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


RegisterSign in


  • Access throughyour institution

Article preview

  • Abstract
  • References (12)
  • Cited by (36)
  • Recommended articles (6)

Pattern Recognition

Volume 25, Issue 5,

May 1992

, Pages 483-496

Author links open overlay panel


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


  • E. BribiescaArithmetic operations among shapes using shape numbers

    Pattern Recognition


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

    Pattern Recognition


  • D.H. Ballard et al.

    Computer Vision


  • N.J. Hicks

    Notes on Differential Geometry


  • 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) 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 2n 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

Recommended articles (6)

  • Research article

    Thoracoscopic or Open Surgery for Pulmonary Metastasectomy: An Observer Blinded Study

    The Annals of Thoracic Surgery, Volume 98, Issue 2, 2014, pp. 466-470

    Video-assisted thoracic surgery (VATS) resection of pulmonary metastases has long been questioned because radiologically undetected parenchymal lesions may be missed when bimanual palpation is restricted to the portholes. Technology, however, has improved and advanced VATS resections are now performed routinely worldwide. This prompted us to conduct a prospective observer-blinded study on pulmonary metastasectomy.

    Eligible patients with oligometastatic pulmonary disease on computed tomography (CT) underwent high-definition VATS, with digital palpation by 1 surgical team and subsequent immediate thoracotomy during the same anesthesia by a different surgical team, with bimanual palpation and resection of all palpable nodules. Preoperative CT evaluations and surgical results were blinded. Primary endpoints were number and histopathology of detected nodules.

    During a 3-year period 89 consecutive patients, with newly developed nodules suspicious of lung metastases from previous cancers in colon-rectum (n= 59), kidney (n= 15), and other malignancies (n= 15) were included, with a total of 140 suspicious nodules visible on CT. During VATS, 122 nodules were palpable (87%). All nodules were identified during thoracotomy, where 67 additional and unexpected nodules were also identified; 22 were metastases (33%), 43 (64%) were benign lesions, and 2 (3%) were primary lung cancers.

    In patients operated for nodules suspicious of lung metastases, a substantial number of additional nodules were detected during thoracotomy despite advancements in CT imaging and VATS technology. Many of these nodules were malignant and would have been missed if VATS was used exclusively. Consequently, we considered VATS inadequate if the intention is to resect all pulmonary metastases during surgery.

  • Research article

    Unsupervised morphological segmentation based on affixality measurements

    Pattern Recognition Letters, Volume 84, 2016, pp. 127-133

    In this paper, we present a method for unsupervised morphological segmentation for multi-slot morphology based on affixality measurements. These measurements quantify three linguistic characteristics of affixes: (1) they combine with many low frequency word-bases (high combinatorial capacity), (2) although they are relatively few, they help to maximize the size of a lexicon (economy principle), i.e. speakers know more words by remembering fewer morphological items, and (3) they are very frequent, so they contain less information than word-bases (entropy), i.e. borders between affixes and stems can be detected by finding entropy peaks. Several experiments combining these measurements were conducted to find the best way to apply them to data. The best strategy consists in successive segmentation when the average of the affixality measurements surpasses a threshold of 0.5. Also, we compared this strategy with some state-of-the-art methods for unsupervised morphological segmentation (Morfessor and ParaMor). Our method outperformed these methods, when tested in a hand-made corpus. Results indicate that our proposal is competitive at least for the morphological segmentation of Spanish words.

  • Research article

    Irreversible Electroporation of the Femoral Neurovascular Bundle: Imaging and Histologic Evaluation in a Swine Model

    Journal of Vascular and Interventional Radiology, Volume 26, Issue 8, 2015, pp. 1212-1220.e1

    To evaluate imaging, histologic changes, and safety of irreversible electroporation (IRE) on the femoral neurovascular bundle in a swine model.

    The study was approved by the institutional animal ethics committee. IRE was performed on the right femoral neurovascular bundle of 9 swine, which were subsequently sacrificed at 24 hours (n = 4, acute group), 7 days (n = 4, subacute group), or 21 days (n = 1, delayed group). Clinical observation, computed tomography (CT), and pathologic examination were carried out.

    After the procedure, 7 of 9 subjects were able to stand and walk, and the remaining 2 subjects could eventually do so within 1 week. The femoral vessels were patent on CT and gross examination. There was microscopic evidence of venous thrombosis in 75% of the subacute group. Except for mild perineural inflammation observed in 1 subject in the subacute group, the femoral nerves were intact on gross and histologic examination. Significant damage to the surrounding muscle and soft tissue was identified on CT and histology, manifesting as necrosis, hematoma, and inflammation.

    The ablative effect of IRE on muscle and soft tissue manifested as necrosis, hemorrhage, and inflammation. Histologic changes were observed in the perineural tissue and veins in a few subjects. The clinical implication of such changes and safety of clinical use of IRE for lesions encasing the neurovascular bundle in humans are yet to be determined.

  • Research article

    Infrarenal Abdominal Aortic Aneurysm Endovascular Treatment: Long-term Results From a Single-Center Experience in an Unselected Patient Population

    Annals of Vascular Surgery, Volume 67, 2020, pp. 274-282

    The aim of the present study was to evaluate early-, mid-, and long-term outcomes in an unselected population of patients treated for abdominal aortic aneurysms (AAAs) by endovascular aneurysm repair (EVAR) with different commercially available off-the-shelf devices.

    A retrospective study was conducted on a prospectively compiled computerized database on patients presenting an infrarenal AAA treated between January 2008 and December 2015 in a high-volume Italian tertiary referral Center. Demographic, clinical, and specific morphological features were considered as potentially influencing the outcomes and the type of the implanted device. Outcome measures were procedure-related reintervention, AAA-related, and all-cause mortality rates at 30-day, 12-month, and long-term follow-up. Reinterventions considered for the analysis were AAA rupture, graft infection, type I or III endoleaks, type II endoleaks with sac enlargement > 5mm, graft stenosis or occlusions, procedures related to renal or visceral ischemia, and reintervention for access vessel injury.

    Of 498 EVAR procedures performed for elective infrarenal AAA treatment during the entire study period, 479 patients were enrolled, the mean age was 73.5±7.34years (range 51-91), and 416 (86.84%) were men. The mean maximum AAA diameter was 52.02±8.04mm (range 39–90.2), a maximum AAA diameter ≥59mm was recorded in 107 patients (22.33%), and an aortic neck length was <10mm in 137 (28.60%). Technical success was achieved in all patients. At a mean follow-up of 52.97±26.16months (range 1-120), overall reintervention and death rates were 8.14% and 20.04%, respectively, without AAA-related deaths. At univariate analysis, hypertension was the only demographical variable found to be associated with higher risk of reintervention, P=0.04 (OR: 2.34; CI 95%: 1.00–5.42). Furthermore, male sex (P=0.02; OR: 2.62; CI 95%: 1.09–6.27) and chronic renal insufficiency (P=0.003; OR: 2.08; CI 95%: 1.27–3.42) were associated with higher mortality rates. AAA diameter ≥59mm was statistically associated with a higher rate of both reintervention and mortality: P<0.001 (OR: 9.05; CI 95%: 4.52–18.11) and <0.001 (4.00; 2.46–6.49), respectively.

    Our experience seems to suggest that EVAR could be safely and effectively performed in an unselected patients’ population, with encouraging results up to a ten-year follow-up.

  • Research article

    Methylation effect on the ohmic resistance of a poly-GC DNA-like chain

    Physics Letters A, Volume 380, Issue 42, 2016, pp. 3559-3563

    We determine, by using a tight-binding model Hamiltonian, the characteristic current–voltage (IxV) curves of a 5-methylated cytosine single strand poly-GC DNA-like finite segment, considering the methyl groups attached laterally to a random fraction of the cytosine basis. Striking, we found that the methylation significantly impacts the ohmic resistance (R) of the DNA-like segments, indicating that measurements of R can be used as a biosensor tool to probe the presence of anomalous methylation.

  • Research article

    Computation of Empowerment for an Autonomous Underwater Vehicle

    IFAC-PapersOnLine, Volume 49, Issue 15, 2016, pp. 81-87

    The paper addresses the computation of the information-theoretic empowerment measure for a simplified vertical plane dynamic model of an eFolaga autonomous underwater vehicle. Empowerment can be used to measure how much influence a vehicle has on its behavior, and it identifies desirable states of the vehicle, hence it can act as an intrinsic cost function to use during a mission. Online empowerment computation can be exploited within complex autonomous robotics missions. The proposed approach is being developed in the framework of a H2020 underwater robotics research project (Widely scalable Mobile Underwater Sonar Technology) to improve the autonomy of autonomous underwater vehicles for acoustic seismic applications.

View full text

Copyright © 1992 Published by Elsevier B.V.

Top Articles
Latest Posts
Article information

Author: Dan Stracke

Last Updated: 11/14/2022

Views: 6288

Rating: 4.2 / 5 (63 voted)

Reviews: 86% of readers found this page helpful

Author information

Name: Dan Stracke

Birthday: 1992-08-25

Address: 2253 Brown Springs, East Alla, OH 38634-0309

Phone: +398735162064

Job: Investor Government Associate

Hobby: Shopping, LARPing, Scrapbooking, Surfing, Slacklining, Dance, Glassblowing

Introduction: My name is Dan Stracke, I am a homely, gleaming, glamorous, inquisitive, homely, gorgeous, light person who loves writing and wants to share my knowledge and understanding with you.