Abstract
We propose a novel curvature estimation algorithm which is capable of estimating the curvature of digital curves and two-dimensional curved image structures. The algorithm is based on the conformal projection of the curve or image signal to the two-sphere. Due to the geometric structure of the embedded signal the curvature may be estimated in terms of first order partial derivatives in ℝ3. This structure allows us to obtain the curvature by just convolving the projected signal with the appropriate kernels. We show that the method performs an implicit plane fitting by convolving the projected signals with the derivative kernels. Since the algorithm is based on convolutions its implementation is straightforward for digital curves as well as images. We compare the proposed method with differential geometric curvature estimators. It turns out that the novel estimator is as accurate as the standard differential geometric methods in synthetic as well as real and noise perturbed environments.
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References
Hermann, S., Klette, R.: Global Curvature Estimation for Corner Detection. Technical report, The University of Auckland, New Zealand (2005)
Williams, D.J., Shah, M.: A Fast Algorithm for Active Contours and Curvature Estimation. CVGIP: Image Underst. 55(1), 14–26 (1992)
Morse, B., Schwartzwald, D.: Isophote-based Interpolation. In: International Conference on Image Processing, vol. 3, p. 227 (1998)
Lachaud, J.O., Vialard, A., de Vieilleville, F.: Fast, Accurate and Convergent Tangent Estimation on Digital Contours. Image Vision Comput. 25(10), 1572–1587 (2007)
Coeurjolly, D., Miguet, S., Tougne, L.: Discrete Curvature Based on Osculating Circle Estimation. In: Arcelli, C., Cordella, L.P., Sanniti di Baja, G. (eds.) IWVF 2001. LNCS, vol. 2059, pp. 303–312. Springer, Heidelberg (2001)
Hermann, S., Klette, R.: Multigrid Analysis of Curvature Estimators. In: Proc. Image Vision Computing, New Zealand, pp. 108–112 (2003)
Klette, R., Rosenfeld, A.: Digital Geometry: Geometric Methods for Digital Picture Analysis. Morgan Kaufmann, San Francisco (2004)
Wietzke, L., Fleischmann, O., Sommer, G.: 2D Image Analysis by Generalized Hilbert Transforms in Conformal Space. In: Forsyth, D., Torr, P., Zisserman, A. (eds.) ECCV 2008, Part II. LNCS, vol. 5303, pp. 638–649. Springer, Heidelberg (2008)
do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice-Hall, Englewood Cliffs (1976)
Needham, T.: Visual Complex Analysis. Oxford University Press, USA (1999)
Zayed, A.: Handbook of Function and Generalized Function Transformations. CRC, Boca Raton (1996)
Kanatani, K.: Statistical Optimization for Geometric Computation: Theory and Practice. Elsevier Science Inc., New York (1996)
Lindeberg, T.: Scale-space Theory in Computer Vision. Springer, Heidelberg (1993)
Felsberg, M., Sommer, G.: The Monogenic Scale-space: A Unifying Approach to Phase-based Image Processing in Scale-space. Journal of Mathematical Imaging and vision 21(1), 5–26 (2004)
Gander, W., Golub, G.H., Strebel, R.: Least-Squares Fitting of Circles and Ellipses. BIT (4), 558–578 (1994)
Coope, I.D.: Circle Fitting by Linear and Nonlinear Least Squares. Journal of Optimization Theory and Applications 76(2), 381–388 (1993)
Romeny, B.M.: Geometry-Driven Diffusion in Computer Vision. Kluwer Academic Publishers, Norwell (1994)
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Fleischmann, O., Wietzke, L., Sommer, G. (2010). A Novel Curvature Estimator for Digital Curves and Images. In: Goesele, M., Roth, S., Kuijper, A., Schiele, B., Schindler, K. (eds) Pattern Recognition. DAGM 2010. Lecture Notes in Computer Science, vol 6376. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15986-2_45
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DOI: https://doi.org/10.1007/978-3-642-15986-2_45
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