《Variational Methods in OptimizationPP22》.pdf

Supplemental Material for ELEC 603 Spring 2006 Variational Methods in Optimization References 1. I. M. Gelfand and S. V. Formin, Calculus of Variations, Dover Publications Inc., 2000. 2. L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D ., vol. 60, pp. 259-268, 1992. 3. L. Rudin and S. Osher, “Total variation based image restoration with free local constraints,” Proc. ICIP, pp. 31-35, Austin, TX., 1994. 4. S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” SIAM Journal on Multiscale Modeling and Simulation, vol. 4, no. 2, pp. 460-489, 2005. 5. Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, University Lecture Series, vol. 22, American Mathematical Society, Providence, RI., 2001. 6. UCLA Reports: /applied/cam/index.html 1. Elements of Calculus of Variations 1.1 Functionals Roughly speaking, a functional is a “function of function”. We all know the definition of function, sayf (x) – which is taken to mean a correspondence that assigns a definite (real) number for each variable value x in a certain domain. Similarly, a functional is a correspondence that assigns a real number to eachfunction (such as a curve, or a surface, etc.) that belongs to some class of functions. For example, for each curve that connects two fixed points a and b, the length of the curve is a functional. As another example, the time required for a mass point traveling down along a smooth and monotonically decreasing plane curve with two fixed ends is a functional. From these examples, we see that a functional is defined over a certain class of functions. 1.2 Variational Problems: Examples Example 1 A smooth plane curve that joins points p = (a, A) and p = (b, B) c