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：摘要：本文提出了一种新颖方法，用于计算多连通开放曲面的双射密度均衡拟共形（DEQ）展平映射。在传统密度均衡映射中，形状变形完全由预设的密度分布约束（定义为单位面积人口数）驱动，而映射的双射性及局部几何畸变则不受控。此外，现有方法主要聚焦于单连通开放曲面，尚未涉及拓扑结构更复杂的曲面。本文方法通过将密度扩散过程构建为拟共形流克服上述问题，使得能够有效控制局部几何畸变，并通过求解涉及映射Beltrami系数的能量最小化问题保证映射的双射性。为实现多连通曲面的最优参数化，我们开发了迭代方案，同步优化目标平面圆形域的形状及其上的密度均衡拟共形映射。此外，本文方法可嵌入地标约束以实现特征一致性对齐，并能自然推广至单连通开放曲面。通过调整预设人口密度，可生成具有不同预期特性的多样化曲面展平映射。该方法已在合成与真实数据上完成测试，在计算机图形学与医学影像的多种应用中展现出有效性。