A Reparameterisation Based Approach to Geodesic Constrained Solvers for Curve Matching

• Published in: International journal of computer vision Vol. 99; no. 1; pp. 103 - 121
• Main Authors: Cotter, Colin J., Clark, Allan, Peiró, Joaquim
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.08.2012; Springer; Springer Nature B.V
• Subjects: Algorithms; Analysis; Applied sciences; Artificial Intelligence; Computer Imaging; Computer Science; Computer science; control theory; systems; Constraints; Correspondence; Deformation; Exact sciences and technology; Image Processing and Computer Vision; Matching; Mathematical analysis; Mathematical models; Mathematical optimization; Nonlinearity; Numerical analysis; Pattern Recognition; Pattern Recognition and Graphics; Pattern recognition. Digital image processing. Computational geometry; Principal components analysis; Registration; Studies; Topological manifolds; Velocity; Vision; Vision systems; Computational anatomy; Geodesic shooting; Curve registration; Diffeomorphism; Image registration; Equation of motion; Anatomy; Modeling; Particle method; Point estimation; Geodesic; Variational calculus; Hamiltonian mechanics; Mesh method; Hamiltonian; Numerical convergence; Numerical algorithm
• ISSN: 0920-5691; 1573-1405
• DOI: 10.1007/s11263-012-0520-0

We present a numerical algorithm for a new matching approach for parameterisation independent diffeomorphic registration of curves in the plane, targeted at robust registration between curves that require large deformations. This condition is particularly useful for the geodesic constrained approach in which the matching functional is minimised subject to the constraint that the evolving diffeomorphism satisfies the Hamiltonian equations of motion; this means that each iteration of the nonlinear optimisation algorithm produces a geodesic (up to numerical discretisation). We ensure that the computed solutions correspond to geodesics in the shape space by enforcing the horizontality condition (conjugate momentum is normal to the curve). Explicitly introducing and solving for a reparameterisation variable allows the use of a point-to-point matching condition. The equations are discretised using the variational particle-mesh method. We provide comprehensive numerical convergence tests and benchmark the algorithm in the context of large deformations, to show that it is a viable, efficient and accurate method for obtaining geodesics between curves.