An invariant shape representation using the anisotropic Helmholtz equation - PubMed (original) (raw)

An invariant shape representation using the anisotropic Helmholtz equation

A A Joshi et al. Med Image Comput Comput Assist Interv. 2012.

Abstract

Analyzing geometry of sulcal curves on the human cortical surface requires a shape representation invariant to Euclidean motion. We present a novel shape representation that characterizes the shape of a curve in terms of a coordinate system based on the eigensystem of the anisotropic Helmholtz equation. This representation has many desirable properties: stability, uniqueness and invariance to scaling and isometric transformation. Under this representation, we can find a point-wise shape distance between curves as well as a bijective smooth point-to-point correspondence. When the curves are sampled irregularly, we also present a fast and accurate computational method for solving the eigensystem using a finite element formulation. This shape representation is used to find symmetries between corresponding sulcal shapes between cortical hemispheres. For this purpose, we automatically generate 26 sulcal curves for 24 subject brains and then compute their invariant shape representation. Left-right sulcal shape symmetry as measured by the shape representation's metric demonstrates the utility of the presented invariant representation for shape analysis of the cortical folding pattern.

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Figures

Fig. 1

(a) Automatic atlas to subject registration and parameterization of cortical surfaces and sulcal curves; and (b) geodesic curvature flow refinement of sulcal curves

Fig. 2

(a) Inferior frontal sulcus highlighted in red; (b) first four color coded GPS coordinates; (c) GPS representation of a sulcus plotted from end to end.

Fig. 3

Three representative sulci from left and right hemispheres and the point correspondence between them.

Fig. 4

Shape symmetry measure of the sulci plotted on a smooth representation of an individual cortical surface. The black regions on the curves indicate that a significant symmetry was not found for those points.

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An invariant shape representation using the anisotropic Helmholtz equation - PubMed (original) (raw)