An important problem in image analysis is the recovery of images corrupted by noise or blurring effect. Another instance of this problem is the separation of an image f into u+v, where u is piecewise smooth and v is oscillatory. The first half of the class studies the Bayesian and the variational approaches to this problem. In the second half, we will study the modeling of the oscillatory component v. This will lead us to a discussion of scales and local scales in images.

Book References:

1. Mathematical Problems in Image Processing by G. Aubert and P. Kornprobst.
2. Oscillating Patterns in Image Processing and Nonlinear Evolution Equations by Y. Meyer.
3. Functions of Bounded Variation and Free Discontinuity Problems by L. Ambrosio, N. Fusco and D. Pallara.
4. Sigularity Integrals and Differentiability Properties of Functions by E.M. Stein.

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• Week Jan 21-25, 2008
  Topics: Basic probability theory;
  Notes: week1.pdf

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• Week Jan 28 - Feb. 1, 2008
  Topics: degradation models for images; Bayesian method for reconstructing images corrupted by Gaussian and Poisson noise.
  Notes: week2.pdf

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• Week Feb 4-8, 2008
  Topics: Banach spaces; the associate and dual spaces; weak and weak* topology.

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• Week Feb 11-15, 2008
  Topics: The space of functions of bounded variation. Energy methods for restoring images corrupted by Gaussian and Poisson noise; existence and uniqueness; characterizations of minimizers.
  References:
  1. L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms. pdf
  2. L. Rudin and S. Osher, Total variation based image restoration with free local constraints. pdf

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• Week Feb 18-22, 2008
  Topics: PDE-based methods, non-linear and anisotropic diffusions.
  References:
  1. P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion. pdf
  2. F. Catte, P.-L. Lions, J.-M. Morel and T. Coll, Image selective smoothing and edge detection by nonlinear diffusion. I. pdf
  3. L. Alvarez, P.-L. Lions, and J.-M. Morel, Image selective smoothing and edge detection by nonlinear diffusion. II. pdf
  4. N. Sochen, R. Kimmel and R. Malladi, A general framework for low level vision. pdf

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• Week Feb 25-29, 2008
  Topics: Learning the statistics of natural images.
  References:
  1. J. Huang and D. Mumford, Statistics of Natural Images and Models. pdf
  2. M. Green, Statistics of Images, the TV Algorithm of Rudin-Osher-Fatemi for Image Denoising and an Improved Denoising Model. pdf

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• Week March 3-7, 2008
  Topics: Modeling Oscillations in images via generalized functions; Sobolev, Besov and Triebel-Lizorkin function spaces.

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• Week March 24-28, 2008
  Topics: Texture decomposition models; existence and uniqueness; characterizations of minimizers.

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• Week March 31 - April 4, 2008
  Topics: Scale-space theory; local scales.

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• Week April 7-11, 2008
  Topics: Local scales (cont.).

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• Week April 14-18, 2008
  Topics: Presentations.

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• Week April 21-25, 2008
  Topics: Presentations.