Recent increase in the availability of warped images projected onto a
manifold (e.g., omnidirectional spherical images), coupled with the success of
higher-order assignment methods, has sparked an interest in the search for
improved higher-order matching algorithms on warped images due to projection.
Although currently, several existing methods "flatten" such 3D images to use
planar graph / hypergraph matching methods, they still suffer from severe
distortions and other undesired artifacts, which result in inaccurate matching.
Alternatively, current planar methods cannot be trivially extended to
effectively match points on images warped onto manifolds. Hence, matching on
these warped images persists as a formidable challenge. In this paper, we pose
the assignment problem as finding a bijective map between two graph induced
simplicial complexes, which are higher-order analogues of graphs. We propose a
constrained quadratic assignment problem (QAP) that matches each p-skeleton of
the simplicial complexes, iterating from the highest to the lowest dimension.
The accuracy and robustness of our approach are illustrated on both synthetic
and real-world spherical / warped (projected) images with known ground-truth
correspondences. We significantly outperform existing state-of-the-art
spherical matching methods on a diverse set of datasets.

本文介绍了一种针对弯曲（如球形）图像的高级匹配算法，使用简单复合复杂度计算每个 p - 基元的同构，在已知真实匹配的情况下测试一系列数据集并超越现有的方案。

基于单纯复合的图像变形到流形上的点对应

Simplicial Complex based Point Correspondence between Images warped onto  Manifolds

We show that a tessellation generated by a small number of random affine
hyperplanes can be used to approximate Euclidean distances between any two
points in an arbitrary bounded set $T$, where the random hyperplanes are
generated by subgaussian or heavy-tailed normal vectors and uniformly
distributed shifts. We derive quantitative bounds on the number of hyperplanes
needed for constructing such tessellations in terms of natural metric
complexity measures of $T$ and the desired approximation error. Our work
extends significantly prior results in this direction, which were restricted to
Gaussian hyperplane tessellations of subsets of the Euclidean unit sphere.
As an application, we obtain new reconstruction results in memoryless one-bit
compressed sensing with non-Gaussian measurement matrices. We show that by
quantizing at uniformly distributed thresholds, it is possible to accurately
reconstruct low-complexity signals from a small number of one-bit quantized
measurements, even if the measurement vectors are drawn from a heavy-tailed
distribution. Our reconstruction results are uniform in nature and robust in
the presence of pre-quantization noise on the analog measurements as well as
adversarial bit corruptions in the quantization process. Moreover we show that
if the measurement matrix is subgaussian then accurate recovery can be achieved
via a convex program.

本文研究利用少数随机仿射超平面生成的分割可近似在任意有界集合 $T$ 内两点间的欧几里得距离，从而得出这种生成方式需要的超平面数量，以及将其应用于一比特压缩感知方案并证明了其鲁棒性以及通过凸优化可以实现精准重建。

非高斯超平面镶嵌及鲁莽一位压缩感知

Non-Gaussian Hyperplane Tessellations and Robust One-Bit Compressed  Sensing

Given a subset K of the unit Euclidean sphere, we estimate the minimal number
m = m(K) of hyperplanes that generate a uniform tessellation of K, in the sense
that the fraction of the hyperplanes separating any pair x, y in K is nearly
proportional to the Euclidean distance between x and y. Random hyperplanes
prove to be almost ideal for this problem; they achieve the almost optimal
bound m = O(w(K)^2) where w(K) is the Gaussian mean width of K. Using the map
that sends x in K to the sign vector with respect to the hyperplanes, we
conclude that every bounded subset K of R^n embeds into the Hamming cube {-1,
1}^m with a small distortion in the Gromov-Haussdorf metric. Since for many
sets K one has m = m(K) << n, this yields a new discrete mechanism of dimension
reduction for sets in Euclidean spaces.

研究了如何利用随机超平面对区域进行均匀的镶嵌，并利用超平面将欧氏空间中的有界集嵌入哈明空间中，从而实现对欧氏空间中集合的离散化降维。