Theeda, Prasad and Sastry, Challa Subrahmanya
(2018)
Sparse Description of Linear Systems and
Application in Tomography.
PhD thesis, Indian Institute of Technology Hyderabad.
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Abstract
The area of compressed sensing (CS) deals with finding sparse descriptions to some linear
systems. The premise in providing such descriptions is based on the fact that many natural
data sets can be represented in terms of a far fewer coefficients than their actual dimension in
some basis or redundant dictionary. In particular, CS finds sparse solutions or approximations
(i.e, the solutions with very few nonzero components) to the matrix equations of type y = Ax
or y � Ax. The matrix A (of size m � N) and the vector y may have different meanings
in different applications. For instance, if A has elements generated as sinusoidal functions, y
represents the vector of Fourier coefficients of x.
In applications that involve compressed data acquisition, the associated reconstruction problem
boils down to dealing with an underdetermined system (i.e, m < N). While in applications
such as denoising and system identification, one often encounters overdetermined systems (i.e,
m > N). Both the scenarios are in sharp contrast mathematically, in the sense that the case
m > N in general does not admit any solution, while the case m < N in general admits infinitely
many solutions. For various applications as mentioned above, the sparse solutions or
approximations to both types of systems are needed.
The thesis is concerned with underdetermined (UD) as well as overdetermined (OD) systems.
In the UD case, our objectives center around determining a preconditioner that improves
the sparse recovery properties of the associated system matrix, generated by certain classes of
frames. In particular, we provide sufficient conditions that ensure guaranteed improvement in
sparse recovery properties which preconditioning brings in. In view of the potential of sparsity
based methods, we address interior as well as incomplete data problems in Computed Tomography
(CT). In the OD case, nevertheless, we provide a sparse approximation to y � Ax via a
new matrix factorization method. A detailed comparison with popular solvers such as the OMP
and LASSO is also included.
Computed Tomography (CT) is one of the significant research areas in the field of medical
image analysis. As Xrays used in CT are harmful to human bodies, it is necessary to reduce
the dosage while maintaining good quality in reconstruction. The reduced dosage problem
via the discretization of Radon transform translates into an underdetermined system of linear
equations. The associated sensing matrix, generated from the Radon transform, is very large in
size and is not known to satisfy the sparse recovery properties. Our work attempts to establish
the sparse recovery properties of Radon sensing matrix via careful choice of angular and radial
parameters. This is in contrast to the existing results that remain focused on improving sparse
solvers as applicable to CT. In addition to analyzing the Radon sensing matrix for its sparse
recovery properties, we determine a preconditioner that is capable of being used in such large
scale problems as in CT. The contributions of the thesis may be summarized as follows:
� A convex optimization technique, along with analytical guarantees, for designing the
vii
preconditioner that yields a sensing matrix with improved sparse recovery properties1
� Analysis of sparse recovery properties of Radon sensing matrix and its application in
CT. In particular, the preconditioner improving the sparse recovery properties of Radon
matrix (A) is obtained from the following optimization problem:
min
kATXA
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IITH Creators: 
IITH Creators  ORCiD 

Sastry, Challa Subrahmanya  UNSPECIFIED 

Item Type: 
Thesis
(PhD)

Uncontrolled Keywords: 
Phase representation Theory, Compressive Sensing, Computed Tomography, Fame Theory 
Subjects: 
Mathematics 
Divisions: 
Department of Mathematics 
Depositing User: 
Team Library

Date Deposited: 
19 Jul 2018 09:09 
Last Modified: 
21 Sep 2019 06:09 
URI: 
http://raiith.iith.ac.in/id/eprint/4284 
Publisher URL: 

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