Représentations d ’ images pour la reconnaissance de formes

One of the main requirements in many signal processing applications is to have a “meaningful representation” in which signal’s characteristics are readily apparent. For example, for recognition, the representation should highlight salient features; for denoising, it should efficiently separate signal and noise; and for compression, it should capture a large part of signal using only a few coefficients. Interestingly, despite these seemingly different goals, good performance of signal processing applications generally has roots in the appropriateness of the adopted representations. Representing a signal involves the design of a set of elementary generating signals, or a dictionary of atoms, which is used to decompose the signal. For many years, dictionary design has been pursued by many researchers for various fields of applications: Fourier transform was proposed to solve the heat equation; Radon transform was created for the reconstruction problem; wavelet transform was developed for piece-wise smooth, one-dimensional signals with a finite number of discontinuities; and contourlet transform was designed to efficiently represent two-dimensional signals made of smooth regions separated by smooth boundaries, etc. For the developed dictionaries up to the present time, they can be roughly classified into two families: mathematical models of the data and sets of realizations of the data. Dictionaries of the first family are characterized by analytical formulations, which can sometimes be fast implemented. The representation coefficients of a signal in one dictionary are obtained by performing signal transform. Dictionaries of the second family, which are often general overcomplete, deliver greater flexibility and the ability to adapt to specific signal data. They are the results of much more recent dictionary designing approaches where dictionaries are learned from data for their representation. The existence of many dictionaries naturally leads to the problem of selecting the most appropriate one for the representation of signals in a certain situation. The selected dictionary should have distinguished and beneficial properties which are preferable in the targeted applications. Speaking differently, it is the actual application that controls the selection of dictionary, not the reverse. In the framework of this thesis, three types of dictionaries, which correspond to three types of transforms/representations, will be studied for their applicability in some image analysis and pattern recognition tasks. They are the Radon transform, unit disk-based moments, and sparse representation. The Radon transform and unit disk-based moments are for invariant pattern recognition problems, whereas sparse representation for image denoising, separation, and classification problems. This thesis contains a number of theoretical contributions which are accompanied by numerous validating experimental results. For the Radon transform, it discusses possible directions that can be followed to define invariant pattern descriptors, leading to the proposal of two descriptors that are totally invariant to rotation, scaling, and translation. For unit disk-based moments, it presents a unified view on strategies that have been used to define unit disk-based orthogonal moments, leading to the proposal of four generic polar harmonic moments and strategies for their fast computation. For sparse representation, it uses sparsity-based techniques for denoising and separation of graphical document images and proposes a representation framework that balances the three criteria sparsity, reconstruction error, and discrimination power for classification.