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We define an alternating zeta function of a digraph D, and give its determinant expression. We present a decomposition formula for the alternating zeta function of a group covering of D. Furthermore, we introduce an alternating L-function of D, and present a determinant expression of it. As a corollary, we present a decomposition formula for the alternating zeta function of a group covering of D by its alternating L-functions.

We define a weighted alternating zeta function of a digraph D and give its determinant expression. We present a decomposition formula for the weighted alternating zeta function of a group covering ...

We define a correlated random walk (CRW) induced from the time evolution matrix (the Grover matrix) of the Grover walk on a graph $G$, and present a formula for the characteristic polynomial of the transition probability matrix of this CRW by using a determinant expression for the generalized weighted zeta function of $G$. As an application, we give the spectrum of the transition probability matrices for the CRWs induced from the Grover matrices of regular graphs and semiregular bipartite graphs. Furthermore, we consider another type of the CRW on a graph.

Recently, the Ihara zeta function for the finite graph has been extended to infinite one by Clair and Chinta et al. In this paper, we obtain the same expressions by a different approach from their analytical method. Our new approach is to take a suitable limit of a sequence of finite graphs via the Konno–Sato theorem. This theorem is related to explicit formulas of characteristic polynomials for the evolution matrix of the Grover walk. The walk is one of the most well-investigated quantum walks which are quantum counterpart of classical random walks. We call the relation between the Grover walk and the zeta function based on the Konno–Sato theorem “Grover/Zeta Correspondence” here.

Recently, Gnutzmann and Smilansky presented a formula for the bond scattering matrix of a graph with respect to a Hermitian matrix. We present another proof for this Gnutzmann and Smilansky's formula by a technique used in the zeta function of a graph. Furthermore, we generalize Gnutzmann and Smilansky's formula to a regular covering of a graph. Finally, we define an $L$-fuction of a graph, and present a determinant expression. As a corollary, we express the generalization of Gnutzmann and Smilansky's formula to a regular covering of a graph by using its $L$-functions., 21 pages. arXiv admin note: substantial text overlap with arXiv:1211.4719

A necessary and sufficient conditions for certain class of periodic unitary transition operators to have eigenvalues are given. Applying this, it is shown that Grover walks in any dimension has both of $\pm 1$ as eigenvalues and it has no other eigenvalues. It is also shown that the lazy Grover walks in any dimension has $1$ as an eigenvalue, and it has no other eigenvalues. As a result, a localization phenomenon occurs for these quantum walks. A general criterion for the existence of eigenvalues can be applied also to certain quantum walks of Fourier type. It is shown that the two-dimensional Fourier walk does not have eigenvalues and hence it is not localized at any point. Some other topics such as Grover walks on the triangular lattice, products and deformations of Grover walks are also discussed., Comment: 17 pages

We define a zeta function of a graph by using the time evolution matrix of a general coined quantum walk on it, and give a determinant expression for the zeta function of a finite graph. Furthermore, we present a determinant expression for the zeta function of an (infinite) periodic graph., 14 pages

We focus on the three-state quantum walk(QW) in one dimension. In this paper, we give the stationary measure in general condition, originated from the eigenvalue problem. Firstly, we get the transfer matrices by our new recipe, and solve the eigenvalue problem. Then we obtain the general form of the stationary measure for concrete initial state and eigenvalue. We also show some specific examples of the stationary measure for the three-state QW. One of the interesting and crucial future problems is to make clear the whole picture of the set of stationary measures., Comment: 12 pages

An image taken under extremely low illumination is modeled as obeying the Poissonian-Gaussian distribution. This paper proposes a method to recover a higher-quality color moving-image sequence from Poissonian-Gaussian observations. The method performs virtual multiplex imaging, formed as a series of pixel binning and redundant subsampling, on the input sequence to increase its statistical reliability, denoises the multiple subsampled image sequences given by the redundant subsampling, and removes image blurs due to the pixel binning by super-resolution deburring to integrate the multiple denoised image sequences into an image sequence. Computer simulations show that the method successfully recovers a relatively-high-quality image sequence.

As a basic tool for deriving sparse representation of a color image from its atomic-decomposition with a redundant dictionary, the authors have recently proposed a new kind of shrinkage technique, viz. color shrinkage, which utilizes inter-channel color dependence directly in the three primary color space. Among various schemes of color shrinkage, this paper particularly presents the soft color-shrinkage and the hard color-shrinkage, natural extensions of the classic soft-shrinkage and the classic hard-shrinkage respectively, and shows their advantages over the existing shrinkage approaches where the classic shrinkage techniques are applied after a color transformation such as the opponent color transformation. Moreover, this paper presents the applications of our color-shrinkage schemes to color-image processing in the redundant tight-frame transform domain, and shows their superiority over the existing shrinkage approaches.

It is a very important and intriguing problem in digital image-processing to decompose an input image into intuitively convincible image-components such as a structure component and a texture component, which is an inherently nonlinear problem. Recently, several numerical schemes to solve the nonlinear image-decomposition problem have been proposed. The use of the nonlinear image-decomposition as a pre-process of several image-processing tasks will possibly pave the way to solve difficult problems posed by the classic approach of digital image-processing. Since the new image-processing approach via the nonlinear image-decomposition treats each separated component with a processing method suitable to it, the approach will successfully attain target items seemingly contrary to each other, for instance invisibility of ringing artifacts and sharpness of edges and textures, which have not attained simultaneously by the classic image-processing approach. This paper reviews quite recently developed state-of-the-art schemes of the nonlinear image-decomposition, and introduces some examples of the decomposition-and-processing approach.

The authors describe a technique for selectively sharpening only edges that were blurred due to various causes without making interference factors such as random noise more conspicuous. The selective sharpening technique described in this paper is implemented as an iterative updating nonlinear algorithm obtained by discretizing a type of nonlinear reaction-diffusion time-evolution equations. The update terms of this algorithm consist of a quadratic nonlinear smoothing term, reaction term, and overshooting term. By performing quantitative evaluation experiments, the authors showed that the proposed technique has significantly better selective sharpness enhancement capabilities than existing sharpness enhancement techniques such as the peaking method. In addition, the authors introduced a processing parameter adaptive control method in the proposed technique and applied it to the suppression of breathing distortions in video sequences. Breathing distortions, which are video sequence distortions that often appear in old films, are time-varying blurring distortions that occur due to temporal variations of the focus. © 2004 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 87(5): 33–47, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.10086

The wavelet transform is a practical method for multiresolution analysis. Previously, methods for describing the transformed signal using zero-crossing information and additional complementing information (zero-crossing representation) have been studied. The zero-crossing representation, however, presents problems in that the physical interpretation of the applied additional information is not clear, and reconstruction of the original signal from the zero-crossing representation involves iterations of nonlinear operations. Other problems include the fact that extension to the multidimensional case is difficult and that the computational complexity is considerable. In this paper, the authors propose a new zero-crossing representation that uses the additional information with clear physical interpretation and has a form that is easy to handle in signal processing. A method is proposed to reconstruct the original signal from the zero-crossing representation. The method uses iterative linear operations with a theoretical guarantee of convergence and has lower computional complexity than the conventional method. The proposed zero-crossing representation is extended to the two-dimensional case and is applied to an actual image. It is experimentally verified that the reconstructed image is a good approximation to the original signal. Furthermore, the amount of additional information in the zero-crossing representation is reduced. The quality degradation of the reconstructed image is examined. © 1998 Scripta Technica, Electron Comm Jpn Pt 3, 82(1): 68–78, 1999

We propose a new sparse-representation approach to super-resolution color-image interpolation. Whether an image to be interpolated actually undergoes blurs or not, the approach assumes that the image is blurred with a certain smoothing filter, and it first performs both deblurring and interpolation simultaneously so as to produce a provisional high-resolution image on the quasi blur-hypothesis, and then it blurs the provisional high-resolution image with the hypothesized blurring filter to produce a finally interpolated color-image. The quasi blur-hypothesis is introduced to enhance the spatially-regularization efficacy of the data-fidelity constraint in the variational problem of the super-resolution deblurring-and-interpolation. If a proper smoothing filter is chosen as the hypothesized blurring filter, the quasi blur-hypothesis approach can achieve state-of-the-art interpolation performance.

This paper extends the total-variation (TV) semi-norm of a scalar-valued image to a new color TV semi-norm of a color image, in the definition of which interdependence among the three primary color channels is taken into account. This paper defines a 12-D color-gradient operator that treats primary color signals and color-difference signals collectively; and then with the color-gradient operator this paper defines three kinds of color-TV semi-norms: isotropic, semi-isotropic, and anisotropic color TV semi-norms. By introducing dual definitions of the color-TV semi-norms, this paper reformulates color-TV variational denoising problems, and solves their dual problems with the projected gradient solver. Experimental simulations of color-image denoising demonstrate that the color TV semi-norms achieve selective noise-removal much more efficiently than the classic TV semi-norm, and show the merits and demerits among the three color TV semi-norms.

To remove signal-dependent noise of a digital color camera, we propose a new denoising method with our hard color-shrinkage in the tight-frame grouplet transform domain. The classic hard-shrinkage works well for monochrome-image denoising. To utilize inter-channel color dependence, a noisy image undergoes the color transformation from the RGB to the luminance-and-chrominance color space, and the luminance and the chrominance components are separately denoised; but this approach cannot cope with actual signal-dependent noise. To utilize the noise's signal-dependencies, we have constructed the hard color-shrinkage where the interchannel color dependence is directly utilized in the RGB color space. The hard color-shrinkage alleviates denoising artifacts, and improves picture quality of denoised images.

Alter, Durand and Froment introduced the total-variation (TV) minimization approach to the artifact-free JPEG decoding, which is referred to as the ADF decoding method [1]. They formulated the decoding problem as the constrained TV restoration problem, in which the TV seminorm of its restored color image is minimized under the constraint that each DCT coefficient of the restored color image should be in the quantization interval of its corresponding DCT coefficient of the JPEG-compressed data. This paper proposes a new restoration approach to the JPEG decoding. Instead of the TV regularization, our new JPEG-decoding method employs a shrinkage operation in the redundant DCT domain, to mitigate degradations caused by the JPEG coding. Our new method not only can selectively suppress ringing artifacts near color edges, but also can efficiently eliminate blocking artifacts in originally smoothly-varying image regions, where the blocking artifacts are very noticeable. Through decoding simulations, we experimentally show that our new decoding method can reduce JPEG-coding artifacts more effectively than the ADF decoding method.

A monochrome-image iterative deblurring method with the classical soft-shrinkage scheme in the shift-invariant Haar wavelet transform domain was recently proposed by R. H. Chan et al [1]. Extending the method, we present a new iterative sharpening-demosaicing method with the shift-invariant Haar wavelet transform and our previously proposed soft color-shrinkage [5] utilizing inter-channel color cross-correlations. Our new iterative method performs demosaicing and sharpening simultaneously in the wavelet transform domain. Simulation results demonstrate that our new method works as efficiently as our previously proposed sharpening-demosaicing method with the TV regularization in the spatial image-domain [2].

This paper extends the BV-L1 variational nonlinear image-decomposition approach, useful for image processing, to a genuine color-image decomposition approach. For utilizing inter-channel color cross-correlations, we introduce TV norms of color differences and TV norms of color sums into the BV-L1 energy functionals to be minimized, and then derive denoising-type decomposition-algorithm with an over-complete wavelet transform, by applying the Besov-norm approximation to the variational problem. Our method decomposes a noisy color image without producing undesirable low-frequency colored artifacts in its separated BV-component, and achieves desirable high-quality color-image decomposition, robust against colored random noise. We apply our decomposition method to color-image denoising.

To remove signal-dependent noise of a digital color camera, we present a new soft color-shrinkage scheme for color-image denoising in a wavelet transform domain. The classic soft-shrinkage scheme works well for monochrome-image denoising; to utilize inter-channel color cross-correlations, a noisy image undergoes the color-transformation from the RGB to the luminance-and-chrominance color space, and the luminance and the chrominance components are separately denoised. However, this color-denoising approach cannot cope with actual signal-dependent noise. To utilize the noise's signal-dependencies, we present an iterative soft color-shrinkage scheme where the inter-channel color cross-correlations are directly utilized in the RGB color space, and theoretically study its convergence property. Our color-shrinkage scheme alleviates denoising artifacts, and improves picture quality of denoised images.

In a digital camera, several factors cause signal-dependency of additive noise. Many denoising methods have been proposed, but many of them do not necessarily work well for actual signal-dependent noise. To solve the problem of removing signal-dependent noise of a digital color camera, this paper presents a denoising approach via nonlinear image-decomposition. As a pre-process, we employ the BV- L 1 nonlinear image-decomposition variational model. This variational model decomposes an input image into three components: a structural component corresponding to a cartoon image approximation collecting geometrical image features, a texture component corresponding to fine image textures, and a residual component. Each separated component is denoised with a denoising method suitable to it. For an image taken with a digital color camera under the condition of high ISO sensitivity, the BV- L 1 model removes its signal-dependent noise to a large extent from its separated structural component, in which geometrical image features are well preserved, but the structural component sometimes suffers color-smear artifacts. To remove those color-smear artifacts, we apply the sparse 3D transform-domain collaborative filtering to the separated structural component. On the other hand, the texture component and the residual component are rather contaminated with noise, and the effects of noise are selectively removed from them with our proposed color shrinkage denoising schemes utilizing inter-channel color crosscorrelations. Our method achieves efficient denoising and selectively removes signal-dependent noise of a digital color camera.

Previously we presented a TV-based super-resolution sharpening-demosaicing method. Our previous method makes it possible to restore frequency components higher than the Nyquist frequency, and to interpolate color signals effectively while preserving their sharp color edges, without producing ringing artifacts along the edges. However, since our previous method applies the TV regularization separately to each primary color channel, as side effects it sometimes produces false color artifacts and/or zipper artifacts along sharp color edges. To remedy the drawback, in addition to the TV regularization of each primary color signal, we introduce the TV regularization of color difference signals such as G-R, and that of color sum signals such as G+R, into the TV-based super-resolution sharpening-demosaicing method. Near sharp color edges, correct interpolation provides the smallest TV norms of color difference signals or the smallest TV norms of color sum signals. Unlike our previous method, our new method jointly interpolates the three primary color channels. We compare demosaicing performance of our new method with the state-of-the-art demosaicing methods. In the evaluations, we consider a noise-free case and a noisy case. For both cases our new method achieves the best performance, and for the noisy case our new method considerably outperforms the state-of-the-art methods.

This paper presents a new image-interpolation approach where one can adjust edge sharpness and texture intensity according to one’s taste. This approach is composed of the three stages. At the first stage, with the BV-G image-decomposition variational model, an image is represented as a product of its two components so that its separated structural component may correspond to a cartoon image-approximation and its separated texture components may collect almost all oscillatory variations representing textures, and the texture co mponent can be amplified or attenuated according to user’s taste. At the second stage, each separated component is interpolated with an interpolation method suitable to it. Since the structural component keeps sharp edges, its proper interpolation method is a TV-regularization super-resolution interpolation method that can restore frequency components higher than the Nyquist frequency and remove sample-hold blurs without producing ringing artifacts near edges. The texture component is an oscillatory function, and its proper interpolation method is a smoothne ss-regularization super-resolution interpolation method that can restore continuous variations and remove the blurs. At the final stage, the two interpolated components are combined. The approach enlarges images without not only blurring edges but also destroying textures, and removes blurs caused by the sample-hold and/or the optical low-pass filter without producing ringing artifacts. Keywords: Image decomposition, variational problem, interpolation, deblurring, super-resolution

Previously we presented a TV (total variation)-based sharpening-demosaicing method. Our previous method interpolates color signals effectively while preserving sharp color edges, without producing ringing artifacts along the edges. However, since our previous method applies the TV regularization separately to each primary color channel, it sometimes produces color artifacts along sharp color edges. To remedy the drawback, we introduce the TV regularization of color differences and color sums into our previous method. Unlike our previous method, our new method jointly interpolates the three primary color channels. We compare demosaicing performance of our new method with the state-of-the- art demosaicing methods. In the evaluations, we consider a noise- free case and a noisy case. For both cases our new method achieves the best performance, and for the noisy case our new method considerably outperforms the state-of-the-art methods.

This paper presents a new image-processing (IP) pipeline adopting an image-processing approach via nonlinear image-decomposition, for a digital color camera. This new IP pipeline is composed of four stages. At the first stage, with the multiplicative BV-G image-decomposition method, each primary color channel of observed raw color data mosaicked with the Bayer color filter array is decomposed as a product of two components. Its structural component corresponds to a cartoon signal-approximation and its texture component collects almost all oscillatory variations representing textures. At the second stage, each separated component is demosaicked with an interpolation method suitable to it. For demosaicing of the structural component, this paper employs our color-TV-regularization super-resolution deblurring-demosaicing method that can remove image blurs without producing ringing artifacts near edges. For interpolation of the texture component gathering most of observation noise, this paper employs a basic averaging-type demosaicing method that is robust against observation noise. At the third stage, white balancing, color enhancement and inverse gamma correction are applied to only the demosaicked structural component. At the final stage, the two processed components are combined. Our proposed IP pipeline succeeds in achieving color interpolation, denoising and deblurring, simultaneously.

A spherically symmetric probability distribution has proven to be a good probability model of normalized AC transform coefficients. For this distribution, quantizers having the property that their codewords are arranged on the surface of concentric hyperspheres will show excellent performance with their low encoding complexity. Permutation codes, previously developed by Berger et al. , having this property do not require any multiplication for encoding and can quantize a very high dimensional input efficiently. Permutation codes, however, cannot show satisfactory performance for high bit rates or low dimensions and require unfeasible operation precision for encoding a codeword's index. To cope with these problems, extending the concept of permutation codes, we developed new improved permutation codes and formed an algorithm that involves only integer operations with limited operation precision for encoding a codeword's index. Furthermore, we incorporated improved permutation codes into discrete cosine transform image coding. The simulation results demonstrate that improved permutation codes can quantize high frequency transform coefficients efficiently.

For high-efficiency image compression, previously, an SVD (singular value decomposition)-based coder was developed using vector quantization, called SVD-VQ. This paper proposes an improved quantization SVD-VQ scheme. For every input subblock, the SVD-VQ coder scalar-quantizes a singular value and vector-quantizes two singular vectors, separately. The SVD-VQ decoder reproduces a subblock as the product of these quantization outputs, but does not necessarily produce a reconstruction with the minimum distortion in an image space. This paper develops a quantization scheme where the minimum-distortion reconstruction is always provided in the original image space and presents its design algorithm. The improved SVD-VQ shows A/N performance improvement of 0.5 - 1.0 dB over the conventional SVD-VQ, and is similar in performance to the adaptive DCT (discrete cosine transform) coder.

To remove signal-dependent noise of a digital camera, we present a denoising approach via nonlinear image-decomposition. In the approach, at the first decomposition stage, multiplicative image-decomposition is performed, and a noisy image is represented as a product of its two components so that its structural component corresponding to a cartoon image approximation may not be corrupted by the noise and its texture component may collect almost all the noise. At the successive denoising stage, the structural component is used instead of the unknown true signal value, to adapt the soft-thresholding-type denoising manipulation of the texture component to the signal dependency of the noise. At the final image-synthesis stage, one combines the separated structure component with the denoised texture component to reproduce a denoised image. The approach selectively removes the signal-dependent noise without not only blurring sharp edges but also destroying visually important textures.

In a digital camera, several factors cause signal-dependency of additive noise. Many denoising methods have been proposed, but unfortunately most of them do not work well for the actual signal-dependent noise. To solve the problem of removing the signal-dependent noise of a digital camera, we present a denoising approach via the nonlinear imagedecomposition. In the nonlinear decomposition-and-denoising approach, at the first nonlinear image-decomposition stage, multiplicative image-decomposition is performed, and a noisy image is represented as a product of its two components so that its structural component corresponding to a cartoon approximation of the noisy image may not be corrupted by the noise and its texture component may collect almost all the noise. At the successive nonlinear denoising stage, intensity of the separated structural component is utilized instead of the unknown true signal value, to adapt the soft-thresholding-type denoising manipulation of the texture component to the signal dependency of the noise. At the final image-synthesis stage, the separated structure component is combined with the denoised texture component, and thus a sharpness-improved denoised image is reproduced. The nonlinear decomposition-and-denoising approach can selectively remove the signal-dependent noise of a digital camera without not only blurring sharp edges but also destroying visually important textures.

A motion deblurring method with spatio-temporal regularization is presented. In conventional motion deblurring, the best image for each frame is restored individually. The aim of proposed method is that the all frames achieve the same level of smoothness, and an extended total variation is introduced to the deconvolution approach. Our method can be applied to a video clip acquired by handheld camera with camera shakes. Experimental results show that the strongly blurred images are deblurred sufficiently and the temporal change is suppressed.

The Retinex theory deals with separation of irradiance and reflectance from an image. Recently, Kimmel and others proposed a variational framework that unifies the previous Poisson-equation-type Retinex algorithms developed by Horn and others, and presented a separation algorithm with the time-evolution of a linear diffusion process. However, this separation algorithm cannot achieve physically rational separation, if true irradiance varies among color channels. To cope with this problem, we introduce a nonlinear diffusion process into the time-evolution. Moreover, we introduce an approach to treat all color channels collectively. Furthermore, we apply our separation algorithm to a high quality chroma key where a color of each pixel in the foreground frame are adaptively corrected through transformation of the separated irradiance.

The DCI (direct color-imaging) approach is promising as an acquisition scheme of color data with high spatial-resolution. The DCI sensor has three photo-sensing layers along its depth direction. Although each pixel has three color channels, their spectral sensitivities are overlapped with each other and the approach has a problem in color separation. To cope with this problem, we present a hybrid color-imaging approach between the DCI approach and the CFA (color-filter-array) approach. Our hybrid approach uses the DCI sensor, and pastes green and magenta color filters on pixels' surfaces according to a checkered pattern. The use of the green-magenta CFA improves color separation, but its sensed color channels have different spectra from the primary color-matching functions. Moreover, its sensed three color channels are sub-sampled according to the checkered pattern. To recover a primary tri-stimulus image with full spatial-resolution, not only color transformation of the sensed color channels but also their interpolation are needed. We present image recovery methods for the color transformation and the interpolation. Our hybrid color-imaging approach using the methods achieves higher spatial-resolution than the CFA approach, and shows the best color-reproduction among the three color-imaging approaches.

This paper presents an adaptable interpolation approach that can adjust edge sharpness and texture intensity to reconstruct a high quality image according to user's taste in picture quality. Our interpolation approach first resolves an input image I into its skeleton image U and its texture generator V and its residual image D such that I = U·V + D, and then interpolates each of the three components independently with a proper interpolation method suitable to each. The skeleton image is a bounded-variation function meaning a cartoon approximation of I, and interpolated with a super-resolution deblurring-oversampling method that interpolates sharp edges without producing ringing artifacts. The texture generator is an oscillatory function representing regular distinct textures, and interpolated with a standard linear interpolation algorithm. The residual image is a function with some randomness, and interpolated with a statistical re-sampling interpolation algorithm.

The Retinex theory was first proposed by Land, and deals with separation of irradiance from reflectance in an observed image. The separation problem is an ill-posed problem. Land and others proposed various Retinex separation algorithms. Recently, Kimmel and others proposed a variational framework that unifies the previous Retinex algorithms such as the Poisson-equation-type Retinex algorithms developed by Horn and others, and presented a Retinex separation algorithm with the time-evolution of a linear diffusion process. However, the Kimmel's separation algorithm cannot achieve physically rational separation, if true irradiance varies among color channels. To cope with this problem, we introduce a nonlinear diffusion process into the time-evolution. Moreover, as to its extension to color images, we present two approaches to treat color channels: the independent approach to treat each color channel separately and the collective approach to treat all color channels collectively. The latter approach outperforms the former. Furthermore, we apply our separation algorithm to a high quality chroma key in which before combining a foreground frame and a background frame into an output image a color of each pixel in the foreground frame are spatially adaptively corrected through transformation of the separated irradiance. Experiments demonstrate superiority of our separation algorithm over the Kimmel's separation algorithm.

Once image motion is accurately estimated, we can utilize those motion estimates for image sharpening and we can remove motion blurs. First, for the motion de-blurring, this paper presents a model-based PDE method that minimizes the regularized energy functional defined with a spatially variant model of motion blurs. Unlike the case of spatially invariant image blurs, the minimization of the energy functional cannot be achieved in a closed non-iterative way, and we derive its iterative algorithm. The standard regularization method uses a square function to measure energy of its solution function, and employs the energy functional composed of the data-fidelity energy term to measure a deviation of a solution function from the assumed model of motion blurs and the regularization energy term to impose smoothness constraints on a solution function. However, the standard variational method is not proper for the motion de-blurring, because it is sensitive to model errors, and occurrence of errors are inevitable in motion estimation. To improve the robustness against the model errors, we employ a nonlinear robust estimation function for measuring energy to be minimized. Secondly, this paper experimentally compares the model-based PDE method with our previously presented model-free PDE method that does not need any accurate blur model. In the model-error-free case the model-based PDE method outperforms the model-free PDE method, whereas in the model-error case the latter works better than the former.

The optical low-pass filter used for a digital camera is formed by combining two types of doubly refractive crystal device. Since the filter cannot sharply cut off high frequency components and it reduces frequency components under the Nyquist frequency, images projected on the imaging surface are blurred. This paper presents a demosaicking method that simultaneously removes image blurs caused by the optical low-pass filter. Our sharpening-demosaicking method employs the Landweber-type iterative algorithm. Since the Bayer's RGB color filter array is not necessarily proper for our method, we study another better color filter array, the WRB filter array where the W-filtering means that all the visible light passes through it. Our sharpening-demosaicking approach is formulated as the least square problem, but it has multiple solutions. To avoid the ambiguity, we introduce the frequency-band limitation corresponding to the mosaicking pattern of color filters, into the iterative algorithm.

We previously presented a method of selective image sharpening based on a coupled nonlinear diffusion process that was composed of a nonlinear diffusion term, a fidelity term and an isotropic peaking term. It could sharpen only blurred edges without increasing noise visibility. This paper extends our method to the removal of blurs in images due to image motion. Motion blur is not only shift-variant but also anisotropic. To adapt our method to motion de-blurring, we replaced our isotropic peaking term, with an anisotropic peaking term steered in direction of motion at each pixel. We then devised discrete calculus to adapt it to the direction of motion. Through experiments with a test sequence of moving images containing artificial motion-blurs, we quantitatively evaluated sharpening performance. Our new method with the anisotropic peaking term performed better than our prototypal method with the isotropic peaking term, and was robust against errors in the direction of motion.

Most digital color cameras use doubly refractive crystal devices as an optical low-pass filter. The filter reduces frequency components lower than the Nyquist frequency, and images are blurred. We previously presented a demosaicking method that simultaneously removes blurs caused by the optical low-pass filter. For the sharpening-demosaicking approach, the Bayer's RGB color filter array is not necessarily proper, and we studied another color-filter array, namely the WRB filter array, where the W-filtering means that all the visible light passes through it. Our prototypal sharpening-demosaicking method employed the iterative algorithm, and restored only spatial frequency components of color signals lower than the Nyquist frequency corresponding to the mosaicking pattern of the W filters. However, the same recovery problem is solved by a non-iterative method in the Discrete Fourier Transform domain. Moreover, our prototypal method often produced ringing artifacts near sharp color edges. To suppress those artifacts, we introduce the total-variation-based super-resolution into the sharpening-demosaicking approach. This super-resolution approach restores spatial frequency components higher than the Nyquist frequency from observed blurry spatial frequency components so that without producing ringing artifacts it can demosaic and sharpen images while preserving 1D image structures that intensity values are almost constant along the edge direction.

We previously presented a demosaicking method that simultaneously removes blurs caused by an optical low-pass filter used in a digital camera. For the sharpening-demosaicking approach, the RGB color filter array is not necessarily proper, and we studied another color-filter array, namely the WRB filter array, where the W-filtering means that all visible light passes through it. Our prototypal method employed the iterative algorithm, and restored only spatial frequency components lower than the Nyquist frequency corresponding to the mosaicking pattern. First, this paper formulates the same recovery problem in the DFT domain more explicitly. Moreover, to suppress ringing artifacts often produced by our prototypal method near sharp color edges, we introduce the total-variation-based super-resolution into the sharpening-demosaicking approach. This super-resolution approach can demosaic and sharpen images while preserving image structures that intensity values are almost constant along the edge, without producing ringing artifacts.

As an optical low-pass filter, a doubly refractive crystal device is used and performs separation of its incident light into the normal light and the abnormal light shifted to the slightly different direction to the normal light. The actual optical low-pass filter is formed by combining two types of doubly refractive crystal device; the one separates its incident light into two traveling directions horizontally spaced each other by one pixel, and the other does vertically spaced by one pixel. The filter cannot sharply cut off high frequency components and it reduces the frequency components near the Nyquist frequency. Thus, images projected on the imaging surface are blurred. This paper presents a demosaicking method that can simultaneously remove image blurs caused by the optical low-pass filter. Most of the existing demosaicking methods do not try to remove the image blurs, whereas our sharpening approach to the demosaicking employs the Landweber-type iterative algorithm. For our sharpening-demosaicking approach, the Bayer’s pattern of the RGB primary color filter array is not necessarily proper, and hence we study another color-filter pattern, namely the WRB filter array that is preferable to the RGB filter array, where the W-filtering means that all the visible light passes through it. Our mathematical formulation of the sharpening-demosaicking has the form of the least square problem, but there exist multiple different least square solutions. To avoid its ambiguity, in the spatial frequency domain we introduce the pass-band limitation corresponding to the sub-sampling pattern of the mosaicking of color filters, into the iterative algorithm.

Previously we have presented a selective sharpening method for monochrome images. Our method is based on the simultaneous nonlinear reaction-diffusion time-evolution equipped with a nonlinear diffusion term, a reaction term and an overshooting term, and can sharpen degraded edges blurred by several causes without increasing the visibility of random noise. This paper extends our method to selective sharpening of color images. As to the how to extend it, we take into accounts two variations about the treatment of three color components and the selection of the color space. By experiments, we quantitatively evaluate performance of these variations. The best performance is achieved by the collective treatment of color components based on the simultaneous full-nonlinear reaction-diffusion time-evolution, and blurred color edges are sharpened selectively much better than by the existing methods such as the adaptive peaking method.

Previously we have presented a method for selective image sharpness enhancement. Our method is based on the simultaneous nonlinear reaction-diffusion time-evolution equipped with a nonlinear diffusion term, a reaction term and an isotropic peaking term, and it can sharpen only degraded edges blurred by several causes without increasing the visibility of nuisance factors such as random noise. This paper applies our simultaneous nonlinear reaction-diffusion method to removal of image blurs due to image motion. The motion blur is not only shift-variant but also anisotropic. To adapt our simultaneous nonlinear reaction-diffusion method for motion de-blurring, we replace the isotropic Laplacian operator, included in the isotropic peaking term of our prototypal method, with the anisotropic operator considering the direction of the estimated image motion. Preliminarily experiments using artificially generated test images show that our method achieves excellent motion de-blurring.

Previously we have presented a PDE-based method for selective image sharpening. Our method works as the simultaneous nonlinear-diffusion process composed of a nonlinear-diffusion term, a fidelity term and a peaking term, and it sharpens blurred edges while smoothing out noisy variations. However, our method has the problem that it does not satisfactorily sharpen complex image-structures such as T-shaped edges and textures. This paper copes with the problem, and improves selective image sharpening. As the preprocess of our PDE-based method, this paper introduces a step to classify each pixel into two categories on the basis of mid-scale image-features contained in the image-gradient field. The classification results are then utilized to preset the parameters characterizing our PDE-based method spatially varyingly.

Previously we have presented a method for selective sharpness enhancement of monochrome images. Our method is based on the simultaneous nonlinear reaction-diffusion time-evolution equipped with a nonlinear diffusion term, a reaction term and an overshooting term, and it can sharpen only degraded edges blurred by several causes without increasing the visibility of nuisance factors such as random noise. This paper extends our method to selective sharpening of color images. As to the how to extend it, we take into accounts some variations about the treatment of three color components and the selection of the color space. By experiments, we quantitatively evaluate performance of these variations. Among them, the collective treatment of color components based on the simultaneous full-nonlinear reaction-diffusion time-evolution achieves the best performance, and sharpens blurred color edges selectively much better than the existing sharpness enhancement methods such as the adaptive peaking method.© (2003) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

n existing single solid-state color image sensors, three primary color filter arrays are pasted on the surface of photo detectors according to the Bayeris pattern. To utilize information conveyed by the incoming light more efficiently, we can employ new color filters whose color components in their passband are overlapped with each other to a larger extent. As the filter array pattern we adopt the Bayeris pattern; but unlike the three primary color filters we use only two color filters, red and blue color filters, and we replace green pixels of the primary color filters with white-black pixels that correspond to pixels on which no light-absorption chemicals are pasted. In this scheme, the key point is to construct a demosaicking method suitable to these color filter arrays. We employ a hybrid demosaicking method that can restrain the occurrence of false color caused by the demosaicking and preserve original hue variations as thoroughly as possible while enhancing the spatial resolution of the restored image. The hybrid demosaicking method first applies the Landweber-type iterative algorithm, equipped with the frequency-band limitation corresponding to the sub-sampling pattern of the white-black pixel array, to the interpolation of white-black pixels, and then performs the interpolation of red and blue pixels with some existing chrominance-preserving-type method such as the gradient-based method. Experiments using test color images demonstrate that our hybrid demosaicking method reproduces a sharpened high-resolution color image without producing noticeable artifacts of false color. Our color image acquisition scheme gives a good compromise between false color occurrence and high-fidelity color reproduction.

To estimate local motion from moving images, many algorithms have been proposed. Although the concept of the existing block-matching algorithm is very simple, it works well even under real noisy conditions, and hence it is often used in various practical application fields such as video processing. However, the existing block-matching method does not work well if a given analysis image block contains multiple image regions moving in different directions, and/or if boundaries of a given analysis image block and object boundaries do not coincide. To solve this problem, we extend the basic concept of the existing block-matching algorithm, and thus form two different algorithms for the analysis of multiple distinct motions in two different complex configurations of local motion. The one algorithm is based on the assumption that there may be multiple distinct image regions undergoing coherent motion within a given analysis image block, while the other algorithm is based on the assumption that multiple different moving image patterns appear superimposed within a given analysis image block. Through computer simulations, we demonstrate the potentiality of the extended block-matching algorithms. >