Over the past decade, the Gr\"obner basis theory and automatic solver
generation have lead to a large number of solutions to geometric vision
problems. In practically all cases, the derived solvers apply a fixed
elimination template to calculate the Gr\"obner basis and thereby identify the
zero-dimensional variety of the original polynomial constraints. However, it is
clear that different variable or monomial orderings lead to different
elimination templates, and we show that they may present a large variability in
accuracy for a certain instance of a problem. The present paper has two
contributions. We first show that for a common class of problems in geometric
vision, variable reordering simply translates into a permutation of the columns
of the initial coefficient matrix, and that -- as a result -- one and the same
elimination template can be reused in different ways, each one leading to
potentially different accuracy. We then prove that the original set of
coefficients may contain sufficient information to train a classifier for
online selection of a good solver, most notably at the cost of only a small
computational overhead. We demonstrate wide applicability at the hand of
generic dense polynomial problem solvers, as well as a concrete solver from
geometric vision.

经过过去十年的发展，Gröbner 基础理论和自动生成求解器已经产生了大量解决几何视觉问题的解决方案。此论文首先展示了不同的变量或单项式排列可以导致不同的消除模板，从而在问题的某个实例中可能导致较大的准确性变化。然后证明了原始系数集中包含足够的信息来训练用于在线选择好的求解器的分类器，尤其只需很小的计算开销。通过通用的密集多项式问题求解器以及具体的几何视觉求解器，我们展示了其广泛适用性。

使用深度学习提升 Groebner 基础求解器的在线稳定性

Online Stability Improvement of Groebner Basis Solvers using Deep  Learning

Deep networks excel in learning patterns from large amounts of data. On the
other hand, many geometric vision tasks are specified as optimization problems.
To seamlessly combine deep learning and geometric vision, it is vital to
perform learning and geometric optimization end-to-end. Towards this aim, we
present BPnP, a novel network module that backpropagates gradients through a
Perspective-n-Points (PnP) solver to guide parameter updates of a neural
network. Based on implicit differentiation, we show that the gradients of a
"self-contained" PnP solver can be derived accurately and efficiently, as if
the optimizer block were a differentiable function. We validate BPnP by
incorporating it in a deep model that can learn camera intrinsics, camera
extrinsics (poses) and 3D structure from training datasets. Further, we develop
an end-to-end trainable pipeline for object pose estimation, which achieves
greater accuracy by combining feature-based heatmap losses with 2D-3D
reprojection errors. Since our approach can be extended to other optimization
problems, our work helps to pave the way to perform learnable geometric vision
in a principled manner. Our PyTorch implementation of BPnP is available on
this http URL

BPnP 是一种全新的网络模块，可以在神经网络的参数更新过程中通过将反向传播的梯度通过透视 n 点（PnP）求解器引导，从而将深度学习和几何视觉进行无缝结合。