Abstract
Conventional splines offer powerful means for modeling surfaces and volumes in three-dimensional Euclidean space. A one-dimensional quaternion spline has been applied for animation purpose, where the splines are defined to model a one-dimensional submanifold in the three-dimensional Lie group. Given two surfaces, all of the diffeomorphisms between them form an infinite dimensional manifold, the so-called diffeomorphism space. In this work, we propose a novel scheme to model finite dimensional submanifolds in the diffeomorphism space by generalizing conventional splines. According to quasiconformal geometry theorem, each diffeomorphism determines a Beltrami differential on the source surface. Inversely, the diffeomorphism is determined by its Beltrami differential with normalization conditions. Therefore, the diffeomorphism space has one-to-one correspondence to the space of a special differential form. The convex combination of Beltrami differentials is still a Beltrami differential. Therefore, the conventional spline scheme can be generalized to the Beltrami differential space and, consequently, to the diffeomorphism space. Our experiments demonstrate the efficiency and efficacy of diffeomorphism splines. The diffeomorphism spline has many potential applications, such as surface registration, tracking and animation.
Highlights
Conventional splines are applied for modeling shapes [1,2,3,4,5,6,7]
The conventional spline scheme can be generalized to the Beltrami differential space, and to the diffeomorphism space
We propose a novel spline scheme of modeling finite dimensional submanifolds in diffeomorphism space by generalizing conventional splines in three dimensional Euclidean space
Summary
Conventional splines are applied for modeling shapes [1,2,3,4,5,6,7]. For example, spline curves are used to model planar curves; bi-variant splines are applied for modeling geometric surfaces embedded in R3 ; and tri-variant splines are used for volumes. The target space Rn can be replaced by an abstract manifold M n , as long as the convex linear combination of two points is well defined in M n. Based on the linear interpolation of quaternions, the quaternion spline can be defined using de Casteljau’s evaluation algorithm [3]. The quaternion spline is a commonly-used spline, which is defined in a finite dimensional Lie group. S2 , the mapping λ f + (1 − λ )g may not be a diffeomorphism This means that direct linear interpolation of diffeomorphisms cannot guarantee a diffeomorphism. This causes the intrinsic difficulty to define splines in diffeomorphism space
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