When modeling directional data, that is, unit-norm multivariate vectors, a
first natural question is to ask whether the directions are uniformly
distributed or, on the contrary, whether there exist modes of variation
significantly different from uniformity. We review in this article a reasonably
exhaustive collection of uniformity tests for assessing uniformity in the
hypersphere. Specifically, we review the classical circular-specific tests, the
large class of Sobolev tests with its many notable particular cases, some
recent alternative tests, and novel results in the high-dimensional low-sample
size case. A reasonably comprehensive bibliography on the topic is provided.

本文介绍了在模拟单位范多元向量，即方向数据时，采用的统一性检验方法，包括传统的基于圆形的检验方法、基于 Sobolev 的检验方法及其他最新的检验方法，针对高维度小样本情况的新结果也得到了探讨，提供了详尽的参考文献。

超球上均匀性测试概述

An overview of uniformity tests on the hypersphere

Structural regularities in man-made environments reflect in the distribution
of their surface normals. Describing these surface normal distributions is
important in many computer vision applications, such as scene understanding,
plane segmentation, and regularization of 3D reconstructions. Based on the
small-variance limit of Bayesian nonparametric von-Mises-Fisher (vMF) mixture
distributions, we propose two new flexible and efficient k-means-like
clustering algorithms for directional data such as surface normals. The first,
DP-vMF-means, is a batch clustering algorithm derived from the Dirichlet
process (DP) vMF mixture. Recognizing the sequential nature of data collection
in many applications, we extend this algorithm to DDP-vMF-means, which infers
temporally evolving cluster structure from streaming data. Both algorithms
naturally respect the geometry of directional data, which lies on the unit
sphere. We demonstrate their performance on synthetic directional data and real
3D surface normals from RGB-D sensors. While our experiments focus on 3D data,
both algorithms generalize to high dimensional directional data such as protein
backbone configurations and semantic word vectors.

本文提出了基于 Bayesian 非参数 von-Mises-Fisher 混合分布的小方差限制的两种新的灵活高效的聚类算法，这些算法自然地与方向数据的几何结构相符，旨在描述人工环境中的表面正常分布，其在计算机视觉应用中具有重要意义，包括场景理解、平面分割和三维重建的规则化。

超球上小方差的非参数聚类

Small-Variance Nonparametric Clustering on the Hypersphere

Directional data arise in various contexts such as oceanography (wave
directions) and meteorology (wind directions), as well as with measurements on
a periodic scale (weekdays, hours, etc.). Our contribution is to introduce a
model-based approach to handle periodic data in the case of measurements taken
at spatial locations, anticipating structured dependence between these
measurements. We formulate a wrapped Gaussian spatial process model for this
setting, induced from a customary linear Gaussian process. We build a
hierarchical model to handle this situation and show that the fitting of such a
model is possible using standard Markov chain Monte Carlo methods. Our approach
enables spatial interpolation (and can accommodate measurement error). We
illustrate with a set of wave direction data from the Adriatic coast of Italy,
generated through a complex computer model.

本研究旨在通过使用基于模型的方法处理定期测量在空间位置的方向数据，提出了一个包含垂直高斯空间过程模型的层次模型，可以进行空间插值，并且可以容纳测量误差，并通过使用标准 Markov Chain Monte Carlo 方法拟合此模型。