| Efficient representations of high-dimensional data such as images, that can essentially describe the data with a few parameters, play a vital role in many problems in signal processing and related fields, ranging from signal compression and denoising to inverse problems. This thesis studies the design and applications of various representation systems for several classes of signals including images and signals on general graphs.;The first half of the thesis deals with two different classes of images that can be considered as signals living on regular grid graph. For the first class of cartoon-like images, which are piecewise smooth away from smooth edges, the optimal sparsity of the contourlet transforms, that provide directional multiresolution representations, is established under a sufficient condition on the directional decay of the contourlet profiles in frequency-domain. This Fourier-based condition does not require directional vanishing moments and therefore opens up an opportunity to design an optimally sparse contourlet filter bank with FIR (finite impulse response) filters.;For the second class of images of a Lambertian object under arbitrary lighting conditions, the well-known spherical harmonic representation is used to approximate the whole class with a low-dimensional linear subspace and to transfer the inverse rendering problem into a special matrix factorization. Our second work is dedicated to solving this factorization in both noiseless and noisy cases using subspace methods.;In the second half of the thesis, the multiresolution representations of signals living on irregular domains, whose discrete topologies are described by weighted graphs, are investigated. A downsampling scheme for signals on general graphs based on maximum spanning trees is discussed in our third work. This framework provides a fast approximation of the max-cut, a criterion for downsampling on graphs, as well as a bipartite graph multiresolution, which is well-suited to the critical-sampling graph wavelet filter banks (GWFBs).;Our fourth work focuses on the compression of a dynamic human body which can be treated as a sequence of signals living on a graph induced by the topology of the mesh representing the body. As we have the freedom to create the underlying graph, a quad subdivision mesh is used to generate a bipartite graph multiresolution for the GWFBs, and a spatial connectivity pattern that can be exploited in a context adaptive entropy coding. |
