Optimization of Surface Registrations Using Beltrami Holomorphic Flow

Abstract

In shape analysis, finding an optimal 1-1 correspondence between surfaces within a large class of admissible bijective mappings is of great importance. Such process is called surface registration. The difficulty lies in the fact that the space of all surface diffeomorphisms is a complicated functional space, making exhaustive search for the best mapping challenging. To tackle this problem, we propose a simple representation of bijective surface maps using Beltrami coefficients (BCs), which are complex-valued functions defined on surfaces with supreme norm less than 1. Fixing any 3 points on a pair of surfaces, there is a 1-1 correspondence between the set of surface diffeomorphisms between them and the set of BCs. Hence, every bijective surface map can be represented by a unique BC. Conversely, given a BC, we can reconstruct the unique surface map associated to it using the Beltrami Holomorphic flow (BHF) method. Using BCs to represent surface maps is advantageous because it is a much simpler functional space, which captures many essential features of a surface map. By adjusting BCs, we equivalently adjust surface diffeomorphisms to obtain the optimal map with desired properties. More specifically, BHF gives us the variation of the associated map under the variation of BC. Using this, a variational problem over the space of surface diffeomorphisms can be easily reformulated into a variational problem over the space of BCs. This makes the minimization procedure much easier. More importantly, the diffeomorphic property is always preserved. We test our method on synthetic examples and real medical applications. Experimental results demonstrate the effectiveness of our proposed algorithm for surface registration.

A Splitting Method for Diffeomorphism Optimization Problem Using Beltrami Coefficients

“…Alternating minimization scheme can then be applied to minimize the the energy functional. The proposed method significantly speed up the previous BHF approach in [1]. It also extends the previous BHF algorithm to Riemann surfaces of arbitrary topologies.…”

“…To tackle with this problem, finding a suitable representation for Diff that facilitates the optimization process is necessary. In [1], the Beltrami coefficient(BC) was proposed to represent an orientation-preserving diffeomorphism. A BC is a complex-valued function defined on S 1 with supreme norm strictly less than 1.…”

“…We usually denote the set of all BCs by B = {μ : S 1 → C : ||μ|| ∞ < 1}. It can be shown that there is an injection : Diff → B from Diff to B under suitable boundary or landmark constraints [1]. In other words, every orientation-preserving diffeomorphism g is associated with a unique BC μ g .…”

“…To solve the DOPs (2) or (3), a gradient descent based method was proposed in [1]. The quasi-conformal map associated to a given perturbed BC, which is needed in deriving the descent direction [such as the second term in (3)], was computed using an integral formula.…”

Computing quasiconformal maps using an auxiliary metric and discrete curvature flow

“…Applications can be found in different areas such as differential equations, topology, Riemann mappings, complex dynamics as well as applied mathematics [4][5][6][7]12,13,17,[42][43][44][45]47]. Despite the rapid development in the theory of QC mapping, the progress on computing QC mappings numerically has been very slow.…”