Grant no. D000246/1.The sparse coding is approximation/representation of signals with the minimum number of
coefficients using an overcomplete set of elementary functions. This kind of approximations/
representations has found numerous applications in source separation, denoising, coding and
compressed sensing. The adaptation of the sparse approximation framework to the coding
problem of signals is investigated in this thesis. Open problems are the selection of appropriate
models and their orders, coefficient quantization and sparse approximation method. Some of
these questions are addressed in this thesis and novel methods developed. Because almost all
recent communication and storage systems are digital, an easy method to compute quantized
sparse approximations is introduced in the first part.
The model selection problem is investigated next. The linear model can be adapted to better
fit a given signal class. It can also be designed based on some a priori information about the
model. Two novel dictionary selection methods are separately presented in the second part
of the thesis. The proposed model adaption algorithm, called Dictionary Learning with the
Majorization Method (DLMM), is much more general than current methods. This generality
allowes it to be used with different constraints on the model. Particularly, two important cases
have been considered in this thesis for the first time, Parsimonious Dictionary Learning (PDL)
and Compressible Dictionary Learning (CDL). When the generative model order is not given,
PDL not only adapts the dictionary to the given class of signals, but also reduces the model
order redundancies. When a fast dictionary is needed, the CDL framework helps us to find a
dictionary which is adapted to the given signal class without increasing the computation cost
so much.
Sometimes a priori information about the linear generative model is given in format of a parametric
function. Parametric Dictionary Design (PDD) generates a suitable dictionary for sparse
coding using the parametric function. Basically PDD finds a parametric dictionary with a minimal
dictionary coherence, which has been shown to be suitable for sparse approximation and
exact sparse recovery.
Theoretical analyzes are accompanied by experiments to validate the analyzes. This research
was primarily used for audio applications, as audio can be shown to have sparse structures.
Therefore, most of the experiments are done using audio signals