Maps are arguably one of the most fundamental concepts used to define and operate on manifold surfaces in differentiable geometry. Accordingly, in geometry processing, maps are ubiquitous and are used in many core applications, such as paramterization, shape analysis, remeshing, and deformation. Unfortunately, most computational representations of surface maps do not lend themselves to manipulation and optimization, usually entailing hard, discrete problems. While algorithms exist to solve these problems, they are problem-specific, and a general framework for surface maps is still in need. In this paper, we advocate considering neural networks as encoding surface maps. Since neural networks can be composed on one another and are differentiable, we show it is easy to use them to define surfaces via atlases, compose them for surface-to-surface mappings, and optimize differentiable objectives relating to them, such as any notion of distortion, in a trivial manner. In our experiments, we represent surfaces by generating a neural map that approximates a UV parameterization of a 3D model. Then, we compose this map with other neural maps which we optimize with respect to distortion measures. We show that our formulation enables trivial optimization of rather elusive mapping tasks, such as maps between a collection of surfaces.
翻译:因此,在几何学处理中,地图是无处不在的,并且用于许多核心应用,例如分层、形状分析、重新模版和变形。不幸的是,大多数地表地图的计算表达方式不适于操纵和优化,通常会产生硬的、分散的问题。虽然算法存在解决这些问题,但它们是有问题的,而且地表图的总体框架仍然需要。在本文中,我们主张将神经网络视为编码地表图。由于神经网络可以相互组成,并且可以不同地用于许多核心应用,例如分层图、形状分析、重新模版和变形等。不幸的是,大多数地表图的计算表达方式并不适于进行操纵和优化,通常会产生硬的、分解的问题。虽然算法存在解决这些问题的方法,但它们是针对具体的问题,而且仍然需要一个一般的地表图框架。在本文中,我们主张将神经网络视为可编码的地表图。由于神经网络可相互组成,因此我们很容易使用这些神经网络来通过图来界定地表表面图,我们通过地表图进行最精确的绘制,以最优化的方式在地图上进行。