Intrinsic brain surface conformal mapping using a variational method

We developed a general method for global conformal parameterizations based on the structure of the cohomology group of holomorphic one-forms with or without boundaries. For genus zero surfaces, our algorithm can find a unique mapping between any two genus zero manifolds by minimizing the harmonic energy of the map. In this paper, we apply the algorithm to the cortical surface matching problem. We use a mesh structure to represent the brain surface. Further constraints are added to ensure that the conformal map is unique. Empirical tests on MRI data show that the mappings preserve angular relationships, are stable in MRIs acquired at different times, and are robust to differences in data triangulation, and resolution. Compared with other brain surface conformal mapping algorithms, our algorithm is more stable and has good extensibility.

[1]  Philip L. Bowers,et al.  Coordinate systems for conformal cerebellar flat maps , 2000, NeuroImage.

[2]  Mark Meyer,et al.  Interactive geometry remeshing , 2002, SIGGRAPH.

[3]  A. Dale,et al.  High‐resolution intersubject averaging and a coordinate system for the cortical surface , 1999, Human brain mapping.

[4]  Ulrich Pinkall,et al.  Computing Discrete Minimal Surfaces and Their Conjugates , 1993, Exp. Math..

[5]  Tony DeRose,et al.  Multiresolution analysis of arbitrary meshes , 1995, SIGGRAPH.

[6]  A. Toga,et al.  A SURFACE-BASED TECHNIQUE FOR WARPING 3-DIMENSIONAL IMAGES OF THE BRAIN , 2000 .

[7]  Guido Gerig,et al.  Surface parametrization and shape description , 1992, Other Conferences.

[8]  David A. Rottenberg,et al.  Quasi-Conformally Flat Mapping the Human Cerebellum , 1999, MICCAI.

[9]  Ron Kikinis,et al.  Conformal Geometry and Brain Flattening , 1999, MICCAI.

[10]  Bruno Lévy,et al.  Least squares conformal maps for automatic texture atlas generation , 2002, ACM Trans. Graph..

[11]  Hiromasa Suzuki,et al.  Three-dimensional geometric metamorphosis based on harmonic maps , 1998, The Visual Computer.

[12]  Muge M. Bakircioglu,et al.  Curve matching on brain surfaces using frenet distances , 1998, Human brain mapping.

[13]  Guillermo Sapiro,et al.  Conformal Surface Parameterization for Texture Mapping , 1999 .

[14]  S. Yau,et al.  Lectures on Harmonic Maps , 1997 .

[15]  Alla Sheffer,et al.  Parameterization of Faceted Surfaces for Meshing using Angle-Based Flattening , 2001, Engineering with Computers.

[16]  Paul M. Thompson,et al.  A framework for computational anatomy , 2002 .

[17]  S. Yau,et al.  Global conformal surface parameterization , 2003 .

[18]  Michael I. Miller,et al.  On The Geometry and Shape of Brain Sub-Manifolds , 1997, Int. J. Pattern Recognit. Artif. Intell..

[19]  Paul M. Thompson,et al.  A surface-based technique for warping three-dimensional images of the brain , 1996, IEEE Trans. Medical Imaging.

[20]  Sean S. B. Moore,et al.  FFTs for the 2-Sphere-Improvements and Variations , 1996 .

[21]  N. Vilenkin.  Special Functions and the Theory of Group Representations , 1968 .

[22]  Shing-Tung Yau,et al.  Computing Conformal Structure of Surfaces , 2002, Commun. Inf. Syst..

[23]  Paul M. Thompson,et al.  Detecting Disease-Specific Patterns of Brain Structure Using Cortical Pattern Matching and a Population-Based Probabilistic Brain Atlas , 2001, IPMI.