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.

翻译：本文提出了一种计算多连通开曲面双射保密度拟共形（DEQ）展平映射的新方法。在传统保密度映射中，形变完全由预设的密度分布约束（定义为每单位面积的人口数）驱动，而映射的双射性与局部几何畸变不受控。此外，现有方法主要关注单连通开曲面，无法处理拓扑结构更复杂的曲面。本文方法通过将密度扩散过程形式化为拟共形流克服了这些问题，通过求解涉及映射 Beltrami 系数的能量最小化问题，能够有效控制局部几何畸变并保证映射的双射性。为实现多连通曲面的最优参数化，我们提出了一个迭代方案，同时优化目标平面圆形区域形状及其上的保密度拟共形映射。此外，本文方法可融入地标约束以保持特征对齐一致性，并能自然适用于单连通开曲面。通过改变预设人口分布，可获得具有不同期望特性的多种曲面展平映射。该方法在合成数据和真实数据上的实验验证了其在计算机图形学和医学成像等多类应用中的有效性。