Systems for spectral multiplexing, rendering, and demultiplexing of source images

0001The present invention relates to a system or systems for spectrally multiplexing a plurality of source images so as to provide a composite image, rendering the composite image, and demultiplexing of such a composite image to recover one or more of the source images.

0002Spectral multiplexing, as used herein, refers to a process for encoding plural source images in a composite image. Composite image rendering refers to a process for rendering the composite image in a physical form. Spectral demultiplexing refers to a process for recovering at least one of the encoded source images from the rendered composite image, such that the recovered source image is made distinguishable from, or within, the composite image, by subjecting the rendered composite image to a narrow-band illuminant that is preselected to reveal the source image.

0003Accordingly, the present invention is directed to methods and apparatus for spectrally-encoding plural source images and for providing the spectrally-encoded plural source images in a composite image, for rendering the composite image in a physical form, or for recovering at least one of the encoded source images from the rendered composite image such that the recovered source image is made distinguishable. That is, when the rendered composite image is subjected to illumination by one of the narrow band illuminants for which a source image was encoded, the source image becomes visually detectable by an observer. An illuminant that is designed to particularly interact with a given colorant is said to be complementary, and vice versa.

0004In one preferred embodiment of the invention as defined in claim 9 or 10, the step of encoding of the plural source images includes encoding at least a first source image that is encoded so as to be a background image having image content that is visually discernible on the substrate during a plurality of modes of intended illumination, and a second source image that is encoded so as to be a narrow band image having image content that is visually discernible on the substrate during a single mode of intended illumination, and the embodiment further comprises the step of: <ul id="ul0001" list-style="none" compact="compact"><li>recovering the first and second encoded source images from the rendered composite image, such that the recovered first and second source images are made distinguishable by subjecting the rendered composite image to a narrow-band illuminant that is preselected to reveal the second source image, and such that the image content of the second source image is discernible and thus achieves visual prominence with respect to the first source image.</li></ul>

0005In a further preferred embodiment of the invention as defined in claims 9 or 10, the step of encoding of the plural source images further comprises the step of encoding the plurality of source images in a composite image according to an image-dependent dynamic range determination so as to provide a maximum usable contrast in at least one recovered source image. <ul id="ul0002" list-style="none"><li>Figure 1 represents reflectance spectra for a white paper substrate and colorants in the form of Cyan, Magenta, Yellow, and Black dyes (at 100% density) operable in a dye sublimation printer.</li><li>Figure 2 represents the relative radiance spectra for the red, green, blue primaries generated by a typical cathode ray tube (CRT).</li><li>Figure 3 is a block diagram of systems for spectral multiplexing and demultiplexing of plural source images, and for rendering a composite image having therein at least one encoded source image, constructed according to the invention.</li><li>Figure 4 is a simplified schematic diagram of methods operable in the system of Figure 3 for spectrally multiplexing first and second source images in a composite image, rendering the composite image with use of respective first and second colorants, and for demultiplexing the rendered composite image.</li><li>Figure 5 is a schematic simplified representation of the spectral multiplexing system of Figure 3, in which an image processing unit and associated peripheral devices and subsystems are employed.</li><li>Figure 6 is a simplified schematic representation of the spectral demultiplexing system of Figure 3, in which a controller and associated peripheral devices and subsystems are employed.</li><li>Figure 7 is a schematic representation illustrating the dominance of a cyan image subjected to illumination by white light.</li><li>Figure 8 is a schematic representation illustrating the operation of gray component replacement (GCR) in the production of a rendered composite image, wherein the density of a cyan image when subjected to white light may be increased in comparison to the density of the cyan image when subjected to red light.</li><li>Figure 9 is a rendered composite image, wherein first and second source images were encoded in a composite image and the composite image was rendered in cyan and yellow colorants, wherein the first and second source images are intended for subsequent recovery when subjected to red and blue illuminants, respectively.</li><li>Figure 10 is a rendered composite image created with a 80% GCR fraction, wherein the appearance of the rendered composite image under red and blue illuminants is substantially identical to the appearance of the rendered composite image provided in Figure 9, and wherein the rendered composite image appears under white light to be more confused.</li><li>Figures 11 and 12 are rendered composite images each of which were rendered using GCR in formation of a composite image rendered in cyan and magenta colorants.</li><li>Figure 13 is a rendered composite image wherein GCR was utilized to incorporate first and second source images intended for recovery under blue and red illumination plus a third source image intended for recovery under white light illumination.</li><li>Figure 14 is a rendered composite image wherein a random variation between 0 and 80% in the GCR fraction was implemented over square blocks of pixels.</li><li>Figure 15 is a rendered composite image that includes a background image.</li></ul>

0006Under normal viewing illumination, the eye adapts to the <i>white-point,</i> which usually corresponds to blank paper with the highest reflectance and different colors can be seen by the eye for prints made with different colorant combinations. However, under relatively narrow band illumination, such as that obtained from a phosphor excited by a single gun of a CRT monitor, the eye is unable to distinguish color. Images viewed under narrow band illumination therefore appear to have only varying levels of gray and little or no chroma. Since the absorption characteristics of each of a plurality of colorants will differ in different spectral bands, the respective reflectance (or density) of each colorant when subjected to a series of differing narrow band illuminants will also appear to have varying levels of gray.

0007The present invention accordingly exploits the interaction between certain narrow band illuminants and their corresponding (complementary) colorants (especially the colorants typically used for printing), and the manner in which the eye detects images illuminated with illuminants having narrow band spectral power distributions. The methodology described herein may be generalized to apply to an arbitrary number of illuminants and colorants, and for the purpose of simplicity the invention is described with reference to the cyan, magenta, yellow, and black colorants commonly used in color printing applications, and to the narrow-band red, green, and blue illuminants commonly generated by CRT-based light sources. This description thus makes reference to the handling of monochromatic and color source images encoded according to an array of colorants such as the CMYK color primaries. However, it will be apparent to one of ordinary skill in the art that there are alternative spectral schemes to be employed in the spectral multiplexing of the invention. An alternative would include a color system that employs primary colorants other than CMYK for color representations, such as systems that use RGB primaries or high-fidelity colorants such as orange and green. Still another alternative would be to employ the invention in a system that processes different types of multi-spectral data, such as source images encoded with respect to narrow band colorants responsive to illuminants generated from ultraviolet or infrared light sources.

0008As the present invention is directed to the multiplexing or demultiplexing of at least one source image encoded in a composite image, the composite image may be defined in a spectrally multiplexed (SM) image plane. This plane may have any number of different patterns of pixels, with a primary characteristic being that the plane is spectrally multiplexed. In general, at each location in the SM plane, a pixel value one or more spectral components may be present, and which spectral component is present depends on the gray level of the corresponding pixel in one of the source image planes. (The invention may also have applications to SM planes in which each pixel includes color values representative of color separation image data from more than one source image plane.)

0009The general theory of the invention may be understood with reference to a rendering device in the form of a color hardcopy output device, such as a printer, and to a mathematical framework that employs nomenclature similar to that used in conventional color imaging. Consider a color hardcopy output device with M colorants. Prints from this device are to be viewed under <i>N</i> different illuminants, {<i>L</i>_<i>i</i>}<maths id="math0001" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext mathvariant="italic">N</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext><mtext>=1</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0001.tif" /></maths>. The luminance characterization of the printer under the <i>K</i> viewing lamps is given by the relation between the control values {<i>A</i>_<i>j</i>}<maths id="math0002" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext mathvariant="italic">M</mtext></mrow><mrow><mtext mathvariant="italic">j</mtext><mtext>=1</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0002.tif" /></maths> used for each of the <i>M</i> colorants at a given pixel location and the luminance produced at the given pixel location under each of the <i>N</i> illuminants. This can be denoted as the set of <i>N</i> functions, where <i>i=1,2,....N</i>:<maths id="math0003" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">f</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow></msub><mtext>(</mtext><msub><mrow><mtext mathvariant="italic">A</mtext></mrow><mrow><mtext>1</mtext></mrow></msub><mtext>,</mtext><msub><mrow><mtext mathvariant="italic">A</mtext></mrow><mrow><mtext>2</mtext></mrow></msub><mtext>,...</mtext><msub><mrow><mtext mathvariant="italic">A</mtext></mrow><mrow><mtext mathvariant="italic">M</mtext></mrow></msub><mtext>) = luminance of region</mtext><mspace linebreak="newline" /><mtext> with colorant control values </mtext><msub><mrow><mtext mathvariant="italic">A</mtext></mrow><mrow><mtext>1</mtext></mrow></msub><mtext>,</mtext><msub><mrow><mtext mathvariant="italic">A</mtext></mrow><mrow><mtext>2</mtext></mrow></msub><mtext>,...</mtext><msub><mrow><mtext mathvariant="italic">A</mtext></mrow><mrow><mtext mathvariant="italic">M</mtext></mrow></msub><mtext> under ith illumination </mtext><msub><mrow><mtext mathvariant="italic">L</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow></msub></mrow></math><img file="EP1408473A2_D0003.tif" /></maths>

0010In the following description, we assume that a control value of 0 for a given colorant represents no printing of that colorant. This convention is not a requirement for the invention and is only adopted for notational simplicity.

0011The description herein is limited to the case of luminance characterization alone, because under narrow band illumination the eye primarily sees differences of luminance and is unable to distinguish most color differences. Note that luminance as described here agrees in concept with its standard usage, i.e., as a measure of the perceived light energy; however, it's definition is not limited to the conventional usage and is expanded to comprehend the special viewing situations also described herein. In particular, under narrow band illumination, specific visual effects may influence the perception of a source image. A specific instance of this is the Purkinje effect that causes increased sensitivity in the blue region of the spectrum at low light levels, which may be of particular relevance for viewing under blue light and CRT illumination in general. Some of the advanced concepts from photometry and colorimetry that are required in such situations are described for instance in G. Wyszecki and W.S. Stiles, <i>Color Science: Concepts and Methods, Quantitative Data and Formulae, 2</i>^<i>nd</i><i> Edition,</i> John Wiley and Sons (1982).

0012The methods of the present invention are directed to the multiplexing, rendering, and recovery via demultiplexing of a source image encoded in a composite image. We assume that the one or more source images to be recovered are described by the spatial luminance distributions desired under each of the illuminants (although, in the alternative, any other equivalent specification that can be transformed to luminance/density may be used.) Thus, there are <i>N</i> images specified, with <i>Y</i>_<i>i</i>(<i>x</i>,<i>y</i>) being the desired luminance values that we wish to produce under the <i>ith</i> illuminant <i>L</i>_<i>i</i> where <i>x, y</i> denote the two spatial coordinates. For the purposes of simplifying the notation in the following discussion, the spatial dependence is sometimes dropped in the following description with the understanding that the discussion applies to each pixel location independently.

0013To examine the basic methodology symbolically, consider a simplified example of a composite image rendered in cyan and yellow colorants. In the simplified example below, additivity of "RGB" densities is assumed. This is for the purposes of simple illustration of the principles only and not intended to restrict the invention; in those situations where this approximation is invalid, more precise assumptions can be made. In this example: C, M, Y, K and R, G, B will respectively denote the colorants and illuminants; a superscript will denote illuminant; and a subscript will denote a colorant. Let: <ul id="ul0003" list-style="none" compact="compact"><li><i>d</i>^<i>R</i> = density of the image perceived under R illumination,</li><li><i>d</i>^<i>B</i> = density of the image under B,</li><li><i>d</i>_<i>C</i>^<i>R</i> = density C separation under R,</li><li><i>d</i>_<i>C</i>^<i>B</i> = density C separation under B,</li><li><i>d</i>_<i>Y</i>^<i>R</i> = density Y separation under R,</li><li><i>d</i>_<i>Y</i>^<i>B</i> = density Y separation under B.</li></ul>

0014When illuminated with a R or B illuminant, the total density perceived can be approximated as,<maths id="math0004" num=""><math display="block"><mrow><msup><mrow><mtext mathvariant="italic">d</mtext></mrow><mrow><mtext mathvariant="italic">R</mtext></mrow></msup><mtext>(</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>) = </mtext><msub><mrow><mtext mathvariant="italic">d</mtext></mrow><mrow><mtext mathvariant="italic">C</mtext></mrow></msub><msup><mrow><mtext mathvariant="italic"> </mtext></mrow><mrow><mtext mathvariant="italic">R</mtext></mrow></msup><mtext>(</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>) + </mtext><msub><mrow><mtext mathvariant="italic">d</mtext></mrow><mrow><mtext mathvariant="italic">Y</mtext></mrow></msub><msup><mrow><mtext mathvariant="italic"> </mtext></mrow><mrow><mtext mathvariant="italic">R</mtext></mrow></msup><mtext>(</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>) ≈ </mtext><msub><mrow><mtext mathvariant="italic">d</mtext></mrow><mrow><mtext mathvariant="italic">C</mtext></mrow></msub><msup><mrow><mtext mathvariant="italic"> </mtext></mrow><mrow><mtext mathvariant="italic">R</mtext></mrow></msup><mtext>(</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>)</mtext></mrow></math><img file="EP1408473A2_D0004.tif" /></maths><maths id="math0005" num=""><math display="block"><mrow><msup><mrow><mtext mathvariant="italic">d</mtext></mrow><mrow><mtext mathvariant="italic">B</mtext></mrow></msup><mtext>(</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>) = </mtext><msub><mrow><mtext mathvariant="italic">d</mtext></mrow><mrow><mtext mathvariant="italic">C</mtext></mrow></msub><msup><mrow><mtext mathvariant="italic"> </mtext></mrow><mrow><mtext mathvariant="italic">B</mtext></mrow></msup><mtext>(</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>) + </mtext><msub><mrow><mtext mathvariant="italic">d</mtext></mrow><mrow><mtext mathvariant="italic">Y</mtext></mrow></msub><msup><mrow><mtext mathvariant="italic"> </mtext></mrow><mrow><mtext mathvariant="italic">B</mtext></mrow></msup><mtext>(</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>) ≈ </mtext><msub><mrow><mtext mathvariant="italic">d</mtext></mrow><mrow><mtext mathvariant="italic">Y</mtext></mrow></msub><msup><mrow><mtext mathvariant="italic"> </mtext></mrow><mrow><mtext mathvariant="italic">B</mtext></mrow></msup><mtext>(</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>)</mtext></mrow></math><img file="EP1408473A2_D0005.tif" /></maths>

0015Accordingly, this methodology exploits the characteristically low density of a colorant when subjected to a first illuminant and the characteristically high density exhibited by the same colorant when subjected to a second, differing illuminant. Thus, at least one perceptibly distinct source image (that is encoded in the rendered composite image by use of the particular colorant), will be imperceptible (or nearly so) to an observer when subjected to the first illuminant, but perceptibly distinguishable to an observer when illuminated by the second illuminant. Upon perception of the source image by an observer, the source image may be comprehended and the information embedded in the composite image, or the composite image itself, is thereby readily comprehended.

0016The example presented above assumed that colorant interactions can be entirely ignored. This assumption is not true with most practical colorants and additional considerations are therefore required.

0017Consider the case of a rendered composite image that is produced by using C and M colorants for subsequent illumination under red and green illuminants. For simplicity, in our illustration below we assume additivity for the red, green, blue band densities, as the general case for situations where this approximation does not hold is described subsequently. A first source image may be recovered primarily from the cyan component of a composite image, and a second source image may be recovered primarily from the magenta component; however, unwanted absorption by these colorants are preferably compensated to avoid artifacts discernible by an observer. The total density under red illumination at pixel location (x,y) can be approximated as<maths id="math0006" num=""><math display="block"><mrow><msup><mrow><mtext>d</mtext></mrow><mrow><mtext>R</mtext></mrow></msup><msub><mrow><mtext>(x,y) = d</mtext></mrow><mrow><mtext>C</mtext></mrow></msub><msup><mrow><mtext> </mtext></mrow><mrow><mtext>R</mtext></mrow></msup><msub><mrow><mtext>(x,y) + d</mtext></mrow><mrow><mtext>M</mtext></mrow></msub><msup><mrow><mtext> </mtext></mrow><mrow><mtext>R</mtext></mrow></msup><mtext>(x,y)</mtext></mrow></math><img file="EP1408473A2_D0006.tif" /></maths> and the total density under green illumination is<maths id="math0007" num=""><math display="block"><mrow><msup><mrow><mtext>d</mtext></mrow><mrow><mtext>G</mtext></mrow></msup><msub><mrow><mtext>(x,y) = d</mtext></mrow><mrow><mtext>M</mtext></mrow></msub><msup><mrow><mtext> </mtext></mrow><mrow><mtext>G</mtext></mrow></msup><msub><mrow><mtext>(x,y) + d</mtext></mrow><mrow><mtext>C</mtext></mrow></msub><msup><mrow><mtext> </mtext></mrow><mrow><mtext>G</mtext></mrow></msup><mtext>(x,y)</mtext></mrow></math><img file="EP1408473A2_D0007.tif" /></maths> where d_U^V (x,y) represents the visual density under illuminant V due to colorant U at pixel location (x,y).

0018The terms d_M^R(x,y) and d_C^G(x,y) represent the unwanted absorption. In the simplest case, it can be assumed that a colorant's absorption under its complementary illuminant is used for two purposes: 1 ) to recover the desired image and 2) to compensate for unwanted absorption by the other colorant(s) present in the composite image. So a magenta colorant may be used to produce the desired image to be seen under green illumination and to compensate for the unwanted absorption of the cyan colorant; a cyan colorant may be used to produce the desired image under red illumination and to compensate for unwanted absorption of magenta under red illumination.

0019The portion that is used to compensate for the unwanted absorption should combine with the unwanted absorption to result in a constant spatial density so as to make it "disappear". Let d1_C^R(x,y) represent the portion of Cyan density that is used to compensate for the unwanted absorption of Magenta under red, which is determined by<maths id="math0008" num=""><math display="block"><mrow><msub><mrow><mtext>d1</mtext></mrow><mrow><mtext>C</mtext></mrow></msub><msup><mrow><mtext> </mtext></mrow><mrow><mtext>R</mtext></mrow></msup><msub><mrow><mtext>(x,y) + d</mtext></mrow><mrow><mtext>M</mtext></mrow></msub><msup><mrow><mtext> </mtext></mrow><mrow><mtext>R</mtext></mrow></msup><msup><mrow><mtext>(x,y) = constant = q</mtext></mrow><mrow><mtext>R</mtext></mrow></msup></mrow></math><img file="EP1408473A2_D0008.tif" /></maths>

0020The remaining density contribution of Cyan under red illumination is d2_C^R(x,y) = d_C^R(x,y) - d1_C^R(x,y). Note that the total density can be written in terms of these components as<maths id="math0009" num=""><img file="EP1408473A2_D0009.tif" /></maths>

0021Therefore the overall visual density under red illumination corresponds a constant background density of q^R with the spatially varying density pattern of d2_C^R(x,y) superimposed. This spatially varying pattern is the one that is seen under red illumination and should therefore represent the first multiplexed image that is to be seen under red illumination.

0022In a similar manner the density contribution of magenta under green illumination can be decomposed into a component d1_M^G(x,y) that is used to compensate for the unwanted absorption of cyan under green illumination, given by<maths id="math0010" num=""><math display="block"><mrow><msub><mrow><mtext>d1</mtext></mrow><mrow><mtext>M</mtext></mrow></msub><msup><mrow><mtext> </mtext></mrow><mrow><mtext>G</mtext></mrow></msup><msub><mrow><mtext>(x,y) + d</mtext></mrow><mrow><mtext>C</mtext></mrow></msub><msup><mrow><mtext> </mtext></mrow><mrow><mtext>G</mtext></mrow></msup><msup><mrow><mtext>(x,y) = constant = q</mtext></mrow><mrow><mtext>G</mtext></mrow></msup></mrow></math><img file="EP1408473A2_D0010.tif" /></maths> and the remaining component<maths id="math0011" num=""><math display="block"><mrow><msub><mrow><mtext>d2</mtext></mrow><mrow><mtext>M</mtext></mrow></msub><msup><mrow><mtext> </mtext></mrow><mrow><mtext>G</mtext></mrow></msup><msub><mrow><mtext>(x,y) = d</mtext></mrow><mrow><mtext>M</mtext></mrow></msub><msup><mrow><mtext> </mtext></mrow><mrow><mtext>G</mtext></mrow></msup><msub><mrow><mtext>(x,y) - d1</mtext></mrow><mrow><mtext>M</mtext></mrow></msub><msup><mrow><mtext> </mtext></mrow><mrow><mtext>G</mtext></mrow></msup><mtext>(x,y)</mtext></mrow></math><img file="EP1408473A2_D0011.tif" /></maths> which satisfies<maths id="math0012" num=""><img file="EP1408473A2_D0012.tif" /></maths>

0023Therefore the overall visual density under green illumination corresponds to a constant background density of K^G with the spatially varying density pattern of d2_C^R(x,y) superimposed. This spatially varying pattern is the one that is seen under red illumination and should therefore represent the second multiplexed image that is to be seen under green illumination.

0024Since the terms d2_C^R(x,y) and d2_M^G(x,y) represent the visual variations in density corresponding to the two multiplexed images, we would like to maximize their dynamic range. Since colorants can only add positive density, this requirement translates to minimizing the terms q^R and q^G subject to meeting the required equations and the physical constraint that colorants can only add positive density. We would therefore like to determine the smallest feasible values of q^R and q^G for which the above equations are feasible.

0025For the purpose of further illustration we use a first order approximation, that the amount of colorant added to compensate for unwanted absorption of the other colorant, itself only contributes a negligible amount of unwanted absorption (because of its small value). This assumption implies that the component of Magenta used to offset unwanted absorption of Cyan contributes negligibly to unwanted absorption under green and the component of Cyan used to offset unwanted absorption of Magenta contributes negligibly to unwanted absorption under blue. This assumption is used for illustration only, in practice, one can iteratively determine the appropriate amounts to account for higher-order effects or use an appropriate model/LUT as outlined subsequently in this disclosure. With this simplifying assumption, the range achievable for the desired spatially varying pattern d2_C^R(x,y) under red illumination is between q^R and d_C^R(x,y) with a total density variation or dynamic range of d_C^R(x,y) - q^R. Likewise the total density range available under green illumination is d_M^G(x,y) - q^G.

0026One set of feasible values for the terms q^R and q^G can be determined as:<maths id="math0013" num=""><math display="block"><mrow><msup><mrow><mtext>q</mtext></mrow><mrow><mtext>R</mtext></mrow></msup><msub><mrow><mtext> = max( d</mtext></mrow><mrow><mtext>M</mtext></mrow></msub><msup><mrow><mtext> </mtext></mrow><mrow><mtext>R</mtext></mrow></msup><msub><mrow><mtext>(x,y)) = d</mtext></mrow><mrow><mtext>M</mtext></mrow></msub><msup><mrow><mtext> </mtext></mrow><mrow><mtext>R</mtext></mrow></msup><mtext>(255) = max density for Magenta under red illuminant</mtext></mrow></math><img file="EP1408473A2_D0013.tif" /></maths><maths id="math0014" num=""><math display="block"><mrow><msup><mrow><mtext>q</mtext></mrow><mrow><mtext>G</mtext></mrow></msup><msub><mrow><mtext> = max(d</mtext></mrow><mrow><mtext>C</mtext></mrow></msub><msup><mrow><mtext> </mtext></mrow><mrow><mtext>G</mtext></mrow></msup><msub><mrow><mtext>(x,y)) = d</mtext></mrow><mrow><mtext>C</mtext></mrow></msub><msup><mrow><mtext> </mtext></mrow><mrow><mtext>G</mtext></mrow></msup><mtext>(255) = max density for Cyan under green illuminant</mtext></mrow></math><img file="EP1408473A2_D0014.tif" /></maths>

0027This can be thought of as follows: the background density under red light q^R is equal to the maximum unwanted density that one can have from Magenta. The Cyan density component d1_C^R(x,y) is designed carefully so that the combination of Cyan and Magenta at each pixel has a density q^R, this can be achieved by putting no Cyan where Magenta is 100% (255 digital count) and appropriate amounts of Cyan to make up the density to q^R at pixels which have less than 100% Magenta. A similar argument applies to the Magenta density component d1_M^G(x,y) that compensates for the unwanted absorption of Cyan under red illumination.

0028With the notation and terminology defined earlier, the general multi-illuminant imaging problem reduces to the following mathematical problem:

0029Given <i>N</i> luminance values {<i>Y</i>_<i>i</i>}<maths id="math0015" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext mathvariant="italic">N</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext><mtext>=1</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0015.tif" /></maths> corresponding to the desired luminance values under the <i>N</i> different illuminants, determine a set of control values for the <i>M</i> colorants {<i>B</i>_<i>j</i>}<maths id="math0016" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext mathvariant="italic">M</mtext></mrow><mrow><mtext mathvariant="italic">j</mtext><mtext>=1</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0016.tif" /></maths> to be used in printing a pixel, such that for all <i>i=1,2,....N :</i><maths id="math0017" num="(1)"><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">f</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow></msub><mtext>(</mtext><msub><mrow><mtext mathvariant="italic">B</mtext></mrow><mrow><mtext>1</mtext></mrow></msub><mtext>,</mtext><msub><mrow><mtext mathvariant="italic">B</mtext></mrow><mrow><mtext>2</mtext></mrow></msub><mtext>,...</mtext><msub><mrow><mtext mathvariant="italic">B</mtext></mrow><mrow><mtext mathvariant="italic">M</mtext></mrow></msub><mtext>) = luminance of pixel under ith illumination </mtext><msub><mrow><mtext mathvariant="italic">L</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow></msub><msub><mrow><mtext mathvariant="italic"> = Y</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow></msub></mrow></math><img file="EP1408473A2_D0017.tif" /></maths>

0030Typically, for <i>N>M</i> (number of image specifications > number of colorants) the system is over-determined and has a solution only under severe constraints on the {<i>Y</i>_<i>i</i>}<maths id="math0018" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext mathvariant="italic">K</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext><mtext>=1</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0018.tif" /></maths> luminance values limiting its utility in illuminant multiplexed imaging. Even if <i>N</i> ≤ <i>M</i> (number of image specifications ≤ number of colorants), the system of N equations presented in (1) above has a solution (corresponding to realizable device control values {<i>B</i>_<i>j</i>}<maths id="math0019" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext mathvariant="italic">M</mtext></mrow><mrow><mtext mathvariant="italic">j</mtext><mtext>=1</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0019.tif" /></maths>) only in a limited region of luminance values, which we refer to as the gamut for the spectrally multiplexed imaging problem:<maths id="math0020" num="(2)"><math display="block"><mrow><mtext mathvariant="italic">G</mtext><mtext> = gamut achievable for illuminant multiplexed imaging</mtext><mspace linebreak="newline" /><mtext> = { Y ∈ </mtext><msubsup><mrow><mtext mathvariant="italic">R</mtext></mrow><mrow><mtext>+</mtext></mrow><mrow><mtext mathvariant="italic">K</mtext></mrow></msubsup><mtext> such that system (1) has a realizable solution}</mtext></mrow></math><img file="EP1408473A2_D0020.tif" /></maths> where <b>Y</b> = [<i>Y</i>₁,<i>Y</i>₂,....<i>Y</i>_<i>N</i>], denotes the vector of luminance values under the <i>N</i> illuminants, and <i>R</i>₊ is the set of nonnegative real numbers. For specified <i>N</i>-tuples of luminance values within the gamut G, there is a set of realizable control values such that a pixel printed with the control values produces the required luminance values under the given illuminants. Vice versa, N-tuples of luminance values outside the gamut G cannot be created using any realizable control values. The situation is analogous to the limited color gamut encountered in color reproduction. It is necessary to include a gamut mapping step in the spectral multiplexing described herein to ensure that the source images are limited to the gamut of the system before attempting to reproduce them. The gamut mapping may be image independent or image dependent, where the term image is used to imply the set of desired source images recoverable under the different illuminants. In addition, the set of images to be multiplexed may be designed to take into account the gamut limitations and produce the best results with those gamut limitations.

0031Once the source images to be multiplexed have been mapped to the achievable gamut G, the problem of reproduction reduces to the determination of the control values for each of the M colorants for each pixel. This corresponds to an inversion of the system of equations in (1) and in a manner similar to color calibration, the inverse could be pre-computed and stored in <i>N</i>-dimensional look-up tables (LUTs), with one LUT one per colorant (or alternately, a single N-dimensional LUT with M outputs).

0032In practice, the function in (1) itself needs to be determined through measurements of the device response by printing a number of patches with different M-tuples of control values and measuring them suitably to obtain the luminance under the different illuminants. The full spectrum of the patches may be measured for instance on a spectrophotometer from which the luminances may be computed using the spectral power distribution of the different illuminants and the visual luminance sensitivity function. The visual luminance sensitivity function might incorporate adjustments for the appropriate light level that account for visual phenomena such as the Purkinje effect. See for instance, G. Wyszecki and W. S. Stiles, <i>Color Science: Concepts and Methods, Quantitative Data and Formulae,</i> 2^nd Ed., 1982, John Wiley and Sons, Inc., New York, NY, in particular pages 406-409. Several simplifications can be incorporated into the general framework above. Suppose first, that N=M and the colorants and lights are such that colorant i absorbs only illuminant <i>L</i>_<i>i</i> and is completely transparent to all other colorants, then we have<maths id="math0021" num=""><img file="EP1408473A2_D0021.tif" /></maths>

0033The system of equations in (1) then reduces to M independent nonlinear equations one for each colorant under the corresponding illumination:<maths id="math0022" num="(4)"><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">g</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow></msub><msub><mrow><mtext mathvariant="italic">(B</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow></msub><mtext mathvariant="italic">)</mtext><mtext> = </mtext><msub><mrow><mtext mathvariant="italic">Y</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow></msub><mtext>   </mtext><mtext mathvariant="italic">i=1,2,...N</mtext></mrow></math><img file="EP1408473A2_D0022.tif" /></maths>

0034The achievable gamut can be defined as follows. Let:<maths id="math0023" num=""><img file="EP1408473A2_D0023.tif" /></maths><maths id="math0024" num=""><img file="EP1408473A2_D0024.tif" /></maths><maths id="math0025" num=""><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">h</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow></msub><mtext> = [</mtext><msubsup><mrow><mtext mathvariant="italic">g</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>min</mtext></mrow></msubsup><mtext>, </mtext><msubsup><mrow><mtext mathvariant="italic">g</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>max</mtext></mrow></msubsup><mtext>] = the interval of luminances from </mtext><msubsup><mrow><mtext mathvariant="italic">g</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>min</mtext></mrow></msubsup><mtext> to </mtext><msubsup><mrow><mtext mathvariant="italic">g</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext>max</mtext></mrow></msubsup></mrow></math><img file="EP1408473A2_D0025.tif" /></maths> where<maths id="math0026" num="(5)"><math display="block"><mrow><mtext mathvariant="italic">i=1,2,...N</mtext></mrow></math><img file="EP1408473A2_D0026.tif" /></maths><maths id="math0027" num="(6)"><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">G</mtext></mrow><mrow><mtext>1</mtext></mrow></msub><mtext> = achievable gamut under assumption of one illuminant (6) interacting with only one colorant = </mtext><msub><mrow><mtext mathvariant="italic">h</mtext></mrow><mrow><mtext>1</mtext></mrow></msub><mtext> × </mtext><msub><mrow><mtext mathvariant="italic">h</mtext></mrow><mrow><mtext>2</mtext></mrow></msub><mtext> × ... × </mtext><msub><mrow><mtext mathvariant="italic">h</mtext></mrow><mrow><mtext mathvariant="italic">N</mtext></mrow></msub></mrow></math><img file="EP1408473A2_D0027.tif" /></maths>

0035In other words, the achievable gamut is the product set of these individual luminance intervals. Note that the assumption in Eq. (6) is that the complete interval between the max and min limits can be realized without any "gaps" which would typically be expected with physical colorants. (For a definition of a product set, see for instance, Friedman, <i>The Foundations of Modern Analysis</i>, Dover, 1982, New York, NY.)

0036Under the assumption of one illuminant interacting with only one colorant, the multi-illuminant imaging characterization problem reduces significantly. Instead of requiring <i>N</i>-dimensional LUTs only one-dimensional LUTs - one per colorant are needed. The value of each colorant may be determined by the luminance under the corresponding illumination alone.

0037In practice, the assumption of one illuminant interacting with only one colorant does not hold for typical colorants. However, if the strongest interactions are between the <i>i</i>th illuminant and the <i>i</i>th colorant with other interactions having a smaller magnitude, the achievable gamut is a reduced <i>N</i>-dimensional region that is contained in <i>G</i>_<i>1</i>. Note that the situation of using cyan, magenta, and yellow colorants with red, green, and blue lights for illumination corresponds to this case, where the cyan interacts most with red, magenta with green, and yellow with blue. Note also that the use of a black colorant that (typically) absorbs all illuminants almost equally, does not satisfy the requirement of strong interaction with only one illuminant. In practice this implies that a black colorant should be viewed as an additional colorant, i.e., if one colorant is black we should have:<maths id="math0028" num=""><math display="block"><mrow><mtext mathvariant="italic">N</mtext><mtext> = number of illuminants = number of images ≤ number of colorants -1 = </mtext><mtext mathvariant="italic">M</mtext><mtext>-1</mtext></mrow></math><img file="EP1408473A2_D0028.tif" /></maths>

0038Black may, however, be used with other colorants in special situations (as is described in the examples below) and can help improve achievable gamut (i.e., improve dynamic range), simplify computation, and reduce cost.

0039The general technique described earlier requires a measurement of the device response in the M-dimensional input space of device control values, and the final characterization may be embodied in the form of multi-dimensional LUTs with N-dimensional inputs. In several cases, the measurement and storage/computation requirements for multi-illuminant color imaging can be significantly reduced by using simple models of the output processes. One useful model is to assume that the visual densities follow an additive model, i.e.,<maths id="math0029" num=""><img file="EP1408473A2_D0029.tif" /></maths> where<maths id="math0030" num=""><img file="EP1408473A2_D0030.tif" /></maths> (Traditionally, densities are defined as logarithms to the base 10, any other base can also be used in practice as it changes the densities only by a scale factor and does not impact any of the other mathematical development.) Note as per our convention, the control values {0,0,...,0} represent an blank paper substrate and therefore <i>f</i>_<i>i</i>(0,0,...0) represents the luminance of the paper substrate under the ith illuminant, and the logarithmic terms represent paper normalized visual densities. The additive model for visual densities is motivated by the Beer-Bouguer law for transparent colorant materials and the assumption of relatively narrow band illumination, for which the additive nature of spectral density implies the approximation above is a valid one. The model also often provides a reasonable approximation for halftone media where the assumptions do not strictly hold. (For a more detailed background, see: F. Grum and C. J. Bartleson, Ed., <i>Optical Radiation Measurements: Color Measurement,</i> vol. 2, 1983, Academic Press, New York, NY or G. Sharma and H.J. Trussell, "Digital Color Imaging", <i>IEEE Transactions on Image Processing,</i> vol. 6, No. 7, pp. 901-932, July 1997.) Full computations using a spectral density model might be performed if necessary to improve the model accuracy, this would be potentially advantageous in a situation where the illuminating lights are not strictly narrow band.

0040The terms<maths id="math0031" num=""><img file="EP1408473A2_D0031.tif" /></maths> represent the paper normalized visual density of a patch printed with the <i>j</i>th colorant alone and no other colorants, with the control value for the <i>j</i>th colorant set as <i>A</i>_<i>j</i>. Therefore the additive density model proposed above allows the determination of the visual density of any patch based on the visual density of control patches of individual colorants. This reduces significantly the number of measurements required. Measurements of "step-wedges" of the individual colorants (for which other colorants are not printed) allow one to determine the functions <i>d</i>_<i>i</i>(<i>A</i>_<i>j</i>) <i>i</i>=1,2,...<i>N</i>, <i>j</i>=1,2,...<i>M</i>, from which the complete device characterization function can be determined using Eq. (8).

0041Using the above model, the system of equations in (1) reduces to:<maths id="math0032" num=""><img file="EP1408473A2_D0032.tif" /></maths>

0042The equations in (9) represent a system of <i>K</i> nonlinear equations in <i>M</i> variables (<i>B</i>₁,<i>B</i>₂,...<i>B</i>_<i>M</i>). The functions <i>d</i>_<i>i</i>(<i>A</i>_<i>j</i>) are available from the measurements of the "step-wedges" and the above equations can be solved for the control values <i>B</i>_<i>j</i> for luminance values within the gamut <i>G</i>, which was defined earlier. For points outside the gamut, the equations may be solved in an approximate sense providing a (less-controlled) form of gamut mapping.

0043Further simplification of these equations is possible by assuming that the densities in different spectral bands are linearly related, i.e.,<maths id="math0033" num="(10)"><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">d</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow></msub><mtext>(</mtext><mtext mathvariant="italic">C</mtext><msubsup><mrow><mtext>)=α</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow><mrow><mtext mathvariant="italic">j</mtext></mrow></msubsup><msub><mrow><mtext mathvariant="italic">d</mtext></mrow><mrow><mtext mathvariant="italic">j</mtext></mrow></msub><mtext>(</mtext><mtext mathvariant="italic">C</mtext><mtext>)   </mtext><mtext mathvariant="italic">i=1,2,...N</mtext></mrow></math><img file="EP1408473A2_D0033.tif" /></maths> where α<maths id="math0034" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext mathvariant="italic">j</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0034.tif" /></maths>= <i>d</i>_<i>i</i>(<i>C</i>)/<i>d</i>_<i>j</i>(<i>C</i>) is the proportionality factor relating the visual density for the <i>j</i>th colorant under the <i>i</i>th illuminant to the visual density for the <i>j</i>th colorant under the <i>j</i>th illuminant and is assumed to be independent of the colorant value C, and α<maths id="math0035" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext mathvariant="italic">j</mtext></mrow><mrow><mtext mathvariant="italic">j</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0035.tif" /></maths>= 1, Thus the convention adopted in Eq. (10) is that the density of the <i>j</i>th colorant under all other illuminants is referenced to its density under the <i>j</i>th illuminant itself, which is not strictly a requirement of our model but is chosen because it results in a simplification of the notation alternate conventions could also be equivalently used. Equation (10) is also motivated by the Beer-Bouguer law for transparent colorant materials and the assumption of relatively narrow band illuminants. (For a more detailed background, refer to: F. Grum and C. J. Bartleson, Ed., <i>Optical Radiation Measurements: Color Measurement,</i> vol. 2, 1983, Academic Press, New York, NY or G. Sharma and H.J. Trussell, "Digital Color Imaging", <i>IEEE Transactions on Image Processing,</i> vol. 6, No. 7, pp. 901-932, July 1997.) Even though a number of colorants and marking processes do not follow the Beer-Bouguer law exactly, in practice, Eq. (10) often provides a reasonably accurate empirical model for measured data and may be used for the purposes of the current invention. With the simplification of (10) the system of equations in (9) reduces to a linear system of equations:<maths id="math0036" num=""><img file="EP1408473A2_D0036.tif" /></maths> which can be written in matrix-vector notation as<maths id="math0037" num="(12)"><math display="block"><mrow><mtext>Ad=t</mtext></mrow></math><img file="EP1408473A2_D0037.tif" /></maths> where A is the <i>N</i>x<i>M</i> matrix whose ij th element is α<maths id="math0038" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext mathvariant="italic">j</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0038.tif" /></maths>, <b>d</b> is <i>M</i>x1 the vector whose <i>j</i>th component is <i>d</i>_<i>j</i>(<i>B</i>_<i>j</i>) and <b>t</b> is the <i>N</i>x1 vector whose <i>i</i>th component is log(<i>Y</i>_<i>i</i>/ <i>Y</i><maths id="math0039" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>0</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0039.tif" /></maths>).

0044The system of linear equations can be solved to determine a value of <b>d</b> , which provides the desired luminance values under the different illuminants (corresponding to the multiplexed images). The individual components of <b>d</b> , i.e., the <i>d</i>_<i>j</i>(<i>B</i>_<i>j</i>) values can then be used with the visual density response for the <i>j</i>th colorant under the <i>j</i>th illuminant to determine the control value corresponding to the <i>j</i>th colorant, i.e., <i>B</i>_<i>j</i>. This process is analogous to inverting a 1-D TRC. Repeating the process for each colorant provides the complete set of colorant control values required by {<i>B</i>_<i>j</i>}<maths id="math0040" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext mathvariant="italic">M</mtext></mrow><mrow><mtext mathvariant="italic">j</mtext><mtext>=1</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0040.tif" /></maths> that produce the desired set of luminance values under the different illuminants.

0045Note that if N=M, the above set of equations has a unique solution provided A is invertable, which is normally the case for typical colorants and illuminants. The solution in this case is obtained simply by inverting the matrix <b>A</b> . Furthermore, if the colorants and illuminants can be ordered in correspondence, i.e., colorant i absorbs illuminant i most and the other illuminants to a lesser extent, then α<maths id="math0041" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext mathvariant="italic">j</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0041.tif" /></maths> ≤ α<maths id="math0042" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext mathvariant="italic">j</mtext></mrow><mrow><mtext mathvariant="italic">j</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0042.tif" /></maths> = 1, for all <i>i=1,2...N,</i> i.e., the matrix <b>A</b> is square with the elements along the diagonal as the largest along each row, which is often desirable from a numerical stability standpoint. If <i>M>N</i> the system of equations will have multiple mathematical solutions, and the choice of a particular solution may be governed by additional criteria. One example of a criterion for choosing among the multiple mathematical solutions is feasibility, a feasible solution being a set of density values that can be realized with the range of colorant control values exercisable.

0046The model inherent in Eq. (12) can also be used to determine suitable approximations to the achievable gamut <i>G</i> and can be of assistance in performing gamut mapping. Typically, the density curves <i>d</i>_<i>j</i>(<i>C</i>) are monotonically increasing functions of the colorant control value C and the achievable range of densities for the <i>j</i>th colorant under the <i>j</i>th illuminant is between <i>d</i><maths id="math0043" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>min</mtext></mrow><mrow><mtext mathvariant="italic">j</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0043.tif" /></maths>= <i>d</i>_<i>j</i>(0) = 0 and <i>d</i><maths id="math0044" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>max</mtext></mrow><mrow><mtext mathvariant="italic">j</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0044.tif" /></maths>= <i>d</i>_<i>j</i>(<i>C</i><maths id="math0045" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>max</mtext></mrow><mrow><mtext mathvariant="italic">j</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0045.tif" /></maths>), where <i>C</i><maths id="math0046" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>max</mtext></mrow><mrow><mtext mathvariant="italic">j</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0046.tif" /></maths> is the maximum control value for the <i>j</i>th colorant. The achievable gamut assuming the model of Eq. (12) is valid is<maths id="math0047" num=""><img file="EP1408473A2_D0047.tif" /></maths> where <b>d</b>^min is an <i>M</i>x1 vector whose <i>j</i>th component is <i>d</i><maths id="math0048" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>min</mtext></mrow><mrow><mtext mathvariant="italic">j</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0048.tif" /></maths>= 0, and d^m^<i>ax</i> is an <i>M</i>x1 vector whose <i>j</i>th component is <i>d</i><maths id="math0049" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>max</mtext></mrow><mrow><mtext mathvariant="italic">j</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0049.tif" /></maths>, y is an <i>N</i>x1 vector whose <i>i</i>th component represents the luminance under the <i>i</i>th illuminant <i>L</i>_<i>i</i>, and y⁰ is a <i>N</i>x1 vector whose <i>i</i>th component represents the paper luminance under the <i>i</i>th illuminant. The inequalities, the division, and the logarithm in the right hand side of Eq. (13) are understood to be applicable on a term-by-term basis for the vectors. The <i>N</i> images to be produced under the <i>N</i> illuminants provide a <i>N</i>-tuple for each pixel location corresponding to the desired luminance values at that pixel location under the <i>N</i> illuminants. The <i>N</i>-tuples corresponding to all the pixel locations must lie within the gamut G defined earlier in order for the image to be producible using the given colorants and illuminants. If images specified for multiplexing do not satisfy this constraint some form of gamut mapping is necessary. A simple image-independent gamut mapping scheme may be defined as follows. First, ranges of luminance values under the different illuminants are determined such that all possible values within these ranges lie within the gamut G. This is mathematically equivalent to stating that we determine a set of <i>N</i>-intervals <i>S</i>_<i>i</i> = [<i>Y</i><maths id="math0050" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>min</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0050.tif" /></maths>,<i>Y</i><maths id="math0051" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>max</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0051.tif" /></maths>], <i>i=1,2,...N</i> such that the product set of these intervals is contained within the gamut G, i.e.,<maths id="math0052" num="(14)"><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">S</mtext></mrow><mrow><mtext>1</mtext></mrow></msub><mtext> × </mtext><msub><mrow><mtext mathvariant="italic">S</mtext></mrow><mrow><mtext>2</mtext></mrow></msub><mtext> × </mtext><msub><mrow><mtext mathvariant="italic">S</mtext></mrow><mrow><mtext>3</mtext></mrow></msub><mtext> ×...× </mtext><msub><mrow><mtext mathvariant="italic">S</mtext></mrow><mrow><mtext mathvariant="italic">N</mtext></mrow></msub><mtext> ⊆ </mtext><mtext mathvariant="italic">G</mtext></mrow></math><img file="EP1408473A2_D0052.tif" /></maths>

0047The gamut mapping may then be performed on an image independent basis by mapping the set of requested luminance values under the <i>i</i>th illuminant to the interval <i>S</i>_<i>i</i> = [<i>Y</i><maths id="math0053" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>min</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0053.tif" /></maths>,<i>Y</i><maths id="math0054" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>max</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0054.tif" /></maths>] by some (typically monotonous) function. The interval <i>S</i>_<i>i</i> determines the luminance dynamic range achieved under the <i>i</i>th illuminant. Since there are typically multiple choices of the sets {<i>S</i>_<i>i</i>}<maths id="math0055" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext mathvariant="italic">N</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext><mtext>=1</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0055.tif" /></maths> for which Eq. (14) is valid, one method for selecting the intervals may be by using a max min optimization where we maximize the minimum dynamic range achievable. Mathematically, this approach can be described as follows: Select the sets {<i>S</i>_<i>i</i>}<maths id="math0056" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext mathvariant="italic">N</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext><mtext>=1</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0056.tif" /></maths>) such that min_<i>i</i><i> f</i>(<i>S</i>_<i>i</i>) is maximized, where <i>f</i>(<i>S</i>_<i>i</i>) is some suitably chosen function that measures the contrast achieved corresponding to the luminance range <i>S</i>_<i>i</i>. Examples of suitable choices of the function <i>f</i>() are simple luminance ratio i.e., <i>f</i>(<i>S</i>_<i>i</i>) = <i>Y</i><maths id="math0057" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>max</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0057.tif" /></maths>/ <i>Y</i><maths id="math0058" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>min</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0058.tif" /></maths>, or density range <i>f</i>(<i>S</i>_<i>i</i>) = log(<i>Y</i><maths id="math0059" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>max</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0059.tif" /></maths>/ <i>Y</i><maths id="math0060" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>min</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0060.tif" /></maths>), or CIE lightness range <i>f</i>(<i>S</i>_<i>i</i>) = <i>L</i>*(<i>Y</i><maths id="math0061" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>max</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0061.tif" /></maths>)-<i>L</i>*(<i>Y</i><maths id="math0062" num=""><math display="inline"><mrow><mfrac linethickness="0"><mrow><mtext>max</mtext></mrow><mrow><mtext mathvariant="italic">i</mtext></mrow></mfrac></mrow></math><img file="EP1408473A2_D0062.tif" /></maths>), where <i>L</i>*() is the CIE lightness function. (See for instance, G. Wyszecki and W. S. Stiles, <i>Color Science: Concepts and Methods, Quantitative Data and Formulae,</i> 2^nd Ed., 1982, John Wiley and Sons, Inc., New York, NY.) Note that the choice of the density range as the function in the max-min optimization along with the model of Eq.(13) reduces this to a linear max-min optimization problem with box constraints that can be solved using numerical optimization schemes.

0048The method proposed above is independent of the images to be used in creating a composite image; this method however is based on worst case assumptions and may sacrifice some dynamic range in the recovered image.

0049Accordingly, the aforementioned methods for encoding and rendering of composite images may also utilize an image-dependent mapping that avoids such a loss in dynamic range. By determining the terms K^R and K^G in an image-dependent fashion, the dynamic range of the recovered source image can be improved.

0050In particular, we have found that regions of a particular rendered composite image that are intended to appear dark under plural illuminants will, in most instances, need little or no compensation for unwanted absorptions. Accordingly, a better overlap between the dark regions under plural illuminants can be achieved through suitable design of the source images, or by shifting the images appropriately.

0051Hence, an improved dynamic range can be obtained by using:<maths id="math0063" num=""><math display="block"><mrow><msup><mrow><mtext>q</mtext></mrow><mrow><mtext>R</mtext></mrow></msup><msub><mrow><mtext> = max</mtext></mrow><mrow><mtext>over all (x,y)</mtext></mrow></msub><msub><mrow><mtext>( d</mtext></mrow><mrow><mtext>M</mtext></mrow></msub><msup><mrow><mtext> </mtext></mrow><mrow><mtext>R</mtext></mrow></msup><msub><mrow><mtext>( i</mtext></mrow><mrow><mtext>G</mtext></mrow></msub><msub><mrow><mtext>(x,y) ) - d</mtext></mrow><mrow><mtext>C</mtext></mrow></msub><msup><mrow><mtext> </mtext></mrow><mrow><mtext>R</mtext></mrow></msup><msub><mrow><mtext>( i</mtext></mrow><mrow><mtext>R</mtext></mrow></msub><mtext>(x,y) ), 0 )</mtext></mrow></math><img file="EP1408473A2_D0063.tif" /></maths><maths id="math0064" num="(2)"><math display="block"><mrow><msup><mrow><mtext>q</mtext></mrow><mrow><mtext>G</mtext></mrow></msup><msub><mrow><mtext> = max</mtext></mrow><mrow><mtext>over all (x,y)</mtext></mrow></msub><msub><mrow><mtext> ( d</mtext></mrow><mrow><mtext>C</mtext></mrow></msub><msup><mrow><mtext> </mtext></mrow><mrow><mtext>G</mtext></mrow></msup><msub><mrow><mtext>(i</mtext></mrow><mrow><mtext>R</mtext></mrow></msub><msub><mrow><mtext>(x,y)) - d</mtext></mrow><mrow><mtext>M</mtext></mrow></msub><msup><mrow><mtext> </mtext></mrow><mrow><mtext>G</mtext></mrow></msup><msub><mrow><mtext>( i</mtext></mrow><mrow><mtext>G</mtext></mrow></msub><mtext>(x,y)), 0 )</mtext></mrow></math><img file="EP1408473A2_D0064.tif" /></maths> where i_R(x,y) represents the digital count for the image desired under red illumination, and i_G(x,y) represents the digital count for the image desired under green illumination, and d_U^V( t ) represents the visual density under illuminant V for colorant U at a digital count t. Note that q^R and q^G in Eq. (2) are guaranteed to be smaller than the values in Eq. (1) thereby ensuring a higher dynamic range for the desired density variations d2_C^R(x,y) and d2_M^G(x,y) corresponding to the rendered composite images, a condition that may be advantageous to image quality to design or shift the rendered composite images to have overlapping dark regions, which can minimize contrast degradation. This spatial effect can be seen in Eq. (2), where a spatial transformation of an image i can yield a reduced constant floor density K. Spatial optimization can be performed either computationally or by design.

0052The argument developed above can be extended to the case of images using C/M and K. By setting:<maths id="math0065" num=""><math display="block"><mrow><msub><mrow><mtext>q</mtext></mrow><mrow><mtext>e</mtext></mrow></msub><msup><mrow><mtext> = min (q</mtext></mrow><mrow><mtext>R</mtext></mrow></msup><msup><mrow><mtext>, q</mtext></mrow><mrow><mtext>G</mtext></mrow></msup><mtext>)</mtext></mrow></math><img file="EP1408473A2_D0065.tif" /></maths> and using a spatial value of K(x,y) such that the density corresponding to K(x,y)<maths id="math0066" num=""><math display="block"><mrow><msub><mrow><mtext>d( K(x,y) ) = max( q</mtext></mrow><mrow><mtext>e</mtext></mrow></msub><msub><mrow><mtext> - d</mtext></mrow><mrow><mtext>C</mtext></mrow></msub><msup><mrow><mtext>​</mtext></mrow><mrow><mtext>R</mtext></mrow></msup><msub><mrow><mtext>( i</mtext></mrow><mrow><mtext>R</mtext></mrow></msub><msub><mrow><mtext>(x,y)), q</mtext></mrow><mrow><mtext>e</mtext></mrow></msub><msub><mrow><mtext>- d</mtext></mrow><mrow><mtext>M</mtext></mrow></msub><msup><mrow><mtext>​</mtext></mrow><mrow><mtext>G</mtext></mrow></msup><msub><mrow><mtext>( i</mtext></mrow><mrow><mtext>G</mtext></mrow></msub><mtext>(x,y))</mtext></mrow></math><img file="EP1408473A2_D0066.tif" /></maths> the K takes up part of the responsibility for the unwanted absorptions increasing the dynamic range available for the desirable images.

0053Such dynamic range determinations relevant to the production of rendered composite images that employ cyan and yellow colorants, and of rendered composite images that employ cyan, yellow, and black colorants, can be developed in a manner similar to the above determinations for the rendered composite images that employ cyan and magenta colorants and rendered composite images that employ cyan, magenta, and black colorants. The determinations for the production of rendered composite images that employ cyan and yellow colorants is made simpler because the unwanted absorptions of yellow can be neglected.

0054Figure 3 illustrates a system 100 operable in a first mode for spectrally multiplexing a plurality of source images to form a composite image, in a second mode for rendering the composite image, or in a third mode for demultiplexing the spectrally multiplexed composite image so as to recover at least one of the plurality of source images for advantageous viewing by an observer.

0055As shown in Figure 3, a plurality of disparate source image arrays 11-1, 11-2, ...11-N are presented to an image input device 20 in a spectral multiplexing system 101. Image input device 20 may be equipped to receive plural monochromatic images or a combination of monochromatic and multichromatic images. Image input device 20 may include an image capture device such as a digital scanner coupled to a random access memory, or any type of analog or digital camera coupled to a storage means such as a computer memory or a magnetic or optical recording medium. Image input device 20 may also include means for receiving an image that had previously been stored in a random access memory, on video tape, or a laser-encoded disk, etc., or for receiving an image created by a computer image generator, or an image encoded in an appropriate format and transmitted on a network.

0056The illustrative representation of the plural source images in respective image arrays received by the image input device 20 in this example includes a first source image 12-1 represented in a first source image array 11-1 and a second source image 12-2 represented in a second source image array 11-2. The system 101 can optionally receive <i>N</i> source images which are represented in a respective image arrays. In this exemplary embodiment of the invention, disparate pictorial source images are employed and at least one of the plural source images is intended for ultimate recovery (via spectral demultiplexing) from a composite image.

0057Once the source image data is received in the input image device 20, it is presented to a spectral multiplexer 30, which encodes a data representation of a composite of at least the first and second source images, so as to provide a composite image 32 on an spectrally multiplexed (SM) image plane. Such encoding may proceed in one embodiment with mapping for every pixel location, or by mapping in localized areas rather than specific pixels, to the composite image 32, so as to multiplex the information necessary for encoding of each corresponding pixel located in each source image.

0058Next, according to operation of a composite image rendering system 102, data representative of the composite image is provided to a rendering device 40, which can be connected to the spectral multiplexer 30 by any one of a variety of suitable means for transmitting or storing electronic information. The rendering device 40 records the composite image 32 on a substrate 44 with use of a predetermined array of narrow band colorants, so as to form a rendered composite image 42. The rendered composite image 42 is thereby fixed on the substrate 44.

0059The rendered composite image 42 is available for viewing in ambient light by an observer 70. Although the rendered composite image 42 is representative of data encoded in the spectrally multiplexed plane using the method of the invention, the rendered composite image 42 typically exhibits a confused appearance under conventional ambient lighting conditions; at least one of the source images 12-1, 12-2, etc. is thus difficult or impossible to distinguish under conventional ambient lighting conditions. A particular source image is made difficult or impossible to distinguish until a demultiplexer 50 is operated to selectively illuminate the composite image 42 in a manner sufficient to reveal the desired source image. Alternatively, one or more of the source images may be encoded so as <i>avoid</i> visual confusion and therefore be visually apparent in the rendered composite image when the rendered composite image is subjected to ambient or wide band illumination, and become confused or difficult to detect when the rendered composite image is subjected to a complementary narrow band illuminant.

0060According to operation of a spectral demultiplexing system 103, a particular source image (as shown in Figure 3, source image 12-1) may be recovered and made distinguishable within the composite image 42. In the embodiment illustrated in Figure 3, the output of the demultiplexer 50 is directed to an observer 70 using the method of the invention. The recovered image is then distinguishable by the observer 70 as one substantially identical with, or a close approximation of, the particular source image 12-1 initially provided to the image input device 20.

0061Recovery of a particular source image will be understood to generally proceed according to an exemplary embodiment of the spectral demultiplexing system 103 as follows. The substrate 44 is positioned with respect to an illuminant source operable within the demultiplexer 50, such that a narrow band illuminant generated by the demultiplexer 50 illuminates the composite image 42 so as to subject the array of colorants in the rendered composite image 42 to the selected illuminant. As a result of the rendered composite image 42 thus being controllably and selectively illuminated by at least one illuminant, a desired source image is then detectable. In the illustrated embodiment, the desired source image is made visually distinguishable to an observer 70. The desired source image 12-1, now recovered, is thereby susceptible to comprehension by an observer 70.

0062Accordingly, by virtue of the aforementioned interaction of a colorant and its corresponding illuminant, and due to the visual response of the observer 70 to this particular interaction, each encoded source image may be present as a confused, or distinguishable, image depending upon the objective of the demultiplexing operation.

0063Figure 4 is a simplified schematic diagram of exemplary embodiments of spectral multiplexing, rendering, and spectral demultiplexing methods 61, 62, 63, respectively. In step 61 for multiplexing plural source images, a first source image 71 and a second source image 72 are provided to the multiplexer 30, which outputs a composite image data file to a rendering device 40. The output of the rendering device 40 is substrate 90 which has incorporated therein a composite image 92. The original source image 71 is rendered as a pattern using a first colorant; in the illustrated embodiment, a cyan ink or toner is chosen. The second source image 72 is rendered as a pattern using a second colorant; in the illustrated embodiment, a magenta ink or toner is chosen. (As there is typically some overlap in absorption bands between practical narrow band colorants, the two source images are preferably encoded in step 61 to account for the absorption that will occur when plural colorants are utilized to produce the composite image.)

0064In a rendering step 62, the composite image specifies patterns in cyan and magenta colorants that are accordingly rendered on a substrate 90 to form the rendered composite image 92. Those skilled in the art will appreciate that certain portions of the two patterns may be co-located and other portions are relatively spatially distinct. Nonetheless, in certain embodiments of the present invention, visual recognition of at least one of the source images in the composite image may be made difficult or impossible due to the confusion between source images that are encoded in the composite image.

0065In step 63 for demultiplexing the rendered composite image 92, the substrate 90 having the rendered composite image 92 fixed thereon is illuminated by the demultiplexer 50. Controlled illumination of the substrate 90 according to a first mode 51 of illumination causes the first source image 71 to achieve a particular level of density with respect to the remainder of the composite image and thus the first source image 71 becomes detectable on the substrate 90. Alternatively, controlled illumination of the substrate 90 according to a second mode 52 of illumination causes the second source image 72 to be similarly detectable on the substrate 90. In the illustrated embodiments, the first source image 71 and the second source image 72 are therefore selectably distinguishable on the substrate 90.

0066Figure 5 illustrates a schematic simplified representation of the spectral multiplexing system 101 of Figure 3, in which an image processing unit 130 and associated peripheral devices and subsystems are employed. An image input terminal 120 may include an image capture device 122 such as a scanner, digital camera, or image sensor array; a computer image generator 124 or similar device that converts 2-D data to an image; or an image storage device 126 such as a semiconductor memory or a magnetic, optical, or magneto-optical data storage device. The image input terminal 120 derives or delivers digital image data in the form of, for example, plural monochromatic image files, wherein the picture elements or "pixels" of each image are defined at some gray value. For example, the input terminal 120 may be employed to derive an electronic representation of, for example a document or photograph from image capture device 122, in a format related to the physical characteristics of the device, and commonly with pixels defined at m bits per pixel. If a color document, the image is defined with two or more separation bitmaps, usually with identical resolution and pixel depth. Image data from the input terminal 120 is directed to an image processing unit (IPU) 130 for processing so as to be encoded to create a composite image. It will be recognized that the data representing one or more source images is spectrally encoded by the image processing unit 130 to provide secondary image data representative of a composite image suitable for subsequent rendering.

0067The image processing unit 130 may include image memory 132 which receives input image data from image input terminal 120 or from another suitable image data source, such as an appropriately programmed general purpose computer (not shown) and stores the input image data in suitable devices such as random access memory (RAM). Image processing unit 130 commonly includes processor 134. The input image data may be processed via a processor 134 to provide image data representative of plural source images defined on respective source image planes in accordance with the present invention. For example, image data signals in the form of RGB or black and white (B/W) images may be processed, and the luminance information derived therefrom may be used to provide data representative of a source image. Image data signals presented in other formats are similarly processed: image data signals in, for example the L*a*b format, may be processed to obtain data representing a source image from the lightness channel. Image data signals that are already formatted in grayscale are generally usable without further processing.

0068Operation of the image processing unit 130 may proceed according to one or more image processing functions 138, 139 so as to encode the source image data into the composite image file as described hereinabove. Processing may include a color conversion which, if necessary, may be implemented to convert a three component color description to the printer-specific four or more component color description, and may include a halftoner which converts a c bit digital image signals to d bit digital image signals, suitable for driving a particular printer, where c and d are integer values. In certain embodiments, additional functions may include one or more of color space transformation, color correction, gamut mapping, and under color removal (UCR)/gray component replacement (GCR) functions. Control signals and composite image output data are provided to an interface 136 for output from the image processing unit 130.

0069The image processing unit 130 may be embodied as an embedded processor, or as part of a general purpose computer. It may include special purpose hardware such as for accomplishing digital signal processing, or merely represent appropriate programs running on a general purpose computer. It may also represent one or more special purpose programs running on a remote computer.

0070Figure 6 is a simplified schematic representation of the spectral demultiplexing system 103 of Figure 3, in which a controller and associated peripheral devices and subsystems are employed to present one or more recovered source images 171, 172. Figure 6 shows a controller 150 connected to a illuminant source 160 that is operable for subjecting the composite image 42 on substrate 44 to first and second predefined illuminants 161, 162. Firstly, as illustrated with reference to the rendered composite image 42 on substrate 44, under conventional ambient lighting and in the absence of an illuminant 161, 162, only the composite image 42 is distinguishable and no source image is detected. However, upon activation of the source 160 so as to provide the first predefined illuminant 161, the recovered source image 171 becomes detectable to an observer 170. Alternatively, the mode of operation of the source 160 may be switched so as to provide a second predefined illuminant 162, whereupon the composite image 42 is instead subjected to the second illuminant 162, and the recovered source image 172 becomes detectable.

0071In its simplest form the controller 150 may be constructed as a manually-operable illuminant selector switch. Alternatively, as illustrated, the controller 150 may be provided in the form of a computer-based control device having an interface 156 connected to source 160, which offers programmable control of the operation of the illuminant source 160. The controller 150 may thus be operated to cause selective activation and deactivation of the illuminant source 160 so as to provide one or more selected fields of illumination 162. Such control may, for example, the accomplished via manual operation of the illuminant source 160 by a human operator, or by programmable control afforded by a computer or similar means.

0072The controller 150 is operable for accomplishing tasks such as activation, deactivation, or sequencing of the illuminant source 160, setting illuminant intensity, illuminant frequency, etc.. Embodiments of the controller 150 benefit from operation of a programmable control system comprising standard memory 152 and processor 154. The controller 150 may be employed, for example, for supplying uniform R or G or B screen images to the interface 156 for subsequent display on the illuminant source 160 when the latter is constructed in the form of a CRT monitor.

0073Operation of the illuminant source 160 by the controller 150 may proceed according to certain sequenced control functions so as to provide, for example, controlled operation of the illuminant source 160 to afford a field of illumination that varies according to selective characteristics such as a sequential activation and deactivation of selected narrow band illuminants, or of controlled operation of the intensity of same; or with interactive control according to intervention by an operator of the particular sequence, intensity, or duration of the illuminants. As noted above, the rendered composite image may be constructed to have a plurality of source images encoded therein; for example, of at least first and second patterns of respective first and second colorants. The rendered composite image may be subjected to a temporal sequencing of illumination by respective first and second narrow band illuminants, thus allowing a respective one of the first and second recovered source images 171, 172 to be sequentially distinguishable.

0074As mentioned, the illuminant source 160 may be provided in the form of a CRT monitor having a screen positionable with respect to the substrate 44 for generating the requisite field of illumination sufficient to illuminate the rendered composite image 42. By way of example, the following description gives an example of control settings for generating a controlled field of illumination from a CRT monitor under the control of a desktop computer using a document presentation application, such as Microsoft PowerPoint. On a blank slide, a rectangle object is created that completely covers the extent of the landscape page. Using the menu selections, the user selects "Format AutoShape", then "Colors and Lines". A custom slide color may be specified. For example, to create a red slide, set Red to 255, Green to 0, Blue to 0. To create a green slide, set Red to 0, Green to 255, Blue to 0. To create a blue slide, set Red to 0, Green to 0, Blue to 255.

0075A gray component replacement technique may be applied to the darkness in a recovered image in the areas common to the different colorants when subjected to their complementary illuminants. For example, when composing and rendering a composite image, black can be used to replace a portion of the cyan colorant in the areas of the common darkness that appear under red light and to the yellow colorant in the areas of the common darkness that appear under blue light. Common image darkness produced with this black component will be more perceptible under white light in comparison to the darkness observed under illumination by a cyan illuminant or a yellow illuminant. Inclusion of GCR in the encoding and rendering of a composite image will provide a rendered composite image having a more confused appearance under white light.

0076In another aspect of this practice, the GCR fraction employed in this process can be modulated spatially to still further increase or decrease the desired level of confusion in the composite image.

0077Furthermore, the GCR fraction employed in this process can be modulated spatially so as to encode an additional, low resolution source image in the composite image. When the resulting rendered composite image is subjected to white light illumination, the additional, low-resolution image is visually discernible.

0078Using the cyan/yellow colorant example from above, the white light illumination problem may be written as<maths id="math0067" num="(3)"><math display="block"><mrow><msup><mrow><mtext mathvariant="italic">d</mtext></mrow><mrow><mtext mathvariant="italic">W</mtext></mrow></msup><mtext>(</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>)= </mtext><msub><mrow><mtext mathvariant="italic">d</mtext></mrow><mrow><mtext mathvariant="italic">C</mtext></mrow></msub><msup><mrow><mtext mathvariant="italic"> </mtext></mrow><mrow><mtext mathvariant="italic">W</mtext></mrow></msup><mtext>(</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>) </mtext><msub><mrow><mtext mathvariant="italic">+ d</mtext></mrow><mrow><mtext mathvariant="italic">Y</mtext></mrow></msub><msup><mrow><mtext mathvariant="italic"> </mtext></mrow><mrow><mtext mathvariant="italic">W</mtext></mrow></msup><mtext>(</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>) </mtext><msub><mrow><mtext mathvariant="italic">≈ d</mtext></mrow><mrow><mtext mathvariant="italic">C</mtext></mrow></msub><msup><mrow><mtext mathvariant="italic"> </mtext></mrow><mrow><mtext mathvariant="italic">W</mtext></mrow></msup><mtext>(</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>)</mtext></mrow></math><img file="EP1408473A2_D0067.tif" /></maths>

0079Cyan has a much higher density under white light compared to the density of yellow under white light, so the cyan image may be understood to dominate the appearance of a rendered composite image under white light.

0080Whereas the typical GCR technique uses common density of colorants under the same illuminant, the contemplated method for implementing GCR in the encoding and rendering of a composite image uses the common density of colorants under different illuminants. This common density is considered herein to the cross-illuminant-common density.

0081Continuing with the cyan/yellow colorant example, one may select a fractional (frac) amount of common density that will be used for black (K) addition and for cyan and yellow (C, Y) subtraction. Assume a printer linearized in density, the amount of colorant, and the density of the colorant under the complementary illuminant, in a synonymous fashion. Let<maths id="math0068" num="(4)"><math display="block"><mrow><msub><mrow><mtext mathvariant="italic">d</mtext></mrow><mrow><mtext mathvariant="italic">K</mtext></mrow></msub><mtext>(</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>) = frac * min [</mtext><msub><mrow><mtext mathvariant="italic">d</mtext></mrow><mrow><mtext mathvariant="italic">CR</mtext></mrow></msub><mtext>(</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>), </mtext><msub><mrow><mtext mathvariant="italic">d</mtext></mrow><mrow><mtext mathvariant="italic">YB</mtext></mrow></msub><mtext>(</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>)]</mtext></mrow></math><img file="EP1408473A2_D0068.tif" /></maths>

0082This amount of density will be subtracted from <i>d</i>_<i>CR</i> to yield <i>d</i>_<i>CR</i>_-GCR, and from <i>d</i>_<i>YB</i> to yield <i>d</i>_<i>YB</i>_-GCR and the K separation will be added to the composite image. To first-order, the density of the perceived images are as follows<maths id="math0069" num="(5)"><math display="block"><mrow><msup><mrow><mtext mathvariant="italic">d</mtext></mrow><mrow><mtext mathvariant="italic">R</mtext></mrow></msup><mtext mathvariant="italic">(x, y</mtext><mtext>) = </mtext><msub><mrow><mtext mathvariant="italic">d</mtext></mrow><mrow><mtext mathvariant="italic">C</mtext></mrow></msub><msup><mrow><mtext mathvariant="italic"> </mtext></mrow><mrow><mtext mathvariant="italic">R</mtext></mrow></msup><msub><mrow><mtext> </mtext></mrow><mrow><mtext>-GCR</mtext></mrow></msub><mtext>(</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>) </mtext><msub><mrow><mtext mathvariant="italic">+ d</mtext></mrow><mrow><mtext mathvariant="italic">Y</mtext></mrow></msub><msup><mrow><mtext mathvariant="italic"> </mtext></mrow><mrow><mtext mathvariant="italic">R</mtext></mrow></msup><msub><mrow><mtext> </mtext></mrow><mrow><mtext>-GCR</mtext></mrow></msub><mtext> (</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>) </mtext><msub><mrow><mtext mathvariant="italic">+ d</mtext></mrow><mrow><mtext mathvariant="italic">K</mtext></mrow></msub><mtext>(</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>) ≈ </mtext><msub><mrow><mtext mathvariant="italic">d</mtext></mrow><mrow><mtext mathvariant="italic">C</mtext></mrow></msub><msup><mrow><mtext mathvariant="italic"> </mtext></mrow><mrow><mtext mathvariant="italic">R</mtext></mrow></msup><msub><mrow><mtext> </mtext></mrow><mrow><mtext>-GCR</mtext></mrow></msub><mtext> (</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>) </mtext><msub><mrow><mtext mathvariant="italic">+ d</mtext></mrow><mrow><mtext mathvariant="italic">K</mtext></mrow></msub><mtext>(</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>) </mtext><msub><mrow><mtext mathvariant="italic">= d</mtext></mrow><mrow><mtext mathvariant="italic">C</mtext></mrow></msub><msup><mrow><mtext mathvariant="italic"> </mtext></mrow><mrow><mtext mathvariant="italic">R</mtext></mrow></msup><mtext>(</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>)</mtext></mrow></math><img file="EP1408473A2_D0069.tif" /></maths><maths id="math0070" num="(6)"><math display="block"><mrow><msup><mrow><mtext mathvariant="italic">d</mtext></mrow><mrow><mtext mathvariant="italic">B</mtext></mrow></msup><mtext>(</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>) </mtext><msub><mrow><mtext mathvariant="italic">= d</mtext></mrow><mrow><mtext>C</mtext></mrow></msub><msup><mrow><mtext> </mtext></mrow><mrow><mtext>B</mtext></mrow></msup><msub><mrow><mtext> </mtext></mrow><mrow><mtext>-GCR</mtext></mrow></msub><mtext> (</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>) </mtext><msub><mrow><mtext mathvariant="italic">+ d</mtext></mrow><mrow><mtext mathvariant="italic">Y</mtext></mrow></msub><msup><mrow><mtext mathvariant="italic"> </mtext></mrow><mrow><mtext mathvariant="italic">B</mtext></mrow></msup><msub><mrow><mtext> </mtext></mrow><mrow><mtext>-GCR</mtext></mrow></msub><mtext> (</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>) </mtext><msub><mrow><mtext mathvariant="italic">+ d</mtext></mrow><mrow><mtext mathvariant="italic">K</mtext></mrow></msub><mtext>(</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>) ≈ </mtext><msub><mrow><mtext mathvariant="italic">d</mtext></mrow><mrow><mtext mathvariant="italic">Y</mtext></mrow></msub><msup><mrow><mtext mathvariant="italic"> </mtext></mrow><mrow><mtext mathvariant="italic">B</mtext></mrow></msup><msub><mrow><mtext> </mtext></mrow><mrow><mtext>-GCR</mtext></mrow></msub><mtext> (</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>) </mtext><msub><mrow><mtext mathvariant="italic">+ d</mtext></mrow><mrow><mtext mathvariant="italic">K</mtext></mrow></msub><mtext>(</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>) </mtext><msub><mrow><mtext mathvariant="italic">= d</mtext></mrow><mrow><mtext mathvariant="italic">Y</mtext></mrow></msub><msup><mrow><mtext mathvariant="italic"> </mtext></mrow><mrow><mtext mathvariant="italic">B</mtext></mrow></msup><mtext>(</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>)</mtext></mrow></math><img file="EP1408473A2_D0070.tif" /></maths><maths id="math0071" num="(7)"><math display="block"><mrow><msup><mrow><mtext mathvariant="italic">d</mtext></mrow><mrow><mtext mathvariant="italic">W</mtext></mrow></msup><mtext>(</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>)= </mtext><msub><mrow><mtext mathvariant="italic">d</mtext></mrow><mrow><mtext mathvariant="italic">C</mtext></mrow></msub><msubsup><mrow><mtext mathvariant="italic"> </mtext></mrow><mrow><mtext>-GCR</mtext></mrow><mrow><mtext mathvariant="italic">W</mtext></mrow></msubsup><mtext> (</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>) + </mtext><msub><mrow><mtext mathvariant="italic">d</mtext></mrow><mrow><mtext mathvariant="italic">Y</mtext></mrow></msub><msubsup><mrow><mtext mathvariant="italic"> </mtext></mrow><mrow><mtext>-GCR</mtext></mrow><mrow><mtext mathvariant="italic">W</mtext></mrow></msubsup><mtext> (</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>) </mtext><msub><mrow><mtext mathvariant="italic">+ d</mtext></mrow><mrow><mtext mathvariant="italic">K</mtext></mrow></msub><msub><mrow><mtext>​</mtext></mrow><mrow><mtext>-GCR</mtext></mrow></msub><mtext> (</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>) ≈ </mtext><msub><mrow><mtext mathvariant="italic">d</mtext></mrow><mrow><mtext mathvariant="italic">C</mtext></mrow></msub><msubsup><mrow><mtext mathvariant="italic"> </mtext></mrow><mrow><mtext>-GCR</mtext></mrow><mrow><mtext mathvariant="italic">W</mtext></mrow></msubsup><mtext> (</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>) + </mtext><msub><mrow><mtext mathvariant="italic">d</mtext></mrow><mrow><mtext mathvariant="italic">K</mtext></mrow></msub><mtext> (</mtext><mtext mathvariant="italic">x</mtext><mtext>, </mtext><mtext mathvariant="italic">y</mtext><mtext>).</mtext></mrow></math><img file="EP1408473A2_D0071.tif" /></maths>

0083Note that under white light, a fraction of the cross-illuminant-common density d_K, now appears. This additional component yields a white light image that appears more confusing than the image described by Eq. (3). The example Illustrated in Figure 7 is repeated in Figure 8 with 80% GCR (frac = 0.8). Figure 8 shows that the density under white light differs more from the red light density image in the GCR image compared to the non-GCR image illustrated in Figure 7. In addition to this density effect, the composite image encoded and rendered with GCR has an additional hue effect that is not illustrated in Figures 7 and 8. That is, under white light, the regions with different amounts of cyan, magenta, and black also exhibit different hues, Thus adding to the confusion.

0084A composite image encoded in rendered using this GCR method will be discussed in Examples 1 and 2, below, wherein Example 1 uses cyan and yellow in a manner similar to the above first-order description, and Example 2 uses cyan and magenta.

0085The aforementioned GCR method may be implemented to encode and render an additional, low-resolution source image by use of a black (K) colorant. In that case, the fractional GCR component frac is given a spatial dependence according to the additional low-resolution source image. Example 3 discusses such a low-resolution image encoded according to the fraction of GCR that is applied.

0086A key consideration is that a colorant will absorb some light from a non-complementary illuminant, and thus it will be somewhat discernible under that illuminant. To effectively suppress this appearance of a residual image, one may calibrate the perceived density for each colorant and illuminant, and one may encode the source images so as to compensate for such spurious absorption.

0087Figure 9 is a rendered composite image wherein two source images are encoded in cyan and yellow that are respectively designed for viewing under red and blue illumination. The image uses a small amount of black (K) to compensate for unwanted absorptions by cyan (C) in the blue region (so as to make the cyan image less than discernible under blue illumination and to recover the source image in the presence of a yellow illuminant). The use of K greatly increases the dynamic range available for encoding the source image. Note that when this rendered composite image is viewed under white light, the image in cyan (C) dominates the other confused images and the source image encoded in yellow (Y) is hardly visible.

0088Figure 10 is a rendered composite image created with a 80% GCR fraction, wherein the appearance of the rendered composite image under red and blue illumination is substantially identical to the appearance of the rendered composite image provided in Figure 9, but the rendered composite image in Figure 10 under white light is more confused due to the application of GCR. One can discern a varying pattern of density and hue that is not immediately apparent as a single image.

0089Although the magenta colorant has the highest density under green light (e.g. <i>d</i>_<i>MG</i><i>(255)=1.38)</i>, the cyan density under green is high also (e.g. <i>d</i>_<i>CG</i><i>(255)=0.53</i>), which makes it discernible under green light. On the other hand, although the magenta density under blue light is slightly lower than under green light (e.g. <i>d</i>_<i>MB</i><i>(255)=0.98</i>), the cyan density under blue is much lower than under green (e.g. <i>d</i>_<i>CB</i><i>(255)=0.24</i>). Accordingly, a rendered composite image that employs a cyan colorant for representing an encoded source image will recover that source image under red light; similarly, an encoded source image represented in a magenta colorant is best recover under blue light. A small proportion of black colorant may be added to the cyan colorant to make the respective source image less discernible under blue light. A small proportion of black colorant may be added to the magenta colorant to make the respective source image less discernible under red light. Accordingly, Figures 11 and 12 are rendered composite images having encoded therein first and second source images intended for recovery under red and blue illuminants, respectively. Figure 11 was rendered in cyan and magenta colorants without the application of GCR. Figure 12 was rendered in cyan, magenta, and black colorants with the application of GCR. In comparing Figures 11 and 12 under white light, one may see that rendered composite image in Figure 12 appears more confused under white light in comparison to the rendered composite image in Figure 11.

0090Figure 13 is a composite image having encoded therein first and second source images intended for recovery under blue and red illumination wherein GCR has been utilized in the rendering of a composite image in cyan and yellow colorants, and a third source image that is encoded in the composite image for recovery under white light illumination. The amount of GCR is spatially varied in accordance with the image content of the third source image. In the rendered composite image of Figure 13, the image content of the third source image is a binary pattern in the shape of the "digital X" (a trademark of Xerox Corporation), with use of a 80% GCR fraction in the regions of the composite image that correspond to the image content of the third source image, and no GCR was implemented in the remaining regions of the composite image. When the image is subjected to red or blue light, a respective one of the first and second source images is recovered. Under white light, the third source image is discernible.

0091In alternative embodiments, the contemplated third source image may include or be restricted to image content that is encoded for detection primarily or exclusively by automated instrumentation (i.e. image content that is designed to be machine-readable rather than human-readable).

0092In still other embodiments, the third source image may be encoded as a grayscale image by use of a suitable halftoning technique.

0093Figure 14 is a rendered composite image that exemplifies an additional application of the contemplated GCR technique, wherein the GCR fraction was varied randomly between 0 and 80% over square blocks of pixels. Note that the resulting rendered composite image will reveal the encoded source images under illumination by red and blue illuminants but image confusion is evident in the rendered composite image when subjected to white light. The level of image confusion may be further optimized by choosing the image alignments with respect to the particular application of the GCR technique. The image confusion may be increased when the frequency content of the GCR matches that of the dominant image and the dark regions in the two encoded images align so as to allow a selected amount of variation in GCR.

0094Figure 15 exemplifies an embodiment of a rendered composite image wherein the rendered composite image includes at least first and second source images, wherein the first source image is encoded in a black (K) colorant so as to be a "background" source image discemible on the substrate during plural modes of intended illumination, and at least a second, narrow band source image that is encoded and rendered with use of a narrow band colorant. The background source image is discernible during the plural modes of intended illumination, and especially during a white light mode of illumination.

0095As indicated in Figure 15, the image content of the narrow band source image is spatially separate from the image content of the background source image. Upon the onset of a particular narrow band illumination, the second source image is made to exhibit a contrast that is substantially equal to that of the background image or to that of the substrate. The image content of the narrow band source image thus respectively becomes apparent alongside the image content of the background image, or becomes less discernible than the image content of the background image, depending upon the particular intensity of the narrow band illuminant.

0096In another feature of the invention, the colorants selected for rendering the composite image may be chosen for imparting a "tinted" or "hand-colored" appearance under white light to the rendered composite image, such that the rendered composite image offers visual interest to an observer even before the onset of the mode of illumination of the narrow band illuminant. Furthermore, and as illustrated in Figure 15, a third source image, in the form of an additional narrow band source image is optionally included for still more visual interest.

0097As represented by Figure 15, a yellow colorant will be expected to appear as a yellow tint under white illuminant and neutral under a blue illuminant. Practical examples of a yellow-tinted background image may be constructed using yellow and black toner according to the following considerations:<maths id="math0072" num=""><math display="block"><mrow><msub><mrow><mtext>yellow tinted image = Y</mtext></mrow><mrow><mtext>B</mtext></mrow></msub><msub><mrow><mtext> (i, j) + K</mtext></mrow><mrow><mtext>Y</mtext></mrow></msub><mtext>(i, j)</mtext></mrow></math><img file="EP1408473A2_D0072.tif" /></maths>

0098The cyan colorant will be expected to appear as a cyan tint under white illuminant, and neutral under a red illuminant. Practical examples of a cyan-tinted background image may be constructed using cyan and black toner according to the following considerations:<maths id="math0073" num=""><math display="block"><mrow><msub><mrow><mtext>cyan tinted image = C</mtext></mrow><mrow><mtext>R</mtext></mrow></msub><msub><mrow><mtext>(i,j) + K</mtext></mrow><mrow><mtext>C</mtext></mrow></msub><mtext>(i,j)</mtext></mrow></math><img file="EP1408473A2_D0073.tif" /></maths> where C_R is the C density under red illuminant, and K_C is set by the density C_DmaxR - C_R(i,j) where C_DmaxR is the max density of cyan under red illuminant.

0099In the illustrated examples, black (K) is used as the colorant for the background image, but this example is not limiting, as other colorants can be selected for use in the composite image for generating the background image.

0100Advantageous use is expected in color printing by various processes including offset lithography, letterpress, gravure, xerography, photography, and any other color reproduction process which uses a defined number of colorants, usually three or four, in various mixtures. Embodiments of the rendering system 102 include apparatus capable of depositing or integrating a defined array of colorants in a substrate, according to the composite image, such that the array of colorants is susceptible to selective reflection or transmission of a selected narrow band illuminant incident thereon. For example, the composite image may be rendered on a transparent film and a desired source image may be recovered when the substrate is backlit by a suitable narrow band illuminant. Examples include hardcopy reprographic devices such as inkjet, dye sublimation, and xerographic printers, lithographic printing systems, silk-screening systems, and photographic printing apparatus; systems for imagewise deposition of discrete quantities of a color on a substrate surface, such as paint, chemical, and film deposition systems; and systems for integration of colorant materials in an exposed surface of a substrate, such as textile printing systems.

0101Embodiments of exemplary substrates include, but are not limited to, materials such as paper, cardboard, and other pulp-based and printed packaging products, glass; plastic; laminated or fibrous compositions; and textiles. Narrow band colorants other than basic CMYK colorants may also be used for this invention.

0102The field of illumination for illuminating a rendered composite image may be provided by a variety of illuminant sources that include a narrow band light source responsive to manual control or to program control according to an illuminant source control signal. Various narrow band light sources may include apparatus for providing filtered sunlight, filtered incandescent, or filtered fluorescent light; coherent light sources such as a solid-state laser or laser diode; projection or image display devices such as those incorporating a cathode ray tube (CRT), flat-panel display (FPD), liquid crystal display (LCD), plasma display, or light emitting diode (LED) and organic light emitting (OLED) arrays. Light sources incorporating a cathode ray tube are advantageous in that they have phosphors that exhibit stable and well-understood spectral characteristics that are sufficiently narrow and complementary to common CMY colorants. In addition, such displays are widely available.