Abstract
In ℝ2, rigid transformations are topology-preserving operations. However, this property is generally no longer true when considering digital images instead of continuous ones, due to digitization effects. In this article, we investigate this issue by studying discrete rigid transformations (DRTs) on ℤ2. More precisely, we define conditions under which digital images preserve their topological properties under any arbitrary DRTs. Based on the recently introduced notion of DRT graph and the classical notion of simple point, we first identify a family of local patterns that authorize topological invariance under DRTs. These patterns are then involved in a local analysis process that guarantees topological invariance of whole digital images in linear time.
The research leading to these results has received funding from the French Agence Nationale de la Recherche (Grant Agreement ANR-2010-BLAN-0205 03).
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Zitová, B., Flusser, J.: Image registration methods: A survey. Image and Vision Computing 21, 977–1000 (2003)
Yilmaz, A., Javed, O., Shah, M.: Object tracking: A survey. ACM Computing Surveys 38, 1–45 (2006)
Jacob, M.A., Andres, E.: On discrete rotations. In: DGCI, Proceedings, pp. 161–174 (1995)
Andres, E.: The Quasi-Shear Rotation. In: Miguet, S., Ubéda, S., Montanvert, A. (eds.) DGCI 1996. LNCS, vol. 1176, pp. 307–314. Springer, Heidelberg (1996)
Nouvel, B., Rémila, E.: Configurations induced by discrete rotations: Periodicity and quasi-periodicity properties. DAM 147, 325–343 (2005)
Thibault, Y., Kenmochi, Y., Sugimoto, A.: Computing upper and lower bounds of rotation angles from digital images. Pattern Recognition 42, 1708–1717 (2009)
Ngo, P., Kenmochi, Y., Passat, N., Talbot, H.: Combinatorial structure of rigid transformations in 2D digital images. To appear in Computer Vision and Image Understanding
Ngo, P., Kenmochi, Y., Passat, N., Talbot, H.: Combinatorial Properties of 2D Discrete Rigid Transformations under Pixel-Invariance Constraints. In: Barneva, R.P., Brimkov, V.E., Aggarwal, J.K. (eds.) IWCIA 2012. LNCS, vol. 7655, pp. 234–248. Springer, Heidelberg (2012)
Kong, T.Y., Rosenfeld, A.: Digital topology: Introduction and survey. Computer Vision Graphics & Image Processing 48, 357–393 (1989)
Couprie, M., Bertrand, G.: New characterizations of simple points in 2D, 3D, and 4D discrete spaces. IEEE Transactions on Pattern Analysis and Machine Intelligence 31, 637–648 (2009)
Khalimsky, E.: Topological structures in computer science. Journal of Applied Mathematics and Simulation 1, 25–40 (1987)
Kovalevsky, V.A.: Finite topology as applied to image analysis. Computer Vision, Graphics & Image Processing 46, 141–161 (1989)
Bertrand, G., Couprie, M., Passat, N.: A note on 3-D simple points and simple-equivalence. Information Processing Letters 109, 700–704 (2009)
Mazo, L., Passat, N., Couprie, M., Ronse, C.: Topology on digital label images. Journal of Mathematical Imaging and Vision 44, 254–281 (2012)
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Ngo, P., Kenmochi, Y., Passat, N., Talbot, H. (2013). Sufficient Conditions for Topological Invariance of 2D Images under Rigid Transformations. In: Gonzalez-Diaz, R., Jimenez, MJ., Medrano, B. (eds) Discrete Geometry for Computer Imagery. DGCI 2013. Lecture Notes in Computer Science, vol 7749. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37067-0_14
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DOI: https://doi.org/10.1007/978-3-642-37067-0_14
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