1. Center of Complex Systems and Visualization
  2. Lehre
  3. Oberseminar

In dem regelmäßig stattfindenden Oberseminar tragen Gäste aus aller Welt über Forschungsarbeiten zu Themen vor, die mit der Arbeit von CeVis und MeVis in Verbindung stehen, und Mitarbeiter von CeVis und MeVis präsentieren ihre neusten Ergebnisse.

C(M)eVis-Oberseminar am 16.5.2002

Datum: Mittwoch, 16. Mai 2002
Zeit: 11.00 Uhr s.t.
Ort: Seminarraum Mandelbrot
Referent: Prof. Dr. Bernd Fischer (Institut für Mathematik, Universität Lübeck)

On fast registration schemes with application to medical imaging

Image registration is one of the most challenging task within digital imaging, in particular in medical imaging. In most applications simple rigid deformations are not satisfactory and complex, non-rigid and non-linear deformations must be employed.

Given a template image$T$ and a reference image$R$, where $T,R: \Omega=[0,1]^d \to {\mathbb{R}}$, the purpose of the registration is to determine a transformation, sometimes called warping, of $T$ onto $R$. Ideally, one wants to determine a displacement field  $u: {\mathbb{R}}^d \to {\mathbb{R}}^d$ such that $T(x-u(x))=R(x)$.

The question is how to find such a mapping$u$.
To start with, we review some of the most promising non-linear registration strategies currently used in medical imaging. Then we show that all
these techniques may be phrased in terms of a variational problem and therefore allow for a unified treatment. More precisely, we consider the following functional

\begin{displaymath}{\cal J}[u] :=\frac{1}{2} \int_\Omega \left(T(x-u(x))-R(x)\right)^2 dx+{\cal S}[u]\end{displaymath}
where ${\cal S}$ denotes a suitable smoothing or regularizing term.  To compute a minimizer of the functional $ {\cal J}$ we apply the calculus of variations and obtain a non-linear partial differential equation: the associated Euler-Lagrange equation. Subsequently we solve this equation by a finite difference approximation accompanied by a time-marching or fixpoint scheme. It should come as no surprise, that the main work in the overall scheme is the repeated solution of a (highly structured) linear system.

In this talk we consider various choices for the smoother ${\cal S}$ and show how to solve the resulting linear system fast and robust by means of direct schemes. Furthermore we demonstrate the performance of the presented approaches for a variety of academic and real life examples.