Area Preserving Brain Mapping

Brain mapping transforms the brain cortical surface to canonical planar domains, which plays a fundamental role in morphological study. Most existing brain mapping methods are based on angle preserving maps, which may introduce large area distortions. This work proposes an area preserving brain mapping method based on Monge-Brenier theory. The brain mapping is intrinsic to the Riemannian metric, unique, and diffeomorphic. The computation is equivalent to convex energy minimization and power Voronoi diagram construction. Comparing to the existing approaches based on Monge-Kantorovich theory, the proposed one greatly reduces the complexity (from n2 unknowns to n ), and improves the simplicity and efficiency. Experimental results on caudate nucleus surface mapping and cortical surface mapping demonstrate the efficacy and efficiency of the proposed method. Conventional methods for caudate nucleus surface mapping may suffer from numerical instability, in contrast, current method produces diffeomorpic mappings stably. In the study of cortical surface classification for recognition of Alzheimer's Disease, the proposed method outperforms some other morphometry features.

[1]  Richard S. Hamilton,et al.  The Ricci flow on surfaces , 1986 .

[2]  Arthur W. Toga,et al.  CoRPORATE: Cortical Reconstruction by Pruning Outliers with Reeb Analysis and Topology-Preserving Evolution , 2011, IPMI.

[3]  Anders M. Dale,et al.  An automated labeling system for subdividing the human cerebral cortex on MRI scans into gyral based regions of interest , 2006, NeuroImage.

[4]  Thomas A. Funkhouser,et al.  Algorithms to automatically quantify the geometric similarity of anatomical surfaces , 2011, Proceedings of the National Academy of Sciences.

[5]  M N Rossor,et al.  Correlation between rates of brain atrophy and cognitive decline in AD , 1999, Neurology.

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

[7]  Karl J. Friston,et al.  Identifying global anatomical differences: Deformation‐based morphometry , 1998 .

[8]  Gallagher Pryor,et al.  Fast Multigrid Optimal Mass Transport for Image Registration and Morphing , 2007, BMVC.

[9]  David A. Rottenberg,et al.  Quantitative evaluation of three cortical surface flattening methods , 2005, NeuroImage.

[10]  B. Chow,et al.  The Ricci flow on surfaces , 2004 .

[11]  Ron Kimmel,et al.  Generalized multidimensional scaling: A framework for isometry-invariant partial surface matching , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[12]  Mark W. Woolrich,et al.  Advances in functional and structural MR image analysis and implementation as FSL , 2004, NeuroImage.

[13]  T. Chan,et al.  Genus zero surface conformal mapping and its application to brain surface mapping. , 2004, IEEE transactions on medical imaging.

[14]  Nick C Fox,et al.  The Alzheimer's disease neuroimaging initiative (ADNI): MRI methods , 2008, Journal of magnetic resonance imaging : JMRI.

[15]  Marie Chupin,et al.  Automatic classi fi cation of patients with Alzheimer ' s disease from structural MRI : A comparison of ten methods using the ADNI database , 2010 .

[16]  F. P. Gardiner,et al.  Quasiconformal Teichmuller Theory , 1999 .

[17]  Paul M. Thompson,et al.  Conformal Slit Mapping and Its Applications to Brain Surface Parameterization , 2008, MICCAI.

[18]  Richard M. Leahy,et al.  Optimization method for creating semi-isometric flat maps of the cerebral cortex , 2000, Medical Imaging: Image Processing.

[19]  L. Kantorovich.  On a Problem of Monge , 2006 .

[20]  A. Dale,et al.  Cortical Surface-Based Analysis II: Inflation, Flattening, and a Surface-Based Coordinate System , 1999, NeuroImage.

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

[22]  Kenneth Stephenson,et al.  Cortical cartography using the discrete conformal approach of circle packings , 2004, NeuroImage.

[23]  Wei Zeng,et al.  3D face matching and registration based on hyperbolic Ricci flow , 2008, 2008 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops.

[24]  D. Chand,et al.  On Convex Polyhedra , 1970 .

[25]  Lei Zhu,et al.  Area-Preserving Mappings for the Visualization of Medical Structures , 2003, MICCAI.

[26]  David Mumford,et al.  2D-Shape Analysis Using Conformal Mapping , 2004, CVPR.

[27]  Y. Brenier.  Polar Factorization and Monotone Rearrangement of Vector-Valued Functions , 1991 .

[28]  Amity E. Green,et al.  Hippocampal, caudate, and ventricular changes in Parkinson's disease with and without dementia , 2010, Movement disorders : official journal of the Movement Disorder Society.

[29]  Jerry L. Prince,et al.  A Geometry-Driven Optical Flow Warping for Spatial Normalization of Cortical Surfaces , 2008, IEEE Transactions on Medical Imaging.

[30]  Lei Zhu,et al.  Optimal Mass Transport for Registration and Warping , 2004, International Journal of Computer Vision.

[31]  Nick C Fox,et al.  The clinical use of structural MRI in Alzheimer disease , 2010, Nature Reviews Neurology.

[32]  D. V. van Essen,et al.  Computerized Mappings of the Cerebral Cortex: A Multiresolution Flattening Method and a Surface-Based Coordinate System , 1996, Journal of Cognitive Neuroscience.

[33]  Jonathan R. Polimeni,et al.  Exact Geodesics and Shortest Paths on Polyhedral Surfaces , 2009, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[34]  A. W. Toga,et al.  3D maps localize caudate nucleus atrophy in 400 Alzheimer’s disease, mild cognitive impairment, and healthy elderly subjects , 2010, Neurobiology of Aging.

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