Some Differential-Geometric Remarks on a Method for Minimizing Constrained Functionals of Matrix-Valued Functions

Abstract

In [3] an approach is given for minimizing certain functionals on certain spaces $${\cal N} = {\rm Maps}(\Omega,N)$$ , where ? is a domain in some Euclidean space and N is a space of square matrices satisfying some extra condition(s), e.g. symmetry and positive-definiteness. The approach has the advantage that in the associated algorithm, the preservation of constraints is built in automatically. One practical use of such an algorithm its its application to diffusion-tensor imaging, which in recent years has been shown to be a very fruitful approach to certain problems in medical imaging. The method in [3] is motivated by differential-geometric considerations, some of which are discussed briefly in [3] and in greater detail in [4]. We describe here certain geometric aspects of this approach that are not readily apparent in [3] or [4]. We also discuss what one can and cannot hope to achieve by this approach.