Sparse and Redundant Representations for Inverse Problems and Recognition

Abstract

Sparse and redundant representation of data enables the

description of signals as linear combinations of a few atoms from

a dictionary. In this dissertation, we study applications of

sparse and redundant representations in inverse problems and

object recognition. Furthermore, we propose two novel imaging

modalities based on the recently introduced theory of Compressed

Sensing (CS).

This dissertation consists of four major parts. In the first part

of the dissertation, we study a new type of deconvolution

algorithm that is based on estimating the image from a shearlet

decomposition. Shearlets provide a multi-directional and

multi-scale decomposition that has been mathematically shown to

represent distributed discontinuities such as edges better than

traditional wavelets. We develop a deconvolution algorithm that

allows for the approximation inversion operator to be controlled

on a multi-scale and multi-directional basis. Furthermore, we

develop a method for the automatic determination of the threshold

values for the noise shrinkage for each scale and direction

without explicit knowledge of the noise variance using a

generalized cross validation method.

In the second part of the dissertation, we study a reconstruction

method that recovers highly undersampled images assumed to have a

sparse representation in a gradient domain by using partial

measurement samples that are collected in the Fourier domain. Our

method makes use of a robust generalized Poisson solver that

greatly aids in achieving a significantly improved performance

over similar proposed methods. We will demonstrate by experiments

that this new technique is more flexible to work with either

random or restricted sampling scenarios better than its

competitors.

In the third part of the dissertation, we introduce a novel

Synthetic Aperture Radar (SAR) imaging modality which can provide

a high resolution map of the spatial distribution of targets and

terrain using a significantly reduced number of needed transmitted

and/or received electromagnetic waveforms. We demonstrate that

this new imaging scheme, requires no new hardware components and

allows the aperture to be compressed. Also, it

presents many new applications and advantages which include strong

resistance to countermesasures and interception, imaging much

wider swaths and reduced on-board storage requirements.

The last part of the dissertation deals with object recognition

based on learning dictionaries for simultaneous sparse signal

approximations and feature extraction. A dictionary is learned

for each object class based on given training examples which

minimize the representation error with a sparseness constraint. A

novel test image is then projected onto the span of the atoms in

each learned dictionary. The residual vectors along with the

coefficients are then used for recognition. Applications to

illumination robust face recognition and automatic target

recognition are presented.