TY - CHAP

T1 - Coordinates-based diffusion over the space of symmetric positive-definite matrices

AU - Gur, Yaniv

AU - Sochen, Nir

N1 - Publisher Copyright:
© 2009, Springer-Verlag Berlin Heidelberg.

PY - 2009

Y1 - 2009

N2 - In some recent applications, the objects of interest are matrices. Diffusion Tensor MRI or in short DTI is an example of such an application. In this case, the image is divided into voxels where each voxel is described by a 3 × 3 symmetric positive-definite (SPD) matrix. In this chapter, we present an intrinsic approach for diffusion over the space of n × n symmetric positive-definite matrices, denoted by Pn. The basis of this framework is the description of Pn as a Riemannian manifold by means of the local coordinates and a natural Riemannian metric. One may choose various coordinate systems to cover the Pn manifold. We show that the analytical calculations, as well as the numerical implementation, become simple by choosing Iwasawa coordinates. Then, we define a G L(n)-invariant metric over Pn with respect to these coordinates. The metric is defined by means of the scalar product on Symn, the space of n × n symmetric matrices. The “image” is described here as a section of a fiber bundle. Then, the metric over Pn is combined with the metric over the twoor three-dimensional image domain in order to form the metric over the section. By means of the Beltrami framework, we define a functional over the space of sections. Variation of this functional leads to a set of ½n(n + 1) coupled equations of motion with respect to the local coordinates on Pn. The solution of these equations defines a structure-preserving flow on this manifold. Finally, we demonstrate this framework on the case of P3 by smoothing of real DTI datasets.

AB - In some recent applications, the objects of interest are matrices. Diffusion Tensor MRI or in short DTI is an example of such an application. In this case, the image is divided into voxels where each voxel is described by a 3 × 3 symmetric positive-definite (SPD) matrix. In this chapter, we present an intrinsic approach for diffusion over the space of n × n symmetric positive-definite matrices, denoted by Pn. The basis of this framework is the description of Pn as a Riemannian manifold by means of the local coordinates and a natural Riemannian metric. One may choose various coordinate systems to cover the Pn manifold. We show that the analytical calculations, as well as the numerical implementation, become simple by choosing Iwasawa coordinates. Then, we define a G L(n)-invariant metric over Pn with respect to these coordinates. The metric is defined by means of the scalar product on Symn, the space of n × n symmetric matrices. The “image” is described here as a section of a fiber bundle. Then, the metric over Pn is combined with the metric over the twoor three-dimensional image domain in order to form the metric over the section. By means of the Beltrami framework, we define a functional over the space of sections. Variation of this functional leads to a set of ½n(n + 1) coupled equations of motion with respect to the local coordinates on Pn. The solution of these equations defines a structure-preserving flow on this manifold. Finally, we demonstrate this framework on the case of P3 by smoothing of real DTI datasets.

UR - http://www.scopus.com/inward/record.url?scp=84955204114&partnerID=8YFLogxK

U2 - 10.1007/978-3-540-88378-4_16

DO - 10.1007/978-3-540-88378-4_16

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AN - SCOPUS:84955204114

SN - 9783319912738

SN - 9783540250326

SN - 9783540250760

SN - 9783540332749

SN - 9783540883777

SN - 9783540886051

SN - 9783642150135

SN - 9783642216077

SN - 9783642231742

SN - 9783642273421

SN - 9783642341403

SN - 9783642543005

T3 - Mathematics and Visualization

SP - 325

EP - 340

BT - Mathematics and Visualization

PB - Springer Heidelberg

ER -