TY - JOUR

T1 - Regularizing flows over lie groups

AU - Gur, Yaniv

AU - Sochen, Nir

PY - 2009/2

Y1 - 2009/2

N2 - In this paper we discuss regularization of images that take their value in matrix Lie groups. We describe an image as a section in a principal bundle which is a fibre bundle where the fiber (the feature space) is a Lie group. Via the scalar product on the Lie algebra, we define a bi-invariant metric on the Lie-group manifold. Thus, the fiber becomes a Riemannian manifold with respect to this metric. The induced metric from the principal bundle to the image manifold is obtained by means of the bi-invariant metric. A functional over the space of sections, i.e., the image manifolds, is defined. The resulting equations of motion generate a flow which evolves the sections in the spatial-Lie-group manifold. We suggest two different approaches to treat this functional and the corresponding PDEs. In the first approach we derive a set of coupled PDEs for the local coordinates of the Lie-group manifold. In the second approach a coordinate-free framework is proposed where the PDE is defined directly with respect to the Lie-group elements. This is a parameterization-free method. The differences between these two methods are discussed. We exemplify this framework on the well-known orientation diffusion problem, namely, the unit-circle S 1 which is identified with the group of rotations in two dimensions, SO(2). Regularization of the group of rotations in 3D and 4D, SO(3) and SO(4), respectively, is demonstrated as well.

AB - In this paper we discuss regularization of images that take their value in matrix Lie groups. We describe an image as a section in a principal bundle which is a fibre bundle where the fiber (the feature space) is a Lie group. Via the scalar product on the Lie algebra, we define a bi-invariant metric on the Lie-group manifold. Thus, the fiber becomes a Riemannian manifold with respect to this metric. The induced metric from the principal bundle to the image manifold is obtained by means of the bi-invariant metric. A functional over the space of sections, i.e., the image manifolds, is defined. The resulting equations of motion generate a flow which evolves the sections in the spatial-Lie-group manifold. We suggest two different approaches to treat this functional and the corresponding PDEs. In the first approach we derive a set of coupled PDEs for the local coordinates of the Lie-group manifold. In the second approach a coordinate-free framework is proposed where the PDE is defined directly with respect to the Lie-group elements. This is a parameterization-free method. The differences between these two methods are discussed. We exemplify this framework on the well-known orientation diffusion problem, namely, the unit-circle S 1 which is identified with the group of rotations in two dimensions, SO(2). Regularization of the group of rotations in 3D and 4D, SO(3) and SO(4), respectively, is demonstrated as well.

KW - Beltrami framework

KW - Differential geometry

KW - Lie groups

KW - PDEs

KW - Tensor-valued image denoising

UR - http://www.scopus.com/inward/record.url?scp=59849104123&partnerID=8YFLogxK

U2 - 10.1007/s10851-008-0127-9

DO - 10.1007/s10851-008-0127-9

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AN - SCOPUS:59849104123

SN - 0924-9907

VL - 33

SP - 195

EP - 208

JO - Journal of Mathematical Imaging and Vision

JF - Journal of Mathematical Imaging and Vision

IS - 2

ER -