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

TL;DR

The paper tackles the challenge of computing bijective density-equalizing maps for multiply-connected open surfaces, where previous methods struggled with bijectivity and local geometric distortion. It introduces a novel DEQ framework that recasts density diffusion as a quasiconformal flow controlled by the Beltrami coefficient $\mu$, and decomposes the map as $f = \tilde f \circ g_0$, enabling joint optimization of the target planar circular domain and the mapping. The core contributions are the DEQ and LDEQ variational models, the geometry-modification and Beltrami-density-equalizing descent (BDED) algorithm, and the iterative scheme that guarantees bijectivity while achieving density equalization; landmark constraints are integrated via a penalty-splitting approach. The method, validated on synthetic and real data, supports applications in surface remeshing, texture mapping, and medical visualization, offering a flexible tool for controlled flattening of complex surfaces with practical impact across graphics and imaging.

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.

Figures (17)

• Figure 1: Bijective density-equalizing quasiconformal (DEQ) maps obtained by our proposed algorithm. Given a multiply-connected open surface and a prescribed density distribution, our method optimizes both the shape of the target 2D circular domain and the flattening map of the surface onto the domain to achieve density equalization, with the bijectivity of the mapping guaranteed by quasiconformal theory. By changing the prescribed density distribution, different effects can be achieved. For instance, we can parameterize a human face (left) onto the plane with the nose shrunk (middle) or enlarged (right) by changing the prescribed density distribution at the nose. Note that the size and position of the inner circular holes are also optimized and hence different in the two parameterization results.
• Figure 2: An illustration of how the Beltrami coefficient $\mu$ determines the quasiconformal distortion of a map $f$. Specifically, the maximal magnification factor is given by $|f_z|(1+|\mu|)$, the maximal shrinkage factor is given by $|f_z|(1-|\mu|)$, and the orientation change is given by $\text{arg}(\mu)/2$.
• Figure 3: An illustration of the density-equalizing maps. The density diffusion creates a velocity field that enlarges regions with higher density and shrinks regions with lower density until the density is equalized.
• Figure 4: An illustration of the geometry modification (GM) algorithm. Given a planar circular domain, we first change the size and position of the inner circular holes based on the density gradient. Then, we reconstruct a smooth quasiconformal map based on the updated boundaries. Here, the dashed curve represents the previous inner boundary, and the solid curve represents its updated position.
• Figure 5: The proposed iterative scheme for computing density-equalizing quasiconformal (DEQ) maps. The geometry modification (GM) method and the Beltrami density-equalizing descent (BDED) method are applied iteratively to improve both the geometry of the target circular domain and the mapping result. Here, each triangle element is color-coded with the input population. The dashed curve in each GM step represents the previous inner boundary, and the solid curve represents its updated position.

Theorems & Definitions (2)

• proof
• proof