A Unified Curvature Definition for Regular, Polygonal, and Digital Planar Curves
Tóm tắt
In this paper, we propose a new definition of curvature, called visual curvature. It is based on statistics of the extreme points of the height functions computed over all directions. By gradually ignoring relatively small heights, a multi-scale curvature is obtained. The theoretical properties and the experiments presented demonstrate that multi-scale visual curvature is stable, even in the presence of significant noise. To our best knowledge, the proposed definition of visual curvature is the first ever that applies to regular curves as defined in differential geometry as well as to turn angles of polygonal curves. Moreover, it yields stable curvature estimates of curves in digital images even under sever distortions. We also show a relation between multi-scale visual curvature and convexity of simple closed curves.
Tài liệu tham khảo
Adamek, T., & Connor, O. (2004). A multiscale representation method for nonrigid shapes with a single closed contour. IEEE Transactions on Circuits and Systems for Video Technology, 14(5), 742–753.
Anderson, M., & Bezdek, J. C. (1984). Curvature and tangential deflection of discrete arcs: A theory based on the commutator of scatter matrix pairs and its application to vertex detection in planar shape data. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 27–40.
Ansari, N., & Delp, E. J. (1991). On detecting dominant points. Pattern Recognition, 24(5), 441–451.
Asada, H., & Brady, M. (1986). The curvature primal sketch. IEEE Transactions on Pattern Analysis and Machine Intelligence, 8, 2–14.
Asian, C., & Tari, S. (2005). An axis-based representation for recognition. In IEEE international conference on computer vision (Vol. 2, pp. 1339–1346).
Bai, X., Latecki, L. J., & Liu, W.-Y. (2007). Skeleton pruning by contour partitioning with discrete curve evolution. IEEE Transaction on Pattern Analysis and Machine Intelligence, 29, 449–462.
Belyaev, A. (2004). Plane and space curves. Curvature. Curvature-based features. Max-Planck-Institut für Informatik.
Bengtsson, A., & Eklundh, J.-O. (1991). Shape representation by multiscale contour approximation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 13(1).
Beus, H. L., & Tiu, S. S. H. (1987). An improved corner detection algorithm based on chain-coded plane curves. Pattern Recognition, 20, 291–296.
Boutin, M. (2000). Numerically invariant signature curves. International Journal of Computer Vision, 40, 235–248.
Calabi, E., Olver, P. J., Shakiban, C., Tannenbaum, A., & Haker, S. (1998). Differential and numerically invariant signature curves applied to object recognition. International Journal of Computer Vision, 26, 107–135.
Cartan, E. (1935). Lamethode du repere mobile, la theorie des groupes continues et les espaces generalises exposes de geometrie No. 5. Paris: Hermann.
Cederberg, R. L. T. (1978). An iterative algorithm for angle detection on digital curves. In International conference on pattern recognition (pp. 576–578). Kyoto, Japan.
Chetverikov, D., & Szabo, Zs. (1999). A simple and efficient algorithm for detection of high curvature points in planar curves. In Proceedings of 23rd workshop of the Austrian pattern recognition group (pp. 175–184).
Coeurjolly, D., Serge, M., & Laure, T. (2001). Discrete curvature based on osculating circles estimation. In Lecture notes in computer science (Vol. 2059, pp. 303–312). Berlin: Springer.
Cong, G., & Parvin, B. (1998). Curve evolution for corner enhancement. In International conference on pattern recognition (Vol. 1, pp. 708–710).
Dudek, G., & Tsotsos, J. K. (1997). Shape representation and recognition from multiscale curvature. Computer Vision and Image Understanding, 68(2), 170–189.
Dunham, J. G. (1986). Optimum uniform piecewise linear approximation of planar curves. IEEE Transactions on Pattern Analysis and Machine Intelligence, 8, 67–75.
Freeman, H., & Davis, L. S. (1977). A corner-finding algorithm for chain-coded curves. IEEE Transactions on Computers C, 26, 297–303.
Gumhold, S. (2004). Designing optimal curves in 2D. CEIG 2004.
Hermann, S., & Klette, R. (2003). Multigrid analysis of curvature estimators. In Image vision and computing, New Zealand (pp. 108–112).
Hermann, S., & Klette, R. (2005). Global curvature estimation for corner detection. In International conference on image and vision computing (IVCNZ), Dunedin/New Zealand (pp. 272–277).
Hermann, S., & Klette, R. (2007). A comparative study on 2d curvature estimators. In International conference on computing: Theory and applications (ICCTA’07).
Katzir, N., Lindenbaum, M., & Porat, M. (1994). Curve segmentation under partial occlusion. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16, 513–519.
Kovalevsky, V. (2001). Curvature in digital 2D images. International Journal of Pattern Recognition and Artificial Intelligence, 15, 1183–1200.
Kruse, B., & Rao, C. V. K. (1978). A matched filtering technique for corner detection. In International conference on pattern recognition (pp. 642–644). Kyoto, Japan
Latecki, L. J., & Rosenfeld, A. (1998). Supportedness and tameness differentialless geometry of plane curves. Pattern Recognition, 31(5), 607–622.
Lewiner, T. (2004). Arc-length based curvature estimator. In SIBGRAPI’04 (pp. 250–257).
Lewiner, T., Gomes, J., Jr., Lopes, H., & Craizer, M. (2004). Curvature estimation: Theory and practice. Prepublicacoes do Departamento de Matematica da PUC-Rio.
Lowe, D. G. (1988). Organization of smooth image curves at multiple scales. In International conference on computer vision (pp. 558–567).
Lu, F., & Milios, E. E. (1991). Optimal local spline approximation of planar shape. In International conference on acoustics, speech, and signal processing (Vol. 4, pp. 2469–2472).
Marji, M. (2003). On the detection of dominant points on digital planar curves. Ph.D. Thesis, June 2003.
Medioni, G., & Yasumoto, Y. (1986). Corner detection and curve representation using cubic B-splines. In IEEE international conference on robotics and automation (Vol. 3, pp. 764–769).
Mokhtarian, F., & Mackworth, A. K. (1992). A theory of multi-scale, curvature-based shape representation for planar curves. IEEE Transactions on Pattern Analysis and Machine Intelligence, 14, 789–805.
Mokhtarian, F., & Suomela, R. (1998). Robust image corner detection through curvature scale space. IEEE Transactions on Pattern Analysis and Machine Intelligence, 20(12), 1376–1381.
Pavlidis, T. (1980). Algorithms for shape analysis and waveforms. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2, 301–312.
Pinheiro, A. M. G., Izquierdo, E., & Ghanhari, M. (2000). Shape matching using a curvature based polygonal approximation in scale-space. In International conference on image processing (Vol. 2, pp. 538–541).
Rattarangsi, A., & Roland, T. (1992). Scale-based detection of corners of planar curves. IEEE Transactions on Pattern Analysis and Machine Intelligence, 14, 430–449.
Rosenfeld, A., & Johnston, E. (1973). Angle detection on digital curves. IEEE Transactions on Computers C, 22, 875–878.
Rosenfeld, A., & Weszka, J. S. (1975). An improved method of angle detection on digital curves. IEEE Transactions on Computers C, 24, 940–941.
Sankar, P. V., & Sharma, C. V. (1978). A parallel procedure for the detection of dominant points on a digital curve. Computer Graphics and Image Processing, 7, 403–412.
Utcke, S. (2003). Error-bounds on curvature estimation. In Lecture note in computer science (Vol. 2695, pp. 657–666). Berlin: Springer.
Wang, Y.-P., Lee, S. L., & Toraichi, K. (1999). Multiscale curvature-based shape representation using B-spline wavelets. IEEE Transactions on Image Processing, 8(11).
Worring, M., & Smeulders, A. W. M. (1993). Digital curvature estimation. CVGIP: Image Understanding, 58, 366–382.
Yuille, A. L. (1989). Zero crossings on lines of curvature. Computer Vision Graphics Image Process, 45, 68–87.