article

Digital straightness: a review

Authors:

Reinhard Klette,

Azriel RosenfeldAuthors Info & Claims

Discrete Applied Mathematics, Volume 139, Issue 1-3

Pages 197 - 230

https://doi.org/10.1016/j.dam.2002.12.001

Published: 30 April 2004 Publication History

Abstract

A digital arc is called 'straight' if it is the digitization of a straight line segment. Since the concept of digital straightness was introduced in the mid-1970s, dozens of papers on the subject have appeared; many characterizations of digital straight lines have been formulated, and many algorithms for determining whether a digital arc is straight have been defined. This paper reviews the literature on digital straightness and discusses its relationship to other concepts of geometry, the theory of words, and number theory.

References

[1]

{1} D.M. Acketa, J.D. Zunic´, On the number of linear partitions of the (m, n)-grid, Inform. Process. Lett. 38 (1991) 163-168.

Digital Library

[2]

{2} T.A. Anderson, C.E. Kim, Representation of digital line segments and their preimages, in: Proceedings of the Seventh International Conference on Pattern Recognition, 1984, pp. 501-504.

[3]

{3} T.A. Anderson, C.E. Kim, Representation of digital line segments and their preimages, Comput. Vision Graphics Image Process. 30 (1985) 279-288.

[4]

{4} E. Andres, R. Acharya, C. Sibata, Discrete analytical hyperplanes, Graphical Models Image Process. 59 (1997) 302-309.

Digital Library

[5]

{5} C. Arcelli, A. Massarotti, Regular arcs in digital contours, Comput. Graphics Image Process. 4 (1975) 339-360.

[6]

{6} C. Arcelli, A. Massarotti, On the parallel generation of straight digital lines, Comput. Graphics Image Process. 7 (1978) 67-83.

[7]

{7} C. Berenstein, D. Lavine, On the number of digital straight line segments, IEEE Trans. Pattern Anal. Mach. Intell. 10 (1988) 880-887.

Digital Library

[8]

{8} J. Berstel, M. Pocchiola, Random generation of finite Sturmian words, LIENS-93-8. Département de Mathématiques et d'Informatique. Ecole Normale Supérieure, LITP Institute Blaise Pascal. 1993.

[9]

{9} G. Bongiovanni, F. Luccio, A. Zorat, The discrete equation of the straight line. IEEE Trans. Comput. 24 (1975) 310-313.

Digital Library

[10]

{10} J. Bresenham, An incremental algorithm tor digital plotting, in: Proceedings of the ACM National Conference, 1963.

[11]

{11} J. Bresenham, Algorithm for computer control of a digital plotter. IBM Systems J. 4 (1965) 25-30.

Digital Library

[12]

{12} R. Brons, Linguistic methods for description of a straight line on a grid. Comput. Graphics Image Process. 2 (1974) 48-62.

[13]

{13} A.M. Bruckstein, Self-similarity properties of digitized straight lines. Contemp. Math. 119 (1991) 1-20.

[14]

{14} S. Chattopadhyay, P.P. Das, A new method of analysis for discrete straight lines, Pattern Recognition Lett. 12 (1991) 747-755.

Digital Library

[15]

{15} D. Coeurjolly, R. Klette, A comparative evaluation of length estimators, in: Proceedings of the 16th International Conference on Pattern Recognition, Vol. IV, 2002, pp. 330-334.

[16]

{16} E.M. Coven, G.A. Hedlund, Sequences with minimal block growth. Math. Systems Theory 7 (1973) 138-153.

[17]

{17} E. Creutzburg, A. Hübler, O. Sýkora, Geometric methods for on-line recognition of digital straight segments, Comput. Artif. Intell. 7 (1988) 253-276.

Digital Library

[18]

{18} E. Creutzburg, A. Hübler, V. Wedler, On-line Erkennung digitaler Geradensegmente in linearer Zeit. in: Proceedings of GEO-BILD'82. Wiss. Beiträge der FSU Jena. 1982, pp. 48-65. (See also: On-line recognition of digital straight line segments, in: Proceedings of the Second International Conference on AI and Information Control Systems of Robots. Smolenice, Czechoslovakia, 1982, pp. 42-46.)

[19]

{19} E. Creutzburg, A. Hübler, V. Wedler, Decomposition of digital arcs and contours into a minimal number of digital straight line segments, in: Proceedings of the Sixth International Conference on Pattern Recognition, 1982, p. 1218.

[20]

{20} I. Debled-Rennesson, J.-P. Reveillès, A linear algorithm for segmentation of digital curves, Internat. J. Pattern Recognition Artif. Intell. 9 (1995) 635-662.

[21]

{21} G. Dettori, An on-line algorithm for polygonal approximation of digitized plane curves, in: Proceedings of the Sixth International Conference on Pattern Recognition, 1982, pp. 739-741.

[22]

{22} H.-U. Döhler, P. Zamperoni, Compact contour codes for convex binary patterns, Signal Process. 8 (1985) 23-39.

[23]

{23} L. Dorst, R.P.W. Duin, Spirograph theory: a framework for calculations on digitized straight lines, IEEE Trans. Pattern Anal. Mach. Intell. 6 (1984) 632-639.

Digital Library

[24]

{24} L. Dorst, A.W.M. Smeulders, Discrete representation of straight lines, IEEE Trans. Pattern Anal. Mach. Intell. 6 (1984) 450-463.

Digital Library

[25]

{25} S. Dulueq, D. Gouyou-Beauchamps, Sur les facteurs des suites de Sturm, Theoret. Comput. Sci. 71 (1991) 381-400.

Digital Library

[26]

{26} A. Fam, J. Sklansky, Cellularly straight images and the Hausdorff metric, in: Proceedings of the Conference on Pattern Recognition and Image Processing, 1977, pp. 242-247.

[27]

{27} J. Feder, Languages of encoded line patterns, Inform. and Control 13 (1968) 230-244.

[28]

{28} J. Françon, J.-M. Schramm, M. Tajine, Recognizing arithmetic straight lines and planes, in: Proceedings of the Discrete Geometry for Computer Imagery, Springer, Berlin, 1996, pp. 141-150.

[29]

{29} H. Freeman, Techniques for the digital computer analysis of chain-encoded arbitrary plane curves, in: Proceedings of the National Electronics Conference, Vol. 17, 1961, pp. 421-432.

[30]

{30} H. Freeman, Boundary encoding and processing, in: B.S. Lipkin, A. Rosenfeld (Eds.), Picture Processing and Psychopictorics, Academic Press, New York, 1970, pp. 241-263.

[31]

{31} M. Gaafar, Convexity verification, block-chords, and digital straight lines, in: Proceedings of the Third International Conference on Pattern Recognition, 1976, pp. 514-518.

[32]

{32} M. Gaafar, Convexity verification, block-chords and digital straight lines, Comput. Graphics Image Process. 6 (1977) 361-370.

[33]

{33} A. Hübler, Diskrete Geometrie für die digitale Bildverarbeitung, Habil.-Schrift, Friedrich-Schiller-Universität Jena, 1989.

[34]

{34} A. Hübler, R. Klette, K. Voss, Determination of the convex hull of a finite set of planar points within linear time, EIK 17 (1981) 121-140.

[35]

{35} S.H.Y. Hung, On the straightness of digital arcs, IEEE Trans. Pattern Anal. Mach. Intell. 7 (1985) 1264 -1269.

[36]

{36} S.H.Y. Hung, T. Kasvand, On the chord property and its equivalences, in: Proceedings of the Seventh International Conference on Pattern Recognition, 1984, pp. 116-119.

[37]

{37} M.C. Irwin, Geometry of continued fractions, Amer. Math. Monthly 96 (1989) 696-703.

Digital Library

[38]

{38} C.E. Kim, On cellular straight line segments, Comput. Graphics Image Process. 18 (1982) 369-391.

[39]

{39} C.E. Kim, Digital convexity, straightness, and convex polygons, IEEE Trans. Pattern Anal. Mach. Intell. 4 (1982) 618-626.

Digital Library

[40]

{40} C.E. Kim, A. Rosenfeld, Digital straightness and convexity, in: Proceedings of the Symposium on Theory of Computing, 1981, pp. 80-89.

[41]

{41} C.E. Kim, A. Rosenfeld, Digital straight lines and convexity of digital regions, IEEE Trans. Pattern Anal. Mach. Intell. 4 (1982) 149-153.

Digital Library

[42]

{42} K. Kishimoto, Characterizing digital convexity and straightness in terms of "length" and "total absolute curvature", Comput. Vision Image Understanding 63 (1996) 326-333.

Digital Library

[43]

{43} H. Klaasman, Some aspects of the accuracy of the approximated position of a straight line on a square grid, Comput. Graphics Image Process. 4 (1975) 225-235.

[44]

{44} R. Klette, Cell complexes through time, in: Proceedings of Vision Geometry IX, SPIE 4117, 2000, pp. 134-145.

[45]

{45} R. Klette, B. Yip, The length of digital curves, Mach. Graphics & Vision 9 (2000) 673-703 (extended version of: R. Klette, V.V. Kovalevsky, B. Yip, Length estimation of digital curves, in: Proceedings of Vision Geometry VIII, SPIE 3811, pp. 117-129).

[46]

{46} D.E. Knuth, The Art of Computer Programming, Vol. 2. Addison-Wesley, Reading, MA, 1969, pp. 309, 316-332.

[47]

{47} J. Koplowitz, M. Lindenbaum, A. Bruckstein, The number of digital straight lines on an N × N grid. IEEE Trans. Inform. Theory 36 (1990) 192-197.

[48]

{48} V.A. Kovalevsky, New definition and fast recognition of digital straight segments and arcs, in: Proceedings of the 10th International Conference on Pattern Recognition, 1990, pp. 31-34.

[49]

{49} W.G. Kropatsch, H. Tockner, Detecting the straightness of digital curves in O(N) steps, Comput. Vision Graphics Image Process. 45 (1989) 1-21.

Digital Library

[50]

{50} H.C. Lee, K.S. Fu, Using the FFT to determine digital straight line chain codes, Comput. Graphics Image Process. 18 (1982) 359-368.

[51]

{51} S.X. Li, M.H. Loew, Analysis and modeling of digitized straight-line segments, in: Proceedings of the Ninth International Conference on Pattern Recognition, 1988, pp. 294-296.

[52]

{52} M. Lindenbaum, Compression of chain codes using digital straight line sequences, Pattern Recognition Lett. 7 (1988) 167-171.

Digital Library

[53]

{53} M. Lindenbaum, A. Bruckstein, On recursive, O(N) partitioning of a digitized curve into digital straight segments, IEEE Trans. Pattern Anal. Mach. Intell. 15 (1993) 949-953.

Digital Library

[54]

{54} M. Lindenbaum, J. Koplowitz, A new parametrization of digital straight lines, IEEE Trans. Pattern Anal. Mach. Intell 13 (1991) 847-852.

Digital Library

[55]

{55} M. Lindenbaum, J. Koplowitz, A. Bruckstein, On the number of digital straight lines on an N × N grid, in: Proceedings of the Conference on Computer Vision and Pattern Recognition, 1988, pp. 610-615.

[56]

{56} J. Loeb, Communication theory of transmission of simple drawings, in: W. Jackson (Ed.), Communication Theory, Butterworths, London, 1953, pp. 317-327.

[57]

{57} M. Lothaire (Ed.). Combinatorics on Words, 2nd Edition, Cambridge University Press, Cambridge, 1987.

[58]

{58} M. Lothaire (Ed.), Algebraic Combinatorics on Words, Cambridge University Press, Cambridge, 2002.

[59]

{59} S. Marchand-Maillet, Y.M. Sharaiha, Discrete convexity, straightness, and the 16-neighborhood, Comput. Vision Image Understanding 66 (1997) 316-329.

Digital Library

[60]

{60} M.D. Mellroy, A note on descrete representation of lines AT & T Techn. J. 64 (1984) 481-490.

[61]

{61} F. Mignosi, On the number of factors of Sturmian words. Theoret. Comput. Sci. 82. (1991) 71-84.

[62]

{62} G.U. Montanari, A note on minimal length polygonal approximation to a digitized contour. Comm. ACM 13 (1970) 41-47.

Digital Library

[63]

{63} G.C. Montanari, A method for obtaining skeletons using a quasi-Euclidean distance. J. ACM. 15 (1968) 600-624.

Digital Library

[64]

{64} M. Morse, G.A. Hedlund, Symbolic dynamics II: Sturmian sequences. Amer. J. Math. 61 (1940) 1-42.

[65]

{65} L. Pavlidis, Structural Pattern Recognition. Springer, New York, 1977.

[66]

{66} G.B. Reggiori, Digital computer transformations for irregular line drawings. Technical Report 403-22. New York University, April 1972.

[67]

{67} J.-P. Reveillès, Géométrie discréte. calcul en nombres entiers et algorithmique. Thèse d'état. Univ. Louis Pasteur. Strasbourg, 1991.

[68]

{68} C. Ronse, A simple proof of Rosenfeld's characterization of digital straight line segments. Pattern Recognition Lett 3 (1985) 323-326.

Digital Library

[69]

{69} C. Ronse, Criteria for approximation of linear and affine functions. Arch. Math. 46 (1986) 371-384.

[70]

{70} A. Rosenfeld, Digital straight line segments. IEEE Trans. Comput. 23 (1974) 1264-1269.

Digital Library

[71]

{71} A. Rosenfeld, C.E. Kim, How a digital computer can tell whether a line is straight. Amer. Math. Monthly 89 (1982) 230-235.

[72]

{72} J. Rothstein, C. Weiman, Parallel and sequential specification of a context sensitive language for straight line grids, Comput. Graphics Image process. 5 (1976) 106-124.

[73]

{73} L.A. Santaló, Complemento a la nota: un teorema sôbre conjuntos de paralelipipedos de artistas paralelas, Publ. Intl. Math. Univ. Nac. Litoral 2 (1940) 49-60.

[74]

{74} Y.M. Sharaiha, P. Garat. A compact chord property for digital arcs. Pattern Recognition 26 (1993) 799 -803.

[75]

{75} S. Shlien, Segmentation of digital curves using linguistic techniques, Comput. Vision. Graphics. Image Process. 22 (1983) 277-286.

[76]

{76} R. Shoueri, R. Benesch, S. Thomas, Note on the determination of a digital straight line from chain codes. Comput. Vision Graphics Image Process. 29 (1985) 133-139.

[77]

{77} J. Sklansky, V. Gonzalez, Fast polygonal approximation of digitized curves. Pattern Recognition 12 (1980) 327-331.

[78]

{78} A.W.M. Smeulders, L. Dorst, Decomposition of descrete curves into piecewise segments in linear time, Contemp. Math. 119 (1991) 169-195.

[79]

{79} I. Stojmenovic, R. Tosic, Digitization schemes and the recognition of digital straight lines, hyperplanes, and flats in arbitary dimensions. Contemp. Math 119 (1991) 197-212.

[80]

{80} K. Voss, Coding of digital straight lines by continued fractions. Comput. Artif. Intell. 10 (1991) 75-80.

Digital Library

[81]

{81} K. Voss, Discrete Images. Objects and Functions in Zn. Springer, Berlin, 1993.

[82]

{82} L. Wu, On the Freeman's conjecture about the chain code of a line, in: Proceedings of the Fifth International Conference on Pattern Recognition, 1980, pp. 32-34.

[83]

{83} L.D. Wu. On the chain code of a line. IEEE Trans. Pattern Anal. Mach. Intell. 4 (1982) 347-353.

[84]

{84} L.D. Wu, F.L. Weng, Chain code for a line segment and formal language, in: Proceedings of the Eighth International Conference on Pattern Recognition, 1986, pp. 1124-1126.

[85]

{85} J. Yuan, C.Y. Suen, An optimal O(n) algorithm for identifying line segments from a sequence of chain codes. Pattern Recognition 28 (1995) 635-646.

Cited By

Saha SBiswas A(2024)A combinatorial technique for generation of digital plane using GCDAnnals of Mathematics and Artificial Intelligence10.1007/s10472-023-09889-492:1(139-167)Online publication date: 1-Jan-2024
https://dl.acm.org/doi/10.1007/s10472-023-09889-4
Laboureix BDebled-Rennesson I(2024)Recognition of Arithmetic Line Segments and Hyperplanes Using the Stern-Brocot TreeDiscrete Geometry and Mathematical Morphology10.1007/978-3-031-57793-2_2(16-28)Online publication date: 15-Apr-2024
https://dl.acm.org/doi/10.1007/978-3-031-57793-2_2
Roussillon T(2023)Combinatorial Generation of Planar SetsJournal of Mathematical Imaging and Vision10.1007/s10851-023-01152-z65:5(702-717)Online publication date: 15-Jul-2023
https://dl.acm.org/doi/10.1007/s10851-023-01152-z
Show More Cited By

Index Terms

Digital straightness: a review

Recommendations

Fast Polygonal Approximation of Digital Curves Using Relaxed Straightness Properties

Several existing digital straight line segment (DSS) recognition algorithms can be used to determine the digital straightness of a given one-pixel-thick digital curve. Because of the inherent geometric constraints of digital straightness, these ...
Digital Planarity of Rectangular Surface Segments

We generalize the concept of evenness which has been developed for digital straight lines. Evenness is a necessary and sufficient condition for a digital arc segment to be a digital straight line segment. We prove that evenness is also a necessary and ...
Consistent digital line segments
SoCG '10: Proceedings of the twenty-sixth annual symposium on Computational geometry

We introduce a novel and general approach for digitalization of line segments in the plane that satisfies a set of axioms naturally arising from Euclidean axioms. In particular, we show how to derive such a system of digital segments from any total ...

Comments

Information & Contributors

Information

Published In

Publisher

Elsevier Science Publishers B. V.

Netherlands

Publication History

Published: 30 April 2004

Author Tags

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

61
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 04 Oct 2024

Other Metrics

View Author Metrics

Citations

Cited By

Saha SBiswas A(2024)A combinatorial technique for generation of digital plane using GCDAnnals of Mathematics and Artificial Intelligence10.1007/s10472-023-09889-492:1(139-167)Online publication date: 1-Jan-2024
https://dl.acm.org/doi/10.1007/s10472-023-09889-4
Laboureix BDebled-Rennesson I(2024)Recognition of Arithmetic Line Segments and Hyperplanes Using the Stern-Brocot TreeDiscrete Geometry and Mathematical Morphology10.1007/978-3-031-57793-2_2(16-28)Online publication date: 15-Apr-2024
https://dl.acm.org/doi/10.1007/978-3-031-57793-2_2
Roussillon T(2023)Combinatorial Generation of Planar SetsJournal of Mathematical Imaging and Vision10.1007/s10851-023-01152-z65:5(702-717)Online publication date: 15-Jul-2023
https://dl.acm.org/doi/10.1007/s10851-023-01152-z
Chiu MKorman MSuderland MTokuyama T(2022)Distance Bounds for High Dimensional Consistent Digital Rays and 2-D Partially-Consistent Digital RaysDiscrete & Computational Geometry10.1007/s00454-021-00349-668:3(902-944)Online publication date: 1-Oct-2022
https://dl.acm.org/doi/10.1007/s00454-021-00349-6
Laboureix BAndres EDebled-Rennesson I(2022)Introduction to Discrete Soft TransformsDiscrete Geometry and Mathematical Morphology10.1007/978-3-031-19897-7_33(422-435)Online publication date: 24-Oct-2022
https://dl.acm.org/doi/10.1007/978-3-031-19897-7_33
Meyron JRoussillon T(2022)Approximation of Digital Surfaces by a Hierarchical Set of Planar PatchesDiscrete Geometry and Mathematical Morphology10.1007/978-3-031-19897-7_32(409-421)Online publication date: 24-Oct-2022
https://dl.acm.org/doi/10.1007/978-3-031-19897-7_32
Ajij MPratihar SNayak SHanne TRoy D(2021)Off-line signature verification using elementary combinations of directional codes from boundary pixelsNeural Computing and Applications10.1007/s00521-021-05854-635:7(4939-4956)Online publication date: 18-Mar-2021
https://dl.acm.org/doi/10.1007/s00521-021-05854-6
Song HPotapov I(2019)Polygon Approximations of the Euclidean Circles on the Square Grid by Broadcasting SequencesDiscrete Geometry for Computer Imagery10.1007/978-3-030-14085-4_35(444-456)Online publication date: 26-Mar-2019
https://dl.acm.org/doi/10.1007/978-3-030-14085-4_35
D'Aniello Ade Luca ADe Luca A(2017)On Christoffel and standard words and their derivativesTheoretical Computer Science10.1016/j.tcs.2016.05.010658:PA(122-147)Online publication date: 7-Jan-2017
https://dl.acm.org/doi/10.1016/j.tcs.2016.05.010
Lachaud JProvençal XRoussillon T(2017)Two Plane-Probing Algorithms for the Computation of the Normal Vector to a Digital PlaneJournal of Mathematical Imaging and Vision10.1007/s10851-017-0704-x59:1(23-39)Online publication date: 1-Sep-2017
https://dl.acm.org/doi/10.1007/s10851-017-0704-x
Show More Cited By

View Options

View options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

Media

Figures

Other

Tables

View Issue’s Table of Contents