Optimally weighted local discriminant bases : theory and applications in statistical signal and image processing

This thesis is concerned with Local Discriminant Basis (LDB) algorithm, its properties, optimization and applications in feature extraction and classification. LDB algorithm targets features extraction from redundant dictionaries such as wavelet packets or local trigonometric bases at low computational complexity. As the main contribution of this thesis, an optimization process is introduced to further improve the accuracy of the overall scheme in applications when a region of interest can be specified by the experts in the field of application (based on LDB selected features) to further characterize signal classes in smaller regions. Audio signal and textured image classifications are practical applications that are studied in this thesis to test the efficiency of optimally weighted local discriminant basis algorithm (OLDB) as a feature extraction scheme. Various properties of the algorithm such as noise behavior and stability analysis are studied from an engineering perspective. The implementation aspects of the algorithm in one dimension are reviewed as well as in two dimensions that serve as implementation guidelines.

Optimally weighted local discriminant bases : theory and applications in statistical signal and image processing

This thesis is concerned with Local Discriminant Basis (LDB) algorithm, its properties, optimization and applications in feature extraction and classification. LDB algorithm targets features extraction from redundant dictionaries such as wavelet packets or local trigonometric bases at low computational complexity. As the main contribution of this thesis, an optimization process is introduced to further improve the accuracy of the overall scheme in applications when a region of interest can be specified by the experts in the field of application (based on LDB selected features) to further characterize signal classes in smaller regions. Audio signal and textured image classifications are practical applications that are studied in this thesis to test the efficiency of optimally weighted local discriminant basis algorithm (OLDB) as a feature extraction scheme. Various properties of the algorithm such as noise behavior and stability analysis are studied from an engineering perspective. The implementation aspects of the algorithm in one dimension are reviewed as well as in two dimensions that serve as implementation guidelines.

Sparse signal recovery via non-convex optimization and overcomplete dictionaries

In this paper, we address the problem of recovering signals from undersampled data where such signals are not sparse in an orthonormal basis, but in an overcomplete dictionary. We show that if the combined matrix obeys a certain restricted isometry property and if the signal is sufficiently sparse, the reconstruction that relies on [Formula: see text] minimization with [Formula: see text] is exact. In addition, under a mild assumption about the dictionary [Formula: see text], we use a similar method [H. Rauhut et al., Compressed sensing and redundant dictionaries, IEEE Trans. Inf. Theory 54(5) (2008) 2210–2219] to derive an estimation of the restricted isometry constant of the composed matrix [Formula: see text]. Finally, the performance of the [Formula: see text] minimization is testified by some numerical examples.