Phase evaluation and segmentation
Michel Rene Nahon, Ronald R. Coifman · 2000
This dissertation is organized in 3 parts. The first part is a study of instantaneous frequencies for one and two-dimensional signals. In 1946, Gabor proposed in [1] to analyze a one dimensional signal via its complexification. We refine this idea and study the Blaschke Factorization to separate frequential and amplitudal information. We demonstrate stability and invariance properties. We describe an iterative algorithm to decompose a signal in an orthogonal basis of finite Blaschke product. An extension of this algorithm is presented for the two dimensional case. In part II, we deal with the problem of textural segmentation. Many applications in image analysis are based directly or indirectly on segmentation. Several algorithms give good results. We study the properties of the pyramidal algorithm developed by J.-M. Morel (and his collaborators), that is based on the Mumford-Shah functional, as its properties interest us. For textured images a preprocessing is mandatory. In [19], a vector image is obtained after filtering with an undecimated wavelet decomposition by Koepfler et al. The problem is now to segment a vector image. The “good filters” have to be chosen to obtain a reasonable segmentation, regions with comparable sizes. We propose an algorithm to select the useful filters in a library for a chosen image. It enables to reduce the computational time and give a more efficient segmentation. We present an extension of the Mumford-Shah functional and show how to reduce the dimension of our vector image. Different results and a counter-example, where the pyramidal algorithm is not optimal, are presented. The last part is devoted to the application of undecimated wavelets. We believe that multi-scale analysis is an important tool for image and signal processing. We represent our data with the wavelets and wavelet packets decomposition. We work with undecimated wavelets since they are grid's independent. We first summarize their properties before showing some applications. The first application is deconvolution, sharpening and smoothing signals using the subspaces obtained with the wavelet packets. The second one is about denoising radar images using the separation of the structures given by the wavelet representation, to solve the troubles generated by the targets. And finally, by extracting the variations at various scales, we show results for the detection of brain activities in functional-MRI.