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

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Abstract

In [3] an approach is given for minimizing certain functionals on certain spaces \({\cal N} = {\rm Maps}(\Omega,N)\), where Ω 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.

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Authors and Affiliations

  1. Department of Mathematics, University of Florida, Gainesville, FL, 32611-8105, USA

    David Groisser

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  1. David Groisser
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Correspondence to David Groisser.

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David Groisser is an Associate Professor of Mathematics at the University of Florida. Dr. Groisser conducts research in pure and applied differential geometry. His current interests include the geometry of shape spaces and applications of differential geometry to image-analysis.

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Groisser, D. Some Differential-Geometric Remarks on a Method for Minimizing Constrained Functionals of Matrix-Valued Functions. J Math Imaging Vis 24, 349–358 (2006). https://doi.org/10.1007/s10851-005-3633-z

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