Geometric Shape Characterisation Based on a Multi-Sweeping Paradigm
<p>Result of the initialisation for the demonstration shape plotted in orange, where cells with <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi>E</mi> </mrow> </semantics></math> are white, <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi>B</mi> </mrow> </semantics></math> are black, while the grey cells indicate <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi>I</mi> </mrow> </semantics></math>.</p> "> Figure 2
<p>Common sweep-line events (the arrow denotes the sweep-line moving direction, while the sequence of red dots belongs to the chains <math display="inline"><semantics> <msub> <mi>L</mi> <mi>i</mi> </msub> </semantics></math>; characteristic positions of sweep-line are marked with characters a–e).</p> "> Figure 3
<p>Results of multi-sweeping, when <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>0</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> (<b>a</b>), <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> (<b>b</b>), <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> (<b>c</b>), <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>90</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> (<b>d</b>), <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>120</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> (<b>e</b>), and <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>150</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> (<b>f</b>). The frontier cells utilised in the sweeping are coloured in blue.</p> "> Figure 4
<p>Chains after filtering, where the red points remain from <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>(</mo> <msup> <mn>0</mn> <mo>∘</mo> </msup> <mo>)</mo> </mrow> </semantics></math>, yellow from <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>(</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> <mo>)</mo> </mrow> </semantics></math>, green from <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>(</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> <mo>)</mo> </mrow> </semantics></math>, orange from <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>(</mo> <msup> <mn>90</mn> <mo>∘</mo> </msup> <mo>)</mo> </mrow> </semantics></math>, blue from <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>(</mo> <msup> <mn>120</mn> <mo>∘</mo> </msup> <mo>)</mo> </mrow> </semantics></math>, and purple from <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>(</mo> <msup> <mn>150</mn> <mo>∘</mo> </msup> <mo>)</mo> </mrow> </semantics></math>.</p> "> Figure 5
<p>Testing shapes: (<b>a</b>) Circle, (<b>b</b>) Square, (<b>c</b>) Rocket, (<b>d</b>) Hand, (<b>e</b>) Airplane, (<b>f</b>) Penguin, (<b>g</b>) Runner, (<b>h</b>) Buddha, (<b>i</b>) Detective, (<b>j</b>) Ballet, (<b>k</b>) Dragon, (<b>l</b>) Cupid.</p> "> Figure 6
<p>Chains produced by the MSCA for <math display="inline"><semantics> <mi mathvariant="script">S</mi> </semantics></math> = <math display="inline"><semantics> <mi mathvariant="italic">Circle</mi> </semantics></math> when: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>0</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>15</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>e</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>f</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>75</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>g</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>90</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>h</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>105</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>i</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>120</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>j</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>135</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>k</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>150</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, and (<b>l</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>165</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>.</p> "> Figure 7
<p>Chains obtained with the MSCA for <math display="inline"><semantics> <mi mathvariant="script">S</mi> </semantics></math> = <math display="inline"><semantics> <mi mathvariant="italic">Cupid</mi> </semantics></math> when: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>0</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>15</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>e</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>f</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>75</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>g</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>90</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>h</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>105</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>i</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>120</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>j</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>135</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, (<b>k</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>150</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>, and (<b>l</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <msup> <mn>165</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>.</p> "> Figure 8
<p>Shapes used for detection of equality; the shapes are denoted by letters (<b>A</b>–<b>P</b>).</p> ">
Abstract
:1. Introduction
- Initialisation, where a shape is inserted into a grid of equally sized cells;
- Processing, where the shape is swept several times with sweep-lines having different slopes; as a result of each sweep, the interior midpoints with respect to the shape boundary are determined and linked into the chains of midpoints;
- Finalisation, where the obtained chains are filtered, vectorised, and normalised. A shape’s characterisation vector is then formed from the polylines, which were obtained by the vectorisation.
- The obtained set of polylines enables the construction of various, application-specific characterisation vectors;
- It handles free-form shapes;
- It processes the shapes containing holes without any modifications in the algorithm;
- It can be parallelised.
2. Related Works
2.1. Sweeping Paradigm
2.2. Characterisation Methods and Skeletons
3. Materials and Methods
- Initialisation;
- Multi-sweeping;
- Finalisation.
3.1. Initialisation
3.2. Multi-Sweeping
- Sorting of geometric objects is not needed as the cells in are organised clearly;
- is not infinite, but bounded by its frontier cells, i.e., , , , , , ;
- does not move from an event to an event, but advances through the consecutive frontier cells.
- Too small values result in many similar (or even equal) chains, which do not contribute additional information to the shape characterisation and slow down the whole process.
- Large values may cause some local feature to be missed if the filtering process, as described in Section 3.3, is applied.
- It is practical that is an integer divisor of .
Algorithm 1 The multi-sweep-line part of MSCA. |
|
Algorithm 2 Algorithm returns the sweep-line’s endpoints. |
|
3.3. Finalisation
- Chain filtering;
- Chain vectorisation;
- Normalisation.
- Chain filtering: Different filtering methods can be designed; however, for demonstration purposes, the following two methods are proposed:
- (a)
- (b)
- The average angle of lines is calculated, determined by the sequential pairs of midpoints . is accepted if is close to being perpendicular in regard to , i.e, if the heuristic, given in (2), is valid.
- Chain vectorisation: Round-off errors in the raster space are, unfortunately, unavoidable. Therefore, it is favourable to vectorise to minimise the effect of the round-off errors in the further characterisation process. The well-known Douglas–Peucker algorithm [44] was applied on . The set of polylines was obtained, which replaced in the further steps of the algorithm.
- Normalisation: The normalisation is performed to make the characterisation of insensitive to scaling or rotation. is transformed into a normalised bounding box according to (3).
3.4. Time Complexity of the Algorithm
4. Experiments
4.1. Demonstration of MSCA on Testing Shapes
4.2. Recognition of Equal Objects
- For each polyline , , its length was calculated according to (4).
- Components of an individual vector were sorted after that in decreasing order.
- , where denotes the cardinality of vectors, and if this condition is true;
- , , where ≈ corresponds to a user-defined 5% tolerance.
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
- Mortenson, M.E. Geometric Modeling; Wileys: New York, NY, USA, 1985. [Google Scholar]
- Hoffmann, C.M. Geometric and Solid Modeling: An Introduction; Morgan Kaufmann Pub.: San Mateo, CA, USA, 1989. [Google Scholar]
- Liu, H.; Motoda, H. Feature Selection for Knowledge Discovery and Data Minimg; Kluwer Academic Publishers: New York, NY, USA, 1998. [Google Scholar]
- de Berg, M.; van Kreveld, M.; Overmars, M.; Schwarzkopf, O. Computational Geometry: Algorithms and Applications; Springer: Berlin, Germany, 1997. [Google Scholar]
- Shamos, M.I.; Hoey, D. Geometric intersection problems. In Proceedings of the 17th Annual Symposium on Foundations of Computer Science (SFCS 1976), Houston, TX, USA, 25–27 October 1976; pp. 208–215. [Google Scholar]
- Ferreira, C.R.; Andrade, M.V.A.; Magalhes, S.V.G.; Franklin, W.R.; Pena, G.C. A Parallel Sweep Line Algorithm for Visibility Computation. In Proceedings of the XIV GEOINFO, Campos do Jordão, Brazil, 24–27 November 2013; pp. 85–96. [Google Scholar]
- Kim, D.S.; Lee, B.; Sugihara, K. A sweep-line algorithm for the inclusion hierarchy among circles. Jpn. J. Ind. Appl. Math. 2006, 23, 127–138. [Google Scholar] [CrossRef]
- Žalik, B.; Jezernik, A.; Rizman Žalik, K. Polygon trapezoidation by sets of open trapezoids. Comput. Graph-UK 2003, 27, 791–800. [Google Scholar] [CrossRef]
- Rizman Žalik, K.; Žalik, B. A sweep-line algorithm for spatial clustering. Adv. Eng. Softw. 2009, 40, 445–451. [Google Scholar] [CrossRef]
- Lukač, N.; Žalik, B.; Rizman Žalik, K. Sweep-hyperplane clustering algorithm using dynamic model. Informatica 2014, 25, 564–580. [Google Scholar] [CrossRef] [Green Version]
- Domiter, V.; Žalik, B. Sweep-line algorithm for constrained Delaunay triangulation. Int. J. Geogr. Inf. Sci. 2008, 22, 449–462. [Google Scholar] [CrossRef]
- Žalik, B. An efficient sweep-line Delaunay triangulation algorithm. Comput. Aided Des. 2005, 37, 1027–1038. [Google Scholar] [CrossRef]
- Fortune, S. A sweepline algorithm for Voronoi diagrams. Algorithmica 1987, 2, 153–174. [Google Scholar] [CrossRef]
- Murtojärvi, M.; Leppänen, V.; Nevalainen, O.S. Determining directional distances between points and shorelines using sweep-line technique. Int. J. Geogr. Inf. Sci. 2009, 23, 355–368. [Google Scholar] [CrossRef]
- Pavlidis, T. A review of algorithms for shape analysis. Comput. Graph. Image Process. 1978, 7, 243–258. [Google Scholar] [CrossRef]
- Loncaric, S. A survey of shape analysis techniques. Pattern Recogn. 1998, 31, 983–1001. [Google Scholar] [CrossRef]
- Hossain, M.D.; Chen, D. Segmentation for Object-Based Image Analysis (OBIA) a review of algorithms and challenges from remote sensing perspective. ISPRS J. Photogramm. 2019, 150, 115–134. [Google Scholar] [CrossRef]
- Burger, W.; Burge, M.J. Principles of Digital Image Processing; Springer: London, UK, 2009. [Google Scholar]
- Solomon, C.; Brekon, T. Fundamentals of Digital Image Processing; Wiley-Blackwell: Chichester, UK, 2011. [Google Scholar]
- Gonzales, R.; Woods, R. Digital Image Processing; Pearson Prentice Hall: Upper Saddle River, NJ, USA, 2017. [Google Scholar]
- Blum, H. A Transformation for Extracting New Descriptors of Shape. In Models for the Perception of Speech and Visual Form; Wathen-Dunn, W., Ed.; MIT Press: Cambridge, MA, USA, 1967; pp. 362–380. [Google Scholar]
- Leborgne, A.; Mille, J.; Tougne, L. Extracting Noise-Resistant Skeleton on Digital Shapes for Graph Matching. In Advances in Visual Computing, Proceedings of the 10th International Symposium, ISVC 2014, Las Vegas, NV, USA, 8–10 December 2014; Bebis, G., Li, B., Yao, A., Liu, Y., Duan, Y., Lau, M., Khadka, R., Crisan, A., Chang, R., Eds.; Lecture Notes in Computer Science 8888 (Part II); Springer: Cham, Switzerland, 2014; pp. 293–302. [Google Scholar]
- Aichholzer, O.; Aurenhammer, F.; Alberts, D.; Gärtner, B. A novel type of skeleton for polygons. J. Univers. Comput. Sci. 1995, 1, 752–761. [Google Scholar]
- Aichholzer, O.; Aurenhammer, F. Straight skeletons for general polygonal figures in the plane. In Proceedings of the Annual International Conference on Computing and Combinatorics (COCOON’96), Hong Kong, 17–19 June 1996; Cai, J.-Y., Wong, C.K., Eds.; Lecture Notes in Computer Science 1090. Springer: Berlin/Heidelberg, Germany, 1996; pp. 117–126. [Google Scholar]
- Smogavec, G.; Žalik, B. A fast algorithm for constructing approximate medial axis of polygons, using Steiner points. Adv. Eng. Softw. 2012, 52, 1–9. [Google Scholar] [CrossRef]
- Giesen, J.; Miklos, B.; Pauly, M.; Wormser, C. The Scale Axis Transform. In Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry (SCG’09), Aarhus, Denmark, 8–10 June 2009; Hershberger, J., Fogel, E., Eds.; ACM: New York, NY, USA, 2009; pp. 106–116. [Google Scholar]
- Kirkpatrick, D.G.; Radke, J.D. A framework for computational morphology. In Computational Geometry, Machine Intelligence and Pattern Recognition; Toussaint, G.T., Ed.; Elsevier: Amsterdam, The Netherlands, 1985; Volume 2, pp. 217–248. [Google Scholar]
- Goh, W.-B. Strategies for shape matching using skeletons. Comput. Vis. Image Underst. 2008, 110, 326–345. [Google Scholar] [CrossRef] [Green Version]
- Ma, C.; Zhang, S.; Wang, A.; Qi, Y.; Chen, G. Skeleton-Based Dynamic Hand Gesture Recognition Using an Enhanced Network with One-Shot Learning. Appl. Sci. 2020, 10, 3680. [Google Scholar] [CrossRef]
- Liu, J.; Wang, G.; Duan, L.; Abdiyeva, K.; Kot, A.C. Skeleton-based Human Action Recognition with Global Context-Aware Attention LSTM Networks. IEEE Trans. Image Process. 2018, 27, 1586–1599. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Tasnim, N.; Islam, M.M.; Baek, J.-H. Deep Learning-Based Action Recognition Using 3D Skeleton Joints Information. Inventions 2020, 5, 49. [Google Scholar] [CrossRef]
- Papadopoulos, K.; Demisse, G.; Ghorbel, E.; Antunes, M.; Aouada, D.; Ottersten, B. Localized trajectories for 2D and 3D action recognition. Sensors 2019, 19, 3503. [Google Scholar] [CrossRef] [Green Version]
- Wang, C. Research on the Detection Method of Implicit Self Symmetry in a High-Level Semantic Model. Symmetry 2020, 12, 28. [Google Scholar] [CrossRef] [Green Version]
- Khanna, N.N.; Jamthikar, A.D.; Gupta, D.; Piga, M.; Saba, L.; Carcassi, C.; Giannopoulos, A.A.; Nicolaides, A.; Laird, J.R.; Suri, H.S.; et al. Rheumatoid arthritis: Atherosclerosis imaging and cardiovascular risk assessment using machine and deep learning-based tissue characterization. Curr. Atheroscler. Rep. 2019, 21, 7. [Google Scholar] [CrossRef] [PubMed]
- Dadoun, H.; Rousseau, A.L.; de Kerviler, E.; Correas, J.M.; Tissier, A.M.; Joujou, F.; Bodard, S.; Khezzane, K.; de Margerie-Mellon, C.; Delingette, H.; et al. Deep learning for the detection, localization, and characterization of focal liver lesions on abdominal US images. Radiol. Artif. Intell. 2022, 4, e210110. [Google Scholar] [CrossRef]
- Yan, X.; Ai, T.; Yang, M.; Yin, H. A graph convolutional neural network for classification of building patterns using spatial vector data. ISPRS J. Photogramm. Remote Sens. 2019, 150, 259–273. [Google Scholar] [CrossRef]
- Bisheh, M.N.; Wang, X.; Chang, S.I.; Lei, S.; Ma, J. Image-based characterization of laser scribing quality using transfer learning. J. Intell. Manuf. 2022, 34, 2307–2319. [Google Scholar] [CrossRef]
- Freeman, H. On the encoding of arbitrary geometric configurations. IRE Trans. Electron. Comput. 1961, EC10, 260–268. [Google Scholar] [CrossRef]
- Bribiesca, E. A new chain code. Pattern Recogn. 1999, 32, 235–251. [Google Scholar] [CrossRef]
- Sánchez-Cruz, H.; Rodríguez-Dagnino, R.M. Compressing bi-level images by means of a 3-bit chain code. Opt. Eng. 2005, 44, 1–8. [Google Scholar]
- Žalik, B.; Mongus, D.; Liu, Y.-K.; Lukač, N. Unsigned Manhattan chain code. J. Vis. Commun. Image Represent. 2016, 38, 186–194. [Google Scholar] [CrossRef]
- Cleary, J.C.; Wyvill, G. Analysis of an Algorithm for Fast Ray Tracing using Uniform Space Subdivision. Vis. Comput. 1988, 4, 65–83. [Google Scholar] [CrossRef]
- Žalik, B.; Clapworthy, G.; Oblonšek, Č. An Efficient Code-Based Voxel-Traversing Algorithm. Comput. Graph. Forum 1997, 16, 119–128. [Google Scholar] [CrossRef]
- Douglas, B.; Peucker, T. Algorithms for the reduction of the number of points required to represent a digitized line or its caricature. Cartographica 1973, 10, 112–122. [Google Scholar] [CrossRef] [Green Version]
No. of | No. of. Holes | No. of Chains | ||
---|---|---|---|---|
Circle | 1068 | 0 | 327 × 327 | 12 |
Square | 1068 | 0 | 327 × 327 | 12 |
Rocket | 1232 | 0 | 343 × 412 | 26 |
Hand | 1860 | 0 | 363 × 352 | 82 |
Airplane | 2260 | 0 | 430 × 431 | 52 |
Penguin | 1968 | 0 | 390 × 484 | 45 |
Runner | 2896 | 0 | 471 × 508 | 56 |
Buddha | 11,366 | 1 | 2648 × 2850 | 46 |
Detective | 14,128 | 2 | 2440 × 2850 | 65 |
Ballet | 16,712 | 1 | 2313 × 2575 | 117 |
Dragon | 26,334 | 2 | 2807 × 2848 | 104 |
Cupid | 25,160 | 4 | 2727 × 2721 | 157 |
Single-Threaded Time (s) | Multi-Threaded Time (s) | |
---|---|---|
Circle | 0.076 | 0.152 |
Square | 0.081 | 0.183 |
Rocket | 0.100 | 0.153 |
Hand | 0.106 | 0.157 |
Airplane | 0.132 | 0.205 |
Penguin | 0.141 | 0.175 |
Runner | 0.195 | 0.220 |
Buddha | 3.508 | 1.798 |
Detective | 4.311 | 1.712 |
Ballet | 4.794 | 1.511 |
Dragon | 9.297 | 2.630 |
Cupid | 11.384 | 2.994 |
A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
A | – 1 | 2 | ✓3 | |||||||||||||
B | – | |||||||||||||||
C | – | ✓ | ✓ | |||||||||||||
D | ✓ | – | ||||||||||||||
E | ✓ | – | ✓ | |||||||||||||
F | – | ✓ | ||||||||||||||
G | – | ✓ | ||||||||||||||
H | – | ✓ | ||||||||||||||
I | – | ✓ | ||||||||||||||
J | – | ✓ | ||||||||||||||
K | ✓ | – | ||||||||||||||
L | ✓ | – | ||||||||||||||
M | ✓ | – | ||||||||||||||
N | ✓ | – | ||||||||||||||
O | ✓ | ✓ | – | |||||||||||||
P | ✓ | – |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Žalik , B.; Strnad , D.; Podgorelec , D.; Kolingerová , I.; Nerat , A.; Lukač , N.; Kohek , Š.; Lukač , L. Geometric Shape Characterisation Based on a Multi-Sweeping Paradigm. Symmetry 2023, 15, 1212. https://doi.org/10.3390/sym15061212
Žalik B, Strnad D, Podgorelec D, Kolingerová I, Nerat A, Lukač N, Kohek Š, Lukač L. Geometric Shape Characterisation Based on a Multi-Sweeping Paradigm. Symmetry. 2023; 15(6):1212. https://doi.org/10.3390/sym15061212
Chicago/Turabian StyleŽalik , Borut, Damjan Strnad , David Podgorelec , Ivana Kolingerová , Andrej Nerat , Niko Lukač , Štefan Kohek , and Luka Lukač . 2023. "Geometric Shape Characterisation Based on a Multi-Sweeping Paradigm" Symmetry 15, no. 6: 1212. https://doi.org/10.3390/sym15061212