High Dimensional Signal Representation

In this thesis we explore the efficiency of signal representations and their robustness in signal reconstruction in three subfields of signal and image processing.

The first result concerns regularity limitation in the construction of directional wavelet bases due to redundancy constraint on the scheme, in an effort to construct “optimal” directional bases with multiresolution and perfect reconstruction proper- ties. We showed that for orthonormal and biorthogonal bases with dilated quincunx downsampling, the wavelets cannot be well localized; however, this regularity limit can be circumvented in a tight frame with dyadic downsampling and a redundancy factor smaller than 2.

The second result introduces a novel framework for patch-based image models combining local structure of patches and nonlocal information in image domain. In particular, we built convolution framelets from local and nonlocal bases, which form a tight frame of the image space and has energy concentration when the local and nonlocal bases are coherent. We applied this framework to reinterpret and improve state-of-the-art low dimensional manifold model.

The final result proposes a new paradigm of phase retrieval, considering signal reconstruction up to a larger equivalence class than a uniform phase shift. It is known that in the classical setting, phase retrieval in infinite or high dimension is inherently unstable. We showed that stability can be achieved, however, for frames of Gabor wavelets or Cauchy wavelets in this new paradigm.