Heat Kernel Smoothing via Laplace-Beltrami Eigenfunctions and Its Application to Subcortical Structure Modeling

We present a new subcortical structure shape modeling framework using heat kernel smoothing constructed with the Laplace-Beltrami eigenfunctions. The cotan discretization is used to numerically obtain the eigenfunctions of the Laplace-Beltrami operator along the surface of subcortical structures of the brain. The eigenfunctions are then used to construct the heat kernel and used in smoothing out measurements noise along the surface. The proposed framework is applied in investigating the influence of age (38-79 years) and gender on amygdala and hippocampus shape. We detected a significant age effect on hippocampus in accordance with the previous studies. In addition, we also detected a significant gender effect on amygdala. Since we did not find any such differences in the traditional volumetric methods, our results demonstrate the benefit of the current framework over traditional volumetric methods.

[1]  K. Amunts,et al.  Cytoarchitectonic mapping of the human amygdala, hippocampal region and entorhinal cortex: intersubject variability and probability maps , 2005, Anatomy and Embryology.

[2]  Bruno Lévy,et al.  Laplace-Beltrami Eigenfunctions Towards an Algorithm That "Understands" Geometry , 2006, IEEE International Conference on Shape Modeling and Applications 2006 (SMI'06).

[3]  Daniel H. Mathalon,et al.  Age-related decline in MRI volumes of temporal lobe gray matter but not hippocampus , 1995, Neurobiology of Aging.

[4]  Ravi Malladi,et al.  Fast Difference Schemes for Edge Enhancing Beltrami Flow and Subjective Surfaces , 2002, VisMath.

[5]  Paul M. Thompson,et al.  A Parameterization-Based Numerical Method for Isotropic and Anisotropic Diffusion Smoothing on Non-Flat Surfaces , 2009, IEEE Transactions on Image Processing.

[6]  Stephen M Smith,et al.  Fast robust automated brain extraction , 2002, Human brain mapping.

[7]  Ravi Malladi,et al.  Fast Difference Schemes for Edge Enhancing Beltrami Flow , 2002, ECCV.

[8]  J A Corsellis,et al.  VARIATION WITH AGE IN THE VOLUMES OF GREY AND WHITE MATTER IN THE CEREBRAL HEMISPHERES OF MAN: MEASUREMENTS WITH AN IMAGE ANALYSER , 1980, Neuropathology and applied neurobiology.

[9]  Edith V. Sullivan,et al.  Preservation of hippocampal volume throughout adulthood in healthy men and women , 2005, Neurobiology of Aging.

[10]  Faith M. Gunning-Dixon,et al.  Sex differences in temporo-limbic and frontal brain volumes of healthy adults. , 2002, Cerebral cortex.

[11]  Karl J. Friston,et al.  Voxel based morphometry of 465 normal adult human brains , 2000, NeuroImage.

[12]  Gabriel Taubin,et al.  Geometric Signal Processing on Polygonal Meshes , 2000, Eurographics.

[13]  E. Bigler,et al.  Hippocampal volume in normal aging and traumatic brain injury. , 1997, AJNR. American journal of neuroradiology.

[14]  Michael Davis,et al.  The amygdala , 2000, Current Biology.

[15]  Karl J. Friston,et al.  A Voxel-Based Morphometric Study of Ageing in 465 Normal Adult Human Brains , 2001, NeuroImage.

[16]  Michael I. Miller,et al.  Multi-structure network shape analysis via normal surface momentum maps , 2008, NeuroImage.

[17]  Paul M. Thompson,et al.  Age effects on hippocampal structural changes in old men: The HAAS , 2008, NeuroImage.

[18]  R. Adler.  On excursion sets, tube formulas and maxima of random fields , 2000 .

[19]  Ron Kimmel,et al.  A general framework for low level vision , 1998, IEEE Trans. Image Process..

[20]  D Le Bihan,et al.  Detection of fMRI activation using Cortical Surface Mapping , 2001, Human brain mapping.

[21]  Moo K. Chung,et al.  Diffusion smoothing on brain surface via finite element method , 2004, 2004 2nd IEEE International Symposium on Biomedical Imaging: Nano to Macro (IEEE Cat No. 04EX821).

[22]  Michael I. Miller,et al.  Smooth functional and structural maps on the neocortex via orthonormal bases of the Laplace-Beltrami operator , 2006, IEEE Transactions on Medical Imaging.

[23]  Moo K. Chung,et al.  Cortical thickness analysis in autism with heat kernel smoothing , 2005, NeuroImage.

[24]  Anders M. Dale,et al.  Consistent neuroanatomical age-related volume differences across multiple samples , 2011, Neurobiology of Aging.

[25]  Karl J. Friston,et al.  Cerebral Asymmetry and the Effects of Sex and Handedness on Brain Structure: A Voxel-Based Morphometric Analysis of 465 Normal Adult Human Brains , 2001, NeuroImage.

[26]  N. Schuff,et al.  Age effects on atrophy rates of entorhinal cortex and hippocampus , 2006, Neurobiology of Aging.

[27]  Brian B. Avants,et al.  Symmetric diffeomorphic image registration with cross-correlation: Evaluating automated labeling of elderly and neurodegenerative brain , 2008, Medical Image Anal..

[28]  Karl J. Friston,et al.  A unified statistical approach for determining significant signals in images of cerebral activation , 1996, Human brain mapping.

[29]  William E. Lorensen,et al.  Marching cubes: A high resolution 3D surface construction algorithm , 1987, SIGGRAPH.

[30]  Ross T. Whitaker,et al.  Geometric surface smoothing via anisotropic diffusion of normals , 2002, IEEE Visualization, 2002. VIS 2002..

[31]  Jitendra Malik,et al.  Scale-Space and Edge Detection Using Anisotropic Diffusion , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[32]  Moo K. Chung,et al.  Deformation-based surface morphometry applied to gray matter deformation , 2003, NeuroImage.

[33]  Alan C. Evans,et al.  A Three-Dimensional Statistical Analysis for CBF Activation Studies in Human Brain , 1992, Journal of cerebral blood flow and metabolism : official journal of the International Society of Cerebral Blood Flow and Metabolism.

[34]  Mads Nielsen,et al.  Computer Vision — ECCV 2002 , 2002, Lecture Notes in Computer Science.

[35]  Hao Zhang,et al.  Spectral Methods for Mesh Processing and Analysis , 2007, Eurographics.

[36]  A. Dale,et al.  Effects of age on volumes of cortex, white matter and subcortical structures , 2005, Neurobiology of Aging.

[37]  P Alvarez,et al.  Memory consolidation and the medial temporal lobe: a simple network model. , 1994, Proceedings of the National Academy of Sciences of the United States of America.

[38]  Guillermo Sapiro,et al.  Direction diffusion , 1999, Proceedings of the Seventh IEEE International Conference on Computer Vision.

[39]  Moo K. Chung,et al.  General multivariate linear modeling of surface shapes using SurfStat , 2010, NeuroImage.

[40]  Alan C. Evans,et al.  A Unified Statistical Approach to Deformation-Based Morphometry , 2001, NeuroImage.