Measure and Image (M16)

Tuomo Valkonen

Photographs and other natural images are usually not smooth maps, they contain edges (dis-continuities) and other non-smooth geometric features that should be preserved by image en-hancement techniques. The correct mathematical modelling of these features involves the spaceof functions of bounded variation and, in consequence, aspects of geometric measure theory.The aim of this course is to provide an introduction to functions of bounded variation and theirapplications in image processing. It will cover the following topics.

Motivation. Why Sobolev spaces are not enough for image processing? Functions ofbounded variation of one variable.

Measure. Refresher on measure theory. Hausdorff measure and rectifiable sets. Functions of bounded variation. Weak convergence and compactness. Poincare inequality.

Co-area formula. Fine properties.

Total variation regularisation. Image denoising. Basic properties of solutions.

Pre-requisites

A basic course in measure theory is strongly recommended, although we do include a quickrefresher to the topic. Basic knowledge of Sobolev spaces and notions of weak convergence infunction spaces are advantageous, but not necessary.

Literature

Introductory reading:

1. A. Friedman, Foundations of Modern Analysis, Dover, 2003.

2. W. Rudin, Real and Complex Analysis, McGraw-Hill, 1987. (Part on measure theory)

3. L. C. Evans, Partial Differential Equations, Americal Mathematical Society, 2010. (Parton Sobolev spaces)

Liteature to complement course material:

1. L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discon-tinuity Problems, Oxford University Press, 2000.

2. G. Aubert, and P. Kornprobst, Mathematical Problems in Image Processing: Partial Dif-ferential Equations and the Calculus of Variations, Springer, 2006.

3. L. C. Evans, and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRCPress, 1991.

Additional support

Three examples sheets will be provided and three associated examples classes will be given. Aaone-hour revision class is planned for the Easter Term.

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