Alessandro Foi

Pointwise Shape-Adaptive DCT for High-Quality Denoising and Deblocking of Grayscale and Color Images

The shape-adaptive discrete cosine transform (SA-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 transforms 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 regions 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

Practical Poissonian-Gaussian Noise Modeling and Fitting for Single-Image Raw-Data

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.

From Local Kernel to Nonlocal Multiple-Model Image Denoising

We review the evolution of the nonparametric regression modeling in imaging from the local Nadaraya-Watson kernel estimate to the nonlocal means and further to transform-domain filtering based on nonlocal block-matching. The considered methods are classified mainly according to two main features: local/nonlocal and pointwise/multipoint. Here nonlocal is an alternative to local, and multipoint is an alternative to pointwise. These alternatives, though obvious simplifications, allow to impose a fruitful and transparent classification of the basic ideas in the advanced techniques. Within this framework, we introduce a novel single- and multiple-model transform domain nonlocal approach. The Block Matching and 3-D Filtering (BM3D) algorithm, which is currently one of the best performing denoising algorithms, is treated as a special case of the latter approach.

Image denoising with block-matching and 3D filtering

We present a novel approach to still image denoising based on effective filtering in 3D transform domain by combining sliding-window transform processing with block-matching. We process blocks within the image in a sliding manner and utilize the block-matching concept by searching for blocks which are similar to the currently processed one. The matched blocks are stacked together to form a 3D array and due to the similarity between them, the data in the array exhibit high level of correlation. We exploit this correlation by applying a 3D decorrelating unitary transform and effectively attenuate the noise by shrinkage of the transform coefficients. The subsequent inverse 3D transform yields estimates of all matched blocks. After repeating this procedure for all image blocks in sliding manner, the final estimate is computed as weighed average of all overlapping blockestimates. A fast and efficient algorithm implementing the proposed approach is developed. The experimental results show that the proposed method delivers state-of-art denoising performance, both in terms of objective criteria and visual quality.

Nonlocal Transform-Domain Filter for Volumetric Data Denoising and Reconstruction

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.

Image restoration by sparse 3D transform-domain collaborative filtering

We propose an image restoration technique exploiting regularized inversion and the recent block-matching and 3D filtering (BM3D) denoising filter. The BM3D employs a non-local modeling of images by collecting similar image patches in 3D arrays. The so-called collaborative filtering applied on such a 3D array is realized by transformdomain shrinkage. In this work, we propose an extension of the BM3D filter for colored noise, which we use in a two-step deblurring algorithm to improve the regularization after inversion in discrete Fourier domain. The first step of the algorithm is a regularized inversion using BM3D with collaborative hard-thresholding and the seconds step is a regularized Wiener inversion using BM3D with collaborative Wiener filtering. The experimental results show that the proposed technique is competitive with and in most cases outperforms the current best image restoration methods in terms of improvement in signal-to-noise ratio.

Optimal Inversion of the Anscombe Transformation in Low-Count Poisson Image Denoising

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.

Optimal Inversion of the Generalized Anscombe Transformation for Poisson-Gaussian Noise

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.

Video Denoising, Deblocking, and Enhancement Through Separable 4-D Nonlocal Spatiotemporal Transforms

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.

Noise estimation and removal in MR imaging: The variance-stabilization approach

We develop optimal forward and inverse variance-stabilizing transformations for the Rice distribution, in order to approach the problem of magnetic resonance (MR) image filtering by means of standard denoising algorithms designed for homoskedastic observations.