We formalize compressed sensing image reconstruction as an optimization problem, incorporating penalization of the spectral representation of images. Leveraging the original formulation of the Alternating Direction Method of Multipliers (ADMM), we introduce the innovative 3F-PnP algorithm. This algorithm integrates three filters: two deep learning neural network-based filters and the spectral BM3D denoiser, implemented through plug-and-play modules. Additionally, we show that the partial solutions of the ADMM optimization correspond precisely to the analysis and synthesis stages of the BM3D filter. Through numerical comparative analysis against ten state-of-the-art methods, we demonstrate the superiority of our algorithm in terms of improved accuracy and faster convergence rates.

The hyperspectral broadband phase retrieval is developed for a scenario where both object and modulation phase masks are spectrally varying. The proposed iterative algorithm is based on a complex domain version of the alternating direction method of multipliers (ADMM) and the novel Spectral Proximity Operators derived for Gaussian and Poissonian multiple intensity observations. These proximity operators solve two problems. First, the complex-domain spectral components of the object are extracted from the total intensity observations calculated as the sums of the spectral intensities of diffractive patterns. Second, noisy observations are filtered, compromising noisy intensity observations and their predicted counterparts. The simulation and physical tests confirm that the broadband hyperspectral phase retrieval in the proposed formulation can be successfully resolved.

Hyperspectral (HS) imaging retrieves information from data obtained across a wide spectral range of spectral channels. The object to reconstruct is a 3D cube, where two coordinates are spatial and third one is spectral. We assume that this cube is complex-valued, i.e. characterized spatially frequency varying amplitude and phase. The observations are squared magnitudes measured as intensities summarized over spectrum. The HS phase retrieval problem is formulated as a reconstruction of the HS complex-valued object cube from Gaussian noisy intensity observations. The derived iterative algorithm includes the original proximal spectral analysis operator and the sparsity modeling for complex-valued 3D cubes. The efficiency of the algorithm is confirmed by simulation tests.

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.

Single image super-resolution (SISR) is an ill-posed problem aiming at estimating a plausible high-resolution (HR) image from a single low-resolution image. Current state-of-the-art SISR methods are patch-based. They use either external data or internal self-similarity to learn a prior for an HR image. External data-based methods utilize a large number of patches from the training data, while self-similarity-based approaches leverage one or more similar patches from the input image. In this paper, we propose a self-similarity-based approach that is able to use large groups of similar patches extracted from the input image to solve the SISR problem. We introduce a novel prior leading to the collaborative filtering of patch groups in a 1D similarity domain and couple it with an iterative back-projection framework. The performance of the proposed algorithm is evaluated on a number of SISR benchmark data sets. Without using any external data, the proposed approach outperforms the current non-convolutional neural network-based methods on the tested data sets for various scaling factors. On certain data sets, the gain is over 1 dB, when compared with the recent method A+. For high sampling rate (x4), the proposed method performs similarly to very recent state-of-the-art deep convolutional network-based approaches.

This paper addresses interferometric phase image estimation, i.e., the estimation of phase modulo-2π images from sinusoidal 2π-periodic and noisy observations. These degradation mechanisms make interferometric phase image estimation a quite challenging problem. We tackle this challenge by reformulating the true estimation problem as a sparse regression, often termed sparse coding, in the complex domain. Following the standard procedure in patch-based image restoration, the image is partitioned into small overlapping square patches, and the vector corresponding to each patch is modeled as a sparse linear combination of vectors, termed the atoms, taken from a set called dictionary. Aiming at optimal sparse representations, and thus at optimal noise removing capabilities, the dictionary is learned from the data that it represents via matrix factorization with sparsity constraints on the code (i.e., the regression coefficients) enforced by the ℓ1norm. The effectiveness of the new sparse-coding-based approach to interferometric phase estimation, termed the SpInPHASE, is illustrated in a series of experiments with simulated and real data where it outperforms the state-of-the-art.

The paper addresses interferometric phase image estimation, that is, the estimation of phase modulo-2π images from sinusoidal 2π-periodic and noisy observations. These degradation mechanisms make interferometric phase image estimation a challenging problem. We tackle this challenge by reformulating the true estimation problem as a sparse regression in the complex domain. Following the standard procedure in patch-based image restoration, the image is partitioned into small overlapping square patches. BM3D algorithm equipped with high order SVD (HOSVD) is used to form complex domain frames suitable to sparse representations of the complex-valued data. HOSVD applied to the groups of BM3D data enables the design of spatially variant and data adaptive orthonormal complex domain transforms. The effectiveness of the new sparse coding based approach to interferometric phase estimation, termed Interferometric PHASE via Block matching and High order SVD (InPHASE-BHS) is illustrated in a series of simulation experiments where it outperforms the state-of-the-art.

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.

A family of the block matching 3-D (BM3D) algorithms for various imaging problems has been recently proposed within the framework of nonlocal patchwise image modeling , . In this paper, we construct analysis and synthesis frames, formalizing BM3D image modeling, and use these frames to develop novel iterative deblurring algorithms. We consider two different formulations of the deblurring problem, i.e., one given by the minimization of the single-objective function and another based on the generalized Nash equilibrium (GNE) balance of two objective functions. The latter results in the algorithm where deblurring and denoising operations are decoupled. The convergence of the developed algorithms is proved. Simulation experiments show that the decoupled algorithm derived from the GNE formulation demonstrates the best numerical and visual results and shows superiority with respect to the state of the art in the field, confirming a valuable potential of BM3D-frames as an advanced image modeling tool.

Recently within the framework of nonlocal patch-wise estimation a family of the Block Matching 3-D (BM3D) algorithms has been developed for various imaging problems [1], [2]. In [3] we demonstrated that BM3D modeling allows frame interpretation and constructed the corresponding analysis and synthesis frames which we call BM3D-frames. In this paper we use the BM3D-frames to develop an image deblurring algorithm based on the alternating optimization of two objective functions corresponding to decoupled denoising and deblurring operations. Compared to the standard approach based on minimization of a single objective function, decoupling allows to achieve both better reconstruction performance and essential simplification of the algorithm. Simulated experiments demonstrate numerical and visual superiority of the proposed algorithm over the current state-of-the-art methods confirming the advantage of the BM3D-frames as an image modeling tool.

We consider reconstruction of a wave field distribution in an input/object plane from data in an output/diffraction (sensor) plane. For the forward propagation of the wave field the matrix form of the discrete diffraction transform (DDT) originated in and is used. This ¿matrix DDT¿ is aliasing free and precise for pixel-wise invariant object and sensor plane distributions. A contribution of this paper concerns a study of the backward wave field propagation as an inverse problem for the diffraction kernel. The analysis of the conditioning of the transfer DDT matrices is presented in order to find when the perfect reconstruction of the object wavefield distribution is possible. This condition number can be used as an indicator of the accuracy of the wave field reconstruction. Simulation experiments show that the developed inverse propagation algorithm demonstrates an improved accuracy as compared with the standard convolutional and discrete Fresnel transform algorithms.

The problem of estimating information symbols transmitted through non-Gaussian multipath CDMA wireless channels with a phased array is considered. First, a robust detector based on the minimax robust estimation theory is proposed. This proposed robust technique has a decreased sensitivity of the estimates with respect to an actually unknown distribution of random noises and interferences. Then, the estimation accuracy of information signals is analysed by Monte Carlo simulation of this direction-of-arrival estimation-based robust detector in multipath scenario.

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.

The local polynomial approximation (LPA) is a nonparametric regression technique with pointwise estimation in a sliding window. We apply the LPA of the argument of cos and sin in order to estimate the absolute phase from noisy wrapped phase data. Using the intersection of confidence interval (HCI) algorithm, the window size is selected as adaptive pointwise varying. This adaptation gives the phase estimate with the accuracy close to optimal in the mean squared sense. For calculations, we use a Gauss-Newton recursive procedure initiated by the phase estimates obtained for the neighboring points. It enables tracking properties of the algorithm and its ability to go beyond the principal interval [-pi, pi] and to reconstruct the absolute phase from wrapped phase observations even when the magnitude of the phase difference takes quite large values. The algorithm demonstrates a very good accuracy of the phase reconstruction which on many occasion overcomes the accuracy of the state-of-the-art algorithms developed for noisy phase unwrap. The theoretical analysis produced for the accuracy of the pointwise estimates is used for justification of the HCI adaptation algorithm.

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.

In this paper, new robust M-estimation techniques are developed for combating impulsive noise and multiple-access interference in communication systems. Power functions of the proposed robust estimators are obtained by minimizing a nonquadratic residual function derived from the Huber's minimax robust estimation theory. Maximum peaks of these power functions are used for estimation of communication signals as well as direction of arrival. A strong advantage of the proposed robust M-estimation algorithms is a decreased sensitivity of the estimates with respect to an actual unknown distribution of random noises and interferences. Simulation results demonstrate that the proposed robust algorithms, the robust median, and robust Huber estimators, offer significant performance gain over the conventional and minimum-variance distortionless-response estimators, with the best results given by the robust Huber estimator.

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.

We propose a novel nonparametric regression metthod for deblurring noisy images. The method is based on the local polynomial approximation (LPA) of the image and the paradigm of intersecting confidence intervals (ICI) that is applied to define the adaptive varying scales (window sizes) of the LPA estimators. The LPA-ICI algorithm is nonlinear and spatially adaptive with respect to smoothness and irregularities of the image corrupted by additive noise. Multiresolution wavelet algorithms produce estimates which are combined from different scale projections. In contrast to them, the proposed ICI algorithm gives a varying scale adaptive estimate defining a single best scale for each pixel. In the new algorithm, the actual filtering is performed in signal domain while frequency domain Fourier transform operations are applied only for calculation of convolutions. The regularized inverse and Wiener inverse filters serve as deblurring operators used jointly with the LPA-design directional kernel filters. Experiments demonstrate the state-of-art performance of the new estimators which visually and quantitatively outperform some of the best existing methods.

Recently, new efficient algorithms, based on Lepski's approach , have been proposed for spatially adaptive varying scale de-noising. Special statistical rules are exploited in order to select the estimate with the best point-wise varying scale h from a set of test-estimates y/spl circ//sub h/(x),h/spl isin/H. In this paper, a novel multiresolution (MR) nonparametric regression technique is developed. The adaptive algorithm consists of two steps. The first step transforms the data into noisy spectrum coefficients (MR analysis). In the second step, these noisy spectrum is filtered by the thresholding procedure and exploited for estimation (MR synthesis). This nonlinear estimate is built using the test-estimates y/spl circ//sub h/(x) of all scales. Simulation confirms the advanced performance of the new de-noising algorithms based on the MR nonparametric regression.

Recently new efficient algorithms, based on Lepski's approach, have been proposed in mathematical statistics for spatially adaptive varying scale denoising. A common feature of this sort of algorithms is that they form test-estimates different by the scale and special statistical rules are exploited in order to select the estimate with the best pointwise varying scale. In this paper a novel alternative multiresolution (MR) approach is proposed. Instead of selection of the estimate with the best scale a nonlinear estimate is built using all of the test-estimates. The estimation consists of two steps. The first step transforms the data into noisy spectrum coefficients (MR analysis). In the second step, these noisy estimates of the spectrum are filtered and used for estimation (MR synthesis). Simulation confirms an advance performance of the denoising algorithms based on the MR nonparametric regression.

The robust M-periodogram is defined for the analysis of signals with heavy-tailed distribution noise. In the form of a robust spectrogram (RSPEC) it can be used for the analysis of nonstationary signals. A RSPEC-based instantaneous frequency (IF) estimator, with a time-varying window length, is presented. The optimal choice of the window length can resolve the bias-variance tradeoff in the RSPEC-based IF estimation. However, it depends on the unknown nonlinearity of the IF. The algorithm used is able to provide accuracy close to the one that could be achieved if the IF to be estimated were known in advance. Simulations show good accuracy ability of the adaptive algorithm and good robustness property with respect to rare high-magnitude noise values.

An algorithm for image noise-removal based on local adaptive window size filtering is developed. Two features for use in local spatial/transform-domain filtering are suggested. First, filtering is performed on images corrupted not only by additive white noise, but also by image-dependent (e.g. film-grain noise) or multiplicative noise. Second, the used transforms are equipped with a varying adaptive window size obtained by the intersection of confidence intervals (ICI) rule. Finally, we combine all estimates available for each pixel from neighboring overlapping windows by weighted averaging these estimates. Comparison of the algorithm with the known techniques for noise removal from images shows the advantage of the new algorithm, both quantitatively and visually.

Local adaptive image denoising in transform domain is a powerful tool for adapting to unknown smoothness of the images. We propose to perform local adaptive denoising with adaptively varying local transform support size rather than using a transform with fixed size. We use a special rule (intersection of confidence intervals-ICI) to select the optimum window sizes locally. The algorithm provides significant improvements in the de-noising performance.

We describe a novel approach to solve a problem of bandwidth selection for median filtering a signal given with an additive noise. The approach is based on the intersection of confidence intervals ICI rule and gives the algorithm, which is simple to implement and nearly ideal within lnN factor in the point-wise mean squared errors risk for estimating the signal. The ICI rule gives the adaptive varying bandwidths and enables the algorithm to be spatial adaptive in the sense that its quality is close to that which one could achieve if the smoothness of the estimated signal was known in advance.

A windowed local polynomial approximation (LPA) of time-varying directions-of-arrival (DOA) is exploited to derive a new form of beamformer for source localization and tracking in nonstationary environments. The relationship between our LPA-beamformer and conventional beamformer is discussed.

The L-Wigner distributions are defined in order to improve the concentration of signal's time-frequency representation. For a finite distribution order and nonlinear instantaneous frequency (IF) it gives biased IF estimates. In the case of noisy signals the optimal window length and distribution order depend on the noise variance and unknown IF. In this article an adaptive IF estimator with the time-varying and data-driven window length and distribution order is developed. Based on the analysis that has been done here, lower order time-frequency distributions are introduced.

A local adaptive image de-noising method based on local selection of the best wavelet, among a finite set, within a sliding window at each level of decomposition is developed. The proposed method suggests certain advantages in terms of de-noising efficiency and detail preservation especially when the image includes different regions which may be efficiently represented by different bases or a priori information on the image is limited. This work concerns the method of using an adaptively varying base and the best wavelet selection rule. The method is implemented and comparative results are submitted.

The local polynomial approximation (LPA) of noisy data is considered with the new adaptive procedure for varying bandwidth selection. The algorithm is simple to implement and nearly optimal within lnN factor in the point-wise risk for estimating the function and its derivatives. The adaptive varying bandwidth enables the algorithm to be spatial adaptive over a wide range of the classes of functions in the sense that its quality is close to that which one could achieve if the smoothness of the estimated function was known in advance. It is shown that the cross-validation adjustment of the threshold parameter of the algorithm significantly improves its accuracy. In particular, simulation demonstrates that the adaptive algorithm with the adjusted threshold parameter performs better than the wavelet estimators.

Estimation of the instantaneous frequency and its derivatives is considered for a harmonic signal with a time-varying phase and time-invariant amplitude. The asymptotic minimax lower bound is derived for the mean squared error of estimation provided that the phase is an arbitrary m-times piecewise differentiable function of time. It is shown that this lower bound is different only in a constant factor from the upper bound for the mean squared errors of the local polynomial periodogram with optimal window size.

A method for minimization of the mean square error (MSE) of the instantaneous frequency estimation using time-frequency distributions, in the case of a discrete optimization parameter, is presented. It does not require a knowledge of the estimation bias. The method is illustrated on adaptive window width determination in the Wigner distribution.

The local polynomial approximation of time-varying phase is used in order to estimate the instantaneous frequency and its derivatives for a complex-valued harmonic signal given by discrete-time observations with a noise. The considered estimators are high-order nonparametric generalizations of the short-time Fourier transform and the Wigner-Ville distribution. The asymptotic variance and bias of the estimates are obtained.

The estimation of the time-varying IF /spl Omega/(t) and amplitude A(t) of harmonic signals is considered. The nonparametric local polynomial approximation (LPA) is used in order to develop two new estimators, which enable one to estimate /spl Omega/(t), /spl Omega//sup (1)/(t), ..., /spl Omega/((m/sub /spl omega//-1))(t) and A(t), A/sup (1)/(t), ..., A((m/sub A/))(t), where m/sub /spl omega// and m/sub A/ are the degrees of the LPA of the phase and the amplitude. The a priori amplitude information about /spl Omega/(t), A(t) and their derivatives can be incorporated in order to improve the estimates. The estimators of the IF and the amplitude have forms of generalized periodograms and Fourier transforms respectively, where a polynomial function appears in the degree of the complex exponent usual for the Fourier transform. The asymptotic bias and covariance are obtained for the new estimators. In particular, it is shown that one of the new estimators ensures complete compensation of the disturbing influence of the time-varying amplitude for the estimation of the IF and vice versa.