TY - GEN

T1 - Denoising tensors via Lie group flows

AU - Gur, Y.

AU - Sochen, N.

PY - 2005

Y1 - 2005

N2 - The need to regularize tensor fields arise recently in various applications. We treat in this paper tensors that belong to matrix Lie groups. We formulate the problem of these SO(N) flows in terms of the principal chiral model (PCM) action. This action is defined over a Lie group manifold. By minimizing the PCM action with respect to the group element, we obtain the equations of motion for the group element (or the corresponding connection). Then, by writing the gradient descent equations we obtain the PDE for the Lie group flows. We use these flows to regularize in particular the group of N-dimensional orthogonal matrices with determinant one i.e. SO(N). This type of regularization preserves their properties (i.e., the orthogonality and the determinant). A special numerical scheme that preserves the Lie group structure is used. However, these flows regularize the tensor field isotropically and therefore discontinuities are not preserved. We modify the functional and thereby the gradient descent PDEs in order to obtain an anisotropic tensor field regularization. We demonstrate our formalism with various examples.

AB - The need to regularize tensor fields arise recently in various applications. We treat in this paper tensors that belong to matrix Lie groups. We formulate the problem of these SO(N) flows in terms of the principal chiral model (PCM) action. This action is defined over a Lie group manifold. By minimizing the PCM action with respect to the group element, we obtain the equations of motion for the group element (or the corresponding connection). Then, by writing the gradient descent equations we obtain the PDE for the Lie group flows. We use these flows to regularize in particular the group of N-dimensional orthogonal matrices with determinant one i.e. SO(N). This type of regularization preserves their properties (i.e., the orthogonality and the determinant). A special numerical scheme that preserves the Lie group structure is used. However, these flows regularize the tensor field isotropically and therefore discontinuities are not preserved. We modify the functional and thereby the gradient descent PDEs in order to obtain an anisotropic tensor field regularization. We demonstrate our formalism with various examples.

UR - http://www.scopus.com/inward/record.url?scp=33646555804&partnerID=8YFLogxK

U2 - 10.1007/11567646_2

DO - 10.1007/11567646_2

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AN - SCOPUS:33646555804

SN - 3540293485

SN - 9783540293484

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 13

EP - 24

BT - Variational, Geometric, and Level Set Methods in Computer Vision - Third International Workshop, VLSM 2005, Proceedings

T2 - 3rd International Workshop on Variational, Geometric, and Level Set Methods in Computer Vision, VLSM 2005

Y2 - 16 October 2005 through 16 October 2005

ER -