A Multigrid and Multiscale Approach to Image Matching
Ulrich Clarenz (GerhardMercatorUniversität 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 ,
find a deformation
,
d=2,3, which maps
onto
in the sense that
is strongly related to
.
This matching problem is known to be illposed. 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.
