3D surface classification is a fundamental problem in computer vision and computational geometry. Surfaces can be classified by different transformation groups. Traditional classification methods mainly use topological transformation groups and Euclidean transformation groups. This paper introduces a novel method to classify surfaces by conformal transformation groups. Conformal equivalent class is refiner than topological equivalent class and coarser than isometric equivalent class, making it suitable for practical classification purposes. For general surfaces, the gradient fields of conformal maps form a vector space, which has a natural structure invariant under conformal transformations. We present an algorithm to compute this conformal structure, which can be represented as matrices, and use it to classify surfaces. The result is intrinsic to the geometry, invariant to triangulation and insensitive to resolution. To the best of our knowledge, this is the first paper to classify surfaces with arbitrary topologies by global conformal invariants. The method introduced here can also be used for surface matching problems.
In this work, we introduce two sets of algorithms inspired by the ideas from modern geometry. One is computational conformal geometry method, including harmonic maps, holomorphic 1-forms and Ricci flow. The other one is optimization method using affine normals.
This paper is motivated by the problem of subdividing a prismatic mesh to a tetrahedral mesh (without inserting Steiner points) so as to not only match arbitrarily prescribed boundary conditions but also allow arbitrary topologies in the base mesh. We explore all possible combinations of these two factors, and propose a complete solution to this 3D problem by converting it to an equivalent 2D graph problem, called cutting flow problem. For each case, we not only prove the sufficient and necessary condition for the existence of solutions, but also provide linear and provable algorithms to compute a solution whenever there is one.
Brain Cortical surface registration is required for inter-subject studies of functional and anatomical data. Harmonic mapping has been applied for brain mapping, due to its existence, uniqueness, regularity and numerical stability. In order to improve the registration accuracy, sculcal landmarks are usually used as constraints for brain registration. Unfortunately, constrained harmonic mappings may not be diffeomorphic and produces invalid registration. This work conquer this problem by changing the Riemannian metric on the target cortical surface to a hyperbolic metric, so that the harmonic mapping is guaranteed to be a diffeomorphism while the landmark constraints are enforced as boundary matching condition. The computational algorithms are based on the Ricci flow method and hyperbolic heat diffusion. Experimental results demonstrate that, by changing the Riemannian metric, the registrations are
Structured light system using a digital video projector is increasingly used for a 3-D shape measurement because of its digital nature. However, the nonlinear gamma of the projector causes the projected fringe patterns to be non-sinusoidal, which results in phase error therefore shape measurement error. Previous work showed that, by using a small look-up-table (LUT), this type of phase error can be reduced significantly for a three-step phase-shifting algorithm. In this research, we prove that this type of phase error compensation method is not limited to a three-step phase-shifting algorithm. It is generic for any phase-shifting algorithm. The phase error compensation algorithm is able to theoretically eliminate the phase error caused by the gamma of the projector completely. It is based on our finding that in phase domain, the phase error due to the projector's gamma is preserved for arbitrary object's surface