Programme - Workshop on symmetries, inverse problems and image processing

Workshop on symmetries, inverse problems and image processing

13-15 January, Linz, Austria

Johann Radon Institute for Computational and Applied Mathematics (RICAM)

Altenbergerstrasse 69, Linz, A-4040

Tel: +43 (0)732 2468 5211, Fax: +43 (0)732 2468 5212

The meeting will take place in the seminar room HF 136

Programme

January 13

9:30 - 10:00 Opening of the workshop

10:00 - 11:00 Peter Olver - Symmetry in computer vision

Abstract: This talk will survey various applications of symmetry methods in computer vision, including denoising, segmentation, and object recognition based on differential invariants and joint differential invariants.

11:00 - 11:30 Coffee Break (RICAM)

11:30 - 12:30 Heinz W. Engl - Iterative regularization methods for nonlinear inverse problems

12:30 - 14:00 Lunch

14:00 - 15:00 Martin Burger - The choice of regularization functionals in imaging and inverse problems

Abstract: We discuss the construction of variational and iterative regularization methods with arbitrary convex regularization functionals for imaging and inverse problems. The freedom in the choice of the regularization functional offered by such a general construction induces the question after the right choice in order to achieve certain special tasks (and possibly to preserve certain invariances). For diffusion filtering tasks, such a question has been partly answered (also using symmetries), but for novel applications such as impainting or processing of graphics, new classes of functionals are used (and still needed), so that the talk will finally conclude with questions related to the use of symmetries.

15:00-15:30 Herbert Egger - Some aspects in iterative regularization

Abstract: Some aspects in modifying and/or accelerating iterative regularization algorithms for ill-posed problems, including symmetry reduction, are discussed and are illustrated in application to a deblurring problem in image processing.

15:30 - 16:00 Coffee Break (RICAM)

16:00-17:00 Arieh Iserles - Highly oscillatory quadrature and its applications (.pdf file)

Abstract: In this talk I will review recent advances, joint with Syvert Nørsett, in understanding and implementing methods for quadrature with highly oscillatory kernels. We develop two methods, one based on an asymptotic expansion and the other on interpolation, that afford very precise approximation in the presence of high oscillation and critical points, in one or more dimensions. Time allowing, I will describe some of the applications of these methods to Fredholm equations of the second kind and ordinary and partial differential equations with rapidly oscillating solutions.

January 14

9:30 - 10:30 Elena Kartaschova - On factorization of linear partial differential operators. Generic case

Abstract: Constructive algorithm for finding conditions of absolute factorization for bivariate LPDO into linear factors is presented. For cases of LPDO order 2 and 3 the corresponding conditions are written out explicitely. Some illustrative examples, inclusive well-known E. Landau example, are discussed.

10:30 - 11:00 Coffee Break (RICAM)

11:00 - 12:00 Peter Clarkson - Special polynomials associated with rational solutions of the Painlev\'e equations and

applications to soliton equations - Part I

(.pdf file)

Abstract In this talk I shall discuss special polynomials associated with rational solutions for the second, third and fourth and fifth Painlev\'e equations (PII - PV) and also special polynomials associated with rational solutions of soliton equations which are solvable by the inverse scattering method, including the Korteweg-de Vries, modified Korteweg-de Vries, nonlinear Schrödinger and Boussinesq equations.

The Painlev\'e equations are six nonlinear ordinary differential equations that have been the subject of much interest in the past twenty-five years, which have arisen in a variety of physical applications. Further they may be thought of as nonlinear special functions. Rational solutions of the Painleve equations are expressible in terms of the logarithmic derivative of certain special polynomials. For PII these polynomials are known as the Yablonskii-Vorob'ev polynomials, first derived in the 1960's by Yablonskii and Vorob'ev. The locations of the roots of these polynomials is shown to have a highly regular triangular structure in the complex plane. The analogous special polynomials associated with rational solutions of PIII, PIV and PV, and with algebraic solutions of PIII and PV, are described and it is shown that their roots also have a highly regular structure and other properties of the polynomials will be discussed.

It is well known that soliton equations have symmetry reductions which reduce them to the Painleve equations. Hence rational solutions of soliton equations arising from symmetry reductions of the Painlev\'e equation can be expressed in terms of the aforementioned special polynomials. Also the motion of the poles of the rational solutions of the Korteweg-de Vries equation is described by a constrained Calogero-Moser system describes the motion of the poles of rational solutions of the Korteweg-de Vries equation, as shown by Airault, McKean, and Moser in 1977. The motion of the poles of more general rational solutions of equations in the Korteweg-de Vries and modified Korteweg-de Vries hierarchies, and the motion of zeroes and poles of rational and rational-oscillatory solutions of the nonlinear Schrödinger equation will be discussed.

12:00 - 13:30 Lunch

13:30 - 14:30 Willem Adriaan de Graaf - Constructing fauthful representations of Lie algebras of characteristic 0

Abstract: I will describe an effective version of Ado's theorem. i.e., an algorithm that produces a faithful matrix representation of a given finite-dimensional Lie algebra of characteristic 0.

14:30 - 15:30 Arieh Iserles - On a Lie - Poisson system and its Lie algebra (.pdf file)

Abstract: In this talk, based on joint work with Tony Bloch, we consider differential equations of the form X'=[N,X^2], where X(0) is a symmetric matrix, while N is skew symmetric. Such flows can be considered as an outcome of two distinct group actions: by similarity (hence they are isospectral) and by congruence. We prove that they are endowed with a Poisson structure. Hence, they correspond to a flow along an orbit of the dual to the free Lie algebra generated by their structure constants. We thus seek a faithful representation of the underlying Lie algebra and attain it by using methods of matrix analysis and numerical linear algebra.

15:30-16:00 Sheehan Olver - TBA

17:00 - Conference dinner at the Pöstlingberg-Schlössl restaurant

January 15

9:30 - 10:00 Erik Hillgarter - Lie symmetry analysis of a special energy equation

Abstract : We present a Lie symmetry analysis of the following special pipe flow energy equation

u_{yy}+u_{xx}+(A-Bx^2)u_{y}+x^(-1) u_{x}=C-Dx^2,
where u=u(x,y) and A,B,C<0, D>0 are constant parameters. This equation has been studied in fluid mechanics since decades, but no symbolic solution was known. Known solutions could be used to study the dependence of solutions on the parameters and to solve boundary value problems. We give a point symmetry classification of this equation, investigate invariant solutions and families of transformed solutions.

10:00 - 11:00 Peter Olver - Symmetry in numerical analysis

Abstract : In this talk, I will show how to exploit symmetry in the design of numerical algorithms. Multi-space serves as a new foundation for geometric numerical integration. The method of moving frames shows how to invariantize any numerical integration scheme, e.g. Runge--Kutta, leading to large classes of symmetry-preserving numerical approximations to differential invariants and invariant differential equations.

11:00 - 11:30 Coffee Break (RICAM)

11:30 - 12:30 Peter Clarkson - Special polynomials associated with rational solutions of the Painlev\'e equations and

applications to soliton equations - Part II

(.pdf file)

12:30 - 13:00 Nicoleta Bila - Invariant geometric flows

Abstract : I am going to talk about some applications of the theory of symmetry groups to image processing. This talk will be based on some recent results by Olver, Sapiro and Tannenbaum. We are interested in studying particular invariant flows for a given Lie group.

13:00 - 13:30 Ralf Hemmecke - Differential Elimination

13:30 Refreshments and closing of the workshop

Last modified on 12 January 2005