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.
- 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.
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.
Download the slides for
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.