Bijective Parameterization with Free Boundaries

When displaying 3D surfaces onto computer screens, additional information is often mapped onto the surface to enhance the quality of the rendering. Surface parameterization generates a correspondence, or mapping, between the 3D surface and 2D parameterization space. This mapping has many applications in computer graphics, but in most cases cannot be performed without introducing large distortions in the 2D parameterization. Along with problems of distortion, the mapping of the 2D space to 3D for many applications can be invalidated if the property of bijectivity is violated. While there is previous research guaranteeing bijectivity, these methods must constrain or modify the boundary of the 2D parameterization. This dissertation, describes a fully automatic method for generating guaranteed bijective surface parameterizations from triangulated 3D surfaces. In particular, a new isometric distortion energy metric is introduced preventing local folds of triangles in the parameterization as well as a barrier function that prevents intersection of the 2D boundaries. By using a computationally efficient isometric metric energy, the dissertation achieves fast and comparable optimization times to previous methods. The boundary of the parameterization is free to change shape during the optimization to minimize distortion. A new optimization approach is introduced called singularity aware optimization and in conjunction with an interior point approach and barrier energy functions guarantee bijectivity. This optimization framework is then modified to allow for an importance weighting allowing for customizable and more efficient texel usage

Bijective Density-Equalizing Quasiconformal Map for Multiply-Connected Open Surfaces

This paper proposes a novel method for computing bijective density-equalizing quasiconformal (DEQ) flattening maps for multiply-connected open surfaces. In conventional density-equalizing maps, shape deformations are solely driven by prescribed constraints on the density distribution, defined as the population per unit area, while the bijectivity and local geometric distortions of the mappings are uncontrolled. Also, prior methods have primarily focused on simply-connected open surfaces but not surfaces with more complicated topologies. Our proposed method overcomes these issues by formulating the density diffusion process as a quasiconformal flow, which allows us to effectively control the local geometric distortion and guarantee the bijectivity of the mapping by solving an energy minimization problem involving the Beltrami coefficient of the mapping. To achieve an optimal parameterization of multiply-connected surfaces, we develop an iterative scheme that optimizes both the shape of the target planar circular domain and the density-equalizing quasiconformal map onto it. In addition, landmark constraints can be incorporated into our proposed method for consistent feature alignment. The method can also be naturally applied to simply-connected open surfaces. By changing the prescribed population, a large variety of surface flattening maps with different desired properties can be achieved. The method is tested on both synthetic and real examples, demonstrating its efficacy in various applications in computer graphics and medical imaging

Variance-Minimizing Transport Plans for Inter-surface Mapping

International audienceWe introduce an effcient computational method for generating dense and low distortion maps between two arbitrary surfaces of same genus. Instead of relying on semantic correspondences or surface parameterization, we directly optimize a variance-minimizing transport plan between two input surfaces that defines an as-conformal-as-possible inter-surface map satisfying a user-prescribed bound on area distortion. The transport plan is computed via two alternating convex optimizations, and is shown to minimize a generalized Dirichlet energy of both the map and its inverse. Computational efficiency is achieved through a coarse-tone approach in diffusion geometry, with Sinkhorn iterations modified to enforce bounded area distortion. The resulting inter-surface mapping algorithm applies to arbitrary shapes robustly, with little to no user interaction

Variance-Minimizing Transport Plans for Inter-surface Mapping

We introduce an efficient computational method for generating dense and low distortion maps between two arbitrary surfaces of same genus. Instead of relying on semantic correspondences or surface parameterization, we directly optimize a variance-minimizing transport plan between two input surfaces that defines an as-conformal-as-possible inter-surface map satisfying a user-prescribed bound on area distortion. The transport plan is computed via two alternating convex optimizations, and is shown to minimize a generalized Dirichlet energy of both the map and its inverse. Computational efficiency is achieved through a coarse-to-fine approach in diffusion geometry, with Sinkhorn iterations modified to enforce bounded area distortion. The resulting inter-surface mapping algorithm applies to arbitrary shapes robustly, with little to no user interaction