Abstract
This paper informs about number-theoretical and geometrical estimates of worst-case bounds for quantization errors in calculating features such as moments, moment based features, or perimeters in image analysis, and about probability-theoretical estimates of error bounds (e.g. standard deviations) for such digital approximations. New estimates (with proofs) and a review of previously known results are provided.
Similar content being viewed by others
References
L. O'Gorman, “Subpixel precision of straight-edged sets for registration and measurement,” IEEE Trans. PAMI, Vol. 18, pp. 746-751, 1996.
M. Gruber and K.-Y. Hsu, “Moment-based image normalization with high noise-tolerance,” IEEE Trans. PAMI, Vol. 19, pp. 136-139, 1997.
R.M. Haralick and L.G. Shapiro, Computer and Robot Vision, Vol. II, Addison-Wesley: Reading, Massachusetts, 1993.
C.S. Ho, “Precision of digital vision systems,” IEEE Trans. PAMI, Vol. 5, pp. 593-691, 1983.
M. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Inf. Theory, Vol. 8, pp. 179-187, 1962.
M.N. Huxley, “The area within a curve,” Proc. Indian. Acad. Sci., Vol. 97, pp. 111-116, 1987.
M.N. Huxley, “Exponential sums and lattice points,” Proc. London Math. Soc., Vol. 60, pp. 471-502, 1990.
A. Ivić, An Introduction in Analytical Number Theory (in Serbian), Izdavaćka knjižarica Zorana Stojanovića: Novi Sad, 1996.
H. Iwaniec and C.J. Mozzochi, “On the divisor and circles problems,” J. Number Theory, Vol. 29, pp. 60-93, 1988.
R. Jain, R. Kasturi, and B.G. Schunck, Machine Vision, McGraw-Hill: New York, 1995.
C. Jordan, “Remarques sur les intégrales deéfinies,” Journal de Mathématiques, 4e série, T., Vol. 8, pp. 69-99, 1892.
B. Kamgar-Parsi and B. Kamgar-Parsi, “Evaluation of quantization error in computer vision,” IEEE Trans. PAMI, Vol. 11, pp. 929-940, 1989.
D.G. Kendall, “On the number of lattice points inside a random oval,” Quart. J. Math. Oxford, Vol. 19, pp. 1-26, 1948.
R. Klette, “Approximation and representation of 3D objects,” in Advances in Digital and Computational Geometry, R. Klette, A. Rosenfeld, and F. Sloboda (Eds.), Springer: Singapore, 1998, pp. 161-194.
R. Klette, V. Kovalevsky, and B. Yip, “Length estimation of digital curves,” in “Vision Geometry VIII,” Denver, Vol. 3811, 1999, pp. 117-128.
R. Klette and J. Žunić, “Errors in calculated moments of convex sets using digital images,” in “Vision Geometry VIII,” Denver, Vol. 3811, 1999, pp. 105-116.
R. Klette and J. Žunić, “Digital approximation of moments of convex sets,” Graphical Models and Image Processing. Vol. 61, pp. 274-298, 1999.
V. Kovalevsky and S. Fuchs, “Theoretical and experimental analysis of the accuracy of perimeter estimates,” in Robust Computer Vision, W. Förstner and S. Ruwiedel (Eds.), Wichmann: Karlsruhe, pp. 218-242, 1992.
H. Minkowski, Geometrie der Zahlen, Teubner: Leipzig, 1910.
A. Rosenfeld, “Digital straight line segments,” IEEE Trans. Computers, Vol. 23, pp. 1264-1269, 1974.
L.A. Santalo, Integral Geometry and Geometrical Probability, Addison-Wesley: London, 1976.
W. Scherrer, “Ein Satz über Gitter und Volumen,” Mathematische Annalen, Vol. 86, pp. 99-107, 1922.
D. Shen and H.H.S. Ip, “Generalized affine invariant image normalization,” IEEE Trans. PAMI, Vol. 19, pp. 431-440, 1997.
F. Sloboda, B. Zatko, and J. Stoer, “On approximation of planar one-dimensional continua,” in Advances in Digital and Computational Geometry, R. Klette, A. Rosenfeld, and F. Sloboda (Eds.), Springer: Singapore, 1998, pp. 113-160.
F. Sloboda, B. Zatko, and R. Klette, “On the topology of grid continua,” in Proc. Vision Geometry VII, SPIE Vol. 3454, San Diego, 20-22 July, 1998, pp. 52-63.
H.P.F. Swinnerton-Dyer, “The number of lattice points on a convex curve,” J. Number Theory, Vol. 6, pp. 128-135, 1974.
C.-H. Teh and R.T. Chin, “On digital approximation of moment invariants,” Comp. Vis. Graph. Image Proc., Vol. 33, pp. 318-326, 1986.
K. Voss, “Digitalisierungseffekte in der automatischen Bildverarbeitung,” EIK, Vol. 11, pp. 469-477, 1975.
K. Voss, “Digitization effects in image analysis,” Acta Stereologica, Vol. 6, pp. 145-147, 1987.
K. Voss, Discrete Images, Objects, and Functions in Z n, Springer: Berlin, 1993.
K. Voss and H. Süsse, Adaptive Modelle und Invarianten für zweidimensionale Bilder, Shaker: Aachen, 1995.
J. Žunić, “Discrete moments of the circles,” in Proc. 8th Int. Conf. Discrete Geometry for Computer Imagery, Marne-la-Vallée, France, March 1999. LNCS, Vol. 1568, Springer, 1999, pp. 41-49.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Klette, R., Žunić, J. Multigrid Convergence of Calculated Features in Image Analysis. Journal of Mathematical Imaging and Vision 13, 173–191 (2000). https://doi.org/10.1023/A:1011289414377
Issue Date:
DOI: https://doi.org/10.1023/A:1011289414377