On the Farey sequence and its augmentation for applications to image analysis

Amin, A. and Fischer, S. (2000). A document skew detection method using the Hough transform, Pattern Analysis and Applications3(3): 243–253.10.1007/s100440070009Open DOISearch in Google Scholar

Attneave, F. (1954). Some informational aspects of visual perception, Psychological Review61(3): 183–193.10.1037/h0054663Open DOISearch in Google Scholar

Bhowmick, P. and Bhattacharya, B.B. (2007). Fast polygonal approximation of digital curves using relaxed straightness properties, IEEE Transactions on Pattern Analysis and Machine Intelligence29(9): 1590–1602.10.1109/TPAMI.2007.1082Open DOISearch in Google Scholar

Buzer, L. (2009). Optimal simplification of polygonal chains for subpixel-accurate rendering, Computational Geometry42(1): 45–59.10.1016/j.comgeo.2008.03.002Search in Google Scholar

Cao, Y., Wang, S. and Li, H. (2003). Skew detection and correction in document images based on straight-line fitting, Pattern Recognition Letters24(12): 1871–1879.10.1016/S0167-8655(03)00010-2Open DOISearch in Google Scholar

Charrier, E. and Buzer, L. (2009). Approximating a real number by a rational number with a limited denominator: A geometric approach, Discrete Applied Mathematics157(16): 3473–3484.10.1016/j.dam.2009.03.005Search in Google Scholar

Chaudhuri, B.B. and Pal, U. (1997). Skew angle detection of digitized Indian script documents, IEEE Transactions on Pattern Analysis and Machine Intelligence19(2): 182–186.10.1109/34.574803Open DOISearch in Google Scholar

Chou, C., Chu, S. and Chang, F. (2007). Estimation of skew angles for scanned documents based on piecewise covering by parallelograms, Pattern Recognition40(2): 443–455.10.1016/j.patcog.2005.10.030Open DOISearch in Google Scholar

Chung, K.L., Liao, P.H. and Chang, J.M. (2008). Novel efficient two-pass algorithm for closed polygonal approximation based on LISE and curvature constraint criteria, Journal of Visual Communication and Image Representation19(4): 219–230.10.1016/j.jvcir.2008.01.004Open DOISearch in Google Scholar

Cormen, T.H., Leiserson, C.E. and Rivest, R.L. (2000). Introduction to Algorithms, Prentice Hall of India, New-Delhi.Search in Google Scholar

Das, A.K. and Chanda, B. (2001). A fast algorithm for skew detection of document images using morphology, International Journal on Document Analysis and Recognition4(2): 109–114.10.1007/PL00010902Open DOISearch in Google Scholar

Das, S., Halder, K., Pratihar, S. and Bhowmick, P. (2010). Properties of Farey sequence and their applications to digital image processing, 4th International Conference on Information Processing, Bangalore, India, pp. 71–81.Search in Google Scholar

Devaney, R.L. (1999). The Mandelbrot set, the Farey tree, and the Fibonacci sequence, The American Mathematical Monthly106(04): 289–302.10.1080/00029890.1999.12005046Search in Google Scholar

Dinesh, R. and Guru, D.S. (2009). Non-parametric adaptive approach for the detection of dominant points on boundary curves based on non-symmetric region of support, International Journal of Image and Graphics9(4): 541–557.10.1142/S0219467809003575Open DOISearch in Google Scholar

Graham, R., Knuth, D. and Patashnik, O. (1994). Concrete Mathematics, Addison-Wesley, London.Search in Google Scholar

Hardy, G.H. and Wright, E.M. (1968). An Introduction to the Theory of Numbers, Oxford University Press, New York, NY.Search in Google Scholar

Hinds, S., Fisher, J. and D’Amato, D.P. (1990). A document skew detection method using run length encoding and the Hough transform, Proceedings of the International Conference on Pattern Recognition, Los Alamitos, CA, USA, pp. 464–468.10.1109/ICPR.1990.118147Open DOISearch in Google Scholar

Hu, H. and Yan, H. (1997). Polygonal approximation of digital curves based on the principles of perceptual organization, Pattern Recognition30(5): 701–718.10.1016/S0031-3203(96)00105-7Open DOISearch in Google Scholar

Jiang, H.-F., Han, C.-C. and Fan, K.-C. (1997). A fast approach to the detection and correction of skew documents, Pattern Recognition Letters18(7): 675–686.10.1016/S0167-8655(97)00032-9Open DOISearch in Google Scholar

Klette, R. and Rosenfeld, A. (2004). Digital Geometry: Geometric Methods for Digital Picture Analysis, Morgan Kaufmann, San Francisco, CA.10.1016/B978-155860861-0/50005-5Search in Google Scholar

Knuth, D. E. (1997). The Art of Computer Programming, Vol. 2, 3rd Edn., Addison-Wesley, Boston, MA.Search in Google Scholar

Koutroumbas, K.D. (2012). Piecewise linear curve approximation using graph theory and geometrical concepts, IEEE Transactions on Image Processing21(9): 3877–3887.10.1109/TIP.2012.2199507Open DOISearch in Google Scholar

Kumar, M.P., Goyal, S., Jawahar, C.V. and Narayanan, P.J. (2002). Polygonal approximation of closed curves across multiple views, 3rd Indian Conference on Computer Vision, Graphics and Image Processing, Ahmadabad, India, pp. 317–322.Search in Google Scholar

Le, D.S., Thoma, G.R. and Wechsler, H. (1994). Automatic page orientation and skew angle detection for binary document images, Pattern Recognition27(10): 1325–1344.10.1016/0031-3203(94)90068-XSearch in Google Scholar

Li, S., Shen, Q. and Sun, J. (2007). Skew detection using wavelet decomposition and projection profile analysis, Pattern Recognition Letters28(5): 555–562.10.1016/j.patrec.2006.10.002Open DOISearch in Google Scholar

Liu, H., Latecki, L. and Liu, W. (2008). A unified curvature definition for regular, polygonal, and digital planar curves, International Journal of Computer Vision80(1): 104–124.10.1007/s11263-008-0131-ySearch in Google Scholar

Manjunath, V.N., Kumar, G.H. and Shivakumara, P. (2006). Skew detection technique for binary document images based on Hough transform, International Journal of Information and Communication Engineering3(7): 493–499.Search in Google Scholar

Masood, A. (2008). Dominant point detection by reverse polygonization of digital curves, Image and Vision Computing26(5): 702–715.10.1016/j.imavis.2007.08.006Open DOISearch in Google Scholar

Melkman, A. and O’Rourke, J. (1988). On polygonal chain approximation, in G.T. Toussaint (Ed.), Computational Morphology, North-Holland, Amsterdam, pp. 87–95.10.1016/B978-0-444-70467-2.50012-6Search in Google Scholar

Mikolajczyk, K. and Schmid, C. (2005). A performance evaluation of local descriptors, IEEE Transactions on Pattern Analysis and Machine Intelligence27(10): 1615–1630.10.1109/TPAMI.2005.188Search in Google Scholar

Mokhtarian, F. and Mohanna, F. (2002). Content-based video database retrieval through robust corner tracking, IEEE Workshop on Multimedia Signal Processing, St. Thomas, Virgin Islands, USA, pp. 224–228.Search in Google Scholar

Neumann, R. and Teisseron, G. (2002). Extraction of dominant points by estimation of the contour fluctuations, Pattern Recognition35(7): 1447–1462.10.1016/S0031-3203(01)00145-5Open DOISearch in Google Scholar

Neville, E.H. (1950). The Farey Series of Order 1025, Cambridge University Press, Cambridge.Search in Google Scholar

Nguyen, T.P. and Debled-Rennesson, I. (2011). A discrete geometry approach for dominant point detection, Pattern Recognition44(1): 32–44.10.1016/j.patcog.2010.06.022Open DOISearch in Google Scholar

Nikiel, S. (2007). A proposition of mobile fractal image decompression, International Journal of Applied Mathematics and Computer Science17(1): 129–136, DOI: 10.2478/v10006-007-0012-5.10.2478/v10006-007-0012-5Open DOISearch in Google Scholar

O’Connell, K.J. (1997). Object-adaptive vertex based shape coding method, IEEE Transactions on Circuits and Systems for Video Technology7(1): 251–255.10.1109/76.554440Open DOISearch in Google Scholar

Parvez, M.T. and Mahmoud, S.A. (2010). Polygonal approximation of digital planar curves through adaptive optimizations, Pattern Recognition Letters31(13): 1997–2005.10.1016/j.patrec.2010.06.007Open DOISearch in Google Scholar

Pătraşcu, C.E. and Pătraşcu, M. (2004). Computing order statistics in the Farey sequence, Symposium on Algorithmic Number Theory, Burlington, VT, USA, pp. 358–366.Search in Google Scholar

Pavlidis, T. and Zhou, J. (1991). Page segmentation by white streams, International Conference on Document Analysis and Recognition, Saint-Malo, France, pp. 945–953.Search in Google Scholar

Pawlewicz, J. and Pătraşcu, M. (2009). Order statistics in the Farey sequences in sublinear time and counting primitive lattice points in polygons, Algorithmica55(2): 271–282.10.1007/s00453-008-9221-zOpen DOISearch in Google Scholar

Prasad, D.K., Leung, M.K.H., Quek, C. and Cho, S. (2012). A novel framework for making dominant point detection methods non-parametric, Image and Vision Computing30(11): 843–859.10.1016/j.imavis.2012.06.010Open DOISearch in Google Scholar

Pratihar, S. and Bhowmick, P. (2009). A thinning-free algorithm for straight edge detection in a gray-scale image, International Conference on Advances on Pattern Recognition, Kolkata, India, pp. 341–344.Search in Google Scholar

Pratihar, S. and Bhowmick, P. (2010). Vectorization of thick digital lines using Farey sequence and geometric refinement, ICVGIP, Chennai, India, pp. 518–525.Search in Google Scholar

Pratihar, S. and Bhowmick, P. (2011). Skew correction of engineering drawings by digital-geometric analysis of Farey ranks, International Conference on Image Information Processing (ICIIP), Shimla, India, pp. 1–6.Search in Google Scholar

Pratihar, S., Bhowmick, P., Sural, S. and Mukhopadhyay, J. (2013). Skew correction of document images by rank analysis in Farey sequence, International Journal of Pattern Recognition and Artificial Intelligence27(7): Article ID 1353004.10.1142/S0218001413530042Search in Google Scholar

Ray, B.K. and Ray, K.S. (1992). An algorithm for detection of dominant points and polygonal approximation of digitized curves, Pattern Recognition Letters13(12): 849–856.10.1016/0167-8655(92)90084-DSearch in Google Scholar

Ray, K.S. and Ray, B.K. (2013). Polygonal approximation of digital curve based on reverse engineering concept, International Journal of Image and Graphics13(4): Article ID 1350017.10.1142/S0219467813500174Search in Google Scholar

Rosin, P.L. (1997). Techniques for assessing polygonal approximation of curves, IEEE Transactions on Pattern Analysis and Machine Intelligence19(6): 659–666.10.1109/34.601253Open DOISearch in Google Scholar

Rosin, P.L. and West, G.A.W. (1988). Detection of circular arcs in images, 4th Alvey Vision Conference, Manchester, UK, pp. 259–263.Search in Google Scholar

Rosin, P.L. and West, G.A.W. (1995). Non-parametric segmentation of curves into various representations, IEEE Transactions on Pattern Analysis and Machine Intelligence17(12): 1140–1153.10.1109/34.476507Open DOISearch in Google Scholar

Routledge, N. (2008). Computing Farey series, Mathematical Gazette92(523): 55–62.10.1017/S002555720018252XSearch in Google Scholar

Sarkar, B., Singh, L.K. and Sarkar, D. (2004). A genetic algorithm-based approach for detection of significant vertices for polygonal approximation of digital curves, International Journal of Image and Graphics4(2): 223–239.10.1142/S0219467804001385Open DOISearch in Google Scholar

Schroeder, M. (2006). Fractions: Continued, Egyptian and Farey, in M.R. Schroeder (Ed.), Number Theory in Science and Communication, Springer, Berlin/Heidelberg, pp. 55–86.10.1007/3-540-26598-8_5Search in Google Scholar

Singh, C., Bhatia, N. and Kaur, A. (2008). Hough transform based fast skew detection and accurate skew correction methods, Pattern Recognition41(12): 3528–3546.10.1016/j.patcog.2008.06.002Open DOISearch in Google Scholar

Srihari, S.N. and Govindraju, V. (1989). Analysis of textual images using the Hough transform, Machine Vision Applications2(3): 141–153.10.1007/BF01212455Search in Google Scholar

Teh, C.H. and Chin, R.T. (1989). On the detection of dominant points on digital curves, IEEE Transactions on Pattern Analysis and Machine Intelligence2(8): 859–872.10.1109/34.31447Search in Google Scholar

Van, T.T. and Le, T.M. (2016). Content-based image retrieval using a signature graph and a self-organizing map, International Journal of Applied Mathematics and Computer Science26(2): 423–438, DOI: 10.1515/amcs-2016-0030.10.1515/amcs-2016-0030Open DOISearch in Google Scholar

Wall, K. and Danielsson, P.-E. (1984). A fast sequential method for polygonal approximation of digitized curves, Computer Vision Graphics and Image Processing28(3): 220–227.10.1016/S0734-189X(84)80023-7Open DOISearch in Google Scholar

Wang, K., Shi, T., Liao, G. and Xia, Q. (2013). Image registration using a point-line duality based line matching method, Journal of Visual Communication and Image Representation24(5): 615–626.10.1016/j.jvcir.2013.04.010Open DOISearch in Google Scholar

Wang, L., Neumann, U. and You, S. (2009). Wide-baseline image matching using line signatures, International Conference on Computer Vision (ICCV), Kyoto, Japan, pp. 1311–1318.Search in Google Scholar

Yan, H. (1993). Skew correction of document images using interline cross-correlation, CVGIP: Graphical Models and Image Processing55(6): 538–543.10.1006/cgip.1993.1041Open DOISearch in Google Scholar

Yin, P.-Y. (2001). Skew detection and block classification of printed documents, Image and Vision Computing19(8): 567–579.10.1016/S0262-8856(00)00098-6Search in Google Scholar

Yin, P.Y. (2003). Ant colony search algorithms for optimal polygonal approximation of plane curves, Pattern Recognition36(8): 1783–1797.10.1016/S0031-3203(02)00321-7Search in Google Scholar

Yin, P.Y. (2004). A discrete particle swarm algorithm for optimal polygonal approximation of digital curves, Journal of Visual Communication and Image Representation15(2): 241–260.10.1016/j.jvcir.2003.12.001Open DOISearch in Google Scholar

Yu, B. and Jain, A.K. (1996). A robust and fast skew detection algorithm for generic documents, Pattern Recognition29(10): 1599–1630.10.1016/0031-3203(96)00020-9Search in Google Scholar

Yuan, B. and Tan, C.L. (2007). Convex hull based skew estimation, Pattern Recognition40(2): 456–475.10.1016/j.patcog.2006.02.016Open DOISearch in Google Scholar

Zhang, L. and Koch, R. (2013). An efficient and robust line segment matching approach based on LBD descriptor and pairwise geometric consistency, Journal of Visual Communication and Image Representation24(7): 794–805.10.1016/j.jvcir.2013.05.006Open DOISearch in Google Scholar

Zhang, Q., Wang, Y. and Wang, L. (2015). Registration of images with affine geometric distortion based on maximally stable extremal regions and phase congruency, Image and Vision Computing36(C): 23–39.10.1016/j.imavis.2015.01.008Search in Google Scholar