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Home Teaching Summer Semester 2015 Nonlinear Multiscale Methods for Image and Signal Analysis (IN3200) (2h, 4 ECTS)

Nonlinear Multiscale Methods for Image and Signal Analysis (IN3200) (2h, 4 ECTS)

Filtering approaches are a fundamental tool in digital image and signal processing. Classical, linear filtering techniques, e.g. Fourier-based filterings, are well understood: One changes the respresentation of the input data via a linear transform, manipulates the coefficients in the new represenation, and finally inverts that transform. Novel nonlinear filtering techniques coming from variational methods (and related approaches) give rise to a novel definition of transformations, multiscale decompositions, and filterings, which we will discuss in this lecture.

In more detail, the lecture will cover the following topics:

  • Multiscale image and signal decompositions using variational methods, scale space flows, and inverse scale space flows
  • Discretization and Implementation of the above approaches
  • Convergence properties of the above approaches
  • Filtering strategies arising from the above approaches
  • Relation of the approaches among each other
  • Relation to a generalized notion of (nonlinear) singular vectors
  • The connection of the inverse scale space flow to optimization techniques
  • Applications in image and signal processing

This lecture is a mathematics as well as a computer science lecture.


News:

  • 24.07: Lecture slides in which the gradient flow solution on singular vectors is complete (=cases by case).
  • 22.07: Added a missing assumption to the solution of sheet 8.
  • 14.07: A final update of the lecture slides, the overview slides, and the solution to the last exercise sheet are online. The overview slides can help you prepare for the exam.
  • 08.07: New lecture slides and the eighth exercise sheet are online.
  • 06.07: The solution to the seventh exercise sheet is online.
  • There will be no lecture on 30.06. Instead, there will be hands-on-code sessions in room 02.05.014. on Monday 6th of July at 2pm and after the lecture on Tuesday 7th of July at 4pm.
  • 24.06: New lecture slides online.
  • 22.06: A new exercise sheet is online.
  • 22.06: Programming solution of Björn for visualizing the subdifferential of TV is online.
  • 19.06: New lecture slides and the solution to the sixth exercise sheet are online. As a homework for this week, try to implement the spectral decomposition based on the variational approach. We will discuss this in more details in the next lecture and will do a hands-on code session in 2,5 weeks from now.
  • 09.06: New lecture slides and the sixth exercise sheet are online.
  • 02.06: New lecture slides are online. Also, I put code for generating 1d and 2d TV singular vectors online. There is no homework sheet this week. I recommend going over the lecture slides again to familiarize yourself with the new definitions of singular vectors, ground states, and the properties of 1-homogeneous functionals.
  • 31.05: The solution to the fifth exercise sheet is online.
  • 20.05: The fifth exercise sheet and new lecture slides are online.
  • 18.05: The solution to the color TV programming exercise is now online.
  • 14.05: Added notes regarding the (partial) proofs of the main theorems of convex analysis.
  • 12.05: The fourth exercise sheet and new lecture slides are online.
  • 12.05: The solution to the third exercise sheet (programming and theory) is online.
  • 07.05: The third exercise sheet and the new lecture slides are online.
Lecture

Location: Room 02.09.023
Time and Date: Tuesday 14:15 - 16:00 Lecturer: Dr. Michael Möller
Start: April 14th
The lecture is held in English.
The exam will be written or oral depending on the number of attendees.

Lecture Material
Suggested Homework
Literature

General reference for convex analysis:

  • R.T. Rockafellar: Convex Analysis.
  • Chapter I of S. Boyd and L. Vandenberghe: Convex Optimization.
  • J.M. Borwein and Q.J. Zhu: Techniques of Variational Analysis.

Particular papers we will discuss:

  • G. Gilboa: A Total Variation Spectral Framework for Scale and Texture Analysis.
  • M. Benning and M. Burger: Ground States and Singular Vectors of Convex Variational Regularization Methods.
  • M. Burger, L. Eckardt, G. Gilboa, M. Moeller: Spectral Decomposition of One-Homogeneous Functionals.

More to be announced.

Last edited 24.07.2015 14:50 by Dr. Michael Möller