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

TL;DR

This paper introduces a novel method for computing bijective density-equalizing quasiconformal maps for multiply-connected open surfaces, controlling local distortions and ensuring bijectivity, with applications demonstrated in graphics and medical imaging.

Contribution

It formulates density diffusion as a quasiconformal flow, enabling control over distortions and bijectivity, and develops an iterative scheme for optimal surface parameterization with landmark constraints.

Findings

Effective control of local geometric distortion.

Guarantees bijectivity of the mapping.

Applicable to both multiply-connected and simply-connected surfaces.

Abstract

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.