| The first part is of theoretical nature: we define Rectified Wavelet Packets, a modified version of Wavelet Packets with perfect frequency localization, belonging to the Schwartz class, and keeping the popular algorithmic structure based on quadrature mirror filters. We then show the duality between Rectified Wavelet Packets and Local Trigonometric functions, and give explicit expressions of the periodized version of Rectified Wavelet Packets. We define also Wavelet Packets on the interval associated to an arbitrary smooth segmentation of the real line. We show that in the case where this construction is combined with Rectified Wavelet Packets, the Local Trigonometric bases are included in the bases of Wavelet Packets on the interval.;In part II, we deal with the problem of extracting information or features out of a library of functions, and more generally how to represent an information (signal, image, or video sequence) in a way that will be optimal for subsequent processing. We present applications such as de-noising of S.A.R. images, de-noising and enhancement of M.R.I. images and video sequences.;The last part is devoted to the application of Wavelet Packets to statistical modeling and discrimination. After a thorough review of Principal Component Analysis, we introduce Multi-Band Karhunen-Loeve bases and present a fast algorithm to compute the related expansions. Then, we review some classical techniques for linear and quadratic discrimination, and present 2 new Wavelet Packet-based discrimination algorithms. The first one, based on greedy approximation, is called Projection Pursuit Discrimination, and the second one, based on Fisher discriminant function, is called Multi-Band Fisher Discrimination. We also present applications of these algorithms to S.A.R. data classification. |
