Math 865


Curriculum Vitae
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Syllabus: Click here for details in a PDF file


Course: Math 865, Topics in Applied Mathematics: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, MWF 12:30pm @CC312


References: (1) Mathematical Problems in Image Processing Partial Differential Equations and the Calculus of Variations Springer: Applied Mathematical Sciences , Vol. 147 Aubert, Gilles, Kornprobst, Pierre, 2nd ed., 2006, ISBN: 978-0-387-32200-1 (2) Image Processing and Analysis Variational, PDE, Wavelet, and Stochastic Methods SIAM, 2005 Tony F. Chan, Jianhong Jackie Shen, ISBN: 0-89871-598-X

bulletTopics: calculus of variations, Euler-Language equation, BV, image denoising, image segmentation, image inpainting, etc
bulletOffice Hours: MW 1:30-2:30pm @MW410 and by appointment
bulletGrading: HW(60%),class participation(10%),and final project(30%)
bullet Policy: Late Homework and project will be no credit.
bulletTentative Schedule: updated regularly



Monday Wednesday Friday
Mar 24


digital image

gray, color

read and write an image


Mar 26


digital image

gray, color

read and write an image

functional derivatives


Mar 28


HW1: on p.21 of the slides
Mar 31

Image Degradation

Image denoising (Chan & Shen 4.5)

Apr 2

ROF model

Apr 4

Numerical implementation for ROF model

Apr 7

Review of functional space(Gilles ch2.)

strong and weak topology


Apr 9

Existence of the minimizer of a functional

make up class: (72 minutes) this Thursday at 5:30pm CH 232


Apr 11

Perona and Malik's Nonlinear Diffusion Model (Chan & Shen 4.6.1)

Apr 14


Apr 16


Apr 18


Apr 21

Scale-Space Theory (Chan & Shen 4.6.2)

Apr 23

Motion blur and Out-of-focus blur (Chan & Shen 5.1)

make up class: (72 minutes) this Thursday at 5:30pm CH 232

Apr 25 Image Debluring: Code


Apr 28 Shock Filter(Aubert Ch3.3.3)

Apr 30

Numerical Scheme for Shock Filter

May 2

conference in SAMSI

May 5

Introduction to image inpainting, interpolation

May 7

PDE based image inpainting techniques

linear inpainting via harmonic extension

make up class: this Thursday at 5:30pm SM 1186

May 9

Masnou and Morel model for inpainting

TV inpainting modelHW4

May 12

Properties of TV inpainting

May 14

TV zoom in

Mean and Medium Interpolation

Third order inpainting, Navier-Stokes Equation

make up class: this Thursday at 5:30pm SM 1186

May 16

Image Segmentation: Mumford-Shah functional


May 19

Chan-Vese piecewise constant model and its extension

CV piecewise constant Code

May 21

Chan-Vese piecewise smooth model

May 23

Edge based models


May 26

Memorial Day (no class)

May 28

Student presentation of final projects

May 30

Student presentation of final projects

final project:

(a)A New Diffusion-Based Variational Model for Image Denoising and Segmentation: FANG LI,CHAOMIN SHEN, and LING PI

(b)Image inpainting using a TV-Stokes equation: Xue-Cheng Tai, Stanley Osher and Randi Holm

(c)A Fast Algorithm for Image Deblurring with Total Variation Regularization: Yilun Wang, Wotao Yin and Yin Zhang

(d)A Multi-resolution Stochastic Level Set Method for Mumford-Shah Image Segmentation Yan Nei Law, Hwee Kuan Lee and Andy M. Yip

(e)Robust Environmental Image Denoising: A. Ben Hamza and Hamid Krim

(f)Global Minimization of the Active Contour Model with TV-Inpainting and Two-phase Denoising: Shingyu Leung and Stanley Osher

(g)Matting through Variational Inpainting Kangyu Ni, Sheshadri Thiruvenkadam, and Tony Chan

(h)Extensions to Total Variation Denoising: Peter Blomgren, Tony F. Chan and Pep Mulet

(i)An Efficient Operator-Splitting Method for Noise Removal in Images D. KRISHNAN, P. LIN and X. C. TAI

(j)A variational approach to reconstructing images corrupted by Poisson noise UCLA CAM Report 05-49 Triet Le, Rick Chartrand, and Thomas J. Asaki

(k)Geometric Surface Smoothing via Anisotropic Diffusion of Normals: Tolga Tasdizen, Ross Whitaker, Paul Burchard, Stanley Osher

(l)On image denoising methods: Antoni Buades,Bartomeu Coll and Jean Michel Morel

(m)Hybrid geodesic region-based curve evolutions for image segmentation: Shawn Lanktona, Delphine Nainb, Anthony Yezzia, and Allen Tannenbauma

or pick images related paper on cam report: