Adaptive nonlinear image enlargement using wavelet transform coefficients

FIELD OF THE INVENTION

[0001] The present invention relates to the field of image processing; more particularly, the present invention relates to edge preserving image interpolation with wavelets.

BACKGROUND OF THE INVENTION

[0002] Resizing of digital images is often performed in digital imaging processing and becomes more important in networks environments that include devices with different dpi resolution. While aliasing and moiré artifacts are the main problems in image reduction, enlargement of images has to deal with the problem of how to introduce high frequency components in order to have the image, in particular edges, not appear too smooth or too blurred. A typical method for enlargement of images is the use of an interpolation filter. This filtering incorporates information from neighboring pixels in order to predict an interpolation value. Commonly used filters as, e.g., in a popular photo-image manipulation application, are bilinear or bicubic interpolation filters. With the use of those filters a perfect step edge cannot be interpolated to produce a perfect step edge at a higher resolution. The interpolated edge will always look a bit blurred.

[0003] A standard method for image interpolation is polynomial interpolation. Depending on the degree of the interpolating polynomial (e.g., linear, quadratic, cubic, etc.), the image looks more or less smooth. The most commonly used technique is referred to as cubic interpolation. An advantage of polynomial methods is their simplicity since they are based on global linear filtering techniques. A disadvantage is that it is not possible to perform an adaptive interpolation, thereby resulting in edges typically being oversmoothed. This is a significant disadvantage in enlargement of documents.

[0004] Other interpolation filters exist, such as Keys filters, that are relatives of polynomial interpolation filters, but have characteristics of unsharp masking filters, i.e. they enhance high frequency content by creating a overshoot-undershoot at edges and an overshoot-undershoot is also created for noise pixel and leads to increase the noise level in the image. Since all these filters operate globally on the entire image, adaptive interpolation is not possible. A trade-off exists between enhancement of edges and suppressing noise in background areas.

[0005] Non-linear interpolation methods exist that operate in the pixel domain and extract edge information from the image and use that information to perform an edge-directed interpolation. One method first computes an edge map of the low resolution image using the Laplacian-of-Gaussian. In a second step, a preprocessing of the low resolution image using the edge information is performed to avoid errors in an estimated high resolution edge map. The third step performs interpolation using the edge information. In smooth areas, a bilinear interpolation is performed. Near edges, interpolated values are replaced by values that keep the sharpness of the edges. At last, an iterative correction step is performed to further improve the interpolation. A typical number of iterations is 10. See Allebach, J., and Wong, P. W., “Edge-directed interpolation,” Proceedings of ICIP'98, pp. 707-710, 1998.

[0006] In another method, local covariance characteristics in the low resolution image are estimated and those estimates are used to perform classical Wiener filtering interpolation. Since local covariances are part of the filter coefficients, a smoothing along edges, but not across edges, is performed. A disadvantage of this method is that isolated dots are not well-preserved after interpolation since they are treated as very short edges. See Li, X., and Orchard, M., “New edge directed interpolation,” Proceedings of ICIP'2000, Vancouver, 2000.

[0007] Compared to the previous two methods, a very simple edge sensitive interpolation method is proposed in Carrato, S., Ramponi, G., and Marsi. S., “A simple edge-sensitive image interpolation filter,” Proceedings of ICIP'96, pp. 711-714, 1996. This technique employs a nonlinear filter to determine the interpolating sample value. In detail, for a one-dimensional signal a local linear interpolation,

<i>x</i>_int=μ_k<i>x</i>_k+(1−μ_k)<i>x</i>_k+1  (1)

[0008] is performed. If x_intis close to 0, the interpolating value is similar to the sample to the right, whereas if x_intis close to 1, the interpolating value is similar to the sample to the left. This placement depends on the smoothness of the low resolution signal in a neighborhood of the interpolating value and is computed via the nonlinearity <maths id="MATH-US-00001" num="1"><math overflow="scroll"><mtable><mtr><mtd><mrow><msub><mi>μ</mi><mi>k</mi></msub><mo>=</mo><mfrac><mrow><mrow><mi>k</mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>-</mo><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>2</mn></mrow></msub></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mn>1</mn></mrow><mrow><mrow><mi>k</mi><mo>(</mo><mrow><msup><mrow><mo>(</mo><mrow><msub><mi>x</mi><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>-</mo><msub><mi>x</mi><mi>k</mi></msub></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo>(</mo><mrow><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>-</mo><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>2</mn></mrow></msub></mrow><mo>)</mo></mrow><mn>2</mn></msup></mrow><mo>)</mo></mrow><mo>+</mo><mn>2</mn></mrow></mfrac></mrow></mtd><mtd><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mtd></mtr></mtable></math><img file="US20030194150A1-20031016-M00001.TIF" id="EMI-M00001" he="22.08465" wi="216.027" img-format="tif" img-content="mf" /><attachments><attachment idref="MATHEMATICA-00001" attachment-type="nb" file="US20030194150A1-20031016-M00001.NB" /></attachments></maths>

[0009] where k is a parameter that controls the edge sensitivity. For k=0, linear interpolation is obtained, while positive values of k cause increased edge sensitivity. An advantage of this nonlinear technique is its simplicity—no iterations are necessary. A disadvantage of this technique is that the parameter k must be tuned and that isolated short edges do not get enlarged and look a bit “squeezed” in the interpolated image. Furthermore, the interpolation of a perfect step edge, e.g., x_k−1=x_k=1, x_k+1=x_k+2=0, is not a perfect step edge anymore: 1, 1, 1, ½, 0, 0, 0. A linear interpolation is performed.

[0010]FIG. 1 is a schematic diagram illustrating a two-dimensional extension of one-dimensional nonlinear interpolation methods. The pixel locations containing “o” in FIG. 1 are representative of pixels of the low resolution image. An extension to two dimensions is performed by applying the one-dimensional method separately to rows and columns of the low resolution image I_low, shown in matrix <b>101</b>, with the results combined into I_comb, shown in matrix <b>102</b>. The missing values are interpolated as averages of interpolation on rows and columns in the combined image I_int, shown in matrix <b>103</b>.

[0011] Several techniques exist that explore multiresolution structures of images in the wavelet domain to extrapolate images. A general approach to edge preserving image interpolation with wavelets is to add an additional high frequency band to the wavelet decomposition of the low resolution image. Some prior art techniques determine the location of an edge by extrapolating extrema of wavelet coefficients across scales, or decomposition levels. This extrapolation typically requires a localization and a least-square fit of the extremes. A problem with those approaches is that the alignment of an edge is never sufficient. For extrapolating smoother images, it is less significant, but rather severe for extrapolation of text. One way to overcome this problem includes iterating on the extrapolation in order to better map the downsampled high resolution image to the original low resolution image. For more information, see Carey, W. K., Chuang, D. B., and Hemami, S. S., “Regularity-Preserving Image Interpolation,” Trans. Image Processing, vol. 8, no. 9, pp. 1293-1297, 1999 and Chang, S. G., Cvetkovic, Z., and Vetterli, M., “Resolution enhancement of images using wavelet transform extrema extrapolation,” Proceedings of ICASSP'95, pp. 2379-2382, 1995.

[0012] In U.S. Pat. No. 5,717,789, entitled, “Image enhancement by non-linear extrapolation in frequency space,”, issued February 1998, the Laplacian Pyramid is used to perform a modified unsharp masking on a smoothly interpolated image. In this case, it is difficult to align a perfect edge appropriately in the interpolated image.

SUMMARY OF THE INVENTION

[0013] A method and apparatus for enlargement and resolution enhancement of images in the wavelet domain is described. In one embodiment, the method comprises receiving a wavelet representation of an image, where the wavelet representation comprises wavelet coefficients, and performing localized adaptive interpolation on the wavelet coefficients in the wavelet domain.

BRIEF DESCRIPTION OF THE DRAWINGS

[0014] The present invention will be understood more fully from the detailed description given below and from the accompanying drawings of various embodiments of the invention, which, however, should not be taken to limit the invention to the specific embodiments, but are for explanation and understanding only.

[0015]FIG. 1 illustrates two-dimensional extension of one-dimensional nonlinear interpolation methods;

[0016]FIG. 2 is a flow diagram illustrating the process for embedding interpolation within an inverse wavelet transform;

[0017]FIGS. 3A, 3B, and <b>3</b>C are flow diagrams illustrating one embodiment of a process for embedding interpolation within an inverse wavelet transform;

[0018]FIG. 4 is a schematic diagram illustrating an apparatus to perform interpolation embedded within an inverse wavelet transform according to one embodiment;

[0019]FIG. 5 is a schematic diagram illustrating an example of adaptive nonlinear image enlargement using wavelet transform coefficients compared with standard bicubic interpolation;

[0020]FIG. 6 is a flow diagram illustrating one embodiment of a process for interpolating a step edge;

[0021]FIG. 7 illustrates a digital copier having a wavelet-based enhancement system;

[0022]FIG. 8 illustrates a process performed for high resolution printing; and

[0023]FIG. 9 is a block diagram of one embodiment of a computer system.

DETAILED DESCRIPTION OF THE INVENTION

[0024] A method and apparatus for adaptive nonlinear image enlargement using wavelet transform coefficients is described. Wavelet transform coefficient naturally divide the image into smooth and edge parts. The present invention performs interpolation in the wavelet domain and uses the information in lowpass and highpass coefficients to automatically perform a smooth interpolation in smooth regions and to predict sharp edges in areas of high frequency.

[0025] The technique described herein is a modified approach to the idea of nonlinear adaptive interpolation by performing adaptive interpolation in the wavelet domain. In one embodiment, this preserves significant sharp edges that are above the noise level (characterized by, for example, a standard deviation a of wavelet coefficients as described below), where edges can be isolated, including short edges, and does not enhance noise pixels. An edge sensitivity parameter k is eliminated from the prior art technique described above and a parameter p that determines the metric distances are measured in is utilized. Moreover, the technique may adapt the algorithm to arbitrary wavelet systems and transforms. In one embodiment, the technique is combined with wavelet-based denoising and enhancement techniques, and is therefore a useful addition to a wavelet sharpening and smoothing (“WSS”) technique. It is also applicable to JPEG 2000 (J2K) compressed images.

[0026] In the following description, numerous details are set forth, such as distances between components, types of molding, etc. It will be apparent, however, to one skilled in the art, that the present invention may be practiced without these specific details. In other instances, well-known structures and devices are shown in block diagram form, rather than in detail, in order to avoid obscuring the present invention.

[0027] Some portions of the detailed descriptions which follow are presented in terms of algorithms and symbolic representations of operations on data bits within a computer memory. These algorithmic descriptions and representations are the means used by those skilled in the data processing arts to most effectively convey the substance of their work to others skilled in the art. An algorithm is here, and generally, conceived to be a self-consistent sequence of steps leading to a desired result. The steps are those requiring physical manipulations of physical quantities. Usually, though not necessarily, these quantities take the form of electrical or magnetic signals capable of being stored, transferred, combined, compared, and otherwise manipulated. It has proven convenient at times, principally for reasons of common usage, to refer to these signals as bits, values, elements, symbols, characters, terms, numbers, or the like.

[0028] It should be borne in mind, however, that all of these and similar terms are to be associated with the appropriate physical quantities and are merely convenient labels applied to these quantities. Unless specifically stated otherwise as apparent from the following discussion, it is appreciated that throughout the description, discussions utilizing terms such as “processing” or “computing” or “calculating” or “determining” or “displaying” or the like, refer to the action and processes of a computer system, or similar electronic computing device, that manipulates and transforms data represented as physical (electronic) quantities within the computer system's registers and memories into other data similarly represented as physical quantities within the computer system memories or registers or other such information storage, transmission or display devices.

[0029] The present invention also relates to apparatus for performing the operations herein. This apparatus may be specially constructed for the required purposes, or it may comprise a general purpose computer selectively activated or reconfigured by a computer program stored in the computer. Such a computer program may be stored in a computer readable storage medium, such as, but is not limited to, any type of disk including floppy disks, optical disks, CD-ROMs, and magnetic-optical disks, read-only memories (ROMs), random access memories (RAMs), EPROMs, EEPROMs, magnetic or optical cards, or any type of media suitable for storing electronic instructions, and each coupled to a computer system bus.

[0030] The algorithms and displays presented herein are not inherently related to any particular computer or other apparatus. Various general purpose systems may be used with programs in accordance with the teachings herein, or it may prove convenient to construct more specialized apparatus to perform the required method steps. The required structure for a variety of these systems will appear from the description below. In addition, the present invention is not described with reference to any particular programming language. It will be appreciated that a variety of programming languages may be used to implement the teachings of the invention as described herein.

[0031] A machine-readable medium includes any mechanism for storing or transmitting information in a form readable by a machine (e.g., a computer). For example, a machine-readable medium includes read only memory (“ROM”); random access memory (“RAM”); magnetic disk storage media; optical storage media; flash memory devices; electrical, optical, acoustical or other form of propagated signals (e.g., carrier waves, infrared signals, digital signals, etc.); etc.

[0032] Wavelet-Based Algorithm

[0033]FIG. 2 is a flow diagram of one embodiment of a process for performing non-linear image enlargement using wavelet transform coefficients. In one embodiment, the process is performed by processing logic that may comprise hardware (e.g., circuitry, dedicated logic, etc.), software (such as is run on a general purpose computer system or dedicated machine), or a combination of both.

[0034] In one embodiment, the process includes embedding interpolation within an inverse wavelet transform. The differences between samples in Eq. (2) above are interpreted in one embodiment as Haar wavelet coefficients. In such a case, the image is assumed to be given in redundant Haar wavelet coefficients. The first level of decomposition is given by lowpass coefficient c_LL(j,i) <b>201</b> and detail coefficients d_LH(j,i) <b>202</b>, d_HL(j,i) <b>203</b> and d_HH(j,i) <b>204</b>. The order of the forward transform is the horizontal transform is performed first followed by the vertical transform. In JPEG 2000, the order is reversed: first the vertical transform is applied followed by application of the horizontal transform. See JPEG2000: ITU-T Rec. T.800-ISO/IEC 15444-1:2000, Information Technology—JPEG2000 Image Coding System. In this case, the inverse transform performs the horizontal transfer in applied followed by the vertical transform.

[0035] Processing logic performs a vertical inverse wavelet transform on columns of c_LL<b>201</b> and d_LH<b>202</b>, with the results being referred to as c_L<b>205</b>.

[0036] Next, processing logic performs a vertical inverse wavelet transform on columns of d_HL<b>203</b> and d_HH<b>204</b> with the result being referred to as d_H<b>206</b>.

[0037] Processing logic places the results c_L<b>205</b> and d_H<b>206</b> on coarse grid images C_L<b>207</b> and D_H<b>208</b>, so that C_L(2j,i)=c_L(j,i), and D_H(2j,i)=d_H(j,i).

[0038] Then, processing logic interpolates columns c_L(2j+1,i) and d_H(2j+1,i) according to

<i>C</i>_L(2<i>j+</i>1,<i>i</i>)=μ_j,i<i>C</i>_L(2<i>j,i</i>)+(1−μ_j,i)<i>C</i>_L(2<i>j+</i>2,<i>i</i>) and   (3)

<i>D</i>_H(2<i>j+</i>1,<i>i</i>)=μ_j,i<i>D</i>_H(2<i>j,i</i>)+(1−μ_j,i)<i>D</i>_H(2<i>j+</i>2,<i>i</i>), where   (4) <maths id="MATH-US-00002" num="2"><math overflow="scroll"><mtable><mtr><mtd><mrow><msub><mi>μ</mi><mrow><mi>j</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>=</mo><mfrac><msup><mrow><mo></mo><mrow><msub><mi>d</mi><mrow><mi>LH</mi><mo>+</mo><mi>HH</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow><mo>,</mo><mi>i</mi></mrow><mo>)</mo></mrow></mrow><mo></mo></mrow><mi>p</mi></msup><mrow><msup><mrow><mo></mo><mrow><msub><mi>d</mi><mrow><mi>LH</mi><mo>+</mo><mi>HH</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mrow><mi>j</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mi>i</mi></mrow><mo>)</mo></mrow></mrow><mo></mo></mrow><mi>p</mi></msup><mo>+</mo><msup><mrow><mo></mo><mrow><msub><mi>d</mi><mrow><mi>LH</mi><mo>+</mo><mi>HH</mi></mrow></msub><mo></mo><mrow><mo>(</mo><mrow><mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow><mo>,</mo><mi>i</mi></mrow><mo>)</mo></mrow></mrow><mo></mo></mrow><mi>p</mi></msup></mrow></mfrac></mrow></mtd><mtd><mrow><mo>(</mo><mn>5</mn><mo>)</mo></mrow></mtd></mtr></mtable></math><img file="US20030194150A1-20031016-M00002.TIF" id="EMI-M00002" he="24.01245" wi="216.027" img-format="tif" img-content="mf" /><attachments><attachment idref="MATHEMATICA-00002" attachment-type="nb" file="US20030194150A1-20031016-M00002.NB" /></attachments></maths>  if |<i>d</i>_LH+HH(<i>j,i</i>)|&gt;ε,|<i>d</i>_LH+HH(<i>j−</i><b>1,</b><i>i</i>)|^p<i>+|d</i>_LH+HH(<i>j+</i><b>1,</b><i>i</i>)|^p<b>≠0</b>  (<b>6</b>)

and <i>d</i>_LH+HH(<i>j−</i>1,<i>i</i>)·<i>d</i>_LH+HH(<i>j+</i>1,<i>i</i>)≧0,  (7)

[0039] ε is the measure of the noise level (e.g., standard deviation of coefficients, median, standard deviation or median of absolute values of coefficients in one region (e.g., one particular band of coefficients at particular levels), standard deviation or median of absolute values of coefficients of the same level at each level, standard deviation or median of absolute values of coefficients of different bands at a first level, standard deviation or median of absolute values of coefficients of different bands at each level, etc.) or is manually set. In one embodiment, the noise level may be characterized by the standard deviation a of wavelet coefficients. In such a case, the threshold indication of the noise level is:

σ{square root}{square root over (2 log N)}

[0040] for N samples.

[0041] Equation(s) is a modified version of Eq.(2) above where the parameter k is eliminated. If the wavelet coefficient to the right is large compared to the one on the left side the interpolating value is placed more to the left side, and vice versa. The parameter p controls whether differences between coefficients are weighted more heavily or less. A good choice for images is p=1. If |d_LH+HH(j,i)|&gt;ε and (|d_LH+HH(j−1,i)|^p+|d_LH+HH(j+1,i)|^p=0 or d_LH+HH(j−1,i)·d_LH+HH(j+1, i)&lt;0) then μ_j,iis set to 1; otherwise, i.e.,

if |<i>d</i>_LH+HH(<i>j,i</i>)|≦ε,μ_j,iis set to 0.5.  (8)

[0042] Processing logic then performs horizontal inverse wavelet transforms on rows of c_L<b>207</b> and d_H<b>208</b>. The result is referred to as i_int<b>209</b>. Processing logic places these samples on a coarse grid image I_INT<b>210</b> by I_INT(j,2i)=i_int(j,i).

[0043] Processing logic then interpolates rows I_INT(j,2i+1) by

<i>I</i>_INT(2<i>j,</i>2<i>i+</i>1)=<i>v</i>_2j,i<i>I</i>_INT(2<i>j,</i>2<i>i</i>)+(1−<i>v</i>_2j,i)<i>I</i>_INT(2<i>j,</i>2<i>i+</i>2),  (9)

<i>I</i>_INT(2<i>j+</i>1,2<i>i+</i>1)=<i>v</i>_2j,i<i>I</i>_INT(2<i>j+</i>1,2<i>i</i>)+(1−<i>v</i>_2j,i)<i>I</i>_INT(2<i>j+</i>1,2<i>i+</i>2),  (10)

[0044] where <maths id="MATH-US-00003" num="3"><math overflow="scroll"><mtable><mtr><mtd><mrow><msub><mi>v</mi><mrow><mrow><mn>2</mn><mo></mo><mi>j</mi></mrow><mo>,</mo><mi>i</mi></mrow></msub><mo>=</mo><mfrac><msup><mrow><mo></mo><mrow><msub><mi>D</mi><mi>H</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mrow><mn>2</mn><mo></mo><mi>j</mi></mrow><mo>,</mo><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo></mrow><mi>p</mi></msup><mrow><msup><mrow><mo></mo><mrow><msub><mi>D</mi><mi>H</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mrow><mn>2</mn><mo></mo><mi>j</mi></mrow><mo>,</mo><mrow><mi>i</mi><mo>-</mo><mn>1</mn></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo></mrow><mi>p</mi></msup><mo>+</mo><msup><mrow><mo></mo><mrow><msub><mi>D</mi><mi>H</mi></msub><mo></mo><mrow><mo>(</mo><mrow><mrow><mn>2</mn><mo></mo><mi>j</mi></mrow><mo>,</mo><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo></mrow><mi>p</mi></msup></mrow></mfrac></mrow></mtd><mtd><mrow><mo>(</mo><mn>11</mn><mo>)</mo></mrow></mtd></mtr></mtable></math><img file="US20030194150A1-20031016-M00003.TIF" id="EMI-M00003" he="24.01245" wi="216.027" img-format="tif" img-content="mf" /><attachments><attachment idref="MATHEMATICA-00003" attachment-type="nb" file="US20030194150A1-20031016-M00003.NB" /></attachments></maths>  if |<i>D</i>_H(2<i>j,i</i>)|&gt;ε,|<i>D</i>_H(2<i>j, i−</i>1)|^p<i>+|D</i>_H(2<i>j,i+</i>1)|^p≠0  (12)

and <i>D</i>_H(2<i>j,i−</i>1)·<i>D</i>_H(2<i>j,i+</i>1)≦0.  (13)

<i>D</i>_H(2<i>j,i−</i>1)·<i>D</i>_H(2<i>j,i+</i>1)&lt;0

[0045] In one embodiment, if |D_H(2j,i)|&gt;ε and (|D_H(2j,i−1)|^p+|D_H(2j,i+1)|^p=0 or D_H(2j,i−1)·D_H(2j,i+1)&lt;0), then v_2j,iis set to 1; otherwise, i.e., if |D_H(2j,i)|≦ε, then v_2j,iis set to 0.5.

[0046] In one embodiment, processing logic clips the image I_INTback to the range of values for its palette. For example, in one embodiment in which the image is a 256 shade grayscale image, the range is clipped back to the range [0 255].

[0047] In equations (6) and (12) above, there is a condition |coefficient|&gt;ε. An alternative to setting the threshold ε can be to use a classifier result that classifies coefficients into classes A and B, and perform operations based on whether, for example, a coefficient is in class A. This could be used, e.g., in halftone areas. If a halftone classifier is used that decides whether a coefficient belongs to halftone area or not, the equations could be written as:

If |d_LH+HL(j,k)| is halftone, |<i>d</i>_LH+HL(<i>j−</i>1<i>,i</i>)<i>v|</i>^p<i>+|d</i>_LH+HL(<i>j+</i>1<i>,i</i>)|^p≠0 and <i>d</i>_LH+HH(<i>j−</i>1<i>,i</i>)·<i>d</i>_LH+HH(<i>j+</i>1,<i>i</i>)≧0

[0048] In general it could be written as

if d_LH+HL(j,k) satisfies condition A, |<i>d</i>_LH+HL(<i>j−</i>1<i>,i</i>)|^p<i>+|d</i>_LH+HL(<i>j+</i>1<i>,i</i>)|^p≠0 <i>D</i>_H(2<i>j,i−</i>1)·<i>D</i>_H(2<i>j,i+</i>1)≧0

[0049] For example, condition A can be

[0050] 1) |coefficient|&gt;threshold, when threshold represent noise level in the image,

[0051] 2) coefficient belongs to halftone area, when classifier for halftone vs. non-halftone is used,

[0052] 3) coefficient belongs to text, when classifier for text vs. non-text is used, or when MRC (Mixed Raster Content) compression scheme is used, or

[0053] 4) coefficient belongs to region-of-interest or to a specific layer in JPEG2000.

[0054] The above process is expressed in different terms below. FIGS. 3A, 3B, and <b>3</b>C are flow diagrams illustrating embedding an interpolation technique into an inverse wavelet transform. FIG. 2 is a flow diagram of one embodiment of a process for performing non-linear image enlargement using wavelet transform coefficients. In one embodiment, the process is performed by processing logic that may comprise hardware (e.g., circuitry, deducted logic, etc.), software (such as is run on a general purpose computer system or dedicated machine), or a combination of both.

[0055] Referring to FIGS. 3A, 3B, and <b>36</b>, in one embodiment, processing logic receives an image given in wavelet coefficients (processing block <b>301</b>). In process block <b>302</b>, processing logic performs a first vertical inverse wavelet transform on columns (j) of a first coefficient matrix of the image (e.g., c_LL(j,i)) and columns (j) of a second coefficient matrix of the image (e.g., d_LH(j,i)). Then, in process block <b>303</b>, processing logic performs a second vertical inverse wavelet transform on columns (j) of a third coefficient matrix of the image (e.g., d_HL(j,i)) and columns (j) of a fourth coefficient matrix of the image (e.g., d_HH(j,i)).

[0056] Next, in process block <b>304</b>, for each column (j) in a result of the first vertical inverse wavelet transform (e.g., c_L(j,i)), processing logic sets a corresponding even-numbered column (2j) in a first vertically coarser grid image (e.g., C_L(2j,i)) equal to the column (j) in the result of the first vertical inverse wavelet transform (e.g., C_L(2j,i)=c_L(j,i)).

[0057] Then, in process block <b>305</b>, for each column (j) in a result of the second vertical inverse wavelet transform (e.g., d_H(j,i)), processing logic sets a corresponding even-numbered column (2j) in a second vertically coarser grid image (e.g., D_H(2j,i)) equal to the column in the result of the second vertical inverse wavelet transform (e.g., D_H(2j,i)=d_H(j,i)).

[0058] Next, in process block <b>306</b>, for each column (j) and each row (i) in a fifth coefficient matrix (e.g., d_LH+HH(j,i)), processing logic sets an element in the column (j) and row (i) of the fifth coefficient matrix equal to a sum of a first addend and a second addend (e.g., d_LH+HH(j,i)=[d_LH(j,i)]+[d_HH(j,i)]x+y). The first addend (x) is an element in a column and a row in the second coefficient matrix (e.g., x=d_LH(j,i)). The column (j) and the row (i) in the second coefficient matrix (e.g., d_LH(j,i)) correspond to the column (j) and the row (i) in the fifth coefficient matrix (e.g., d_LH+HH(j,i)). The second addend (y) is an element in a column and a row in the fourth coefficient matrix (e.g., y=d_HH(j,i)). The column (j) and the row (i) in the fourth coefficient matrix (e.g., d_HH(j,i)) correspond to the column (j) and the row (i) in the fifth coefficient matrix (e.g., d_LH+HH(j,i)).

[0059] Then, in process block <b>307</b>, for each row (i) of each odd-numbered column (2j+1) in the first vertically coarser grid image (e.g., C_L(2j+1,i)), processing logic sets an element in the row of the odd-numbered column equal to a sum of a first addend and a second addend (erg., C_L(2j+1,i)=[μ_j,iC_L(2j,i)]g+[(1−μ_j,i)C_L(2j+2,i)]=x+y). The first addend (x) is a product of a first parameter (μ_j,i) and an element in a same row (i) of an even-numbered column to the left of the odd-numbered column (e.g. x=[μ_j,i][C_L(2j,i)]). The second addend (y) is a product of a difference (z) and an element in a same row (i) of an even-numbered column to the right of the odd-numbered column (e.g., y=[(1−μ_j,i)][C_L(2j+2,i)]=[z][C_L(2j+2,i)]). The difference (z) is equal to one minus the first parameter (e.g. z=1−μ_j,i).

[0060] Next, in process block <b>308</b>, for each row (i) of each odd-numbered column (2j+1) in the second vertically coarser grid image (e.g., D_H(2j+1,i)), processing logic sets an element in the row of the odd-numbered column equal to a sum of a first addend and a second addend (e.g., D_H(2j+1,i)=[μ_j,iD_H(2j,i)]+[(1−μ_j,i)D_H(2j+2,i)]=x+y). The first addend (x) is a product of the first parameter (μ_j,i) and an element in a same row (i) of an even-numbered column to the left of the odd-numbered column (e.g. x=[μ_j,i][D_H(2j,i)]). The second addend (y) is a product of a difference (z) and an element in a same row (i) of an even-numbered column to the right of the odd-numbered column (e.g. y=[z][D_H(2j+2,i)]). The difference (z) is equal to one minus the first parameter (e.g. z=1−μ_j,i).

[0061] Then, in process block <b>309</b>, processing logic performs a horizontal inverse wavelet transform on rows (i) of the first vertically coarser grid image (e.g., C_L(j,i)) and rows (i) of the second vertically coarser grid image (e.g., D_H(j,i)).

[0062] Next, in process block <b>310</b>, for each row (i) in a result of the horizontal inverse wavelet transform (e.g., i_int(j,i)), processing logic sets a corresponding even-numbered row (2i) in a horizontally coarser grid image (e.g., I_INT(j,2i)) equal to the row in the result of the horizontal inverse wavelet transform (e.g., I_INT(j,2i)=i_int(j,i)).

[0063] Then, in process block <b>311</b>, for each even-numbered column (2j) of each odd-numbered row (2i+1) in the horizontally coarser grid image (e.g., I_INT(2j,2i+1)), processing logic sets an element in the even-numbered column of the odd-numbered row equal to a sum of a first addend and a second addend (e.g., I_INT(2j,2i+1)=[v_2j,iI_INT(2j,2i)]+[(1−v_2j,i)I_INT(2j,2i+2)]=x+y). The first addend (x) is a product of a third parameter (v_2j,i) and an element in a same even-numbered column (2j) of an even-numbered row above the odd-numbered row (e.g. x=[v_2j,i][I_INT(2j,2i)]). The second addend (y) is a product of a difference (z) and an element in a same even-numbered column (2j) of an even-numbered row below the odd-numbered row (e.g., y=[z][I_INT(2j,2i+2)]). The difference (z) is equal to one minus the third parameter (e.g., z=1−v_2j,i).

[0064] Finally, in process block <b>312</b>, for each odd-numbered column (2j+1) of each odd-numbered row (2i+1) in the horizontally coarser grid image (e.g., I_INT(2j+1,2i+1)), processing logic sets an element in the odd-numbered column of the odd-numbered row equal to a sum of a first addend and a second addend (e.g., I_INT(2j+1,2i+1)=[v_2j,iI_INT(2j+1,2i)]+[(1−v_2j,i)I_INT(2j+1,2i+2)]=x+y). The first addend (x) is a product of the third parameter (v_2j,i) and an element in a same odd-numbered column (2j+1) of an even-numbered row above the odd-numbered row (e.g. x=[v_2j,i][I_INT(2j+1,2i)]). The second addend (y) is a product of a difference (z) and an element in a same odd-numbered column (2j+1) of an even-numbered row below the odd-numbered row (e.g., y=[Z][I_INT(2j+1,2i+2)]). The difference (z) is equal to one minus the third parameter (e.g., z=1−v_2j,i).

[0065] In one embodiment, processing logic clips the resulting image back to a range, shown in process block <b>313</b>. Process block <b>314</b> illustrates one embodiment of the clipping in which each element in the resulting image having a value below the range being set to a lowest value in the range and each element in the resulting image having a value above the range being set to a highest value in the range. Thus, in one embodiment, the range is a range of 256 shades of gray. In another embodiment, the range is a range of a greater number of shades of gray; for example, 65,536 shades. In yet another embodiment, the image is a color image having three ranges for hues of red, green, and blue, respectively. Each of these ranges is similarly clipped. Thus, the desired image palette is maintained throughout the interpolation.

[0066] In one embodiment, the wavelet coefficients are redundant Haar wavelet coefficients.

[0067] In one embodiment, the first parameter (μ_j,i) and the third parameter (v_2j,i) are set to 0.5.

[0068] In one embodiment, if a condition (k′) is satisfied, the first parameter (μ_j,i) is a numerator divided by a denominator (e.g., μ_j,i=[|d_LH+HH(j+1,i)|^p]/[|d_LH+HH(j−1,i)|^p+|d_LH+HH(j+1,i)|^p]=x/y). The numerator (x) is a first absolute value (|x′|) exponentially raised to a power of a second parameter (e.g., x=[|d_LH+HH(j+1,i)|]^[p]=|x′|^p). The first absolute value (|x′|) is an absolute value of a first element (x′). The first element (x′) is an element in a column (j+1) and row (i) in the fifth coefficient matrix (e.g., x′=d_LH+HH(j+1,i)). The column (j+1) in the fifth coefficient matrix is a column right of a column (j) in the fifth coefficient matrix that corresponds to an even-numbered column (2j) left of the odd-numbered column (2j+1) in the first vertically coarser grid image (and also the odd-numbered column (2j+1) in the second vertically coarser grid image). The row (i) in the fifth coefficient matrix corresponds to the row (i) of the odd-numbered column in the first vertically coarser grid image (and also the row (i) of the odd-numbered column in the second vertically coarser grid image). The denominator (y) is a sum of an addend (z) and the numerator (e.g., y=[|d_LH+HH(j−1,i)|^p]+[|d_LH+HH(j+1,i)|^p]=z+x). The addend (z) is a second absolute value (|z′|) exponentially raised to the power of the second parameter (e.g., z=[|d_LH+HH(j−1,i)|]^[p]=|z′|^p). The second absolute value (|z′|) is an absolute value of a second element (z′). The second element (z′) is an element in a column (j−1) and row (i) in the fifth coefficient matrix (e.g., z′=d_LH+HH(j−1,i)). The column (j−1) in the fifth coefficient matrix is a column left of a column (j) in the fifth coefficient matrix that corresponds to an even-numbered column (2j) left of the odd-numbered column (2j+1) in the first vertically coarser grid image (and also the odd-numbered column (2j+1) in the second vertically coarser grid image). The row (i) in the fifth coefficient matrix corresponds to the row (i) of the odd-numbered column in the first vertically coarser grid image (and also the row (i) of the odd-numbered column in the second vertically coarser grid image).

[0069] In one embodiment, the second parameter (p) is equal to one.

[0070] In one embodiment, the condition (k′) is satisfied if an absolute value is greater than a threshold (e.g., |d_LH+HH(j,i)|&gt;ε) and the denominator is not equal to zero (e.g., |d_LH+HH(j−1,i)|^p+|d_LH+HH(j+1,i)|^p≠0) and a product is not less than zero (e.g., [d_LH+HH(j−1,i)][d_LH+HH(j+1,i)]≧0). The absolute value is an absolute value of a third element. The third element is an element in a column (j) and row (i) in the fifth coefficient matrix (e.g., d_LH+HH(j,i)). The column (j) in the fifth coefficient matrix corresponds to an even-numbered column (2j) left of the odd-numbered column (2j+1) in the first vertically coarser grid image (and also the odd-numbered column (2j+1) in the second vertically coarser grid image). The row (i) in the fifth coefficient matrix corresponds to the row (i) of the odd-numbered column in the first vertically coarser grid image (and also the row (i) of the odd-numbered column in the second vertically coarser grid image). The product is a product of the first element and the second element (e.g., [x′][z′]=[d_LH+HH(j+1,i)][d_LH+HH(j−1,i)]).

[0071] In one embodiment, the first parameter (μ_j,i) is set to one if the absolute value is greater than the threshold (e.g., |d_LH+HH(j,i)|&gt;ε) and the denominator is equal to zero (e.g., |d_LH+HH(j−1,i)|^p+|d_LH+HH(j+1,i)|^p=0).

[0072] In one embodiment, the first parameter (μ_j,i) is set to one if the absolute value is greater than the threshold (e.g., |d_LH+HH(j,i)|&gt;ε) and the product is less than zero (e.g., [d_LH+HH(j−1,i)][d_LH+HH(j+1,i)]&lt;0).

[0073] In one embodiment, if a condition (k′) is satisfied, the third parameter (v_2j,i) is a numerator divided by a denominator (e.g., v_2j,i=[|D_H(2j,i+1)|^p]/[|D_H(2j,i−1)|^p+|D_H(2j,i+1)|^p]=x/y), where k′ is different than the parameter k mentioned above in the prior art. The numerator (x) is a first absolute value (|x′|) exponentially raised to a power of a second parameter (e.g., x=[|D_H(2j,i+1)|]^[p]=|x′|^p). The first absolute value (|x′|) is an absolute value of a first element (x′). The first element (x′) is an element in a row (i+1) and even-numbered column (2j) in the second vertically coarser grid image (e.g., x′=D_H(2j,i+1)). The row (i+1) in the second vertically coarser grid image is a row below a row (i) in the second vertically coarser grid image that corresponds to an even-numbered row (2i) above the odd-numbered row (2i+1) in the horizontally coarser grid image. The even-numbered column (2j) in the second vertically coarser grid image corresponds to the even-numbered column (2j) of the odd-numbered row in the horizontally coarser grid image. The denominator (y) is a sum of an addend (z) and the numerator (e.g., y=[|D_H(2j,i−1)|^p]+[|D_H(2j,i+1)|^p]=z+x). The addend (z) is a second absolute value (|z′|) exponentially raised to the power of the second parameter (e.g., z=[|D_H(2j,i−1)|]^[p]=|z′|^p). The second absolute value (|z′|) is an absolute value of a second element (z′). The second element (z′) is an element in a row (i−1) and even-numbered column (2j) in the second vertically coarser grid image (e.g., z′=D_H(2j,i−1)). The row (i−1) in the second vertically coarser grid image is a row above a row (i) in the second vertically coarser grid image that corresponds to an even-numbered row (2i) above the odd-numbered row (2i+1) in the horizontally coarser grid image. The even-numbered column (2j) in the second vertically coarser grid image corresponds to the even-numbered column (2j) of the odd-numbered row in the horizontally coarser grid image.

[0074] In one embodiment, the condition (k′) is satisfied if an absolute value is greater than a threshold (e.g., |D_H(2j,i)|&gt;ε) and the denominator is not equal to zero (e.g., |D_H(2j,i−1)|^p+|D_H(2j,i+1)|^p≠0) and a product is not less than zero (e.g., [D_H(2j,i−1)][D_H(2j,i+1)]≧0). The absolute value is an absolute value of a third element. The third element is an element in a row (i) and even-numbered column (2j) in the second vertically coarser grid image (e.g., D_H(2j,i)). The row (i) in the second vertically coarser grid image corresponds to an even-numbered row (2i) above the odd-numbered row (2i+1) in the horizontally coarser grid image. The even-numbered column (2j) in the second vertically coarser grid image corresponds to the even-numbered column (2j) of the odd-numbered row in the horizontally coarser grid image. The product is a product of the first element and the second element (e.g., [x′] [z′]=[D_H(2j,i−1)][D_H(2j,i+1)]).

[0075] In one embodiment, the third parameter (v_2j,i) is set to one if the absolute value is greater than the threshold (e.g., |D_H(2j,i)|&gt;ε) and the denominator is not equal to zero (e.g., |D_H(2j,i−1)|^p+|D_H(2j,i+1)|^p≠0).

[0076] In one embodiment, the third parameter (v_2j,i) is set to one if the absolute value is greater than the threshold (e.g., |D_H(2j,i)|&gt;ε) and the product is not less than zero (e.g., [D_H(2j,i−1)][D_H(2j,i+1)]≧0).

[0077] The threshold (ε) may prevent noise pixels, including halftone noise pixels, from being treated as strong edges. In smooth regions where detail coefficients are zero, the interpolation may reduce to linear interpolation.

[0078]FIG. 4 is a block diagram of one embodiment of an apparatus to perform interpolation embedded within an inverse wavelet transform. Referring to FIG. 4, the apparatus <b>400</b> comprises a receiving unit <b>401</b> to receive an image given in wavelet coefficients and a processing unit <b>402</b> coupled with the receiving unit <b>401</b>. The processing unit <b>402</b> performs the functionality described above. In one embodiment, the apparatus shown in FIG. 4 comprises a multifunction machine.

[0079]FIG. 5 is a schematic diagram illustrating an example of adaptive nonlinear image enlargement using wavelet transform coefficients compared with standard bicubic interpolation according to one embodiment. Image <b>501</b> was generated using standard bicubic interpolation. Image <b>502</b> was generated using wavelet-based nonlinear interpolation. In one embodiment, the pixel differences in the prior art algorithm are equivalent to Haar wavelet coefficients of a redundant wavelet transform. In contrast to the prior art algorithm, in one embodiment, the technique set forth herein using a redundant Haar transform interpolates a perfect step edge “exactly,” i.e., the interpolated signal has also a perfect step edge.

[0080]FIG. 6 is a flow diagram illustrating one embodiment of a process for interpolating a step edge. In one embodiment, an image having a perfect step edge is received in process block <b>601</b>, and the perfect step edge is interpolated exactly while the image is enlarged in process block <b>602</b>.

[0081] In one embodiment, the technique is also applied using redundant wavelet transform coefficients for arbitrary wavelet types. It is also used for maximal decimated wavelet transforms and complex wavelet transforms.

[0082] In one embodiment, the technique is extended to include information from various levels of the wavelet decomposition by, e.g., using wavelet coefficients from larger scales for computation of the parameters μ and v.

[0083] In combination with a wavelet sharpening and smoothing (“WSS”) technique, the enlargement may be performed after denoising by thresholding and sharpening/smoothing by resealing of wavelet coefficients. For more information on WSS, see U.S. patent application Ser. No. 09/467,544, entitled “Multiscale Sharpening and Smoothing With Wavelets” filed on Dec. 10, 1999, and assigned to the corporate assignee of the present invention.

[0084] In one embodiment, for the purpose of enlargement of JPEG2000 or similar compressed images, the technique described above is incorporated into the last level of the inverse wavelet transform on the decoder side.

[0085] Applications

[0086] There are a number of additional applications in which the adaptive non-linear image enlargement described herein may be advantageous. For example, adaptive non-linear image enlargement may be used in the enlargement process in a digital copier processing path. Such a system is shown in FIG. 7. Referring to FIG. 7, a forward redundant wavelet transform <b>701</b> is applied to an image. The wavelet transform <b>701</b> may be applied M times to the image, such that there are M levels, where M is greater than 1. After applying the forward redundant wavelet transform <b>701</b>, noise removal, text sharpening, halftone smoothing is applied to the coefficients at image processor <b>702</b>. After such processing, an inverse redundant wavelet transform <b>703</b> is applied to the processed coefficients. In one embodiment, the inverse redundant wavelet transform <b>703</b> is applied M−1 times so that there are M−1 levels. After applying the inverse redundant wavelet transform <b>703</b>, an enlarger <b>704</b> performs the enlargement process described herein. Note that each of these blocks may be implemented in software, hardware or a combination of both.

[0087] The adaptive non-linear image enlargement may be utilized in an upsampling process in which upsampling is done for printing at a higher dpi, e.g., images scanned at 600 dpi and printed at 1200 dpi. Such a system is shown in FIG. 8. Referring to FIG. 8, a one level redundant wavelet transform <b>801</b> is applied to a received digital image at a specific dpi resolution R (<b>800</b>). The coefficients are output to enlarger <b>802</b> which performs the enlargement process described herein. After enlargement, processing is performed for printing at 2R dpi by image processor <b>803</b>. The processing may include gamma correction, halftoning, etc. The output <b>804</b> is a processed image. In this manner, the adaptive non-linear image enlargement process is used to repair the lower resolution image for high resolution printing. Note that each processing element in FIG. 8 may be implemented in hardware, software, or a combination of both.

[0088] Other applications include the use of the adaptive non-linear image enlargement technique described herein for the enlargement of JPEG2000 images. This might require an addition to a JPEG decoder to compensate for the non-linear image enlargement. Another application is digital cameras. Specifically, the adaptive non-linear image enlargement process may be performed for demosaicing in a digital camera.

[0089] An Exemplary Computer System

[0090]FIG. 9 is a block diagram of an exemplary computer system that may perform one or more of the operations described herein. Referring to FIG. 9, computer system <b>900</b> may comprise an exemplary client <b>950</b> or server <b>900</b> computer system. Computer system <b>900</b> comprises a communication mechanism or bus <b>911</b> for communicating information, and a processor <b>912</b> coupled with bus <b>911</b> for processing information. Processor <b>912</b> includes a microprocessor, but is not limited to a microprocessor, such as, for example, Pentium™, PowerPC™, etc.

[0091] System <b>900</b> further comprises a random access memory (RAM), or other dynamic storage device <b>904</b> (referred to as main memory) coupled to bus <b>911</b> for storing information and instructions to be executed by processor <b>912</b>. Main memory <b>904</b> also may be used for storing temporary variables or other intermediate information during execution of instructions by processor <b>912</b>.

[0092] Computer system <b>900</b> also comprises a read only memory (ROM) and/or other static storage device <b>906</b> coupled to bus <b>911</b> for storing static information and instructions for processor <b>912</b>, and a data storage device <b>907</b>, such as a magnetic disk or optical disk and its corresponding disk drive. Data storage device <b>907</b> is coupled to bus <b>911</b> for storing information and instructions.

[0093] Computer system <b>900</b> may further be coupled to a display device <b>921</b>, such as a cathode ray tube (CRT) or liquid crystal display (LCD), coupled to bus <b>911</b> for displaying information to a computer user. An alphanumeric input device <b>922</b>, including alphanumeric and other keys, may also be coupled to bus <b>911</b> for communicating information and command selections to processor <b>912</b>. An additional user input device is cursor control <b>923</b>, such as a mouse, trackball, trackpad, stylus, or cursor direction keys, coupled to bus <b>911</b> for communicating direction information and command selections to processor <b>912</b>, and for controlling cursor movement on display <b>921</b>.

[0094] Another device that may be coupled to bus <b>911</b> is hard copy device <b>924</b>, which may be used for printing instructions, data, or other information on a medium such as paper, film, or similar types of media. Furthermore, a sound recording and playback device, such as a speaker and/or microphone may optionally be coupled to bus <b>911</b> for audio interfacing with computer system <b>900</b>. Another device that may be coupled to bus <b>911</b> is a wired/wireless communication capability <b>925</b> to communication to a phone or handheld palm device.

[0095] Note that any or all of the components of system <b>900</b> and associated hardware may be used in the present invention. However, it can be appreciated that other configurations of the computer system may include some or all of the devices.

[0096] Whereas many alterations and modifications of the present invention will no doubt become apparent to a person of ordinary skill in the art after having read the foregoing description, it is to be understood that any particular embodiment shown and described by way of illustration is in no way intended to be considered limiting. Therefore, references to details of various embodiments are not intended to limit the scope of the claims which in themselves recite only those features regarded as essential to the invention.