arXiv subjects

Michael Elad

Publications and source records attributed to Michael Elad.

115 records · Page 7Linked to original sources

arXiv2011

Generalized Tree-Based Wavelet Transform

In this paper we propose a new wavelet transform applicable to functions defined on graphs, high dimensional data and networks. The proposed method generalizes the Haar-like transform proposed in [1], and it is defined via a hierarchical tree, which is assumed to capture the geometry and structure of the input data. It is applied to the data using a modified version of the common one-dimensional (1D) wavelet filtering and decimation scheme, which can employ different wavelet filters. In each level of this wavelet decomposition scheme, a permutation derived from the tree is applied to the approximation coefficients, before they are filtered. We propose a tree construction method that results in an efficient representation of the input function in the transform domain. We show that the proposed transform is more efficient than both the 1D and two-dimensional (2D) separable wavelet transforms in representing images. We also explore the application of the proposed transform to image denoising, and show that combined with a subimage averaging scheme, it achieves denoising results which are similar to those obtained with the K-SVD algorithm.

cs.CV↗

arXiv2010

RIP-Based Near-Oracle Performance Guarantees for Subspace-Pursuit, CoSaMP, and Iterative Hard-Thresholding

This paper presents an average case denoising performance analysis for the Subspace Pursuit (SP), the CoSaMP and the IHT algorithms. This analysis considers the recovery of a noisy signal, with the assumptions that (i) it is corrupted by an additive random white Gaussian noise; and (ii) it has a K-sparse representation with respect to a known dictionary D. The proposed analysis is based on the Restricted-Isometry-Property (RIP), establishing a near-oracle performance guarantee for each of these algorithms. The results for the three algorithms differ in the bounds' constants and in the cardinality requirement (the upper bound on $K$ for which the claim is true). Similar RIP-based analysis was carried out previously for the Dantzig Selector (DS) and the Basis Pursuit (BP). Past work also considered a mutual-coherence-based analysis of the denoising performance of the DS, BP, the Orthogonal Matching Pursuit (OMP) and the thresholding algorithms. This work differs from the above as it addresses a different set of algorithms. Also, despite the fact that SP, CoSaMP, and IHT are greedy-like methods, the performance guarantees developed in this work resemble those obtained for the relaxation-based methods (DS and BP), suggesting that the performance is independent of the sparse representation entries contrast and magnitude.

stat.ME↗

arXiv2010

Spatially-Adaptive Reconstruction in Computed Tomography Based on Statistical Learning

We propose a direct reconstruction algorithm for Computed Tomography, based on a local fusion of a few preliminary image estimates by means of a non-linear fusion rule. One such rule is based on a signal denoising technique which is spatially adaptive to the unknown local smoothness. Another, more powerful fusion rule, is based on a neural network trained off-line with a high-quality training set of images. Two types of linear reconstruction algorithms for the preliminary images are employed for two different reconstruction tasks. For an entire image reconstruction from full projection data, the proposed scheme uses a sequence of Filtered Back-Projection algorithms with a gradually growing cut-off frequency. To recover a Region Of Interest only from local projections, statistically-trained linear reconstruction algorithms are employed. Numerical experiments display the improvement in reconstruction quality when compared to linear reconstruction algorithms.

cs.CV↗

arXiv2010

Analysis of Basis Pursuit Via Capacity Sets

Finding the sparsest solution $α$ for an under-determined linear system of equations $Dα=s$ is of interest in many applications. This problem is known to be NP-hard. Recent work studied conditions on the support size of $α$ that allow its recovery using L1-minimization, via the Basis Pursuit algorithm. These conditions are often relying on a scalar property of $D$ called the mutual-coherence. In this work we introduce an alternative set of features of an arbitrarily given $D$, called the "capacity sets". We show how those could be used to analyze the performance of the basis pursuit, leading to improved bounds and predictions of performance. Both theoretical and numerical methods are presented, all using the capacity values, and shown to lead to improved assessments of the basis pursuit success in finding the sparest solution of $Dα=s$.

math.NA↗

arXiv2010

The Projected GSURE for Automatic Parameter Tuning in Iterative Shrinkage Methods

Linear inverse problems are very common in signal and image processing. Many algorithms that aim at solving such problems include unknown parameters that need tuning. In this work we focus on optimally selecting such parameters in iterative shrinkage methods for image deblurring and image zooming. Our work uses the projected Generalized Stein Unbiased Risk Estimator (GSURE) for determining the threshold value lambda and the iterations number K in these algorithms. The proposed parameter selection is shown to handle any degradation operator, including ill-posed and even rectangular ones. This is achieved by using GSURE on the projected expected error. We further propose an efficient greedy parameter setting scheme, that tunes the parameter while iterating without impairing the resulting deblurring performance. Finally, we provide extensive comparisons to conventional methods for parameter selection, showing the superiority of the use of the projected GSURE.

cs.CV↗

arXiv2010

On MMSE and MAP Denoising Under Sparse Representation Modeling Over a Unitary Dictionary

Among the many ways to model signals, a recent approach that draws considerable attention is sparse representation modeling. In this model, the signal is assumed to be generated as a random linear combination of a few atoms from a pre-specified dictionary. In this work we analyze two Bayesian denoising algorithms -- the Maximum-Aposteriori Probability (MAP) and the Minimum-Mean-Squared-Error (MMSE) estimators, under the assumption that the dictionary is unitary. It is well known that both these estimators lead to a scalar shrinkage on the transformed coefficients, albeit with a different response curve. In this work we start by deriving closed-form expressions for these shrinkage curves and then analyze their performance. Upper bounds on the MAP and the MMSE estimation errors are derived. We tie these to the error obtained by a so-called oracle estimator, where the support is given, establishing a worst-case gain-factor between the MAP/MMSE estimation errors and the oracle's performance. These denoising algorithms are demonstrated on synthetic signals and on true data (images).

cs.CV↗

arXiv2009

Coherence-Based Performance Guarantees for Estimating a Sparse Vector Under Random Noise

We consider the problem of estimating a deterministic sparse vector x from underdetermined measurements Ax+w, where w represents white Gaussian noise and A is a given deterministic dictionary. We analyze the performance of three sparse estimation algorithms: basis pursuit denoising (BPDN), orthogonal matching pursuit (OMP), and thresholding. These algorithms are shown to achieve near-oracle performance with high probability, assuming that x is sufficiently sparse. Our results are non-asymptotic and are based only on the coherence of A, so that they are applicable to arbitrary dictionaries. Differences in the precise conditions required for the performance guarantees of each algorithm are manifested in the observed performance at high and low signal-to-noise ratios. This provides insight on the advantages and drawbacks of convex relaxation techniques such as BPDN as opposed to greedy approaches such as OMP and thresholding.

math.ST↗