Image restoration and reconstruction using projections onto epigraph set of convex cost fuchtions
This thesis focuses on image restoration and reconstruction problems. These
inverse problems are solved using a convex optimization algorithm based on orthogonal
Projections onto the Epigraph Set of a Convex Cost functions (PESC).
In order to solve the convex minimization problem, the dimension of the problem
is lifted by one and then using the epigraph concept the feasibility sets corresponding
to the cost function are defined. Since the cost function is a convex
function in R
N , the corresponding epigraph set is also a convex set in R
convex optimization algorithm starts with an arbitrary initial estimate in R
and at each step of the iterative algorithm, an orthogonal projection is performed
onto one of the constraint sets associated with the cost function in a sequential
manner. The PESC algorithm provides globally optimal solutions for different
functions such as total variation,
1-norm, 2-norm, and entropic cost functions.
Denoising, deconvolution and compressive sensing are among the applications of
PESC algorithm. The Projection onto Epigraph Set of Total Variation function
(PES-TV) is used in 2-D applications and for 1-D applications Projection onto
Epigraph Set of
1-norm cost function (PES-1) is utilized.
1 algorithm, first the observation signal is decomposed using wavelet or pyramidal decomposition. Both wavelet denoising and denoising methods using the concept of sparsity are based on soft-thresholding. In sparsity-based denoising methods, it is assumed that the original signal is sparse in some transform domain such as Fourier, DCT, and/or wavelet domain and transform domain coefficients of the noisy signal are soft-thresholded to reduce noise. Here, the relationship between the standard soft-thresholding based denoising methods and sparsity-based wavelet denoising methods is described. A deterministic soft-threshold estimation method using the epigraph set of 1-norm cost function is presented. It is
demonstrated that the size of the
1-ball can be determined using linear algebra. The size of the 1-ball in turn determines the soft-threshold. The PESC, PES-TV
and PES-`1 algorithms, are described in detail in this thesis. Extensive simulation
results are presented. PESC based inverse restoration and reconstruction
algorithm is compared to the state of the art methods in the literature.