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18th GAMM-Seminar Leipzig on
Multigrid and related methods for optimization problems

Max-Planck-Institute for Mathematics in the Sciences
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  18th GAMM-Seminar
January, 24th-26th, 2002
 
     
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  Abstract Ulrich Clarenz, Thu, 14.55-15.20 Previous Contents Next  
  A Multigrid and Multiscale Approach to Image Matching
Ulrich Clarenz (Gerhard-Mercator-Universitšt Duisburg)

We present a fast multigrid method for image matching in 2D and 3D, an important optimization problem in computer vision and indispensable for medical imaging application based on different image modalities.

We consider the following problem: Given two image intensity maps , on a domain tex2html_wrap_inline14, find a deformation , d=2,3, which maps onto in the sense that is strongly related to .

This matching problem is known to be ill-posed. There are two types of solution strategies: The ``elastic'' approach, which consists in regularizing the energy or a ``fluid dynamic'' approach interpreting this problem as gradient flow.

We will follow the fluid dynamic approach computing the solution of



for a matching energy E. I.e. using a regularizing metric g on we get . Generally, the representation of the metric in the scalar product allows the interpretation of the above flow equation as .

The most basic example of matching energy to be minimized is .

We propose multigrid operators for the inverse of the representation of the thereby induced metric . Their application oriented smoothing properties will be discussed. Furthermore, to avoid convergence to local minima multiple scales of the images to be matched are considered. Again, these image scales can be generated applying multigrid operators and we propose to resolve the pyramid of scales on a properly chosen pyramid of hierarchical grids.

Examples on 2D and large 3D image matching problems prove the robustness and efficiency of the proposed approach. Future extensions of the gradient approach as a flexible image matching methodology will be outlined.
 

 
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