Invited Speakers

Invited speakers

Fiorella Sgallari, University of Bologna, Italy
Serena Morigi, University of Bologna, Italy
Otmar Scherzer, University of Innsbruck, Austria
Anders Heyden, Malmö University, Sweden
François Louze, Copenhagen University
Gunnar Carlsson, Stanford University
Stephan Didas, Saarland University

Geometrical partial differential equations:

numerics and applications

Bergen 6-7 December 2006

Talks and abstracts:
Ottmar Scherzer, "Total Variation Regularization in Statistics and Imaging" Total variation regularization has been considered both in Imaging and Statistics. In both communities various algorithms have been developed for the efficient solution. It is the goal of this talk to review some of them and unify the view using methods from convex analysis. Since the data in statistics and imaging is different, total variation regularization has to be adopted to different needs. This is a joint work with A. Obereder (Innsbruck) and A. Kovac (Bristol). Francois Louze, “Segmentation driven by Inpainting” In this talk I present a segmentation algorithm based on the idea of background disocclusion. When background is known and an observation is given, a background subtraction can be used in order to build a foreground object detection/segmentation algorithm, and we present a simple one. In the case of interest for us, background is not known, but a prior distribution is assumed to be known instead, which allows to estimate this unknown background via inpainting when foreground location is assumed to be known. Combining it with our simple segmentation algorithm, it provides a "disocclusion quality measure", which attributes a numerical score to a given subregion of the image domain. We derive a segmentation algorithm by optimization of that measure, an optimization performed in the frameworks of control systems governed by PDE's and Shape Derivative tools. I will present a series of examples, and especially examples for Medical Image Analysis.

Stephan Didas, “Integrodifferential equations for image denoising”
We investigate the connection between wavelet shrinkage and integro-differential equations involving nonlinear diffusion filtering on multiple scales. It is well-known that discrete wavelet shrinkage on the finest scale is closely related to nonlinear diffusion filtering. In this talk we consider the extension to multiple scales in the continuous case which introduces additional convolution integrals in the corresponding nonlinear diffusion equation. The convolution kernels as well as the derivative orders in this integro-differential equation depend directly on the type of wavelet used for the shrinkage. The integrals introduce the multiscale approach of the wavelet methods into the framework of nonlinear diffusion filtering and allow for faster evolutions compared to pure PDE filters. Nevertheless numerical examples show that fine-scale details might be longer preserved during the evolution with larger scales. It is further possible to express the convolutions in the evolution equation in terms of power series of differential operators. This transfers the integro-differential equations to equations with pseudodifferential operators.
This is joint work with J. Weickert.

Fiorella Sgallari and Serena Morigi, “Numerical experience in PDE Image and Surface Processing: Part I & II” In these talks we will discuss numerical problems related to the solution of partial differential equations (PDEs) arising in image and surface processing. PDE-based models have been successfully applied to several areas such as edge-preserving image denoising-deblurring, image segmentation, shape reconstruction and other graphic applications. We discuss a new cascadic multiresolution method with a conjugate gradient-type algorithm as basic iterative scheme. In particular, we consider applications to image deblurring and denoising using edge-preserving prolongation operators based on partial differential equations defined by Perona-Malik and total variation-type models. Moreover we will focus on the evolution of surfaces formulated in a Lagrangian framework and we will describe how this framework may be employed in applications such as mesh regularization, morphing and feature preserving surface smoothing.
Gunnar Carlsson, “The topology of point cloud data”
Algebraic topology is a mathematical formalism which allows one to quantify and make precise various kinds of recognition of geometric patterns. In recent years, the methods have been extended to allow one to study spaces from incomplete information, i.e. from finite but large sets of points sampled from them (point clouds). In this talk, we will summarize the mathematics behind this development as well as examples from real world data.

Anders Heyden “Variational problems and level set methods in comptuer vision: theory and applications
Current state of the art suggests the use of variational formulations for solving a variety of computer vision problems. This talk deals with such variational problems which often include the optimization of curves and surfaces. The level set method is used throughout the talk, both as a tool in the theoretical analysis and for constructing practical algorithms. One frequently occurring example is the problem of recovering three-dimensional (3D) models of a scene given only a sequence of images. Other applications such as segmentation are also considered. The talk consists of three parts. The first part contains a review of background material and the level set method. The second part contains a collection of theoretical contributions such as a gradient
descent framework and an analysis of several variational curve and surface problems. The third part contains contributions for applications such as a framework for open surfaces and variational surface fitting to different types of data.

CIPR-CMA workshop