NONLINEAR EVOLUTIONARY PDEs IN IMAGE PROCESSING AND COMPUTER VISION
NONLINEAR EVOLUTIONARY PDEs IN IMAGE PROCESSING AND COMPUTER VISION
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Date
2004-11-24
Authors
liu, kexue
Advisor
Liu, Jian-Guo
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Abstract
Evolutionary PDE-based methods are widely used in image
processing and computer vision. For many
of these evolutionary PDEs, there is little or no theory on the
existence and regularity of solutions, thus there is little
or no understanding on how to implement them effectively to produce
the desired effects. In this thesis work, we study
one class of evolutionary PDEs which appear in the literature and are highly
degenerate.
The study of such second order parabolic PDEs has been carried out by
using semi-group theory and maximum monotone operator in case that the
initial value is in the space of functions of bounded variation. But the
noisy initial image is usually not in this space, it is desirable to
know the solution property under weaker assumption on initial image.
Following the study of time dependent minimal surface problem, we
study the existence and uniqueness of generalized solutions of a class
of second order parabolic PDEs. Second order evolutionary PDE-based
methods preserve edges very well
but sometimes they have undesirable staircase effect. In order to
overcome this drawback, fourth order evolutionary PDEs were proposed
in the literature. Following the same approach, we study the existence and regularity of
generalized solutions of one class of fourth order evolutionary PDEs in
space of functions of bounded Hessian and bounded Laplacian.
Finally, we study some evolutionary PDEs which
do not satisfy the parabolicity condition by adding a regularization
term.
Through the rigorous study of evolutionary PDEs which appear in the
literature of image processing and
computer vision, we provide a solid theoretical foundation for them which helps us better
understand the behaviors and properties of them. The existence and regularity theory is
the first step toward effective numerical scheme. The regularity
results also answer the questions to which function spaces the solutions of
evolutionary PDEs belong and the questions if the processing results
have the desired properties.