Area-Preservation Mapping using Optimal Mass Transport

We present a novel area-preservation mapping/flattening method using the optimal mass transport technique, based on the Monge-Brenier theory. Our optimal transport map approach is rigorous and solid in theory, efficient and parallel in computation, yet general for various applications. By comparison with the conventional Monge-Kantorovich approach, our method reduces the number of variables from O(n2) to O(n), and converts the optimal mass transport problem to a convex optimization problem, which can now be efficiently carried out by Newton's method. Furthermore, our framework includes the area weighting strategy that enables users to completely control and adjust the size of areas everywhere in an accurate and quantitative way. Our method significantly reduces the complexity of the problem, and improves the efficiency, flexibility and scalability during visualization. Our framework, by combining conformal mapping and optimal mass transport mapping, serves as a powerful tool for a broad range of applications in visualization and graphics, especially for medical imaging. We provide a variety of experimental results to demonstrate the efficiency, robustness and efficacy of our novel framework.

[1]  Leonidas J. Guibas,et al.  The Earth Mover's Distance as a Metric for Image Retrieval , 2000, International Journal of Computer Vision.

[2]  Lei Zhu,et al.  Flattening maps for the visualization of multibranched vessels , 2005, IEEE Transactions on Medical Imaging.

[3]  S Ullman,et al.  Shifts in selective visual attention: towards the underlying neural circuitry. , 1985, Human neurobiology.

[4]  Bo Li,et al.  Retraction Note: Texture-guided volumetric deformation and visualization using 3D moving least squares , 2014, The Visual Computer.

[5]  Ron Kimmel,et al.  Texture Mapping Using Surface Flattening via Multidimensional Scaling , 2002, IEEE Trans. Vis. Comput. Graph..

[6]  M. V. D. Panne,et al.  Displacement Interpolation Using Lagrangian Mass Transport , 2011 .

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

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

[9]  Lei Zhu,et al.  An Image Morphing Technique Based on Optimal Mass Preserving Mapping , 2007, IEEE Transactions on Image Processing.

[10]  Hong Qin,et al.  Restricted Trivariate Polycube Splines for Volumetric Data Modeling , 2012, IEEE Transactions on Visualization and Computer Graphics.

[11]  Jing Hua,et al.  Area-Preserving Surface Flattening Using Lie Advection , 2011, MICCAI.

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

[13]  Ron Kikinis,et al.  Nondistorting flattening maps and the 3-D visualization of colon CT images , 2000, IEEE Transactions on Medical Imaging.

[14]  Yi Gao,et al.  Volumetric Mapping of Genus Zero Objects via Mass Preservation , 2012, ArXiv.

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

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

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

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

[19]  Hong Qin,et al.  Surface Mesh to Volumetric Spline Conversion with Generalized Polycubes , 2013, IEEE Transactions on Visualization and Computer Graphics.

[20]  Wei Zeng,et al.  Conformal Magnifier: A Focus+Context Technique with Local Shape Preservation , 2012, IEEE Transactions on Visualization and Computer Graphics.

[21]  William J. Schroeder,et al.  The visualization toolkit user's guide : updated for version 4.0 , 1998 .

[22]  Wenping Wang,et al.  GPU-Assisted Computation of Centroidal Voronoi Tessellation , 2011, IEEE Transactions on Visualization and Computer Graphics.

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

[24]  Jing Hua,et al.  Authalic Parameterization of General Surfaces Using Lie Advection , 2011, IEEE Transactions on Visualization and Computer Graphics.

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

[26]  Franz Aurenhammer,et al.  Power Diagrams: Properties, Algorithms and Applications , 1987, SIAM J. Comput..

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

[28]  Paul M. Thompson,et al.  Genus zero surface conformal mapping and its application to brain surface mapping , 2004, IEEE Transactions on Medical Imaging.

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

[30]  Wei Zeng,et al.  Volumetric colon wall unfolding using harmonic differentials , 2011, Comput. Graph..

[31]  Pierre Alliez,et al.  Computational geometry algorithms library , 2008, SIGGRAPH '08.

[32]  Hong Qin,et al.  Feature-Aware Reconstruction of Volume Data via Trivariate Splines , 2011, PG.