Lorenzo Granai

On the use of a priori information for sparse signal approximations

Recent results have underlined the importance of incoherence in redundant dictionaries for a good behavior of decomposition algorithms like matching and basis pursuit. However, appropriate dictionaries for a given application may not be able to meet the incoherence condition. In such a case, decomposition algorithms may completely fail in the retrieval of the sparsest approximation. This paper studies the effect of introducing a priori knowledge when recovering sparse approximations over redundant dictionaries. Theoretical results show how the use of reliable a priori information (which in this paper appears under the form of weights) can improve the performances of standard approaches such as greedy algorithms and relaxation methods. Our results reduce to the classical case when no prior information is available. Examples validate and illustrate our theoretical statements. EDICS: 2-NLSP

Image compression using an edge adapted redundant dictionary and wavelets

Low bit rate image coding is an important problem regarding applications such as storage on low memory devices or streaming data on the internet. The state of the art in image compression is to use two-dimensional (2-D) wavelets. The advantages of wavelet bases lie in their multiscale nature and in their ability to sparsely represent functions that are piecewise smooth. Their main problem on the other hand, is that in 2-D wavelets are not able to deal with the natural geometry of images, i.e. they cannot sparsely represent objects that are smooth away from regular submanifolds. In this paper we propose an approach based on building a sparse representation of the edge part of images in a redundant geometrically inspired library of functions, followed by suitable coding techniques. Best N-terms non-linear approximations in general dictionaries is, in most cases, a NP-hard problem and sub-optimal approaches have to be followed. In this work we use a greedy strategy, also known as Matching Pursuit to compute the expansion. The residual, that we suppose to be the smooth and texture part, is then coded using wavelets. A rate distortion optimization procedure chooses the number of functions from the redundant dictionary and the wavelet basis.

New dictionary and fast atom searching method for matching pursuit representation of displaced frame difference

Matching pursuit decomposes a signal into a linear expansion of functions selected from a redundant dictionary, isolating the signal structures that are coherent with respect to a given dictionary. In this paper we focus on the Matching Pursuit representation of the displaced frame difference (dfd). In particular, we introduce a new dictionary for matching pursuit that efficiently exploits the signal structures of the dfd. We also propose a fast strategy to find the atoms exploiting the maximum of the absolute value of the error in the motion predicted image and the convergence of the MSE with the rotation of the atoms. Results show that the fast strategy is quite robust when compared to exhaustive search techniques and it improves the results of a suboptimal search strategy based on a genetic algorithm.

Very low bit rate image coding using redundant dictionaries

Very low bit rate image coding is an important problem regarding applications such as storage on low memory devices or streaming data on the internet. The state of the art in image compression is to use 2-D wavelets. The advantages of wavelet bases lie in their multiscale nature and in their ability to sparsely represent functions that are piecewise smooth. Their main problem on the other hand, is that in 2-D wavelets are not able to deal with the natural geometry of images, i.e they cannot sparsely represent objects that are smooth away from regular submanifolds. In this paper we propose an approach based on building a sparse representation of images in a redundant geometrically inspired library of functions, followed by suitable coding techniques. Best N-term non- linear approximations in general dictionaries is, in most cases, a NP-hard problem and sub-optimal approaches have to be followed. In this work we use a greedy strategy, also known as Matching Pursuit to compute the expansion. Finally the last step in our algorithm is an enhancement layer that encodes the residual image: in our simulation we have used a genuine embedded wavelet codec.

R-D analysis of adaptive edge representations

This paper presents a rate-distortion analysis for a simple horizon edge image model. A quadtree with anisotropy and rotation is performed on this kind of image, giving a toy model for a non-linear adaptive coding technique, and its rate-distortion behavior is studied. The effect of refining the quadtree decomposition is also analyzed.

Ridgelet transform applied to motion compensated images

Wavelet transform is a powerful instrument in catching zero-dimensional singularities. Ridgelets are powerful instrument in catching and representing mono-dimensional singularities in bidimensional space. In this paper we propose a hybrid video coder scheme using ridgelet transform for the first approximation of line-edge singularities in displaced frame difference images. We demonstrate the potential of ridgelets and results show substantial improvements when compared to wavelet only based coder.

Ventricular and Atrial Activity Estimation Through Sparse Ecg Signal Decompositions

This paper explores a novel approach for ventricular and atrial activities estimation in electrocardiogram (ECG) signals, based on sparse source separation. Sparse decompositions of ECG over signal-adapted multi-component dictionaries can lead to natural separation of its components. In this work, dictionaries of functions adapted to ventricular and atrial activities are respectively defined. Then, the weighted orthogonal matching pursuit algorithm is used to unmix the two components of ECG signals. Despite the simplicity of the approach, results are very promising, showing the capacity of the algorithm to generate realistic estimations of atrial and ventricular activities

Sparse decomposition over multi-component redundant dictionaries

In many applications - such as compression, de-noising and source separation - a good and efficient signal representation is characterized by sparsity. This means that many coefficients are close to zero, while only few ones have a non-negligible amplitude. On the other hand, real-world signals such as audio or natural images - clearly present peculiar structures. In this paper we introduce a global optimization framework that aims at respecting the sparsity criterion while decomposing a signal over an overcomplete, multi-component dictionary. We adopt a probabilistic analysis which can lead to consider the signal internal structure. As an example that fits this framework, we propose the weighted basis pursuit algorithm, based on the solution of a convex, non-quadratic problem. Results show that this method can provide sparse signal representations and sparse m-terms approximations. Moreover, weighted basis pursuit provides a faster convergence compared to basis pursuit.

Hybrid video coding based on bidimensional matching pursuit

Hybrid video coding combines together two stages: first, motion estimation and compensation predict each frame from the neighboring frames, then the prediction error is coded, reducing the correlation in the spatial domain. In this work, we focus on the latter stage, presenting a scheme that profits from some of the features introduced by the standard H.264/AVC for motion estimation and replaces the transform in the spatial domain. The prediction error is so coded using the matching pursuit algorithm which decomposes the signal over an appositely designed bidimensional, anisotropic, redundant dictionary. Comparisons are made among the proposed technique, H.264, and a DCT-based coding scheme. Moreover, we introduce fast techniques for atom selection, which exploit the spatial localization of the atoms. An adaptive coding scheme aimed at optimizing the resource allocation is also presented, together with a rate-distortion study for the matching pursuit algorithm. Results show that the proposed scheme outperforms the standard DCT, especially at very low bit rates.