6 research outputs found
Image restoration and reconstruction using projections onto epigraph set of convex cost fuchtions
Cataloged from PDF version of article.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
N+1. The
convex optimization algorithm starts with an arbitrary initial estimate in R
N+1
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.
In PES-`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.Tofighi, MohammadM.S
Characteristics of a detail preserving nonlinear filter.
by Lai Wai Kuen.Thesis (M.Phil.)--Chinese University of Hong Kong, 1993.Includes bibliographical references (leaves [119-125]).Abstract --- p.iAcknowledgement --- p.iiTable of Contents --- p.iiiChapter Chapter 1 --- IntroductionChapter 1.1 --- Background - The Need for Nonlinear Filtering --- p.1.1Chapter 1.2 --- Nonlinear Filtering --- p.1.2Chapter 1.3 --- Goal of the Work --- p.1.4Chapter 1.4 --- Organization of the Thesis --- p.1.5Chapter Chapter 2 --- An Overview of Robust Estimator Based Filters Morphological FiltersChapter 2.1 --- Introduction --- p.2.1Chapter 2.2 --- Signal Representation by Sets --- p.2.2Chapter 2.3 --- Robust Estimator Based Filters --- p.2.4Chapter 2.3.1 --- Filters based on the L-estimators --- p.2.4Chapter 2.3.1.1 --- The Median Filter and its Derivations --- p.2.5Chapter 2.3.1.2 --- Rank Order Filters and Derivations --- p.2.9Chapter 2.3.2 --- Filters based on the M-estimators (M-Filters) --- p.2.11Chapter 2.3.3 --- Filter based on the R-estimators --- p.2.13Chapter 2.4 --- Filters based on Mathematical Morphology --- p.2.14Chapter 2.4.1 --- Basic Morphological Operators --- p.2.14Chapter 2.4.2 --- Morphological Filters --- p.2.18Chapter 2.5 --- Chapter Summary --- p.2.20Chapter Chapter 3 --- Multi-Structuring Element Erosion FilterChapter 3.1 --- Introduction --- p.3.1Chapter 3.2 --- Problem Formulation --- p.3.1Chapter 3.3 --- Description of Multi-Structuring Element Erosion Filter --- p.3.3Chapter 3.3.1 --- Definition of Structuring Element for Multi-Structuring Element Erosion Filter --- p.3.4Chapter 3.3.2 --- Binary multi-Structuring Element Erosion Filter --- p.3.9Chapter 3.3.3 --- Selective Threshold Decomposition --- p.3.10Chapter 3.3.4 --- Multilevel Multi-Structuring Element Erosion Filter --- p.3.15Chapter 3.3.5 --- A Combination of Multilevel Multi-Structuring Element Erosion Filter and its Dual --- p.3.21Chapter 3.4 --- Chapter Summary --- p.3.21Chapter Chapter 4 --- Properties of Multi-Structuring Element Erosion FilterChapter 4.1 --- Introduction --- p.4.1Chapter 4.2 --- Deterministic Properties --- p.4.2Chapter 4.2.1 --- Shape of Invariant Signal --- p.4.3Chapter 4.2.1.1 --- Binary Multi-Structuring Element Erosion Filter --- p.4.5Chapter 4.2.1.2 --- Multilevel Multi-Structuring Element Erosion Filter --- p.4.16Chapter 4.2.2 --- Rate of Convergence of Multi-Structuring Element Erosion Filter --- p.4.25Chapter 4.2.2.1 --- Convergent Rate of Binary Multi-Structuring Element Erosion Filter --- p.4.25Chapter 4.2.2.2 --- Convergent Rate of Multilevel Multi-Structuring Element Erosion Filter --- p.4.28Chapter 4.3 --- Statistical Properties --- p.4.30Chapter 4.3.1 --- Output Distribution of Multi-Structuring Element Erosion Filter --- p.4.30Chapter 4.3.1.1 --- One-Dimensional Statistical Analysis of Multilevel Multi-Structuring Element Erosion Filter --- p.4.31Chapter 4.3.1.2 --- Two-Dimensional Statistical Analysis of Multilevel Multi-Structuring Element Erosion Filter --- p.4.32Chapter 4.3.2 --- Discussions on Statistical Properties --- p.4.36Chapter 4.4 --- Chapter Summary --- p.4.40Chapter Chapter 5 --- Performance EvaluationChapter 5.1 --- Introduction --- p.5.1Chapter 5.2 --- Performance Criteria --- p.5.2Chapter 5.2.1 --- Noise Suppression --- p.5.5Chapter 5.2.2 --- Subjective Criterion --- p.5.16Chapter 5.2.3 --- Computational Requirement --- p.5.20Chapter 5.3 --- Chapter Summary --- p.5.23Chapter Chapter 6 --- Recapitulation and Suggestions for Further WorkChapter 6.1 --- Recapitulation --- p.6.1Chapter 6.2 --- Suggestions for Further Work --- p.6.4Chapter 6.2.1 --- Probability Measure Function for the Two-Dimensional Filter --- p.6.4Chapter 6.2.2 --- Hardware Implementation --- p.6.5ReferencesAppendice