  We present a Deep Differentiable Simplex Layer (DDSL) for neural networks for
geometric deep learning. The DDSL is a differentiable layer compatible with
deep neural networks for bridging simplex mesh-based geometry representations
(point clouds, line mesh, triangular mesh, tetrahedral mesh) with raster images
(e.g., 2D/3D grids). The DDSL uses Non-Uniform Fourier Transform (NUFT) to
perform differentiable, efficient, anti-aliased rasterization of simplex-based
signals. We present a complete theoretical framework for the process as well as
an efficient backpropagation algorithm. Compared to previous differentiable
renderers and rasterizers, the DDSL generalizes to arbitrary simplex degrees
and dimensions. In particular, we explore its applications to 2D shapes and
illustrate two applications of this method: (1) mesh editing and optimization
guided by neural network outputs, and (2) using DDSL for a differentiable
rasterization loss to facilitate end-to-end training of polygon generators. We
are able to validate the effectiveness of gradient-based shape optimization
with the example of airfoil optimization, and using the differentiable
rasterization loss to facilitate end-to-end training, we surpass state of the
art for polygonal image segmentation given ground-truth bounding boxes.
