as the radioactive intensity of the body rises, raising the photon count. Thus, a lower value of α may be used when the measured body structure is in a state of high-intensity emittance. For example, a body structure may be characterized by relatively high emittance immediately following the administration of the radiopharmaceutical, and be characterized by lower emittance at a later time.

0144In a second preferred embodiment, a suspected organ target is defined as a low-emittance portion of the reconstruction. In heart imaging, for example, a suspected organ target is defined as a low-emittance portion of the reconstruction, indicating non-functional tissues. A low-emittance portion is characterized by an intensity that is lower than the background intensity by a factor of at least (1+β), where β is a parameter specified by the user.

0145In the preferred embodiment the further views are used immediately for radioactive-emission measurements. The results of the new measurements are then used in another analysis to define new further views for additional measurements. The radioactive-emission measurements may then be said to be performed iteratively.

0146Reference is now made to <figref idref="f0011">Fig. 5a</figref>, which is a simplified flowchart of a iterative method of performing radioactive-emission measurements of a body structure, according to a first preferred embodiment of the present invention. In step 500, radioactive-emission measurements of the body structure are performed at predetermined views. In step 510, an analysis is performed of the previously performed emission measurements. In step 520 a decision is made whether to continue with further measurements. If yes, in step 530 further views are defined based on the analysis. Subsequent iterations continue until the decision to end the emission measurement process. After the first iteration, the analysis performed at a given stage may include consideration of all or on part of the measurements performed during one or more previous iterations, in addition to the new measurements.

0147Reference is now made to <figref idref="f0012">Fig. 5b</figref>, which is a simplified flowchart of a iterative method of performing radioactive-emission measurements of a body structure, according to a second preferred embodiment of the present invention. In the present preferred embodiment, a reconstruction of the body structure is formed in step 505. The analysis step 510 is then performed utilizing data provided by the reconstruction(s).

0148Referring again to <figref idref="f0005 f0006 f0007 f0008">Fig. 2</figref>, preferably, analysis step 210 includes determining an accuracy of the reconstruction. Accuracy is preferably determined by analyzing the variance of the reconstructions formed over multiple iterations. Preferably, further views are defined in step 220 to concentrate on the region for which higher accuracy is required. Regions of the reconstruction having low variance provide a high degree of confidence regarding the accuracy of the reconstruction in the given region (where a portion may include the entirety of the body structure being imaged). Further views may be added to the current measurements until the variance is reduced to a required level.

0149Preferably, analysis step 210 includes determining a resolution of the reconstruction. Resolution is preferably determined by analyzing the full width at half maximum (FWHM) of peak values of the reconstruction. The FWHM is given by the distance between points at which the reconstructions reaches half of a peak value. Preferably, further views are defined in step 220 to concentrate on the region for which higher resolution is required.

0150An additional way to define future views using the reconstruction is on an information-theoretic basis. A quality function expressing an information theoretic measure is defined. The quality function rates the information that is obtainable from the body structure when one or more permissible views are added to current measurement process. Several examples of quality functions based on information-theoretic measures are discussed in detail below. The quality function is used to rate potential further views. The measurement process may then continue at those further views whose addition to the previous views yields a high rating.

0151Reference is now made to <figref idref="f0013">Fig. 6a</figref>, which is a simplified flowchart of a method for dynamically defining further views, according to a first preferred embodiment of the present invention. In step 610 a quality function is provided. The quality function expresses an information-theoretic measure which rates the quality of information obtainable from potential further views. In step 620 a set of further views is selected to maximize the quality function. Preferably the selected further views fulfill certain constraints; for example the further views may be selected from a predefined set or may be located in the vicinity of a region of interest within the body structure.

0152In the abovedescribed reconstruction-based analyses, the quality function is evaluated independently for a single reconstruction of the emission intensity of the body structure. However, quality functions may be defined which calculate the score for a given set in relation to one or more reconstructions and/or emittance models. An emittance model is a representation of a specific radioactive-emission intensity distribution within the volume U, so as to model organ targets, such as hot regions, of a radioactive emission intensity, higher than the background level, regions of low-level radioactive emission intensity, which is nonetheless above the background level, and cold regions, of a radioactive emission intensity, lower than the background level. Given an object or class of objects, emittance models may be devised to reflect expected or typical emission patterns for the given object.

0153Developing an emittance model for a particular body structure involves analyzing known information about the body structure to determine expected emission patterns of the body structure. In order to develop a model of a particular body structure, for example a prostate, many factors may be considered. Physical aspects, such as the size and shape of the prostate and the position of the prostate within the torso may be considered, as well as medical knowledge regarding typical emissions from healthy and diseased prostates. Additional information may concern variations between individuals, such as age, weight, percentage of body fat, and the like.

0154For simplicity, the following discussion describes the evaluation of information-theoretic quality functions based on emittance models only. It is to be understood that at least one of the emittance models is a reconstruction of the body structure based on past measurements. Any remaining emittance models are provided externally, and may be based on general medical knowledge or on information gathered during a previous round of emission measurements of the body structure.

0155Reference is now made to <figref idref="f0014">Fig. 6b</figref>, which is a simplified flowchart of a method for dynamically defining further views, according to a second preferred embodiment of the present invention. The current method differs from the method of <figref idref="f0013">Fig. 6a</figref> by the addition of steps 605-606. In step 605 a set of one or more emittance models is provided (where the set includes one or more reconstructions of the body structure). An emittance model specifies the radiative intensity of each voxel in the body structure. As discussed above, some of the viewing parameters affect the radiative intensity of the voxels in the volume, for example the type of radiophannaceutical and the time since administration of the radiopharmaceutical. Therefore, the enuttance models provided in step 605 preferably correspond to the relevant viewing parameters. In step 606 a collection of possible further views of the body structure is provided. The collection of views includes possible further views for future measurements, preferably based on anatomical and other constraints. Furthermore, the quality function provided in step 610 may utilize multiple emission models.

0156Reference is now made to <figref idref="f0015">Figs. 7a and 7b</figref> which present the principles of modeling of a body structure, in accordance with the present invention. <figref idref="f0015">Fig. 7a</figref> schematically illustrates a body section 720, having a region of interest (ROI) 730. The region of interest 730 is associated with a body structure 740, with a specific radioactive-emission-density distribution, possibly suggestive of a pathological feature 750, termed herein an organ target 750. Additionally, there may be certain physical viewing constraints, associated with the region of interest 730. Model 760, as illustrated in <figref idref="f0015">Fig. 7b</figref>, represents body section 720 as a collection of voxels, having a specified radioactive-emission intensity distribution.

0157It will be appreciated that the model 760 of the region of interest 730 may be based on general medical information of the body structure 740 and common pathological features associated with it. Additionally, the model may be based on information related to a specific patient, such as age, sex, weight, and body type. Furthermore, a structural image, such as by ultrasound or MRI, may be used for providing information about the size and location of the body structure 740 in relation to the body section 720, for generating the model 760.

0158Reference is now made to <figref idref="f0016">Fig. 8</figref>, which shows four models of a prostate emittance model. Three views are given for each of the emittance models shown. Each of the views is a maximum intensity projection (MIP) along a different one of the three axes. For example, in an X-Y projection the intensity of a given point is taken as the maximum intensity for that point along a line parallel to the z-axis. Thus the volume in effect becomes "transparent", and the maximum intensity regions are shown clearly. Variations in emission levels are indicated by differences in shading. The emittance model is shaped as an ellipsoid, the typical shape of a prostate. Since a diseased prostate is generally characterized by one or more hot regions, each of the emittance models shown has a number of high-emittance portions (two hot regions in Models A-C and three hot regions in Model D). Each high-emittance area is an ellipsoid, with a size of approximately 1 cubic centimeter. The intensity of the modeled organ targets varies randomly from two to eight times brighter than the surrounding volume.

0159In the preferred embodiment, one or more of the emittance models contains at least one high-emittance portion (i.e. hot region). A prostate containing a tumor, for example, may be modeled as an ellipsoid volume with one or more high-emittance portions.

0160In the preferred embodiment, one or more of the emittance models contains at least one low-emittance portion. A diseased heart may therefore be modeled as a heart-shaped volume with low-emittance portions.

0161Note that an emittance model need not contain high- or low- emittance portions, but may have a uniform intensity or a slowly varying intensity.

0162Following are two embodiments of quality functions, both of which include considerations of the emission distribution of the volume, in light of the emittance models provided (step 605 of <figref idref="f0014">Fig. 6b</figref>). The information quality given by one or more further views is evaluated in relation to multiple reconstructions and/or emittance models of the body structure. In a first preferred embodiment the quality function is based on an information theoretic measure of separability. In a second preferred embodiment, the quality function is based on an information-theoretic Fisher information measure, for ensuring reliable reconstruction of a radioactive-emission density distribution.

0163In a first preferred embodiment, the quality function implements a separability criterion. Separability is a measure of the extent to which the measurements that are obtained from each pair of models can be distinguished from one another.

0164The concept of separability is illustrated in <figref idref="f0017">Figs. 9a-9c</figref>, each of which shows an emittance model of a given volume 900. Each of the emittance models contains two high-emittance portions (hot regions), 910 and 920, which are located in respectively different locations in each of the models. It can be seen that the hot regions in emittance models 9b and 9c are in similar positions of the volume, as opposed to the hot regions in model 9a. It may therefore be difficult to distinguish between reconstructions of emittance models 9b and 9c. The separability criterion ensures that the selected set includes views which provide reconstructions which distinguish between emittance models 9b and 9c.

0165Letting <b><i>I</i></b> be the emittance model set, a measure for the dissimilarity of any two given densities in <b><i>I</i></b> is defined. Since most state-of-the-art estimating algorithm are aimed at finding ML estimators, in the current example the quality function is based on the likelihood function. The likelihood of an estimator of <b><i>I</i>,</b> given a set of Poissonian measurements y is: <maths id="math0027" num="(19)"><math display="block"><mi>L</mi><mfenced><mi>I</mi></mfenced><mo>=</mo><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mi>t</mi></munder></mstyle><mfenced open="{" close="}"><mo>-</mo><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mi>u</mi></munder></mstyle><msub><mi>φ</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced><mo>⁢</mo><mi>I</mi><mfenced><mi>u</mi></mfenced></mstyle><mo>+</mo><msub><mi>y</mi><mi>t</mi></msub><mspace width="1em" /><mi>ln</mi><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mi>u</mi></munder></mstyle><msub><mi>φ</mi><mi>t</mi></msub><mfenced><mi>u</mi></mfenced><mo>⁢</mo><mi>I</mi><mfenced><mi>u</mi></mfenced></mstyle><mo>-</mo><mi>ln</mi><mfenced><msub><mi>y</mi><mi>t</mi></msub><mo>!</mo></mfenced></mfenced></mstyle></math><img file="EP1908011B1_D0027.tif" /></maths>

0166For separability, it is desired that this measure be different for each <i>I</i> ∈ <b><i>I.</i></b> Since the measure is a random variable that depends on the actual measurements, all possible pairings of emittance models should be examined to ensure that the resulting distributions are separable. A quality function that captures this separability is given by the square of the difference between the means of the distributions normalized by the sum of their variances: <maths id="math0028" num="(20)"><math display="block"><msub><mi>SEPARABILITY</mi><mi mathvariant="normal">Φ</mi></msub><mfenced><msub><mi>I</mi><mn>1</mn></msub><msub><mi>I</mi><mn>2</mn></msub></mfenced><mo>=</mo><mfrac><msup><mfenced open="|" close="|"><mi mathvariant="bold">E</mi><mo>⁢</mo><mi mathvariant="bold-italic">L</mi><mfenced><msub><mi>I</mi><mn>1</mn></msub></mfenced><mo>-</mo><mi mathvariant="bold">E</mi><mo>⁢</mo><mi mathvariant="bold-italic">L</mi><mfenced><msub><mi>I</mi><mn>2</mn></msub></mfenced></mfenced><mn>2</mn></msup><mrow><mi mathvariant="bold">Var</mi><mo>⁢</mo><mi mathvariant="bold-italic">L</mi><mfenced><msub><mi>I</mi><mn>1</mn></msub></mfenced><mo>+</mo><mi mathvariant="bold">Var</mi><mo>⁢</mo><mi mathvariant="bold-italic">L</mi><mfenced><msub><mi>I</mi><mn>2</mn></msub></mfenced></mrow></mfrac></math><img file="EP1908011B1_D0028.tif" /></maths>

0167The expectations and variances in Equation 20 are taken over random measurements y, sampled from the true intensity <i>I</i>₁ (note that the measure is not symmetric).

0168Since the true (unknow) intensity can be any <b><i>I</i></b> ∈ <b><i>I</i></b>, a projection set Φ* that maximizes the worst-case separability is desired. That is: <maths id="math0029" num="(21)"><math display="block"><mi mathvariant="normal">Φ</mi><mo>*</mo><mo>=</mo><msub><mi>arg max</mi><mi mathvariant="normal">Φ</mi></msub><mo>⁢</mo><msub><mi>min</mi><mrow><msub><mi>I</mi><mn>1</mn></msub><mo>,</mo><msub><mi>I</mi><mn>2</mn></msub><mo>∈</mo><mi mathvariant="bold-italic">I</mi></mrow></msub><mspace width="1em" /><msub><mi>SEPARABILITY</mi><mi mathvariant="normal">Φ</mi></msub><mfenced><msub><mi>I</mi><mn>1</mn></msub><msub><mi>I</mi><mn>2</mn></msub></mfenced></math><img file="EP1908011B1_D0029.tif" /></maths>

0169Scoring for separability is based on the minimum separability obtained with a given set of views for all of the possible pairings of emittance models from the set of emittance models, thereby enabling defining a desired resolution in more than one direction, or in more than one portion of the volume. All of the emittance models are modeled on a substantially identical volume. The emittance models preferably differ from one another in the modeled organ targets, where the modeled organ targets are separated by a difference of at least the required resolution (where the displacement which produces the required resolution is denoted delta herein). Substantially identical sets of views are formed from the collection of views, and each of the formed sets is scored with respect to each of the pairs. One of the sets of views is selected, based on the minimum or average score for the plurality of pairs.

0170For example, assume the set of emittance models contains the three models 9a-9c. A separability score is calculated for a given formed set of views by applying Equation 20 to all three pairs 9a/9b, 9a/9c, and 9b/9c. The lowest of the three calculated values is taken as the separability score for the formed set. Once a separability score has been calculated in such manner for each of the formed sets of views, the view set having the highest separability is selected.

0171The separability criterion may be used to ensure that a required resolution is obtained in all or a portion of the body. In a preferred embodiment, view set selection for separability is performed utilizing a set of emittance models consisting of one pair of emittance models having substantially identical volumes but with different modeled organ targets. The modeled organ targets are separated by a delta in a given direction so as to define a required resolution in that direction and portion of the volume U. Substantially identical sets of views are formed from the collection of views, and scored with respect to the pair of emittance models, using a quality function based on the separability criterion, and one of the sets of views is selected based on the separability scores. The selected set is thus the set which provides the optimum resolution in the given direction and in the vicinity of the modeled organ targets.

0172A similar approach may be used to ensure resolution in more than one direction and/or portion of the volume U. Consider for example, a pair of models of substantially identical volumes, as follows: The model of <figref idref="f0018">Fig. 9d</figref>, which schematically illustrates the volume U, having the modeled organ target HS, whose center is at a location (x;y;z)_HS, and the model of <figref idref="f0018">Fig. 9e</figref>, which schematically illustrates the volume U, having the modeled organ target HS', whose center is at a location (x;y;z)_HS'. <figref idref="f0018">Fig. 9e</figref> also shows modeled organ target HS of <figref idref="f0018">Fig. 9d</figref> superimposed over the present emittance model to illustrate that HS' is somewhat displaced from HS, along the x-axis, and the displacement, is denoted as delta1 in the present example. The displacement delta may be measured, for example, in mm.

0173An optimal set of views, from the standpoint of separability, is that which will best distinguish between the model of <figref idref="f0018">Fig. 9d</figref> and that of <figref idref="f0018">Fig. 9e</figref>. Thus, a score, in terms of separability is given for the pair of models, and relates to a resolution as defined by the difference between the models of the pair. In the present example, the difference is delta1 along the x-axis, around the locations of HS and HS', so the score given by the information theoretic measure of separability will relate specifically to a resolution as defined by deltal along the x-axis, around the locations of HS and HS'. Other portions of the volume U and other directions may have different resolutions.

0174For example, consider the model of <figref idref="f0019">Fig. 9f</figref>, which schematically illustrates the volume U, having the modeled organ target HS", whose center is at a location (x;y;z)_HS", and shows HS of <figref idref="f0018">Fig. 9d</figref> and HS' of <figref idref="f0018">Fig. 9e</figref> superimposed over the present emittance model. HS" is displaced from HS, along the z-axis, a displacement delta2. and additionally, HS" is displaced from HS', along the x- and z- axes, a displacement delta3.

0175Scores, in terms of separability, may be given to all the paring combinations, that is the models of <figref idref="f0018">Figs. 9d - 9e</figref>, relating to delta1; the models of <figref idref="f0018 f0019">Figs. 9d -9f</figref>, relating to delta2, and the models of <figref idref="f0018 f0019">Figs. 9e - 9f</figref>, relating to delta3. An optimal set of views may be selected based on its minimum or average scores for all the pairing combinations; for example, the optimal set of views may be that whose average score for all the pairing combinations is the highest. Alternatively, a weighted average may be applied.

0176In a second preferred embodiment, the quality function implements a reliability criterion, which is a measure of how reliably the intensity distribution of a given object may be reconstructed from the sampled views. Since the input to the reconstructed algorithm is a random sample, the output estimator is also random. A desired property of this output is that it be reliable in the sense that similar estimators for different projected samples (i.e. different sets of measurements) of the same input intensity are obtained with high confidence.

0177The Fisher Information, <i>F</i>_φ(<i>I</i>) is a measure, known in the art, which is used to evaluate the expected curvature of the likelihood of a model <b><i>I</i></b> (taken over measurements sampled from the model <b><i>I</i></b>). The Fisher Information is defined as: <maths id="math0030" num="(22)"><math display="block"><msub><mi>F</mi><mi mathvariant="normal">Φ</mi></msub><mfenced><mi>I</mi></mfenced><mo>=</mo><mo>-</mo><mi mathvariant="bold">E</mi><mo>⁢</mo><msup><mo mathvariant="normal">∇</mo><mn>2</mn></msup><mo>⁢</mo><mi>L</mi><mfenced><mi>I</mi></mfenced></math><img file="EP1908011B1_D0030.tif" /></maths>

0178The derivatives are taken with respect to the parameters of the intensity <b><i>I,</i></b> and the expectation is taken with respect to the random counts. Intuitively, a sharper curvature means that a maximum-likelihood estimation algorithm is more likely to produce a low-variance estimator. Indeed, this property is captured by the Cramer-Rao lower bound, which states that the inverse of the Fisher Information is a lower bound for the variance of any unbiased estimator, where an estimator <i>f</i> of the intensity <i>I</i> is unbiased if <b>E</b><i>f</i>(y) = <b><i>I</i></b>

0179The Fisher <i>information</i> provides a value, <i>F</i>_φ(<i>I</i>)_u,y, for each pair of voxels <i>u</i> and <i>v</i>. To provide a single measure for the reliability of the estimator, the average level, worst case, or other reasonable measure may be taken over the voxels. In the current example, the quality function is based on the average Fisher information. Starting with a set of emittance models to be used by the estimating algorithm to provide a reliable estimator, the average of the above measure is maximized over the entire set, defining the selected set Φ* as: <maths id="math0031" num="(23)"><math display="block"><msup><mi mathvariant="normal">Φ</mi><mo>*</mo></msup><mo>=</mo><msub><mi>arg max</mi><mi>φ</mi></msub><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>l</mi><mo>∈</mo><mi>I</mi></mrow></munder></mstyle><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>u</mi><mo>∈</mo><mi>U</mi></mrow></munder></mstyle><msub><mfenced open="[" close="]"><msub><mi>F</mi><mi mathvariant="normal">Φ</mi></msub><mfenced><mi>I</mi></mfenced></mfenced><mrow><mi>u</mi><mo>,</mo><mi>u</mi></mrow></msub></mstyle></mstyle></math><img file="EP1908011B1_D0031.tif" /></maths>

0180Alternately, a set may be chosen to minimize the inverse of the Fisher information, by selecting for: <maths id="math0032" num="(24)"><math display="block"><msup><mi mathvariant="normal">Φ</mi><mo>*</mo></msup><mo>=</mo><msub><mi>arg min</mi><mi>φ</mi></msub><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>l</mi><mo>∈</mo><mi>I</mi></mrow></munder></mstyle><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>u</mi><mo>∈</mo><mi>U</mi></mrow></munder></mstyle><msub><mfenced open="[" close="]"><msub><mi>F</mi><mi mathvariant="normal">Φ</mi></msub><mo>⁢</mo><msup><mfenced><mi>I</mi></mfenced><mrow><mo>-</mo><mn>1</mn></mrow></msup></mfenced><mrow><mi>u</mi><mo>,</mo><mi>u</mi></mrow></msub></mstyle></mstyle></math><img file="EP1908011B1_D0032.tif" /></maths>

0181Since inverting <i>F</i>_φ(<i>I</i>) may be computationally expensive, the └<i>F</i>_φ(<i>I</i>)^-1┘<i>_u,u</i> term in Equation 24 may be replaced with 1/[<i>F</i>_φ(<i>I</i>)]_u,u (thus neglecting the off-diagonal elements of the Fisher Information matrix <i>F</i>φ(<i>I</i>)). Note that Equations 23 and 24 are not mathematically equivalent, and may therefore yield different selected sets.

0182In the preferred embodiment, evaluating the quality function is performed using the reliability criterion, and two or more emittance models are provided, having substantially identical volumes, but different modeled organ targets. Substantially identical sets of views are formed for all the emittance models, and each set is scored for reliability. One of the sets of views is then selected based on the average score for all the of the emittance models.

0183In a further preferred embodiment, a weighted combination of several information theoretic measures is used. For example, a plurality of models may be provided, all having substantially identical dimensions and volumes, as follows: <ol id="ol0006" compact="compact"><li>i. a pair of models with slightly different distributions of radioactive emission sources, as seen in <figref idref="f0020">Figs. 9g</figref> and <figref idref="f0021">9h</figref>, for scoring sets of views in terms of separability;</li><li>ii. a model with a given distribution of radioactive emission sources, as seen in any one of <figref idref="f0020">Figs. 9g</figref>, <figref idref="f0021">9h</figref> or <figref idref="f0022">9i</figref>, for scoring sets of views in terms of reliability.</li></ol>

0184Identical sets of views may be applied to all the models, and each view may be scored in terms of separability and reliability. An optimal set of views may be selected based on a summation of the two scores, or based on a weighted average of the two scores, where the relative weight given to each criterion reflects the relative importance of each measure per the given application.

0185The quality function is preferably defined in accordance with one of the following: worst case effectiveness for the given further view over the volume, average effectiveness for the given further view over the volume, worst case effectiveness for the given further view over the set of emittance models and average effectiveness for the given further view over the set of emittance models.

0186Maximization of the quality function may be performed utilizing any method known in the art such as simulated annealing and gradient ascent. In the simulated annealing (SA) method, each point of the search space is compared to a state of some physical system. The quality function to be maximized is interpreted as the internal energy of the system in that state. Therefore the goal is to bring the system from an arbitrary initial state to a state with the minimum possible energy.

0187The neighbors of each state and the probabilities of making a transition from each step to its neighboring states are specified. At each step, the SA heuristic probabilistically decides between moving the system to a neighboring state s' or staying put in state s. The probabilities are chosen so that the system ultimately tends to move to states of lower energy. Typically this step is repeated until the system reaches and acceptable energy level.

0188Gradient ascent, on the other hand, is based on the observation that if a real-valued function <i>F</i>(<i>x</i>), such as the quality function of the present embodiments, is defined and differentiable in a neighborhood of a point <i>a</i>, then <i>F</i>(<i>x</i>) increases fastest if one goes from a in the direction of the gradient of <i>F</i> at <i>a</i>, Δ<i>F</i>(<i>a</i>). It follows that if: <maths id="math0033" num="(25)"><math display="block"><mi>b</mi><mo>=</mo><mi>a</mi><mo>+</mo><mi mathvariant="normal">γ</mi><mo>∇</mo><mi>F</mi><mfenced><mi>a</mi></mfenced></math><img file="EP1908011B1_D0033.tif" /></maths> for γ > 0 a small enough number, then <i>F</i>(<i>a</i>)≤ <i>F</i>(<i>b</i>). Gradient ascent starts with a guess x₀ for a local maximum of <i>F</i>, and considers the sequence x₀, x₁, x₂,... such that: <maths id="math0034" num="(26)"><math display="block"><msub><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mn mathvariant="italic">1</mn></mrow></msub><mo>=</mo><msub><mi>x</mi><mi>n</mi></msub><mo>+</mo><mi mathvariant="normal">γ</mi><mo>∇</mo><mi>F</mi><mfenced><msub><mi>x</mi><mi mathvariant="normal">n</mi></msub></mfenced><mo>,</mo><mi>n</mi><mo>≥</mo><mn>0.</mn></math><img file="EP1908011B1_D0034.tif" /></maths>

0189Since <i>F</i>(<i>x₀</i>) ≤ <i>F</i>(<i>x₁</i>) ≤ <i>F(x₂)</i> ≤ ..., the sequence (x_n) is expected converges to a local maximum.

0190Preferably, the set of views selected with the quality function is increased by at least one randomly selected view. The randomly selected view(s) increase the probability that the quality of information obtained with the further views is maximized globally rather than locally.

0191As discussed above, selecting the best set of size <i>N</i> from amongst a large set of candidate projections is computationally complex. Since the size of the collection of views and of the required set may be large, a brute force scheme might not be computationally feasible.

0192In an additional preferred embodiment, a so-called "greedy algorithm" is used to incrementally construct larger and larger sets, until the required number of further views is defined. When multiple further views are required, it is computationally complex to maximize the quality function over all possible combinations of further views. The greedy algorithm reduces the computational burden by selecting the further views one at a time. The algorithm starts with a current set of views, and for each iteration determines a single view that yields the maximum improvement of the set score (hence the name "greedy").

0193In theoretical terms, assume ρ(·) is the quality measure we are using for the view selection, and assume without loss of generality that we are trying to maximize this measure. We gradually build a set <i>W</i> of projections as follows. We start with an empty set <i>W</i> =∅, and at every stage choose the projection that maximizes the quality measure when added to the current set: <maths id="math0035" num="(27)"><math display="block"><mi>W</mi><mo>←</mo><msub><mi>arg max</mi><mi mathvariant="italic">Wʹ</mi></msub><mfenced open="{" close="}"><mi mathvariant="normal">ρ</mi><mfenced><mi mathvariant="italic">Wʹ</mi></mfenced><mrow><mo>|</mo></mrow><mi mathvariant="italic">Wʹ</mi><mo>=</mo><mi>W</mi><mo>∪</mo><mfenced open="{" close="}"><mi mathvariant="normal">φ</mi></mfenced><mo>,</mo><mi mathvariant="normal">φ</mi><mo>∈</mo><mi mathvariant="bold">Φ</mi></mfenced></math><img file="EP1908011B1_D0035.tif" /></maths>

0194In other words, during a given iteration, a respective score is calculated for a combination of the previous set with each of the views which is not a member of the current set. The current set is then expanded by adding the view which yielded the highest respective score, and the expanded current set serves as the input to the following iteration. Thus the number of times the scoring function is calculated per iteration drops from iteration to iteration. For a large collection of possible views, the greedy algorithm reduces the total number of computations required for set selection.

0195Reference is now made to <figref idref="f0023">Fig. 10</figref>, which is a simplified flowchart of an iterative "greedy" method for defining further views, according to a preferred embodiment of the present invention. In step 1000 a collection of views of the body structure is provided. The collection of views includes possible further views for future measurements, preferably based on anatomical and other constraints. In step 1010, the set of views used for the previous emission measurements is established as a current set of views. In step 1020 the view set is incrementally increased by a single further view during each iteration, until the required number of further views has been selected.

0196Reference is now made to <figref idref="f0024">Fig. 11</figref>, which is a simplified flowchart of a single iteration of the view selection method of <figref idref="f0023">Fig. 10</figref>, according to a preferred embodiment of the present invention. The method of <figref idref="f0024">Fig. 11</figref> expands the current set of views by a single view. The method begins with a current set of views, which is the predetermined set (step 1010 above) for the first iteration of the greedy algorithm, or the set formed at the end of the previous iteration (step 1120 below) for all subsequent iterations. In step 1100, a respective expanded set is formed for each view not yet in the current set of views. A given view's expanded set contains all the views of the current set of views as well as the given view. In step 1110, a respective score is calculated for each of the expanded sets using the quality function. In step 1120, the view which yielded the highest-scoring expanded set is selected as a further view, to be used for further radioactive emission measurements. Finally, in step 1130, the current set is equated to the highest-scoring expanded set by adding the selected view to the current set. The newly formed current set serves as an input to the subsequent iteration, until the desired number of views is attained.

0197Reference is now made to <figref idref="f0025">Fig. 12</figref>, which is a simplified flowchart of a method for dynamically defining further views,, according to a third preferred embodiment of the present invention. In step 1210, a collection of possible further views for performing radioactive-emission measurements of the body structure is provided. Each of the views is associated with at least one viewing parameter. Preferably the viewing parameters consist of at least one the following: detector unit location, detector unit orientation, collection angle, and measurement duration.

0198In step 1220 at least one quality function is provided. Each quality function is for evaluating sets of views, essentially as described above. A single quality function may be used to select several sets of views, where each set of views contains a different number of views.

0199In step 1230, multiple sets of further views (where a set may include a single further view) are formed from the collection of views, using the quality function(s) provided in step 1220. In a first preferred embodiment, each of the sets is formed using a different one of the quality functions. In an alternate preferred embodiment, one or more of the quality functions are used to form more than one set of views, where sets formed with the same quality function have differing numbers of views.

0200In step 1240, a selected set of views is obtained from the sets formed in step 1230.

0201In a first preferred embodiment, the final set of views is obtained by choosing one of the sets formed in step 1230 using a set selection criterion. For example, a respective set is formed in step 1230 for the separability and reliability criteria independently. A set selection criterion which calculates an overall performance rating for a given set taking both criteria into account is defined, and the formed set with the highest overall rating is selected as the final set.

0202In another preferred embodiment, the selected set of views is obtained by merging the sets formed in step 1230 according to the relative importance of the respective quality function used to form each set.

0203In the preferred embodiment, the method further consists of providing at least one emittance model and/or reconstruction representing the radioactive-emission density distribution of the volume, and of evaluating with at least one of the quality functions of step 1220 is performed in relation to the emittance models.

0204As discussed above, since each view is associated with one or more parameters, the selected set yields a group of parameter values for performing effective detection of the intensity distribution of the body structure. For example, if each view is associated with a view location parameter the selected set defines a set of locations for collecting emission data from an object, in order to provide a high-quality reconstruction of the intensity distribution of the body structure.

0205Reference is now made to <figref idref="f0026">Fig. 13</figref>, which is a simplified block diagram of measurement unit for performing radioactive-emission measurements of a body structure, according to a preferred embodiment of the present invention. Measurement unit 1300 includes probe 1310, analyzer 1320 and view definer 1330. Probe 1310 performs the radioactive-emission measurements of the body structure. Radioactive-emission-measuring probe 1310 preferably comprises several detecting units, which may be of different geometries and different collection angles δ, within a housing (see for example <figref idref="f0001">Fig. 1d</figref>). Preferably, the orientation and/or collection angle of the individual collimators is controllable. Analyzer 1320 analyzes the radioactive-emission measurements obtained from probe 1310. View definer 1330 dynamically defines further views for measurements, based on the analysis provided by analysis unit 1320. The analysis and view definition are performed substantially as described above.

0206The abovedescribed methods for radioactive-emission measurements of a body structure begin by performing measurements at a predetermined set of views. The results of the initial measurements are then analyzed and further views are defined.

0207The initial set of views is preferably selected based on information theoretic measures that quantify the quality of the data fed to the reconstruction algorithm, in order to obtain the best data for reconstructing a three-dimensional image of the body structure.

0208In accordance with the present invention, our approach is delineated by the following process: <ol id="ol0007" compact="compact"><li>i. modeling the body structure or region of interest as a model of a volume U, possibly with one or several modeled organ targets HS, within anatomical constraints AC (see <figref idref="f0015">Figs. 7a-7b</figref>);</li><li>ii. obtaining an optimal and permissible initial set of views for the modeled volume U; and</li><li>iii. initiating the radioactive-emission measurements at the optimal set of views.</li></ol>

0209The following section concentrates on the second step of the process, namely, obtaining the optimal and permissible set of initial views for performing the radioactive-emission measurements of the body structure. The initial predetermined set of views is denoted herein the optimal set of views.

0210We consider here the following problem: Assume that there is a large pool of candidate views to choose from, but due to time restrictions or other restrictions we are limited to a specific number of views <i>N.</i> Which are the best <i>N</i> projections in terms of the quality of the reconstruction? It is further assumed that the pool of projections may be constrained, and hence general sampling theorems (<i>e.g.,</i> Radon Transform) cannot be applied. For instance, we consider a scenario in emission tomography where the detecting unit can be located on top of one face of a given volume but not on the others. In such cases, methods such as EM estimation do not clearly establish what is the best scanning scheme, in order to provide the best reconstruction of the radioactive intensity distribution of the volume.

0211Reference is now made to <figref idref="f0027">Fig. 14</figref>, which is a simplified flowchart of a method for selecting an optimal initial predetermined set of views of a volume to be imaged, according to a preferred embodiment of the present invention. In step 1400, an emittance model of the body structure being imaged is provided. The body structure is modeled as a volume, and is preferably represented as a collection of voxels. In contrast to the use of emittance models for evaluation of the quality function, in the following all of the emittance models are provided prior to performing the radioactive-emission measurements, and thus do not include reconstructions of the body structure based on a current round of measurements. In step 1410, a collection of views of the body structure is provided. Each view reflects a possible data collection option, from which the goal is to select a set of views which together provide the desired image reconstruction quality. Each view is associated with one or more viewing parameters, such as the location and orientation of the detecting unit with respect to the volume, as described in detail above.

0212The view set is preferably tailored to correspond to the actual limitations and constraints likely to be encountered when collecting data for a given object, that is a given body structure. For example, in medical applications the detecting unit can generally view the body structure only from certain distances and orientations, which are determined by the patient's anatomy and the structure of the probe. The view set may therefore be designed to contain views having only those viewing parameter values consistent with attainable distances and orientations. The view set may also be designed to contain views having only those viewing parameter values suitable for a given measurement scenario, for example having identical values for the type of radiophaimaceutical and time since administration.

0213Preferably, the collection of views represents a quantized continuum of views. The view collection thus reflects the detection probability distribution for a detecting unit making periodic measurements while moving along a trajectory.

0214Reference is now made to <figref idref="f0028">Fig. 15</figref>, which shows a non-limiting example of a volume 1500 with two views, V₁ and V₂. Volume 1500 divided into twenty-seven cube-shaped voxels (labeled 1-27). Each view reflects a number of viewing parameters, in the case shown a location (in the XYZ coordinate system), orientation (ω, θ and σ) and collection angle δ. The viewing parameters determine which voxels in volume 1500 are within a detecting unit's collection angle, and generally affect the probability of detection for each voxel. The probability of detection may be affected by additional factors, including the attenuation coefficients within the volume.

0215During data collection, the probability of detection for each voxel is dependent on the parameters outlined above. In the preferred embodiment, a respective projection is calculated for each view, giving the view's detection probability distribution (i.e. the detection probability for each voxel of the volume). For a given view, the associated projection will have significant detection probabilities only for those voxels within the sector defined by the detecting unit's collection angle and location and orientation, as illustrated in conjunction with <figref idref="f0004">Fig. 1j</figref>.

0216The detection probability distribution for each view, that is the group of probabilities for each voxel, for a given view, is calculated according to techniques known in the art for determining the detection probability of a radioactive emission from a given voxel, under the constraints specified by the viewing parameters, for example, based on computer simulations of the geometry and the other viewing parameters, delineated hereinabove.

0217Generally, more distant voxels will have a lower probability of detection than closer voxels along the same line of sight. The volume attenuation coefficient may be constant or may vary over the volume. Thus, different sets of views may be produced for a single sector, by defining volumes with differing attenuation coefficients. For example, bone tissue and muscle tissue have different attenuations. Anatomical knowledge of the body structure being imaged may be used to develop a model of the volume U with a non-uniform attenuation that reflects the expected attenuation of the given body structure.

0218Referring again to <figref idref="f0027">Fig. 14</figref>, in step 1420 a scoring function is provided. The scoring function rates the information that is obtainable from the volume using any set of views (containing at least one view) taken from the collection of views. Evaluation of the scoring function does not require data from previously performed measurements. This is in contrast to the quality function, which rates the information quality on the basis of previous emission measurements as well as by the characteristics of possible further views. Preferably, the scoring function is a function of the detection probability distribution, as given by the projections. Several preferred embodiments of the scoring function, are discussed in detail below.

0219In step 1425, sets of views are formed from the collection of views. A score is then calculated for each of the sets.

0220In step 1430, the scores calculated in step 1425 are used to select one of the formed sets of views as the optimal set of views for the abovedescribed methods of performing radioactive-emission measurements of a body structure. A given scoring function may be used to select a set in a number of ways. In a first preferred embodiment, a required number of views is specified, and the highest scoring set with the specified number of views is selected. In a second preferred embodiment, the user may specify a minimal score which is known to provide satisfactory information quality, and select the smallest set which provides the specified score. However given a large collection of views the required number of calculations may be prohibitive. A third preferred embodiment used to reduce the computational burden is the greedy algorithm embodiment, similar to the abovedescribed method of <figref idref="f0023 f0024">Figs. 10-11</figref>.

0221As discussed above, the scoring function is a measure of the quality of information which may be gathered for the volume using the given set of views. Both the separability and reliability criteria discussed above may be utilized as scoring functions, substantially as described above.

0222In an additional preferred embodiment, the scoring function implements a uniformity criterion, to ensure uniform coverage of the volume. It is often desired to obtain a uniform reconstruction quality among a given set of voxels <i>W</i>⊆<i>U</i>, where <i>W</i> is the set of voxels for which it is desired to obtain uniform detection. Note that by selecting the appropriate <i>W</i>, the uniformity criterion is applied to all or a portion of the body. The uniformity criterion ensures that the spread of the total influence of each element on the set of measurements is as uniform as possible. The uniformity criterion depends only on the collection of views Φ and requires no assumptions on the distribution <i>I</i>.

0223For a set Φ of projections, the total influence of an element, u, is given by ∑_ϕ∈Φ ϕ (<i>u</i>). Normalizing these values to <i>P</i>_Φ(<i>u</i>), such that <maths id="math0036"><math display="inline"><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>n</mi><mo>∈</mo><mi>W</mi></mrow></munder></mstyle><msub><mi>P</mi><mi mathvariant="normal">Φ</mi></msub><mfenced><mi>u</mi></mfenced></mstyle><mo>=</mo><mn>1</mn><mo>,</mo></math><img file="EP1908011B1_D0036.tif" /></maths> a probability measure is obtained for which the entropy <i>H</i>(Φ) can serve as a uniformity measure: <maths id="math0037" num="(28)"><math display="block"><mi>H</mi><mfenced><mi mathvariant="normal">Φ</mi></mfenced><mo>=</mo><mo>-</mo><mstyle displaystyle="false"><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>u</mi><mo>∈</mo><mi>W</mi></mrow></munder></mstyle><msub><mi mathvariant="normal">P</mi><mi mathvariant="normal">Φ</mi></msub></mstyle><mspace width="1em" /><mi>log</mi><mspace width="1em" /><msub><mi mathvariant="normal">P</mi><mi mathvariant="normal">Φ</mi></msub><mfenced><mi>u</mi></mfenced></math><img file="EP1908011B1_D0037.tif" /></maths>

0224The selected set Φ* is the set (containing the required number of views) that satisfies: <maths id="math0038" num="(29)"><math display="block"><mi mathvariant="normal">Φ</mi><mo>*</mo><mo>=</mo><msub><mi>arg max</mi><mi mathvariant="normal">Φ</mi></msub><mo>⁢</mo><mi>H</mi><mfenced><mi mathvariant="normal">Φ</mi></mfenced></math><img file="EP1908011B1_D0038.tif" /></maths>

0225Reference is now made to <figref idref="f0029">Fig. 16</figref>, which illustrates the concept of uniform coverage of a volume. Volume 1600 consists of twenty-seven cube-shaped voxels, including voxels 1610 and 1620. <b>A, B, C</b> and <b>D</b> are three views of volume 1600, showing the detector position and orientation for each view. For the uniformity criterion, it is desired that the overall influence of voxels 1610 and 1620 be approximately equal.

0226Assume that the probabilities of detection are as follows: <tables id="tabl0001" num="0001"><table frame="all"><tgroup cols="3"><colspec colnum="1" colname="col1" colwidth="14mm" /><colspec colnum="2" colname="col2" colwidth="22mm" /><colspec colnum="3" colname="col3" colwidth="22mm" /><thead><row><entry valign="top">View</entry><entry valign="top">Voxel 1610</entry><entry valign="top">Voxel 1620</entry></row></thead><tbody><row><entry><b>A</b></entry><entry>0.6</entry><entry>0</entry></row><row><entry><b>B</b></entry><entry>0.2</entry><entry>0.5</entry></row><row><entry><b>C</b></entry><entry>0</entry><entry>0.3</entry></row><row><entry><b>D</b></entry><entry>0</entry><entry>0.1</entry></row></tbody></tgroup></table></tables>

0227Consider two possible sets of views: set {<b>A, B, C</b>} and set {<b>B, C, D</b>}. For set {<b>A, B, C</b>}, the total contribution of voxel 1610 is 0.8 (0.6+0.2+0) and of voxel 1620 is 0.8 (0+0.5+0.3). Normalizing these values for set {<b>A, B, C</b>} gives a probability set of [0.5,0.5]. For set {<b>B, C, D</b>}<b>,</b> the total contribution of voxels 1610 is 0.2 (0.2+0+0) and of voxel 1620 is 0.9 (0.5+0.3+0.1). Normalizing these values for set for set {<b>B, C, D</b>} gives a probability set of [0.18,0.82]. Thus: <maths id="math0039"><math display="block"><mi mathvariant="normal">H</mi><mfenced><mfenced open="{" close="}"><mi mathvariant="bold">A</mi><mi mathvariant="bold">B</mi><mi mathvariant="bold">C</mi></mfenced></mfenced><mo>=</mo><mo>-</mo><mfenced><mn>0.5</mn><mo>*</mo><msub><mi>log</mi><mn>2</mn></msub><mspace width="1em" /><mn>0.5</mn><mo>+</mo><mn>0.5</mn><mo>*</mo><msub><mi>log</mi><mn>2</mn></msub><mspace width="1em" /><mn>0.5</mn></mfenced><mo>=</mo><mo>-</mo><mfenced><mo>-</mo><mn>0.5</mn><mo>-</mo><mn>0.5</mn></mfenced><mo>=</mo><mn>1</mn></math><img file="EP1908011B1_D0039.tif" /></maths><maths id="math0040"><math display="block"><mi mathvariant="normal">H</mi><mfenced><mfenced open="{" close="}"><mi mathvariant="bold">B</mi><mi mathvariant="bold">C</mi><mi mathvariant="bold">D</mi></mfenced></mfenced><mo>=</mo><mo>-</mo><mfenced><mn>0.18</mn><mo>*</mo><msub><mi>log</mi><mn>2</mn></msub><mspace width="1em" /><mn>0.18</mn><mo>+</mo><mn>0.82</mn><mo>*</mo><msub><mi>log</mi><mn>2</mn></msub><mo>⁢</mo><mn>0.82</mn></mfenced><mo>=</mo><mo>-</mo><mfenced><mo>-</mo><mn>0.44</mn><mo>-</mo><mn>0.07</mn></mfenced><mo>=</mo><mn>0.51</mn></math><img file="EP1908011B1_D0040.tif" /></maths>

0228Set {<b>A, B, C</b>} is thus seen to provide a more uniform coverage of volume 1600 than set {<b>B, C, D</b>}.

0229In an additional preferred embodiment, the greedy algorithm is used to incrementally construct larger and larger sets, until a set containing the required number of views is obtained. As described for <figref idref="f0023 f0024">Figs. 10-11</figref> above, the algorithm starts with an initial set of views (which may be the empty set), and in each iteration adds the view that yields the maximum improvement of the set score, until the required optimal set of views is obtained.

0230In a further preferred embodiment of a method for the selection of an optimal set, multiple view sets are first formed from one or more scoring functions, and then a final selection is made of one of the resulting sets, substantially as described for <figref idref="f0025">Fig. 12</figref> above. Preferably the viewing parameters consist of at least one the following: detector location, detector orientation, viewing angle, material, thickness, collimator length, septa thickness, cell size, detection duration, time of detection, and a type of radiopharmaceutical. Knowledge about the conditions under which a particular set of measurements will be taken may be used to determine the optimal final set of views for a given data collection scenario.

0231The abovedescribed methods may each be embodied as a computer program stored on a computer-readable storage medium. In the preferred embodiment, computer-readable storage medium contains a set of instructions for defining views for radioactive-emission measurements of the body structure. An analysis routine analyzes the radioactive-emission measurements obtained from a radioactive-emission-measuring probe, and a view definition routine dynamically defines further views for measurements, based on the analyzing.

0232The active vision embodiments discussed above may be provided as modifications to improve reconstruction reliability. Reference is now made to <figref idref="f0030">Fig. 17</figref>, which is a simplified flowchart of a method for stabilizing the reconstruction of an imaged volume with dynamic view definition, according to a preferred embodiment of the present invention. <figref idref="f0030">Fig. 17</figref> presents an embodiment in which the modifications to improve reconstruction reliability are performed by the active vision techniques described herein. In step 1700 radioactive emission measurements are performed. In step 1710, the reliability of reconstruction is analyzed, for example utilizing the SVD analysis described above. In step 1720, a determination is made whether measurements are finished, preferably based on the reliability analysis performed in step 1710. If it is determined that modifications are required, further views are defined without ceasing the measurement process, and preferably utilizing active vision techniques. Further measurements are then performed at the newly-defined views, until an adequate reconstruction reliability is achieved or it is decided to tenninate the measurement process for other reasons.

0233The above-described improved reconstruction reliability and active vision techniques enable high-quality reconstruction based on data collected from a limited collection of views. The ability to improve reconstruction reliability, both before and during imaging, reduces the need for repeated imaging to obtain additional measurements. Reconstructing the intensity distribution from a smaller quantity of collected data utilizing a stable reconstruction process simplifies the computational process. Furthermore, these methods enable resolving the current conflict between the relatively large-pixel detectors needed for measurement speed and data processing considerations, with the small-pixel detectors needed until now to obtain a high-resolution reconstruction. The abovedescribed embodiments are particularly suitable for medical imaging purposes, where a high-resolution image is needed and it is desired to minimize the difficulties of the patient undergoing the diagnostic testing or treatment.

0234It is expected that during the life of this patent many relevant detection probes, detector types, radiation-based detection systems, algorithms for reconstruction and analyzing the reliability of reconstruction, and algorithms will be developed and the scope of the corresponding terms are intended to include all such new technologies <i>α priori.</i>

0235It is appreciated that certain features of the invention, which are, for clarity, described in the context of separate embodiments, may also be provided in combination in a single embodiment. Conversely, various features of the invention, which are, for brevity, described in the context of a single embodiment, may also be provided separately or in any suitable subcombination.

0236Although the invention has been described in conjunction with specific embodiments thereof, it is evident that many alternatives, modifications and variations will be apparent to those skilled in the art. Accordingly, it is intended to embrace all such alternatives, modifications and variations that fall within the scope of the appended claims.

0237In addition, citation or identification of any reference in this application shall not be construed as an admission that such reference is available as prior art to the present invention.