Characteristic Number: Theory and Its Application to Shape Analysis

Fan, Xin; Luo, Zhongxuan; Zhang, Jielin; Zhou, Xiaoxu; Jia, Qi; Luo, Daiyun

doi:10.3390/axioms3020202

Cited by 5 publications

(2 citation statements)

References 30 publications

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“…The applications of algebraic curves encompass several domains in the mathematical sciences and in the computational sciences [12,13]. There are interplays between algebraic curves and the concepts of topology that determine various interesting algebraic as well as topological properties [12].…”

Section: Motivationsmentioning

confidence: 99%

A Topological Approach to the Bézout’ Theorem and Its Forms

Bagchi

2023

Symmetry

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The interplays between topology and algebraic geometry present a set of interesting properties. In this paper, we comprehensively revisit the Bézout theorem in terms of topology, and we present a topological proof of the theorem considering n-dimensional space. We show the role of topology in understanding the complete and finite intersections of algebraic curves within a topological space. Moreover, we introduce the concept of symmetrically complex translations of roots in a zero-set of a real algebraic curve, which is called a fundamental polynomial, and we show that the resulting complex algebraic curve is additively decomposable into multiple components with varying degrees in a sequence. Interestingly, the symmetrically complex translations of roots in a zero-set of a fundamental polynomial result in the formation of isomorphic topological manifolds if one of the complex translations is kept fixed, and it induces repeated real roots in the fundamental polynomial as a component. A set of numerically simulated examples is included in the paper to illustrate the resulting manifold structures and the associated properties.

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Section: Motivationsmentioning

confidence: 99%

A Topological Approach to the Bézout’ Theorem and Its Forms

Bagchi

2023

Symmetry

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“…However, the structure of the line neighborhood was sensitive to the inaccurate location of line endpoints. Jia et al [19] proposed a coplanar line-points projective invariant named the characteristic number [20] to undertake the matching. Two line segments were assumed to be coplanar, if the intersection of the two lines was close to one of the endpoints.…”

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confidence: 99%

A Line Matching Method Based on Multiple Intensity Ordering with Uniformly Spaced Sampling

Xing

Wei

Zhang

2020

Sensors

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This paper presents a line matching method based on multiple intensity ordering with uniformly spaced sampling. Line segments are extracted from the image pyramid, with the aim of adapting scale changes and addressing fragmentation problem. The neighborhood of line segments was divided into sub-regions adaptively according to intensity order to overcome the difficulty brought by various line lengths. An intensity-based local feature descriptor was introduced by constructing multiple concentric ring-shaped structures. The dimension of the descriptor was reduced significantly by uniformly spaced sampling and dividing sample points into several point sets while improving the discriminability. The performance of the proposed method was tested on public datasets which cover various scenarios and compared with another two well-known line matching algorithms. The experimental results show that our method achieves superior performance dealing with various image deformations, especially scale changes and large illumination changes, and provides much more reliable correspondences.

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