Triangle meshes remain the most popular data representation for surface
geometry. This ubiquitous representation is essentially a hybrid one that
decouples continuous vertex locations from the discrete topological
triangulation. Unfortunately, the combinatorial nature of the triangulation
prevents taking derivatives over the space of possible meshings of any given
surface. As a result, to date, mesh processing and optimization techniques have
been unable to truly take advantage of modular gradient descent components of
modern optimization frameworks. In this work, we present a differentiable
surface triangulation that enables optimization for any per-vertex or per-face
differentiable objective function over the space of underlying surface
triangulations. Our method builds on the result that any 2D triangulation can
be achieved by a suitably perturbed weighted Delaunay triangulation. We
translate this result into a computational algorithm by proposing a soft
relaxation of the classical weighted Delaunay triangulation and optimizing over
vertex weights and vertex locations. We extend the algorithm to 3D by
decomposing shapes into developable sets and differentiably meshing each set
with suitable boundary constraints. We demonstrate the efficacy of our method
on various planar and surface meshes on a range of difficult-to-optimize
objective functions. Our code can be found online:
this https URL

提出了可微分的表面三角形网格化算法，基于加权德劳内三角形的软松弛，并通过优化顶点和边界约束来拟合目标曲面，成功地解决了现有网格处理和优化技术在连续体上的局限性。

可微分的曲面三角剖分

Differentiable Surface Triangulation

Recent advances in deep learning have significantly pushed the
state-of-the-art in photorealistic video animation given a single image. In
this paper, we extrapolate those advances to the 3D domain, by studying 3D
image-to-video translation with a particular focus on 4D facial expressions.
Although 3D facial generative models have been widely explored during the past
years, 4D animation remains relatively unexplored. To this end, in this study
we employ a deep mesh encoder-decoder like architecture to synthesize realistic
high resolution facial expressions by using a single neutral frame along with
an expression identification. In addition, processing 3D meshes remains a
non-trivial task compared to data that live on grid-like structures, such as
images. Given the recent progress in mesh processing with graph convolutions,
we make use of a recently introduced learnable operator which acts directly on
the mesh structure by taking advantage of local vertex orderings. In order to
generalize to 4D facial expressions across subjects, we trained our model using
a high resolution dataset with 4D scans of six facial expressions from 180
subjects. Experimental results demonstrate that our approach preserves the
subject's identity information even for unseen subjects and generates high
quality expressions. To the best of our knowledge, this is the first study
tackling the problem of 4D facial expression synthesis.

本文研究应用深度学习技术将单张静态图片转化为逼真的三维动画，重点探究四维面部表情的生成。作者运用一种称为深度网格编码器 - 解码器的技术，结合表情识别模型，以高分辨率的四维扫描数据集为基础，成功地实现了对面部表情的高度逼真合成，且具有较好地泛化能力。该研究是首次尝试解决四维面部表情合成的问题。