Optimal mass transport based brain morphometry for patients with congenital hand deformities

Congenital hand deformities (CHD) have attracted increasing research attention in the past few decades. The impacts of CHD on the brain structure, however, are not fully studied to date. In this work, we propose a novel framework to study brain morphometry in CHD patients using Wasserstein distance based on optimal mass transport (OMT) theory. We first employ conformal mapping to map the left and right surface-based functional brain areas to planar rectangles, which pushes the area element on the brain surface to the planar rectangle and incurs the area distortion. A measure is then determined by this area distortion. We further propose a new rectangle domain-based OMT map algorithm. Given two measures on two surfaces, we employ the proposed algorithm to compute a unique OMT map between the two measures encoding the geometric information of left and right surface-based functional brain areas. The transportation cost of this OMT map gives the Wasserstein distance between two surfaces, which intrinsically measures the dissimilarities between two surface-based shapes. Our method is theoretically rigorous and computationally efficient and stable. We finally evaluate the proposed Wasserstein distance-based method on the left and right post-central gyri from the CHD patients and healthy control subjects for analyzing brain cortical morphometry. Experimental results demonstrate the efficiency and efficacy of our method, and shed insightful lights on the study of the brain morphometry for those subjects with CHD.

[1]  Paul A. Yushkevich,et al.  Segmentation, registration, and measurement of shape variation via image object shape , 1999, IEEE Transactions on Medical Imaging.

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

[3]  Qing He,et al.  Semi-automatic 3D segmentation of brain structures from MRI , 2011, Int. J. Data Min. Bioinform..

[4]  Jing Hua,et al.  Deformation similarity measurement in quasi-conformal shape space , 2014, Graph. Model..

[5]  Paul M. Thompson,et al.  MAPPING ABNORMAL SUBCORTICAL BRAIN MORPHOMETRY IN AN ELDERLY HIV+ COHORT. , 2015, Proceedings. IEEE International Symposium on Biomedical Imaging.

[6]  Pierre Alliez,et al.  An Optimal Transport Approach to Robust Reconstruction and Simplification of 2d Shapes , 2022 .

[7]  T. Chan,et al.  Shape analysis with conformal invariants for multiply connected domains and its application to analyzing brain morphology , 2009, CVPR 2009.

[8]  Isaac Harvey,et al.  Classification of congenital anomalies of the hand and upper limb: development and assessment of a new system. , 2013, The Journal of hand surgery.

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

[10]  Mathieu Desbrun,et al.  Blue noise through optimal transport , 2012, ACM Trans. Graph..

[11]  Rastko R. Selmic,et al.  Measuring Distance between Unordered Sets of Different Sizes , 2014, 2014 IEEE Conference on Computer Vision and Pattern Recognition.

[12]  Wei Zeng,et al.  Teichmüller Shape Descriptor and Its Application to Alzheimer’s Disease Study , 2013, International Journal of Computer Vision.

[13]  Allen R. Tannenbaum,et al.  Texture Mapping via Optimal Mass Transport , 2010, IEEE Transactions on Visualization and Computer Graphics.

[14]  Renaud Lebrun,et al.  Visualizing shape transformation between chimpanzee and human braincases , 2007, The Visual Computer.

[15]  L. Holmes,et al.  Anatomic and etiological classification of congenital limb deficiencies , 2011, American journal of medical genetics. Part A.

[16]  Ivo D. Dinov,et al.  A Computational Model of Multidimensional Shape , 2010, International Journal of Computer Vision.

[17]  Shing-Tung Yau,et al.  Computational Conformal Geometry , 2016 .

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

[19]  Charles A Goldfarb,et al.  Congenital hand differences. , 2009, The Journal of hand surgery.

[20]  Timothy F. Cootes,et al.  Shape Discrimination in the Hippocampus Using an MDL Model , 2003, IPMI.

[21]  C. Gillberg,et al.  Children with surgically corrected hand deformities and upper limb deficiencies: self-concept and psychological well-being , 2011, The Journal of hand surgery, European volume.

[22]  Pascale Tremblay,et al.  Structural brain aging and speech production: a surface-based brain morphometry study , 2015, Brain Structure and Function.

[23]  Moo K. Chung,et al.  Tensor-Based Cortical Surface Morphometry via Weighted Spherical Harmonic Representation , 2008, IEEE Transactions on Medical Imaging.

[24]  Nicolas Bonnotte,et al.  From Knothe's Rearrangement to Brenier's Optimal Transport Map , 2012, SIAM J. Math. Anal..

[25]  Mario Giuffrè,et al.  Congenital malformations , 2012, The journal of maternal-fetal & neonatal medicine : the official journal of the European Association of Perinatal Medicine, the Federation of Asia and Oceania Perinatal Societies, the International Society of Perinatal Obstetricians.

[26]  Gallagher Pryor,et al.  3D nonrigid registration via optimal mass transport on the GPU , 2009, Medical Image Anal..

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

[28]  Quentin Mérigot,et al.  A Multiscale Approach to Optimal Transport , 2011, Comput. Graph. Forum.

[29]  Craig Gotsman,et al.  Conformal Flattening by Curvature Prescription and Metric Scaling , 2008, Comput. Graph. Forum.

[30]  Wei Zeng,et al.  Area Preserving Brain Mapping , 2013, 2013 IEEE Conference on Computer Vision and Pattern Recognition.

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

[32]  Antonio Costanzo,et al.  Regulation of Dlx5 and Dlx6 gene expression by p63 is involved in EEC and SHFM congenital limb defects , 2008, Development.

[33]  Alan C. Evans,et al.  Brain size and cortical structure in the adult human brain. , 2008, Cerebral cortex.

[34]  Charles A Goldfarb,et al.  Congenital failure of formation of the upper limb. , 2009, Hand clinics.

[35]  Paul J. Besl,et al.  Method for registration of 3-D shapes , 1992, Other Conferences.

[36]  Richard M. Heiberger,et al.  Data and Statistics , 2015 .

[37]  Xianfeng Gu,et al.  Discrete Surface Ricci Flow , 2008, IEEE Transactions on Visualization and Computer Graphics.

[38]  P. Manske,et al.  Developmental biology and classification of congenital anomalies of the hand and upper extremity. , 2010, The Journal of hand surgery.

[39]  Xin Zhao,et al.  Area-Preservation Mapping using Optimal Mass Transport , 2013, IEEE Transactions on Visualization and Computer Graphics.

[40]  Mark Meyer,et al.  Discrete Differential-Geometry Operators for Triangulated 2-Manifolds , 2002, VisMath.

[41]  David Cohen-Steiner,et al.  Restricted delaunay triangulations and normal cycle , 2003, SCG '03.

[42]  S. Yau,et al.  Variational Principles for Minkowski Type Problems, Discrete Optimal Transport, and Discrete Monge-Ampere Equations , 2013, 1302.5472.

[43]  Erick Jorge Canales-Rodríguez,et al.  Surface-based brain morphometry and diffusion tensor imaging in schizoaffective disorder , 2017, The Australian and New Zealand journal of psychiatry.

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

[45]  E LorensenWilliam,et al.  Marching cubes: A high resolution 3D surface construction algorithm , 1987 .