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Geometric Optimization Problems in Computer Vision. Edition No. 1
VDM Publishing House, May 2008, Pages: 112
We explore various optimization problems which
appear in computer vision and image processing. By exploiting the geometric structure of the underlying search spaces, we develop more efficient and robust ways to find optimal solutions.
We present the Nelder-Mead algorithms on Lie groups and describe its application to medical image registration. A widely used approach to image registration involves finding the transformation that maximizes the mutual information between two images, with the transformation restricted to be either rigid-body (i.e., belonging to SE(3)) or volume-preserving (i.e., belonging to SL(3)). We present coordinate-invariant, geometric versions
of the Nelder-Mead optimization algorithm on the transformation groups SL(3), SE(3), and its various subgroups, that are applicable to a wide class of image registration problems. Because the algorithms
respect the geometric structure of the underlying transformation groups, they are numerically more stable, and exhibit better convergence properties than existing local coordinate-based algorithms. Experimental results demonstrate the improved convergence properties of our geometrics algorithm.
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