Overcomplete representations are attracting interest in signal processing theory, particularly due to their potential to generate sparse representations of signals. However, in general, the problem of finding sparse representations must be unstable in the presence of noise. This paper establishes the possibility of stable recovery under a combination of sufficient sparsity and favorable structure of the overcomplete system. Considering an ideal underlying signal that has a sufficiently sparse representation, it is assumed that only a noisy version of it can be observed. Assuming further that the overcomplete system is incoherent, it is shown that the optimally sparse approximation to the noisy data differs from the optimally sparse decomposition of the ideal noiseless signal by at most a constant multiple of the noise level. As this optimal-sparsity method requires heavy (combinatorial) computational effort, approximation algorithms are considered. It is shown that similar stability is also available using the basis and the matching pursuit algorithms. Furthermore, it is shown that these methods result in sparse approximation of the noisy data that contains only terms also appearing in the unique sparsest representation of the ideal noiseless sparse signal.

In this paper we consider a large number of experiments (some 800 in total) using a variety of different methods for texture segmentation based upon Wavelets. The experiments also consider ten different wavelet filters and use as a testing bed a variety of composite images taken from the Brodatz database. Ground truth for the texture segmentation is known for these images. We present a method for evaluating how well the different textures are separated in the selected feature spaces of these different algorithms, as well as measuring the final performance of the segmentation boundary compared to ground truth. We introduce the two-point correlation function as a performance measure, as well as a tool for selecting features, and show that it can quantify performance in a way that correlates well with ground truth measures. We show that among the methods we have tested, one stands out clearly as superior to all the others, and that the choice of filters plays little role. 1 Introduction ...

This thesis is concerned with the performance measures for the wavelet-based texture segmentation algorithms. After, a brief introduction to wavelets and various texture segmentation algorithms, we present four wavelet detection transformation techniques. We then introduce the distance histogram and the two-point correlation function as quality measures in feature space. We use the distance histogram as a performance measure, as well as a tool for selecting features, and show that it can quantify performance in a way that correlates with ground truth measures. We show the results of 4 different possible wavelet-based feature detection methods combined by 10 wavelet filters. Brodatz images are used as test images. We show that among the methods we have tested, one stands out clearly as superior to all the others, and that the choice of filters plays little role. There are, however, cases where the distance histogram does not indicate the presence of any distinct clusters in the feature ...

Abstract: We present a Bayesian approach for nonparametric curve estimation based on a continuous wavelet dictionary, where the unknown function is modeled by a random sum of wavelet functions at arbitrary locations and scales. By avoiding the dyadic constraints for orthonormal wavelet bases, the continuous overcomplete wavelet dictionary has greater flexibility to adapt to the structure of the data, and leads to sparse representations. The price for this flexibility is the computational challenge of searching over an infinite number of potential dictionary elements. We develop a reversible jump Markov Chain Monte Carlo algorithm which utilizes local features in the proposal distributions and leads to better mixing of the Markov chain. Performance comparison in terms of sparsity and mean square error is carried out on standard wavelet test functions. Results on a non-equally spaced example show that our method compares favorably to methods using interpolation or imputation. Key words and phrases: overcomplete dictionaries; Bayesian inference; wavelets; nonparametric regression; reversible jump Markov chain Monte Carlo; stochastic expansions; 1.