Similarity Metric for Curved Shapes in Euclidean Space

Abstract

In this paper, we introduce a similarity metric for curved shapes that can be described, distinctively, by ordered points. The proposed method represents a given curve as a point in the deformation space, the direct product of rigid transformation matrices, such that the successive action of the matrices on a fixed starting point reconstructs the full curve. In general, both open and closed curves are represented in the deformation space modulo shape orientation and orientation preserving diffeomorphisms. The use of direct product Lie groups to represent curved shapes led to an explicit formula for geodesic curves and the formulation of a similarity metric between shapes by the L 2 -norm on the Lie algebra. Additionally, invariance to reparametrization or estimation of point correspondence between shapes is performed as an intermediate step for computing geodesics. Furthermore, since there is no computation of differential quantities on the curves, our representation is more robust to local perturbations and needs no pre-smoothing. We compare our method with the elastic shape metric defined through the square root velocity (SRV) mapping, and other shape matching approaches.

Object Detection and Classification of Metal Polishing Shaft Surface Defects Based on Convolutional Neural Network Deep Learning

“…(3) mAP assessment [44] Mean average accuracy (mAP) is the most commonly used evaluation criterion in object detection. In common classification problems, recall and precision are the most commonly-used statistics.…”

Deformation Based Curved Shape Representation

“…This fact can be observed by plugging the action of G into (20), in which case it will cancel itself out. As discussed in [15], this property is particularly important in transporting deformation between two similar shapes. To clarify further, we consider below a deformation transportation problem discussed in [54].…”

“…In this subsection, we will present point correspondence estimation for uniformly sampled shapes, as discussed in [15]. The approach restricts shape sampling functions to a group of z-cyclic permutations, given the shapes to be aligned are uniformly sampled.…”

“…In this paper, we elaborate and extend the deformation based curved shape representation, proposed in [15], that does not impose smoothness constraints while leading to a simple and explicit geodesic equation. 1 In [15], a new curved shape representation is formulated on the curve's deformation space, which leads to a much simpler metric with an explicit solution to its geodesic equation. The metric has a close relationship with what is sometimes referred to as effort functional [44], [63].…”

“…Computational cost, on the other hand, is a polynomial function of the sample size. The main contributions of this paper are summarized as follows: 1) A relatively simple yet effective representation for curved shapes, described in [15], is introduced. Particular to this representation, curves are treated as elements of the deformation space which is a finite dimensional Lie group.…”