Multiscale Representation of Deformation via Beltrami Coefficients
Ka Chun Lam, Tsz Ching Ng, Lok Ming Lui · Multiscale Modeling and Simulation · 2017
Analyzing the deformation pattern of an object is crucial in various fields, such as in computer vision and medical imaging. A deformation can be considered as a combination of local and global deformations at different locations. To fully understand and analyze the deformation pattern, extracting deformation components of various scales and locations is necessary. We propose an algorithm for the multiscale decomposition of a bijective deformation using quasi-conformal theories. A deformation of an object can be described as a orientation-preserving homeomorphism of a two-dimensional domain. The mapping is then represented by its associated Beltrami coefficient (BC), which measures the local geometric (conformality) distortion of the deformation. The BC is a complex-valued function defined on the source domain. By applying the wavelet transform on the BC, the BC can be decomposed into different components of different frequencies compactly supported in different subdomains. Quasi-conformal mappings associated to different components of the BC can be reconstructed by solving Beltrami's equation. A multiscale decomposition of the deformation can then be constructed. To validate our proposed algorithm, we test it on synthetic examples as well as real medical data. Experimental results show the efficacy of our proposed model to decompose deformations at multiple scales and locations.