October 31, 2006Thesis Defense, UTK1/30 Variational and Partial Differential Equation Models for Color Image Denoising and Their Numerical Approximation using Finite Element Methods Thesis Defense Miun Yoon

October 31, 2006Thesis Defense, UTK2/30 Digital Image Processing - Image restoration - Image compression -Image segmentation What is “Digital Image Processing”? observed image Stochastic modeling Wavelets Variational & PDE modeling output

October 31, 2006Thesis Defense, UTK3/30 What is an image in Mathematics? Pixel: Picture + Element observed image color image

October 31, 2006Thesis Defense, UTK4/30 Pixel Representations RGB Color Image :256 shades of RGB Gray Image :256 shades of gray- level

October 31, 2006Thesis Defense, UTK5/30 Image Denoising Model original image “unknown” additive noise noisy image

October 31, 2006Thesis Defense, UTK6/30 Gray Image Denoising Total Variational (TV) Model: Rudin, Osher, and Fatemi [Rud92](1992) Constrained minimization problem: Constraints noisy image Error level

October 31, 2006Thesis Defense, UTK7/30 Unconstrained minimization problem: Gradient Flow (TV Flow):

October 31, 2006Thesis Defense, UTK8/30 Regularized Problem Previous Studies - A. Chamnbolle and P. –L. Lions [Cha97]: proved the existence and the uniqueness result for constraint minimization problem and unconstraint minimization problem is equivalent to the constraint minimization for a unique and non-negative - X. Feng and A. Prohl [Fen03]: proved the existence and the uniqueness for the TV flow and regularized problem and an error analysis for the fully discrete finite approximation for the regularized problem

October 31, 2006Thesis Defense, UTK9/30 Weak Formulation

October 31, 2006Thesis Defense, UTK10/30 Semi-Discrete Finite Element Method T h = {K 1,…,K mR } Finite-Dimensional subspace : set of all vertices of the triangulation T h uniquely determined & forms a basis for V h

October 31, 2006Thesis Defense, UTK11/30 Semi-Discrete Finite Element Method Non-linear ODE system in  t 

October 31, 2006Thesis Defense, UTK12/30 Fully Discrete Finite Element Method X. Feng, M. von Oehen and A. Prohl [Fen05]: rate of convergence for the fully discrete finite approximation of the regularized problem

October 31, 2006Thesis Defense, UTK13/30 Numerical Tests I t=0 t=5e-5 t=1e-4t=1.5e-4 t=2e-4

October 31, 2006Thesis Defense, UTK14/30 Numerical Tests II t=0 t=5e-5 t=1e-4t=1.5e-4 t=2e-4

October 31, 2006Thesis Defense, UTK15/30 Numerical Tests III t=0 t=5e-5 t=1e-4t=1.5e-4 t=2e-4

October 31, 2006Thesis Defense, UTK16/30 Color Image Denoising brightness chromaticity color vector TV flow P-harmonic map flow Non-flat feature channel by channel model chromaticity & brightness (CB) Model

October 31, 2006Thesis Defense, UTK17/30 p-harmonic Map Minimizer of E p Euler-Lagrange equation unit sphere p-energy Constrained Minimization Problem constraint p-harmonic map

October 31, 2006Thesis Defense, UTK18/30 p-harmonic Color Image Denoising Model Gradient flow Non-linear Non-convex Non-linear

October 31, 2006Thesis Defense, UTK19/30 Regularization of p-energy nonlinearnonconvex

October 31, 2006Thesis Defense, UTK20/30 Regularized Model p-harmonic map heat flow

October 31, 2006Thesis Defense, UTK21/30 Weak Formulation

October 31, 2006Thesis Defense, UTK22/30 Semi-Discrete Finite Element Method : set of all vertices of the triangulation T h Finite-Dimensional subspace T h = {K 1,…,K mR }

October 31, 2006Thesis Defense, UTK23/30 Semi-Discrete Finite Element Method Non-linear ODE system in  t 

October 31, 2006Thesis Defense, UTK24/30 Semi-Discrete Finite Element Method

October 31, 2006Thesis Defense, UTK25/30 Fully-Discrete Finite Element Method Decomposition of the density function

October 31, 2006Thesis Defense, UTK26/30 Numerical Tests t=0t=2e-4 t=5e-4t=7e-4t=1e-3

October 31, 2006Thesis Defense, UTK27/30 Generalization Generalized model of the p-harmonic map Regularized flow of generalized model

October 31, 2006Thesis Defense, UTK28/30 Numerical Tests I and q=1 t=2e-4 t=5e-4 t=7e-4 t=1e-3

October 31, 2006Thesis Defense, UTK29/30 Numerical Tests II and q=1 t=2e-4 t=5e-4 t=7e-4 t=1e-3

October 31, 2006Thesis Defense, UTK30/30 Numerical Tests III channel-by-channel t=1e-4 t=3e-4 t=5e-4 t=1e-3

31 Appendix