We often wish to classify objects by their shapes. Indeed, the study of shapes is an important part of many scientific fields such as evolutionary biology, structural biology, image processing, and archaeology. The most widely-used method of shape analysis, Geometric Morphometrics, assumes that that the mathematical space in which shapes are represented is linear. However, it has long been known that shape space is, in fact, rather more complicated, and certainly non-linear. Diffeomorphic methods that take this non-linearity into account, and so give more accurate estimates of the distances among shapes, exist but have rarely been applied to real-world problems. Using a machine classifier, we tested the ability of several of these methods to describe and classify the shapes of a variety of organic and man-made objects. We find that one method, the Square-Root Velocity Function (SRVF), is superior to all others, including a standard Geometric Morphometric method (eigenshapes). We also show that computational shape classifiers outperform human experts, and that the SRVF shortest-path between shapes can be used to estimate the shapes of intermediate steps in evolutionary series. Diffeomorphic shape analysis methods, we conclude, now provide practical and effective solutions to many shape description and classification problems in the natural and human sciences.

翻译：事实上,对形状的研究是进化生物学、结构生物学、图像处理和考古学等许多科学领域的一个重要部分。最广泛使用的形状分析方法,即几何光谱测量法,假定形状所代表的数学空间是线性。然而,人们早已知道形状空间事实上比较复杂,当然也非线性。考虑到这种非线性,对形状的研究是许多科学领域的重要部分,因此对形状之间的距离作出更准确的估计,已经存在,但很少应用于现实世界的问题。我们用机器分类器测试了这些方法中用来描述和分类各种有机和人造物体的形状的几种形状的几种方法的能力。我们发现,一种方法,即平极速函数(Sqreare-root Velocity 函数),优于所有其他方法,包括标准的几何光度测法方法(igenshape),我们还表明,计算形状的形状超越了人类专家的形状,而现在的SRVF最短的形状分析法现在也很少适用于现实世界的问题。我们用这些方法来描述各种形状的形状的形状的形状的形状和形状的形状的形状的形状分析,我们可以用了许多步骤和形态的形态的形态的形态的形态的形态分析方法。