## Mathematical Models for Image Analysis

Lecture Meeting Time: MWF 11.00AM - 11.50AM
Lecture Location: MS 6221.

Instructor: Luminita A. Vese
Office: MS 7620-D
Office hours: Mon 3-4, Fri 4-5 (sometimes Fri 3-4), or by appointment.

E-mail: lvese@math.ucla.edu

Class Web page: http://www.math.ucla.edu/~lvese/285j.1.05s

Virtual Office Hours

Course Description:
This seminar is devoted to mathematical models for image analysis.
- Theory topics: calculus of variations, energy minimization, duality theory, Euler-Lagrange equations, optimality conditions, functions of bounded variation, functionals with linear growth and with jumps, geometric non-linear partial differential, viscosity solutions, oscillatory functions.
- Applications: image restoration (denoising, deblurring), image decomposition into cartoon and texture, image segmentation and edge detection, snakes, curve evolution, active contours, level set methods.

The lectures will not follow one particular textbook. The topics presented can be found in research papers or graduate textbooks.

Sample Codes: The best choice for image processing calculations is C++. However, for easy routines, such as reading an image and adding noise, Matlab is a good choice to help you to begin to work with images.

Matlab code to add uniform noise to an image and to compute the SNR (signal-to-noise-ratio):
NoiseSNR1.m for a synthetic image   NoiseSNR2.m for a real image that you can find here Lena.bmp

Matlab code for a denoising algorithm using the Rudin-Osher-Fatemi model, using an implicit fixed point finite difference scheme for the stationary E-L equation. TV_L2 denoising code
The above code uses this scheme for the discretization of the Rudin-Osher-Fatemi model.

Textbook References:
• G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, (Partial Differential Equations and the Calculus of Variations), Springer, 2002.
• I. Ekeland and R. Temam, Convex analysis and variational problems, (Classics in Applied Mathematics 28), SIAM, 1999 (new edition).
• Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, AMS 2001.
• J.-M. Morel and S. Solimini, Variational Methods in Image Segmentation: With Seven Image Processing Experiments (Progress in Nonlinear Differential Equations and Their Applications), Birkhauser 1994.
• S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer-Verlag, 2002.
• J. Sethian, Level Set Methods and Fast Marching Methods : Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge University Press, 1999.
• S. Osher and N. Paragios (Eds), Geometric Level Set Methods in Imaging, Vision, and Graphics, Springer-Verlag Telos, 2003.
• R. Kimmel, Numerical Geometry of Images: Theory, Algorithms, and Applications, Springer-Verlag, 2003.
• L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems (Oxford Mathematical Monographs), Oxford University Press, 2000.
• F. Andreu-Vaillo, V. Caselles, J.M. Mazon, Parabolic Quasilinear Equations Minimizing Linear Growth Functionals, Birkhauser, 2004.
• F. Guichard and J.-M. Morel, Image Analysis and P.D.E.'s, manuscript.

Assignments & Projects:
- All enrolled students will have to solve problems and to do numerical implementations of the methods discussed in class (work in group or teams of two or three students is encouraged).
- The assignments will be balanced between "pencil and paper" problems and numerical implementations.
- However, function of your own background and of your own interests, you can work more on one type of assignments, and less on the other type.
- If you have questions, please come and discuss with me your case and your specific interests.
- Students interested in working on a new research project, proposed by the instructor and with the instructor's advise and help, can do so. The research project can substitute all the assignments.
- Late homework is accepted.

Some publications (full paper access) on image restoration by minimization of regularizing functionals with linear or sublinear growth
• Nonlinear Total Variation Based Noise Removal Algorithms
L. Rudin, S. Osher, E. Fatemi, Physica D: Nonlinear Phenomena, Vol. 60, Issues 1-4, 1992.
• Constrained Restoration and the Recovery of Discontinuities
D. Geman, G. Reynolds, IEEE T on PAMI, Vol. 14, No. 3, 1992.
• Analysis of Bounded Variation Penalty Methods for Ill-Posed Problems,
R. Acar, C.R. Vogel, Inverse problems, vol: 10, iss: 6, pg: 1217, yr: 1994.
• Nonlinear Image Recovery with Half-Quadratic Regularization,
D. Geman, C. Yang, IEEE TIP, Vol. 4, No. 7, 1995.
• Image Recovery via Multiscale Total Variation,
V. Caselles and L. Rudin, Second European Conference on Image Processing, Palma, Spain, September 1995.
• Iterative methods for total variation denoising,
C.R. Vogel, M.E. Oman, SIAM J. on Scientific Computing 17 (1): 227-238, 1996.
• Deterministic edge-preserving regularization in computed imaging
Charbonnier, P.; Blanc-Feraud, L.; Aubert, G.; Barlaud, M.; Image Processing, IEEE Transactions on Volume 6, Issue 2, Feb. 1997 Page(s):298 - 311.
• Image recovery via total variation minimization and related problems
Chambolle A, Lions PL, NUMERISCHE MATHEMATIK 76 (2): 167-188 APR 1997
• A variational method in image recovery
Aubert G., Vese L., SIAM Journal on Numerical Analysis, 34 (5): 1948-1979, Oct 1997.
• A study in the BV space of a denoising-deblurring variational problem
Vese L., Applied Mathematics and Optimization, 44 (2):131-161, Sep-Oct 2001.

Some publications (full paper access) on active contours and snakes.
• Geodesic Active Contours,
V. Caselles, R. Kimmel, G. Sapiro, IJCV 1997.
• Conformal curvature flows: From phase transitions to active vision,
Kichenssamy, Kumar, Olver, Tannenbaum, Yezzi, ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS 134 (3): 275-301 1996.
• A GEOMETRIC MODEL FOR ACTIVE CONTOURS IN IMAGE-PROCESSING,
CASELLES V, CATTE F, COLL T, DIBOS F, NUMERISCHE MATHEMATIK 66 (1): 1-31 OCT 1993
• SHAPE MODELING WITH FRONT PROPAGATION - A LEVEL SET APPROACH,
MALLADI R, SETHIAN JA, VEMURI BC , IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 17 (2): 158-175 FEB 1995
• Active contours without edges,
Chan, T.F.; Vese, L.A., IEEE Transactions on Image Processing, 10 (2), Feb. 2001, pp. 266 -277.
• SNAKES - ACTIVE CONTOUR MODELS,
KASS M, WITKIN A, TERZOPOULOS D, INTERNATIONAL JOURNAL OF COMPUTER VISION 1 (4): 321-331 1987.

Assignment #1: Due on Monday, April 18.
Assignment1.pdf       An original image in bmp format       A noisy image in bmp format

Assignment #2: Due on Friday, May 13.
Assignment2.pdf

Assignment #3: Due on Friday, June 10.
Assignment3.pdf