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Natural Representations for Straight Lines and the Hough Transform on Discrete Arrays
September 1989 (vol. 11 no. 9)
pp. 941-950

The grid of discrete sampled data points in a digital image supports a limited set of lines at angles and displacements 'natural' to that grid. The effect of this implicit line quantization on the parametrization of the Hough transform is presented. The function describing the discrete Hough transforms line-detection sensitivity is derived. Expressions for the orientation, frequency, and popularity of lines in the natural set are given. The results obtained are of importance to data arrays of small size. The distribution of lines in the natural set is also important as it determines the precision and reliability with which straight lines can be measured on a discrete imaging array. From the natural line set concept, a general (a,d) slope/offset straight-line parametrization is developed for which the Hough transform is compact and fast to compute, and which is as easy to interpret as the class (p, theta ) parametrization.

[1] R.O. Duda and P.E. Hart, "Use of the Hough transformation to detect lines and curves in pictures,"Commun. Ass. Comput. Mach., vol. 15, no. 1, pp. 11-15, Jan. 1972.
[2] A. Iannino and S. D. Shapiro, "A survey of the Hough transform and its extensions for curve detection," inProc. IEEE Comput. Soc. Conf. Pattern Recognition and Image Processing, Chicago, IL, 1978, pp. 32-38.
[3] F. M. Rhodes, J. J. Dituri, G. H. Chapman, B. E. Emerson, A. M. Soares, and J. I. Raffel, "A monolithic Hough transform processor based on restructurable VLSI,"IEEE Trans. Pattern Anal. Machine Intell., vol. 10, pp. 106-110, Jan. 1988.
[4] K. Hanahara, T. Maruyama, and T. Uchiyama, "A real-time processor for the Hough transform,"IEEE Trans. Pattern Anal. Machine Intell., vol. 10, pp. 121-125, Jan. 1988.
[5] W. H. Steier and R. K. Shori, "Optical Hough transform,"Appl. Opt., vol. 25, no. 16, pp. 2734-2738, Aug. 1986.
[6] P. Ambs, S. H. Lee, Q. Tian, and Y. Fainman, "Optical implementation of the Hough transform by a matrix of holograms,"Appl. Opt., vol. 25, no. 22, pp. 4039-4045, Nov. 1986.
[7] H. Li, M. A. Lavin, and R. J. LeMaster, "Fast Hough transform,"Comput. Vision Graphics Image Processing, vol. 36, pp. 139-161, 1986.
[8] P. M. Merlin and D. J. Farber, "A parallel mechanism for detecting curves in pictures,"IEEE Trans. Comput., vol. C-24, pp. 96-98, Jan. 1975.
[9] A. Rosenfeld, J. Ornelas, and Y. Hung, "Hough Transform Algorithms for Mesh-Connected SIMD Parallel Processors,"Computer Vision, Graphics and Image Processing, Vol. 41, No. 3, Mar. 1988, pp. 293-305.
[10] M. Cohen and G. T. Toussaint, "On the detection of structures in noisy pictures,"Pattern Recognition, vol. 9, pp. 95-98, 1977.
[11] S. D. Shapiro and A. Iannio, "Geometric constructions for predicting Hough transform performance,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-1, pp. 310-317, July 1979.
[12] T. M. van Veen and F. C. A. Groen, "Discretization errors in the Hough transform,"Pattern Recognition, vol. 14, nos. 1-6, pp. 137- 145, 1981.
[13] C. M. Brown, "Inherent bias and noise in the Hough transform,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-5, pp. 493- 505, Sept. 1983.
[14] H. Maitre, "Contribution to the prediction of performances of the Hough transform,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-8, no. 5, pp. 669-674, Sept. 1986.
[15] E. R. Davies, "Improved localization in a generalized Hough scheme for the detection of straight edges,"Image Vision Comput., vol. 5, no. 4, pp. 279-286, Nov. 1987.
[16] R. S. Wallace, "A modified Hough transform for lines," inProc. IEEE Comput. Sci. Conf. Pattern Recognition Image Processing, San Francisco, CA, 1985, pp. 665-667.
[17] S. J. Gordon and W. P. Seering, "Accuracy issues in measuring quantized images of straight-line features," inIEEE Comput. Soc. Proc. Robotics and Automation Conf., San Franscisco, CA, 1986, pp. 931-936.
[18] L. Dorst and A. W. M. Smeulders, "Discrete representation of straight lines,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-6, pp. 450-463, July 1984.
[19] L. Dorst and R. P. W. Duin, "Spirograph theory: A framework for calculations on digitized straight lines,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-6, pp. 632-638, Sept. 1984.
[20] R. Brons, "Theoretical and linguistic methods for describing straight lines," inFundamentals of Computer Graphics, R. A. Earnshaw, Ed., NATO ASI Series, 1985, pp. 19-57.
[21] R. Krishnaswamy and C. E. Kim, "Digital parallelism, perpendicularity, and rectangles,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-9, pp. 316-321, Mar. 1987.
[22] R. Bellman,Analytic Number Theory: An Introduction. Reading, MA: Benjamin Cummings, 1980, ch. 1, p. 3.
[23] I. D. Svalbe, "Natural representations for straight lines and the Hough transform on discrete arrays," Dep. Comput. Sci. FR-35, Univ. Washington, Seattle, Tech. Rep. 06, Apr. 1987.

Index Terms:
straight lines; discrete sampled data points; digital image; line quantization; line-detection sensitivity; natural set; discrete imaging array; computerised picture processing; transforms
I.D. Svalbe, "Natural Representations for Straight Lines and the Hough Transform on Discrete Arrays," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 11, no. 9, pp. 941-950, Sept. 1989, doi:10.1109/34.35497
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