Numerical recipes in C (2nd ed.): the art of scientific computing
Numerical recipes in C (2nd ed.): the art of scientific computing
Shape recognition using metrics on the space of shapes
Shape recognition using metrics on the space of shapes
Computing the Kantorovich Distance for Images
Journal of Mathematical Imaging and Vision
Computer and Robot Vision
Fast Fluid Registration of Medical Images
VBC '96 Proceedings of the 4th International Conference on Visualization in Biomedical Computing
The Earth Mover''s Distance as a Metric for Image Retrieval
The Earth Mover''s Distance as a Metric for Image Retrieval
Perceptual metrics for image database navigation
Perceptual metrics for image database navigation
Deformable templates using large deformation kinematics
IEEE Transactions on Image Processing
Local Histogram Based Segmentation Using the Wasserstein Distance
International Journal of Computer Vision
Histogram based segmentation using Wasserstein distances
SSVM'07 Proceedings of the 1st international conference on Scale space and variational methods in computer vision
The Monge-Ampère equation: Various forms and numerical solution
Journal of Computational Physics
Computing Color Transforms with Applications to Image Editing
Journal of Mathematical Imaging and Vision
A graph-based method for detecting characteristic phenotypes from biomedical images
ISBI'10 Proceedings of the 2010 IEEE international conference on Biomedical imaging: from nano to Macro
Fast finite difference solvers for singular solutions of the elliptic Monge-Ampère equation
Journal of Computational Physics
Multilevel Algorithms for Large-Scale Interior Point Methods
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
An Efficient Numerical Method for the Solution of the $L_2$ Optimal Mass Transfer Problem
SIAM Journal on Scientific Computing
Mass preserving registration for heart MR images
MICCAI'05 Proceedings of the 8th international conference on Medical image computing and computer-assisted intervention - Volume Part II
Displacement interpolation using Lagrangian mass transport
Proceedings of the 2011 SIGGRAPH Asia Conference
Texture segmentation via non-local non-parametric active contours
EMMCVPR'11 Proceedings of the 8th international conference on Energy minimization methods in computer vision and pattern recognition
Transportation Distances on the Circle
Journal of Mathematical Imaging and Vision
Generalized Monge-Kantorovich Optimization for Grid Generation and Adaptation in $L_{p}$
SIAM Journal on Scientific Computing
SIAM Journal on Numerical Analysis
IWDM'06 Proceedings of the 8th international conference on Digital Mammography
Learning the dynamics of objects by optimal functional interpolation
Neural Computation
Computer Graphics Forum
International Journal of Computer Vision
Modelling Convex Shape Priors and Matching Based on the Gromov-Wasserstein Distance
Journal of Mathematical Imaging and Vision
Tensor-SIFT Based Earth Mover's Distance for Contour Tracking
Journal of Mathematical Imaging and Vision
Numerical solution of the Optimal Transportation problem using the Monge-Ampère equation
Journal of Computational Physics
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Image registration is the process of establishing a common geometric reference frame between two or more image data sets possibly taken at different times. In this paper we present a method for computing elastic registration and warping maps based on the Monge–Kantorovich theory of optimal mass transport. This mass transport method has a number of important characteristics. First, it is parameter free. Moreover, it utilizes all of the grayscale data in both images, places the two images on equal footing and is symmetrical: the optimal mapping from image A to image B being the inverse of the optimal mapping from B to A. The method does not require that landmarks be specified, and the minimizer of the distance functional involved is unique; there are no other local minimizers. Finally, optimal transport naturally takes into account changes in density that result from changes in area or volume. Although the optimal transport method is certainly not appropriate for all registration and warping problems, this mass preservation property makes the Monge–Kantorovich approach quite useful for an interesting class of warping problems, as we show in this paper. Our method for finding the registration mapping is based on a partial differential equation approach to the minimization of the L2 Kantorovich–Wasserstein or “Earth Mover's Distance” under a mass preservation constraint. We show how this approach leads to practical algorithms, and demonstrate our method with a number of examples, including those from the medical field. We also extend this method to take into account changes in intensity, and show that it is well suited for applications such as image morphing.