Abstract
Estimation of differentials of discrete signals is almost mandatory in digital segmentation. We present a new fast method based on convolutions by a mask with a logarithmic number of constant layers. Then we compare it to other multigrid convergent methods in the field such as the Binomial Convolution, the Digital Straight Segment Tangent Estimator, and the Taylor Polynomial Fitting. Our convolution method’s main advantage is its complexity of O(2n.log2(m)), which makes it competitive to the convolution by Fast Fourier Transform (FFT) latest implementation. In the experimental part, we also tested the precision of the first order derivative estimation, its resistance to noise and its convergence rate.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Cooley, J., Tukey, J.: An algorithm for the machine calculation of complex fourier series. Math. Comput. 19(90), 297–301 (1965)
De Vieilleville, F., Lachaud, J.O.: Comparison and improvement of tangent estimators on digital curves. Pattern Recognition 42(8), 1693–1707 (2009)
Esbelin, H., Malgouyres, R., Cartade, C.: Convergence of binomial-based derivative estimation for 2 noisy discretized curves. Theoretical Computer Science 412(36), 4805 (2011)
Fousse, L., Hanrot, G., Lefèvre, V., Pélissier, P., Zimmermann, P.: MPFR: A multiple-precision binary floating-point library with correct rounding. ACM Transactions on Mathematical Software 33(2), 13:1–13:15 (2007)
Frigo, M., Johnson, S.G.: The design and implementation of FFTW3. Proceedings of the IEEE 93(2), 216–231 (2005); Special issue on “Program Generation, Optimization, and Platform Adaptation”
Galassi, M., Davies, J., Theiler, J., Gough, B., Jungman, G., Booth, M., Rossi, F.: GNU scientific library. Network Theory Ltd. (2002)
Heideman, M., Johnson, D., Burrus, C.: Gauss and the history of the fast fourier transform. IEEE ASSP Magazine 1(4), 14–21 (1984)
Johnson, S., Frigo, M.: A modified split-radix fft with fewer arithmetic operations. IEEE Transactions on Signal Processing 55(1), 111–119 (2007)
Kass, M., Witkin, A., Terzopoulos, D.: Snakes: Active contour models. International Journal of Computer Vision 1(4), 321–331 (1988)
Kerautret, B., Lachaud, J.-O.: Curvature estimation along noisy digital contours by approximate global optimization. Pattern Recognition 42(10), 2265–2278 (2009)
Kerautret, B., Lachaud, J.-O., Naegel, B.: Comparison of Discrete Curvature Estimators and Application to Corner Detection. In: Bebis, G., Boyle, R., Parvin, B., Koracin, D., Remagnino, P., Porikli, F., Peters, J., Klosowski, J., Arns, L., Chun, Y.K., Rhyne, T.-M., Monroe, L. (eds.) ISVC 2008, Part I. LNCS, vol. 5358, pp. 710–719. Springer, Heidelberg (2008)
Lachaud, J., Vialard, A., de Vieilleville, F.: Fast, accurate and convergent tangent estimation on digital contours. Image and Vision Computing 25(10), 1572–1587 (2007)
Lindeberg, T.: Scale-space theory in computer vision. Springer (1994)
Lundy, T., Van Buskirk, J.: A new matrix approach to real ffts and convolutions of length 2 k. Computing 80(1), 23–45 (2007)
Ma, Z., Tavares, J., Jorge, R., Mascarenhas, T.: A review of algorithms for medical image segmentation and their applications to the female pelvic cavity. Computer Methods in Biomechanics and Biomedical Engineering 13(2), 235–246 (2010)
Malgouyres, R., Brunet, F., Fourey, S.: Binomial Convolutions and Derivatives Estimation from Noisy Discretizations. In: Coeurjolly, D., Sivignon, I., Tougne, L., Dupont, F. (eds.) DGCI 2008. LNCS, vol. 4992, pp. 370–379. Springer, Heidelberg (2008)
Provot, L., Gérard, Y.: Estimation of the derivatives of a digital function with a convergent bounded error. In: Discrete Geometry for Computer Imagery, pp. 284–295. Springer (2011)
Vialard, A.: Geometrical Parameters Extraction from Discrete Paths. In: Miguet, S., Ubéda, S., Montanvert, A. (eds.) DGCI 1996. LNCS, vol. 1176, pp. 24–35. Springer, Heidelberg (1996)
de Vieilleville, F., Lachaud, J., Feschet, F.: Maximal digital straight segments and convergence of discrete geometric estimators. Image Analysis, 988–997 (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gonzalez, D., Malgouyres, R., Esbelin, HA., Samir, C. (2012). Fast Level-Wise Convolution. In: Barneva, R.P., Brimkov, V.E., Aggarwal, J.K. (eds) Combinatorial Image Analaysis. IWCIA 2012. Lecture Notes in Computer Science, vol 7655. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34732-0_17
Download citation
DOI: https://doi.org/10.1007/978-3-642-34732-0_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-34731-3
Online ISBN: 978-3-642-34732-0
eBook Packages: Computer ScienceComputer Science (R0)