Gaussian Mixture Models (GMM) do not adapt well to curved and strongly
nonlinear data. However, we can use Gaussians in the curvilinear coordinate
systems to solve this problem. Moreover, such a solution allows for the
adaptation of clusters to the complicated shapes defined by the family of
functions. But still, it is challenging to model clusters as closed curves
(e.g., circles, ellipses, etc.). In this work, we propose a density
representation of the closed curve, which can be used to detect the complicated
templates in the data. For this purpose, we define a new probability
distribution to model closed curves. Then we construct a mixture of such
distributions and show that it can be effectively trained in the case of the
one-dimensional closed curves.

本文提出了一种用于探测数据中复杂模板的闭合曲线的密度表示方法，并构建了该方法的混合概率分布，实验证明其在一维封闭曲线情况下的有效性。

闭合曲线的高斯模型

Gaussian model for closed curves

While the equality of differential signatures (Calabi et al, Int. J. Comput.
Vis. 26: 107-135, 1998) is known to be a necessary condition for congruence, it
is not sufficient (Musso and Nicolodi, J. Math Imaging Vis. 35: 68-85, 2009).
Hickman (J. Math Imaging Vis. 43: 206-213, 2012, Theorem 2) claimed that for
non-degenerate planar curves, equality of Euclidean signatures implies
congruence. We prove that while Hickman's claim holds for simple, closed curves
with simple signatures, it fails for curves with non-simple signatures. In the
later case, we associate a directed graph with the signature and show how
various paths along the graph give rise to a family of non-congruent,
non-degenerate curves with identical signatures. Using this additional
structure, we formulate congruence criteria for non-degenerate, closed, simple
curves and show how the paths reflect the global and local symmetries of the
corresponding curve.

该论文研究了简单封闭曲线的等度数特征相等是完全相似的必要条件，但并不是充分条件，在非正常封闭曲线的情况下，节点可以沿着多种路径移动，导致一系列不全等的封闭曲线。