## Variational Methods in Image Processing

Lecture Meeting Time: Mon & Fri 3-4.15pm.
Lecture Location: MS 5138.

Instructor: Luminita A. Vese
Office: MS 7620-D
Office hours: TBD.

E-mail: lvese@math.ucla.edu

Class Web page: http://www.math.ucla.edu/~lvese/285j.2.15f

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 equations, viscosity solutions, oscillatory functions, Sobolev gradients.
- Applications: image restoration (denoising, deblurring), image decomposition into cartoon and texture, image segmentation and edge detection, snakes, curve evolution, active contours, level set methods.

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

Textbook References:
• G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, (Partial Differential Equations and the Calculus of Variations), Springer, 2002 or 2006.
Online access restricted to UC Campuses (2nd 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.
• Book manuscripts by J.-M. Morel and collaborators: http://www.cmla.ens-cachan.fr/Membres/morel.html
• T.F. Chan and J. Shen, Image processing and analysis, SIAM 2005
• R. Malladi (Ed.), Geometric Methods in Bio-Medical Image Processing, Springer 2002.
• A. Blake and A. Zisserman, Visual Reconstruction, The MIT Press Cambridge, Massachusetts, 1987. (online link)
• G. Sapiro, Geometric Partial Differential Equations and Image Processing, Cambridge University Press, 2001.

Assignments & Projects:
Usually biweekly assignments are given containing theoretical and computational exercises. Maybe this time interested students could read instead one paper of their choice and prepare a short report with numerical implementations and a presentation in class (presentation of the method, Euler-Lagrange equation, numerical implementation and experimental results, sensitivity).
- All enrolled students will have to solve problems and do numerical implementations of the methods discussed in class (work in group or teams of two or three students is encouraged), or read a paper and make a presentation in class and prepare a short report.
- 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 and discussing it with the instructor, can do so.
- Late homework (within a few days) 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.
• An Algorithm for Total Variation Minimization and Applications
A. Chambolle, Journal of Mathematical Imaging and Vision 20 (1-2): 89-97, January - March, 2004.
• A review of image denoising algorithms, with a new one",
A. Buades, B. Coll, J.M Morel, Multiscale Modeling and Simulation (SIAM interdisciplinary journal), Vol 4 (2), pp: 490-530, 2005.
• Stochastic relaxation, Gibbs distributions, and the bayesean restoration of images,
S. Geman, D. Geman, IEEE TPAMI 6: 721-741, 1984.
• K. Niklas Nordström, Biased anisotropic diffusion: a unified regularization and diffusion approach to edge detection, Image and Vision Computing, 8(4): 318-327, 1990 (see also http://www.eecs.berkeley.edu/Pubs/TechRpts/1989/CSD-89-514.pdf for more details)

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.
• On the relation between parametric and geometric active contours,
C. Xu, A. Yezzi, J. Prince, Proc. of 34th Asimolar Conf. on Sig. Sys Comp., Pacific Grove CA, October 29, 2000.
• Snakes, Shapes, and Gradient Vector Flow,
C. Xu and J.L. Prince, IEEE Transactions on Image Processing, 7(3):359-369, March 1998.

Review of definitions and properties of BV functions
Time dependent scheme for ROF denoising
Lecture notes on Nonlocal methods (prepared by Xiaoqun Zhang)

An image in three different formats:
image
image
image

Assignments
• Assignment 1
• Assignment 2
• Assignment 3
• Assignment 4
• Assignment 5