Due to environmental and economic constraints inherent to seismic exploration, there are often missing shotpoints and receivers that degrade the resolution of the final seismic image. Hence sophisticated interpolation techniques are required for the recovery of dense and uniform spatial sampling. Recent approaches improve the interpolation by adopting robust models through denoisers. We introduce a 3D shot gather interpolation method that jointly considers a sparse prior and a regularization induced by a multichannel volumetric denoiser. The proposed volumetric regularization uses collaborative filters that perform denoising through transform-domain shrinkage of a group of similar seismic cubes extracted from a land seismic acquisition. This grouping and collaborative filtering paradigm exploit the local correlation present in each cube and the non-local correlation between different cubes. Experiments on theStratton 3D surveyshow that the proposed method can interpolate 3D shot gathers in an orthogonal seismic recording from a swath geometry, outperforming methods based on 2D denoisers and 5D seismic data reconstruction in terms of root mean square error and in the recovery of seismic reflections.

The fusion of a low-spatial-and-high-spectral resolution hyperspectral image (HSI) with a high-spatial-and-low-spectral resolution multispectral image (MSI) allows synthesizing a high-resolution image (HRI), supporting remote sensing applications such as disaster management, material identification, and precision agriculture. Unlike existing variational methods using low-rank regularizations separately, we present an HSI-MSI fusion method promoting various low-rank regularizations jointly. Our method refines the HRI spatial and spectral correlations from the individual HSI and MSI data through the proper plug-and-play (PnP) of a nonlocal patch-based denoiser in the alternating direction method of multipliers (ADMM). Notably, we consider the nonlocal self-similarity, the spectral low-rank, and introduce a rank-one similarity prior. Furthermore, we demonstrate via an extensive empirical study that the rank-one similarity prior is an inherent characteristic of the HRI. Simulations over standard benchmark datasets show the effectiveness of the proposed HSI-MSI fusion outperforming state-of-the-art methods, particularly in recovering low-contrast areas.

3D point clouds commonly contain positional errors which can be regarded as noise. We propose a point cloud denoising algorithm based on aggregation of multiple anisotropic estimates computed on local coordinate systems. These local estimates are adaptive to the shape of the surface underlying the point cloud, leveraging an extension of the Local Polynomial Approximation (LPA) - Intersection of Confidence Intervals (ICI) technique to 3D point clouds. The adaptivity due to LPA-ICI is further strengthened by the dense aggregation with data-driven weights. Experimental results demonstrate state-of-the-art restoration quality of both sharp features and smooth areas.

Collaborative filters perform denoising through transform-domain shrinkage of a group of similar patches extracted from an image. Existing collaborative filters of stationary correlated noise have all used simple approximations of the transform noise power spectrum adopted from methods which do not employ patch grouping and instead operate on a single patch. We note the inaccuracies of these approximations and introduce a method for the exact computation of the noise power spectrum. Unlike earlier methods, the calculated noise variances are exact even when noise in one patch is correlated with noise in any of the other patches. We discuss the adoption of the exact noise power spectrum within shrinkage, in similarity testing (patch matching), and in aggregation. We also introduce effective approximations of the spectrum for faster computation. Extensive experiments support the proposed method over earlier crude approximations used by image denoising filters such as Block-Matching and 3D-filtering (BM3D), demonstrating dramatic improvement in many challenging conditions.

Collaborative filters perform denoising through transform-domain shrinkage of a group of similar blocks extracted from an image. Existing methods for collaborative filtering of stationary correlated noise have all used simple approximations of the transform noise power spectrum adopted from methods which do not employ block grouping. We note the inaccuracies of these approximations and introduce a method for the exact computation and effective approximations of the noise power spectrum. Unlike earlier methods, the calculated noise variances are exact even when noise in one block is correlated with noise in any of the other blocks. We discuss the adoption of the exact noise power spectrum within shrinkage, in similarity testing (block matching), and in aggregation. Extensive experiments support the proposed method over earlier crude approximations used by image denoising filters such as BM3D, demonstrating dramatic improvement in many challenging conditions.

Path tracing produces realistic results including global illumination using a unified simple rendering pipeline. Reducing the amount of noise to imperceptible levels without post-processing requires thousands of samples per pixel (spp), while currently it is only possible to render extremely noisy 1 spp frames in real time with desktop GPUs. However, post-processing can utilize feature buffers, which contain noise-free auxiliary data available in the rendering pipeline. Previously, regression-based noise filtering methods have only been used in offline rendering due to their high computational cost. In this article we propose a novel regression-based reconstruction pipeline, called Blockwise Multi-Order Feature Regression (BMFR), tailored for path-traced 1 spp inputs that runs in real time. The high speed is achieved with a fast implementation of augmented QR factorization and by using stochastic regularization to address rank-deficient feature data. The proposed algorithm is 1.8× faster than the previous state-of-the-art real-time path-tracing reconstruction method while producing better quality frame sequences.

We introduce a paradigm for nonlocal sparsity reinforced deep convolutional neural network denoising. It is a combination of a local multiscale denoising by a convolutional neural network (CNN) based denoiser and a nonlocal denoising based on a nonlocal filter (NLF), exploiting the mutual similarities between groups of patches. CNN models are leveraged with noise levels that progressively decrease at every iteration of our framework, while their output is regularized by a nonlocal prior implicit within the NLF. Unlike complicated neural networks that embed the nonlocality prior within the layers of the network, our framework is modular, and it uses standard pretrained CNNs together with standard nonlocal filters. An instance of the proposed framework, called NN3D, is evaluated over large grayscale image datasets showing state-of-the-art performance.

Denoising is often addressed via sparse coding with respect to an overcomplete dictionary. There are two main approaches when the dictionary is composed of translates of an orthonormal basis. The first, traditionally employed by techniques such as wavelet cycle spinning, separately seeks sparsity w.r.t. each translate of the orthonormal basis, solving multiple partial optimizations and obtaining a collection of sparse approximations of the noise-free image, which are aggregated together to obtain a final estimate. The second approach, recently employed by convolutional sparse representations, instead seeks sparsity over the entire dictionary via a global optimization. It is tempting to view the former approach as providing a suboptimal solution of the latter. In this letter, we analyze whether global sparsity is a desirable property, and under what conditions the global optimization provides a better solution to the denoising problem. In particular, our experimental analysis shows that the two approaches attain comparable performance in case of natural images and global optimization outperforms the simpler aggregation of partial estimates only when the image admits an extremely sparse representation. We explain this phenomenon by separately studying the bias and variance of these solutions, and by noting that the variance of the global solution increases very rapidly as the original signal becomes less and less sparse.

This paper proposes a new method of simulating dose reduction in digital breast tomosynthesis, starting from a clinical image acquired with a standard radiation dose. It considers both signal-dependent quantum and signal-independent electronic noise. Furthermore, the method accounts for pixel crosstalk, which causes the noise to be frequency-dependent, thus increasing the simulation accuracy. For an objective assessment, simulated and real images were compared in terms of noise standard deviation, signal-to-noise ratio (SNR) and normalized noise power spectrum (NNPS). A two-alternative forced-choice (2-AFC) study investigated the similarity between the noise strength of low-dose simulated and real images. Six experienced medical physics specialists participated on the study, with a total of 2 160 readings. Objective assessment showed no relevant trends with the simulated noise. The relative error in the standard deviation of the simulated noise was less than 2% for every projection angle. The relative error of the SNR was less than 1.5%, and the NNPS of the simulated images had errors less than 2.5%. The 2-AFC human observer experiment yielded no statistically significant difference ( =0.84) in the perceived noise strength between simulated and real images. Furthermore, the observer study also allowed the estimation of a dose difference at which the observer perceived a just-noticeable difference (JND) in noise levels. The estimated JND value indicated that a change of 17% in the current-time product was sufficient to cause a noticeable difference in noise levels. The observed high accuracy, along with the flexible calibration, make this method an attractive tool for clinical image-based simulations of dose reduction.

The automatic detection of anomalies, defined as patterns that are not encountered in representative set of normal images, is an important problem in industrial control and biomedical applications. We have shown that this problem can be successfully addressed by the sparse representation of individual image patches using a dictionary learned from a large set of patches extracted from normal images. Anomalous patches are detected as those for which the sparse representation on this dictionary exceeds sparsity or error tolerances. Unfortunately, this solution is not suitable for many real-world visual inspection-systems since it is not scale invariant: since the dictionary is learned at a single scale, patches in normal images acquired at a different magnification level might be detected as anomalous. We present an anomaly-detection algorithm that learns a dictionary that is invariant to a range of scale changes, and overcomes this limitation by use of an appropriate sparse coding stage. The algorithm was successfully tested in an industrial application by analyzing a dataset of Scanning Electron Microscope (SEM) images, which typically exhibit different magnification levels.

We denoise Poisson images with an iterative algorithm that progressively improves the effectiveness of variance-stabilizing transformations (VST) for Gaussian denoising filters. At each iteration, a combination of the Poisson observations with the denoised estimate from the previous iteration is treated as scaled Poisson data and filtered through a VST scheme. Due to the slight mismatch between a true scaled Poisson distribution and this combination, a special exact unbiased inverse is designed. We present an implementation of this approach based on the BM3D Gaussian denoising filter. With a computational cost at worst twice that of the noniterative scheme, the proposed algorithm provides significantly better quality, particularly at low signal-to-noise ratio, outperforming much costlier state-of-the-art alternatives.

The design, optimization, and validation of many image-processing or image-based analysis systems often require testing of the system performance over a dataset of images corrupted by noise at different signal-to-noise ratio (SNR) regimes. A noise-free ground-truth image may not be available, and different SNRs are simulated by injecting extra noise into an already noisy image. However, noise in real-world systems is typically signal dependent, with variance determined by the noise-free image. Thus, the noise to be injected shall also depend on the unknown ground-truth image. To circumvent this issue, we consider the additive injection of noise in variance-stabilized range, where no previous knowledge of the ground-truth signal is necessary. Specifically, we design a special noise-injection operator that prevents the errors on expectation and variance that would otherwise arise when standard variance-stabilizing transformations are used for this task. Thus, the proposed operator is suitable for accurately injecting signal-dependent noise, even to images acquired at very low counts.

Groups of mutually similar image blocks are the key element in nonlocal image processing. In this work, the spatial coordinates of grouped blocks are leveraged in two distinct parts of the transform-domain collaborative filtering within the BM3D algorithm. First, we introduce an adaptive 1-D transform for 3-D collaborative filtering based on sampling 2-D smooth functions at the positions of grouped blocks. This adaptive transform is applied for improved decorrelation of the 2-D spectra of the grouped blocks. Second, we propose a directional sharpening procedure whose strength varies adaptively according to the relative orientation of the transform basis functions with respect to the group coordinates. Experiments confirm the efficacy of the proposed adaptations, for denoising as well as for sharpening of noisy images.

We address the problem of detecting anomalies in images, specifically that of detecting regions characterized by structures that do not conform those of normal images. In the proposed approach we exploit convolutional sparse models to learn a dictionary of filters from a training set of normal images. These filters capture the structure of normal images and are leveraged to quantitatively assess whether regions of a test image are normal or anomalous. Each test image is at first encoded with respect to the learned dictionary, yielding sparse coefficient maps, and then analyzed by computing indicator vectors that assess the conformance of local image regions with the learned filters. Anomalies are then detected by identifying outliers in these indicators. Our experiments demonstrate that a convolutional sparse model provides better anomaly-detection performance than an equivalent method based on standard patch-based sparsity. Most importantly, our results highlight that monitoring the local group sparsity, namely the spread of nonzero coefficients across different maps, is essential for detecting anomalous regions.

We introduce an adaptive Gaussian-Cauchy mixture modeling for the likelihood of pairwise mean/standard-deviation scatter points found when estimating signal-dependent noise. The maximization of the likelihood is used to identify the noise-model parameters, following an adaptive mixture parameter that controls the balance between the Gaussian and the heavy-tailed Cauchy. This renders the estimation robust with respect to outliers, typically present in large quantities among the scatter points from images dominated by texture. The modeling is directly suited to describing also observations subject to clipping, i.e. under- or over-exposure. Experiments on a dataset of badly exposed and highly textured images demonstrate the effectiveness of the adaptive Gaussian-Cauchy mixture likelihood for the accurate estimation of the noise standard-deviation curve.

Traditional point-by-point image similarity metrics, such as the ℓ2-norm, are not always consistent with human perception, especially in textured regions. We consider the problem of identifying textures that are perceptually identical to a query texture; this is important for image retrieval, compression, and restoration applications. Recently proposed structural texture similarity (STSIM) metrics assign high similarity scores to such perceptually identical textured patches, even though they may have significant pixel-wise deviations. We use an STSIM approach that compares a set of statistical patch descriptors through a weighted distance, and, given a dataset of labeled texture images partitioned into classes of perceptually identical patches, we calculate the weights as the variances of each statistic centered around the mean of its class. Experimental results demonstrate that the proposed approach outperforms existing structural similarity metrics and STSIMs as well as traditional point-by-point metrics when assessing texture similarity in both noisy and noise-free conditions.

We consider the estimation of signal-dependent noise from a single image. Unlike conventional algorithms that build a scatterplot of local mean-variance pairs from either small or adaptively selected homogeneous data samples, our proposed approach relies on arbitrarily large patches of heterogeneous data extracted at random from the image. We demonstrate the feasibility of our approach through an extensive theoretical analysis based on mixture of Gaussian distributions. A prototype algorithm is also developed in order to validate the approach on simulated data as well as on real camera raw images.

We propose a framework for the denoising of videos jointly corrupted by spatially correlated (i.e., nonwhite) random noise and spatially correlated fixed-pattern noise. Our approach is based on motion-compensated 3D spatiotemporal volumes, i.e., a sequence of 2D square patches extracted along the motion trajectories of the noisy video. First, the spatial and temporal correlations within each volume are leveraged to sparsify the data in 3D spatiotemporal transform domain, and then the coefficients of the 3D volume spectrum are shrunk using an adaptive 3D threshold array. Such array depends on the particular motion trajectory of the volume, the individual power spectral densities of the random and fixed-pattern noise, and also the noise variances which are adaptively estimated in transform domain. Experimental results on both synthetically corrupted data and real infrared videos demonstrate a superior suppression of the random and fixed-pattern noise from both an objective and a subjective point of view.

In digital imaging, there is often a need to produce estimates of the parameters that define the chosen noise model. We investigate how the mismatch between the estimated and true parameter values affects the stabilization of variance of signal-dependent noise. As a practical application of the general theoretical considerations, we devise a novel approach for estimating Poisson–Gaussian noise parameters from a single image, combining variance-stabilization and noise estimation for additive Gaussian noise. Furthermore, we construct a simple algorithm implementing the devised approach. We observe that when combined with optimized rational variance-stabilizing transformations, the algorithm produces results that are competitive with those of a state-of-the-art Poisson–Gaussian estimator.

This paper presents the evaluation results of the methods submitted to Challenge US: Biometric Measurements from Fetal Ultrasound Images, a segmentation challenge held at the IEEE International Symposium on Biomedical Imaging 2012. The challenge was set to compare and evaluate current fetal ultrasound image segmentation methods. It consisted of automatically segmenting fetal anatomical structures to measure standard obstetric biometric parameters, from 2D fetal ultrasound images taken on fetuses at different gestational ages (21 weeks, 28 weeks, and 33 weeks) and with varying image quality to reflect data encountered in real clinical environments. Four independent sub-challenges were proposed, according to the objects of interest measured in clinical practice: abdomen, head, femur, and whole fetus. Five teams participated in the head sub-challenge and two teams in the femur sub-challenge, including one team who tackled both. Nobody attempted the abdomen and whole fetus sub-challenges. The challenge goals were two-fold and the participants were asked to submit the segmentation results as well as the measurements derived from the segmented objects. Extensive quantitative (region-based, distance-based, and Bland-Altman measurements) and qualitative evaluation was performed to compare the results from a representative selection of current methods submitted to the challenge. Several experts (three for the head sub-challenge and two for the femur sub-challenge), with different degrees of expertise, manually delineated the objects of interest to define the ground truth used within the evaluation framework. For the head sub-challenge, several groups produced results that could be potentially used in clinical settings, with comparable performance to manual delineations. The femur sub-challenge had inferior performance to the head sub-challenge due to the fact that it is a harder segmentation problem and that the techniques presented relied more on the femur's appearance.

When our gaze fixates a point, the visual acuity is maximal at the fixation point (imaged by the fovea, i.e. the central part of the retina) and decreases rapidly towards the periphery of the visual field. This phenomenon is known as foveated vision or foveated imaging. We recently investigated the role of fovation in image filtering and we have shown that the foveated patch distance, i.e. the Euclidean distance between foveated patches, is a valuable feature for the assessment of nonlocal self-similarity. Foveation operators apply spatially variant blur, providing a compact multiscale representation of each image patch. Here, we introduce anisotropic foveation operators that embed directional point-spread functions, and we show that the operators providing the highest denoising quality are characterized by radial orientations. This result is coherent with the orientation preference in the human visual system.

We present an extension of the BM3D filter to volumetric data. The proposed algorithm, BM4D, implements the grouping and collaborative filtering paradigm, where mutually similar d-dimensional patches are stacked together in a (d+1)-dimensional array and jointly filtered in transform domain. While in BM3D the basic data patches are blocks of pixels, in BM4D we utilize cubes of voxels, which are stacked into a 4-D "group." The 4-D transform applied on the group simultaneously exploits the local correlation present among voxels in each cube and the nonlocal correlation between the corresponding voxels of different cubes. Thus, the spectrum of the group is highly sparse, leading to very effective separation of signal and noise through coefficient shrinkage. After inverse transformation, we obtain estimates of each grouped cube, which are then adaptively aggregated at their original locations. We evaluate the algorithm on denoising of volumetric data corrupted by Gaussian and Rician noise, as well as on reconstruction of volumetric phantom data with non-zero phase from noisy and incomplete Fourier-domain (k-space) measurements. Experimental results demonstrate the state-of-the-art denoising performance of BM4D, and its effectiveness when exploited as a regularizer in volumetric data reconstruction.

Many digital imaging devices operate by successive photon-to-electron, electron-to-voltage, and voltage-to-digit conversions. These processes are subject to various signal-dependent errors, which are typically modeled as Poisson-Gaussian noise. The removal of such noise can be effected indirectly by applying a variance-stabilizing transformation (VST) to the noisy data, denoising the stabilized data with a Gaussian denoising algorithm, and finally applying an inverse VST to the denoised data. The generalized Anscombe transformation (GAT) is often used for variance stabilization, but its unbiased inverse transformation has not been rigorously studied in the past. We introduce the exact unbiased inverse of the GAT and show that it plays an integral part in ensuring accurate denoising results. We demonstrate that this exact inverse leads to state-of-the-art results without any notable increase in the computational complexity compared to the other inverses. We also show that this inverse is optimal in the sense that it can be interpreted as a maximum likelihood inverse. Moreover, we thoroughly analyze the behavior of the proposed inverse, which also enables us to derive a closed-form approximation for it. This paper generalizes our work on the exact unbiased inverse of the Anscombe transformation, which we have presented earlier for the removal of pure Poisson noise.

The characteristic errors of many digital imaging devices can be modelled as Poisson-Gaussian noise, the removal of which can be approached indirectly through variance stabilization. The generalized Anscombe transformation (GAT) is commonly used for stabilization, but rigorous studies regarding its unbiased inverse transformation have been neglected. We introduce the exact unbiased inverse of the GAT, show that it is of essential importance for ensuring accurate denoising, and demonstrate that our approach leads to state-of-the-art results. This paper generalizes our earlier work, in which we presented an exact unbiased inverse of the Anscombe transformation for the case of pure Poisson noise removal.

When dealing with motion blur there is an inevitable trade-off between the amount of blur and the amount of noise in the acquired images. The effectiveness of any restoration algorithm typically depends on these amounts, and it is difficult to find their best balance in order to ease the restoration task. To face this problem, we provide a methodology for deriving a statistical model of the restoration performance of a given deblurring algorithm in case of arbitrary motion. Each restoration-error model allows us to investigate how the restoration performance of the corresponding algorithm varies as the blur due to motion develops. Our modeling treats the point-spread-function trajectories as random processes and, following a Monte-Carlo approach, expresses the restoration performance as the expectation of the restoration error conditioned on some motion-randomness descriptors and on the exposure time. This allows to coherently encompass various imaging scenarios, including camera shake and uniform (rectilinear) motion, and, for each of these, identify the specific exposure time that maximizes the image quality after deblurring.

We propose a powerful video filtering algorithm that exploits temporal and spatial redundancy characterizing natural video sequences. The algorithm implements the paradigm of nonlocal grouping and collaborative filtering, where a higher dimensional transform-domain representation of the observations is leveraged to enforce sparsity, and thus regularize the data: 3-D spatiotemporal volumes are constructed by tracking blocks along trajectories defined by the motion vectors. Mutually similar volumes are then grouped together by stacking them along an additional fourth dimension, thus producing a 4-D structure, termed group, where different types of data correlation exist along the different dimensions: local correlation along the two dimensions of the blocks, temporal correlation along the motion trajectories, and nonlocal spatial correlation (i.e., self-similarity) along the fourth dimension of the group. Collaborative filtering is then realized by transforming each group through a decorrelating 4-D separable transform and then by shrinkage and inverse transformation. In this way, the collaborative filtering provides estimates for each volume stacked in the group, which are then returned and adaptively aggregated to their original positions in the video. The proposed filtering procedure addresses several video processing applications, such as denoising, deblocking, and enhancement of both grayscale and color data. Experimental results prove the effectiveness of our method in terms of both subjective and objective visual quality, and show that it outperforms the state of the art in video denoising.

In this paper we consider the restoration of images corrupted by both uniform motion blur and Poissonian noise. We formulate an image formation model that explicitly takes into account the length of the blur point-spread function and the noise level as functions of the exposure time. Further, we present an analysis of the achievable restoration performance by showing how the root mean squared error varies with respect to the exposure time. It turns out that the worst situations are represented by either too short or too long exposure times. In between there exists an optimal exposure time that maximizes the restoration performance, balancing the amount of blur and noise in the observation. We justify such result through a mathematical analysis of the signal-to-noise ratio in Fourier domain; this study is then validated by deblurring synthetic data as well as camera raw data.

The removal of Poisson noise is often performed through the following three-step procedure. First, the noise variance is stabilized by applying the Anscombe root transformation to the data, producing a signal in which the noise can be treated as additive Gaussian with unitary variance. Second, the noise is removed using a conventional denoising algorithm for additive white Gaussian noise. Third, an inverse transformation is applied to the denoised signal, obtaining the estimate of the signal of interest. The choice of the proper inverse transformation is crucial in order to minimize the bias error which arises when the nonlinear forward transformation is applied. We introduce optimal inverses for the Anscombe transformation, in particular the exact unbiased inverse, a maximum likelihood (ML) inverse, and a more sophisticated minimum mean square error (MMSE) inverse. We then present an experimental analysis using a few state-of-the-art denoising algorithms and show that the estimation can be consistently improved by applying the exact unbiased inverse, particularly at the low-count regime. This results in a very efficient filtering solution that is competitive with some of the best existing methods for Poisson image denoising.

We presented an exact unbiased inverse of the Anscombe variance-stabilizing transformation in [M. Mäkitalo and A. Foi, "Optimal inversion of the Anscombe transformation in low-count Poisson image denoising," IEEE Trans. Image Process., vol. 20, no. 1, pp. 99-109, Jan. 2011.] and showed that when applied to Poisson image denoising, the combination of variance stabilization and state-of-the-art Gaussian denoising algorithms is competitive with some of the best Poisson denoising algorithms. We also provided a MATLAB implementation of our method, where the exact unbiased inverse transformation appears in nonanalytical form. Here, we propose a closed-form approximation of the exact unbiased inverse in order to facilitate the use of this inverse. The proposed approximation produces results equivalent to those obtained with the accurate (nonanalytical) exact unbiased inverse, and thus, notably better than one would get with the asymptotically unbiased inverse transformation that is commonly used in applications.

We present a simple and usable noise model for the raw-data of digital imaging sensors. This signal-dependent noise model, which gives the pointwise standard-deviation of the noise as a function of the expectation of the pixel raw-data output, is composed of a Poissonian part, modeling the photon sensing, and Gaussian part, for the remaining stationary disturbances in the output data. We further explicitly take into account the clipping of the data (over- and under-exposure), faithfully reproducing the nonlinear response of the sensor. We propose an algorithm for the fully automatic estimation of the model parameters given a single noisy image. Experiments with synthetic images and with real raw-data from various sensors prove the practical applicability of the method and the accuracy of the proposed model.

We propose an effective color image denoising method that exploits filtering in highly sparse local 3D transform domain in each channel of a luminance-chrominance color space. For each image block in each channel, a 3D array is formed by stacking together blocks similar to it, a process that we call "grouping". The high similarity between grouped blocks in each 3D array enables a highly sparse representation of the true signal in a 3D transform domain and thus a subsequent shrinkage of the transform spectra results in effective noise attenuation. The peculiarity of the proposed method is the application of a "grouping constraint" on the chrominances by reusing exactly the same grouping as for the luminance. The results demonstrate the effectiveness of the proposed grouping constraint and show that the developed denoising algorithm achieves state-of-the-art performance in terms of both peak signal-to-noise ratio and visual quality.

We introduce a new approach to image reconstruction from highly incomplete data. The available data are assumed to be a small collection of spectral coefficients of an arbitrary linear transform. This reconstruction problem is the subject of intensive study in the recent field of "compressed sensing" (also known as "compressive sampling"). Our approach is based on a quite specific recursive filtering procedure. At every iteration the algorithm is excited by injection of random noise in the unobserved portion of the spectrum and a spatially adaptive image denoising filter, working in the image domain, is exploited to attenuate the noise and reveal new features and details out of the incomplete and degraded observations. This recursive algorithm can be interpreted as a special type of the Robbins-Monro stochastic approximation procedure with regularization enabled by a spatially adaptive filter. Overall, we replace the conventional parametric modeling used in CS by a nonparametric one. We illustrate the effectiveness of the proposed approach for two important inverse problems from computerized tomography: Radon inversion from sparse projections and limited-angle tomography. In particular we show that the algorithm allows to achieve exact reconstruction of synthetic phantom data even from a very small number projections. The accuracy of our reconstruction is in line with the best results in the compressed sensing field.

We propose a novel image denoising strategy based on an enhanced sparse representation in transform domain. The enhancement of the sparsity is achieved by grouping similar 2-D image fragments (e.g., blocks) into 3-D data arrays which we call "groups." Collaborative filtering is a special procedure developed to deal with these 3-D groups. We realize it using the three successive steps: 3-D transformation of a group, shrinkage of the transform spectrum, and inverse 3-D transformation. The result is a 3-D estimate that consists of the jointly filtered grouped image blocks. By attenuating the noise, the collaborative filtering reveals even the finest details shared by grouped blocks and, at the same time, it preserves the essential unique features of each individual block. The filtered blocks are then returned to their original positions. Because these blocks are overlapping, for each pixel, we obtain many different estimates which need to be combined. Aggregation is a particular averaging procedure which is exploited to take advantage of this redundancy. A significant improvement is obtained by a specially developed collaborative Wiener filtering. An algorithm based on this novel denoising strategy and its efficient implementation are presented in full detail; an extension to color-image denoising is also developed. The experimental results demonstrate that this computationally scalable algorithm achieves state-of-the-art denoising performance in terms of both peak signal-to-noise ratio and subjective visual quality.

The shape-adaptive discrete cosine transform ISA-DCT) transform can be computed on a support of arbitrary shape, but retains a computational complexity comparable to that of the usual separable block-DCT (B-DCT). Despite the near-optimal decorrelation and energy compaction properties, application of the SA-DCT has been rather limited, targeted nearly exclusively to video compression. In this paper, we present a novel approach to image filtering based on the SA-DCT. We use the SA-DCT in conjunction with the Anisotropic Local Polynomial Approximation-Intersection of Confidence Intervals technique, which defines the shape of the transform's support in a pointwise adaptive manner. The thresholded or attenuated SA-DCT coefficients are used to reconstruct a local estimate of the signal within the adaptive-shape support. Since supports corresponding to different points are in general overlapping, the local estimates are averaged together using adaptive weights that depend on the region's statistics. This approach can be used for various image-processing tasks. In this paper, we consider, in particular, image denoising and image deblocking and deringing from block-DCT compression. A special structural constraint in luminance-chrominance space is also proposed to enable an accurate filtering of color images. Simulation experiments show a state-of-the-art quality of the final estimate, both in terms of objective criteria and visual appearance. Thanks to the adaptive support, reconstructed edges are clean, and no unpleasant ringing artifacts are introduced by the fitted transform.

A spatially adaptive image deblurring algorithm is presented for Poisson observations. It adapts to the unknown image smoothness by using local polynomial approximation (LPA) kernel estimates of varying scale and direction based on the intersection of confidence intervals (ICI) rule. The signal-dependant characteristics of the Poissonian noise are exploited to accurately compute the pointwise variances of the directional estimates. The results show that this accurate pointwise adaptive algorithm significantly improves the image restoration quality.