Department of Mathematics, Utah State University, Logan, UT 84322.
IEEE Trans Pattern Anal Mach Intell. 1984 Jan;6(1):27-40. doi: 10.1109/tpami.1984.4767472.
This paper introduces a new theory for the tangential deflection and curvature of plane discrete curves. Our theory applies to discrete data in either rectangular boundary coordinate or chain coded formats: its rationale is drawn from the statistical and geometric properties associated with the eigenvalue-eigenvector structure of sample covariance matrices. Specifically, we prove that the nonzero entry of the commutator of a piar of scatter matrices constructed from discrete arcs is related to the angle between their eigenspaces. And further, we show that this entry is-in certain limiting cases-also proportional to the analytical curvature of the plane curve from which the discrete data are drawn. These results lend a sound theoretical basis to the notions of discrete curvature and tangential deflection; and moreover, they provide a means for computationally efficient implementation of algorithms which use these ideas in various image processing contexts. As a concrete example, we develop the commutator vertex detection (CVD) algorithm, which identifies the location of vertices in shape data based on excessive cummulative tangential deflection; and we compare its performance to several well established corner detectors that utilize the alternative strategy of finding (approximate) curvature extrema.
本文提出了一种新的平面离散曲线切向挠度和曲率理论。我们的理论适用于矩形边界坐标或链式编码格式的离散数据：其基本原理来自于与样本协方差矩阵特征值-特征向量结构相关的统计和几何特性。具体来说，我们证明了由离散弧段构造的一对散射矩阵的交换子的非零项与它们特征空间之间的角度有关。并且进一步，我们表明，在某些极限情况下，该项与从中提取离散数据的平面曲线的解析曲率成正比。这些结果为离散曲率和切向挠度的概念提供了合理的理论基础；此外，它们为在各种图像处理环境中使用这些思想的算法提供了计算效率高的实现手段。作为一个具体的例子，我们开发了交换子顶点检测（CVD）算法，该算法根据过度累积的切向挠度来确定形状数据中顶点的位置；并将其性能与几种利用寻找（近似）曲率极值的替代策略的成熟角检测器进行了比较。
