# A Geometric structure for two-dimensional shapes and three-dimensional surfaces (2023)

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• References (12)
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## Pattern Recognition

Volume 25, Issue 5,

May 1992

, Pages 483-496

## Abstract

A geometric structure called slope change notation (SCN), which describes two-dimensional (2D) shapes and three-dimensional (3D) surfaces in a discrete representation, is presented. The SCN of a curve is obtained by placing constant-length straight-line segments around the curve (the endpoints of the straight-line segments always touching the curve), and calculating the slope changes between contiguous segments scaled to a continuous range from −1 to 1. The SCN is independent of translation and rotation (due to the fact that slope changes around the curve are used), and optionally, of size. The SCN for 2D shapes is 1D. This is an important characteristic, because shapes with particular characteristics are easily generated by numerical sequences; also, it is possible to perform arithmetic operations among shapes and surfaces. The SCN differs from other chain codes, for instance, Freeman chains (Proc. Natn. Electron. Conf.18, 312–324 (1961)), since the proposed notation does not use a grid (and so depends only on itself); its range of slope changes varies continuously from −1 to 1; its vertices always touch the curve, which produces a better description of the shape; and its discrete elements always have the same length. Using this geometric structure only slope changes are variable; the segments size of any shape is always constant. At the end of the paper a related theory “B”, that allows variable segment size as a function of slope changes, is introduced. These ideas are based on previous work (Pattern Recognition13, 123–137 (1981)) and the solutions to many problems which arose are presented in this paper.

## References (12)

• E. Bribiesca et al.How to describe pure form and how to measure differences in shape using shape numbers

### Pattern Recognition

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• E. BribiescaArithmetic operations among shapes using shape numbers

### Pattern Recognition

(1981)

• R.F.A. Collard et al.Minimization of structural information: a set-theoretical approach

### Pattern Recognition

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• D.H. Ballard et al.

(1982)

• N.J. Hicks

(1964)

• H. Freeman

### Techniques for the digital computer analysis of chain encoded arbitrary plane curves

There are more references available in the full text version of this article.

## Cited by (36)

• Reflection symmetry detection of shapes based on shape signatures

2022, Pattern Recognition

We present two novel shape signature-based reflection symmetry detection methods with their theoretical underpinning and empirical evaluation. LIP-signature and R-signature share similar beneficial properties allowing to detect reflection symmetry directions in a high-performing manner. For the shape signature of a given shape, its merit profile is constructed to detect candidates of symmetry direction. A verification process is utilized to eliminate the false candidates by addressing Radon projections. The proposed methods can effectively deal with compound shapes which are challenging for traditional contour-based methods. To quantify the symmetric efficiency, a new symmetry measure is proposed over the range [0, 1]. Furthermore, we introduce two symmetry shape datasets with a new evaluation protocol and a lost measure for evaluating symmetry detectors. Experimental results using standard and new datasets suggest that the proposed methods prominently perform compared to state of the art.

• Detection of rotational symmetry in curves represented by the slope chain code

2020, Pattern Recognition

We present a new approach based on the Slope Chain Code to determine whether a curve is rotational symmetrical and its order of symmetry. The proposed approach works for open and closed perfectly symmetrical or quasi-symmetrical 2D curves. Simple operations on the SCC and its invariant properties are central to our methodology. To evaluate the proposed methodology, we use 1400 curves from a public database. For the symmetrical/asymmetrical classification task, a recall (R) of 0.86, a balanced accuracy (BA) of 0.92, and a precision (P) of 0.87 were obtained. For the quasi-symmetrical/quasi-asymmetrical classification task, R=0.77, BA=0.83, and P=0.70 were obtained. For the order of rotational symmetry detection task, the following performance was achieved: R=0.97, BA=0.98, and P=0.95 for a symmetrical set of curves, and R=0.98, BA=0.98, and P=0.90 for a quasi-symmetrical set of curves. We conclude our presentation demonstrating the usefulness of our methodology with three practical applications

• The spirals of the Slope Chain Code

2019, Pattern Recognition

Citation Excerpt :

The chain elements produce finite alphabets which allow us to use grammatical techniques for shape classification. Most chain code methods presented in literature [2,10–14] are based on the representation of contour shapes by means of constant straight-line segments at different previous-defined directions. In 2D domain, there are methods to represent 2D curves via chain coding for example in refs. [2,10,12–15].

Generally speaking, a spiral is a 2D curve which winds about a fixed point. Now, we present a new, alternative, and easy way to describe and generate spirals by means of the use of the Slope Chain Code (SCC) [E. Bribiesca, A measure of tortuosity based on chain coding, Pattern Recognition 46 (2013) 716–724]. Thus, each spiral is represented by only one chain. The chain elements produce a finite alphabet which allows us to use grammatical techniques for spiral classification. Spirals are composed of constant straight-line segments and their chain elements are obtained by calculating the slope changes between contiguous straight-line segments (angle of contingence) scaled to a continuous range from $-1$ ($-{180}^{\circ }$) to 1 (180). The SCC notation is invariant under translation, rotation, optionally under scaling, and it does not use a grid. Other interesting properties can be derived from this notation, such as: the mirror symmetry and inverse spirals, the accumulated slope, the slope change mean, and tortuosity for spirals. We introduce new concepts of projective polygonal paths and osculating polygons. We present a new spiral called the SCC polygonal spiral and its chain which is described by the numerical sequence $\frac{2}{n}$ for n ≥ 3, to the best of our knowledge this is the first time that this spiral and its chain are presented. The importance of this spiral and its chain is that this chain is covering all the slope changes of all the regular polygons composed of n edges (n-gons). Also, we describe the chain which generates the spiral of Archimedes. Finally, we present some results of different kind of spirals from the real world, including spiral patterns in shells.

• A chain code for representing high definition contour shapes

2019, Journal of Visual Communication and Image Representation

Citation Excerpt :

Thus, chain code techniques may be a useful tool for representing contour shapes in computer vision and pattern recognition. Most chain code methods presented in literature [1–6] are based on the representation of contour shapes by means of constant straight-line segments at different previous-defined directions. The above-mentioned methods produce low definition contour shapes.

The biggest disadvantage of using chain code techniques is the generation of low definition contour shapes, in this paper we present the Extended Slope Chain Code (ESCC) which is an improvement on the Slope Chain Code (SCC). The ESCC is focused on the representation of high definition contour shapes. Generally speaking, most chain codes hold the length of the straight-line segments which represent the contour shape as a constant. In this case, the contour shapes represented by ESCC are composed of variable segments, which allow us to have a better description of the contour shape. Thus, the length of the segments are a function of the slope changes, i.e. the length of the next segment depends on the value of the slope change at that point. Therefore, the ESCC is continuously adjusting to the curvature requirements of contour shapes, in order to have a better description of contour shapes.

• Mirror symmetry detection in curves represented by means of the Slope Chain Code

2019, Pattern Recognition

Citation Excerpt :

For the sake of completeness we give a concise presentation of the theory behind the SCC for 2D curves. For further explanations and examples of the ideas below, we refer the reader to [17,18] and their references. In this section we introduce a new method to detect the reflection (i.e., mirror) symmetry in 2D curves that takes advantage of the aforementioned properties of the slope chain code representation.

Symmetry is an important feature in natural and man-made objects; particularly, mirror symmetry is a relevant task in fields such as computer vision and pattern recognition. In the current work, we propose a new method to characterize mirror-symmetry in open and closed curves represented by means of the Slope Chain Code. This representation is invariant under scale, rotation, and translation, highly desirable properties for object recognition applications. The proposed method detects symmetries through simple inversion, concatenation and reflection operations on the chains, thus allowing the classification of symmetrical and asymmetrical contours. It also introduces a measure to quantify the degree of symmetry in quasi-mirror-symmetrical objects. Furthermore, it allows the identification of multiple symmetry axes and their location. Results show high performances in symmetrical/asymmetrical classification (0.9 recall, 0.9 accuracy, 0.97 precision) and axes’ detection (0.8 recall, 0.84 accuracy, 0.99 precision). Compared to other methods, the proposed algorithm provides properties such as: global, local, and multiple axes’ detection, as well as the capability to classify symmetrical objects, which makes it adequate for several practical applications, like the three exemplified in the paper.

• 2D tree object representation via the slope chain code

2014, Pattern Recognition

Citation Excerpt :

Characteristics of the proposed method for representing 2D tree objects. This approach is based on the SCC [2]. It was proposed for 2D domain.

A method for representing 2D (two-dimensional) tree objects is described. This representation is based on a chain code, which is called the Slope Chain Code (SCC). Thus, 2D tree objects are described by means of a chain of element strings suitably combined by means of parentheses. These 2D tree objects correspond to naturally existing 2D tree structures. This tree notation preserves the shape of trees (and the shape of their branches), allows us to know their topological and geometrical properties. The proposed notation of 2D tree objects is invariant under translation, rotation and, optionally, under scaling. Also, it is possible to define a unique start vertex for each tree via the unique path in the tree. Using this notation it is possible to obtain the mirror image of any tree with ease. Furthermore, two interesting properties of trees are presented: the accumulated slope and the tortuosity. Tortuosity is a very important property of trees and has many applications in different fields. In order to prove our method for representing 2D tree objects, we obtain some tree descriptors of tree objects and compute their measures of accumulated slope and tortuosity. Finally, we present some examples of 2D trees from the real world about echinoderm species identification.

View all citing articles on Scopus

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