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Rectilinearity Measurements for Polygons
September 2003 (vol. 25 no. 9)
pp. 1193-1200

Abstract—The paper introduces a shape measure intended to describe the extent to which a closed polygon is rectilinear. Other than somewhat obvious measures of rectilinearity (e.g., the sum of the differences of each corner's angle from multiples of 90^\circ), there has been little work in deriving a measure that is straightforward to compute, is invariant under scale, rotation, and translation, and corresponds with the intuitive notion of rectilinear shapes. There are applications in a number of different areas of computer vision and photogrammetry. Rectilinear structures often correspond to human-made objects and are therefore justified as attentional cues for further processing. For instance, in aerial image processing and reconstruction, where building footprints are often rectilinear on the local ground plane, building structures, once recognized as rectilinear, can be matched to corresponding shapes in other views for stereo reconstruction. Perceptual grouping algorithms may seek to complete shapes based on the assumption that the object in question is rectilinear. Using the proposed measure, such systems can verify this assumption.

[1] M. Brady and A.L. Yuille, An Extremum Principle for Shape from Contour IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 6, no. 3, pp. 288-301, 1984.
[2] A. Brunn, U. Weidner, and W. Förstner, Model-Based 2D-Shape Recovery Mustererkennung, G. Sagerer et al., eds., Springer-Verlag, pp. 260-268, 1995.
[3] D. Chen and J. Xu, An Efficient Direct Approach for Computing Shortest Rectilinear Paths Among Obstacles in a Two-Layer Interconnection Model Computational Geometry: Theory and Applications, vol. 18, pp. 155-166, 2001.
[4] R. Collins, C. Jaynes, Y.Q. Cheng, X. Wang, F. Stolle, E. Riseman, and A. Hanson, “The Ascender System: Automated Site Modeling from Multiple Aerial Images,” Computer Vision and Image Understanding, vol. 72, no. 2, pp. 143-162, Nov. 1998.
[5] P.M. Dare, R. Ruskoni, and I.J. Dowman, Algorithm Development for the Automatic Registration of Satellite Images Proc. Image Registration Workshop, NASA Goddard Space Flight Centre, pp. 83-88, 1997.
[6] J.M. Díaz-Baqez and J.A. Mesa, Fitting Rectilinear Polygonal Curves to a Set of Points in the Plane European J. Oper. Res., vol. 130, no. 1, pp. 214-222, 2001.
[7] S. Hyde et al. The Language of Shape. Elsevier, 1997.
[8] J Feldman, Bias toward Regular Form in Mental Shape Spaces J. Experimental Psychology: Human Perception, vol. 26, no. 1, pp. 1-14, 2000.
[9] A.R. Hanson, M. Marengoni, H. Schultz, F. Stolle, and E.M. Riseman, Ascender II: A Framework for Reconstruction of Scenes from Aerial Images Proc. Int'l Workshop Aerial and Spaceborne Imagery, pp. 25-34, 2001.
[10] R.M. Haralick, A Measure for Circularity of Digital Figures IEEE Trans. Systems, Man, and Cybernetics, vol. 4, pp. 394-396, 1974.
[11] Y.C. Hsieh,D.M. Mckeown, Jr.,, and F.P. Perlant,“Performance evaluation of scene registration and stereo matching for cartographic feature extraction,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 14, no. 2, Feb. 1992.
[12] R. Irving and D. McKeown, “Methods for Exploiting the Relationship between Buildings and Their Shadows in Aerial Imagery,” IEEE Trans. Systems, Man, and Cybernetics, vol. 19, no. 6, pp. 1564-1575, Nov./Dec. 1989.
[13] A.K. Jain and A. Vailaya, Shape-Based Retrieval: A Case-Study with Trademark Image Databases Pattern Recognition, vol. 31, no. 9, pp. 1369-1390, 1998.
[14] M. Kass, A. Witkin, and D. Terzopoulos, Snakes: Active Contour Models Proc. Int'l Conf. Computer Vision, pp. 259-268, 1987.
[15] Y. Liow and T. Pavlidis, Use of Shadows for Extracting Buildings in Aerial Images Computer Vision, Graphics, and Image Processing, vol. 49, no. 2, pp. 242-277, Feb. 1990.
[16] S. Mayer, Constrained Optimization of Building Contours from High-Resolution Ortho-Images Proc. Int'l Conf. Image Processing, 2001.
[17] S. Noronha and R. Nevatia, Detection and Modeling of Buildings from Multiple Aerial Images IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 23, no. 5, pp. 501-518, May 2001.
[18] D. Proffitt, The Measurement of Circularity and Ellipticity on a Digital Grid Pattern Recognition, vol. 15, no. 5, pp. 383-387, 1982.
[19] P.L. Rosin, Measuring Rectangularity Machine Vision and Applications, vol. 11, pp. 191-196, 1999.
[20] P.L. Rosin, Shape Partitioning by Convexity IEEE Trans. Systems, Man, and Cybernetics, vol. 30, no. 2, pp. 202-210, 2000.
[21] P.L. Rosin and J. Hervás, Remote Sensing Image Thresholding for Determining Landslide Activity Int'l J. Remote Sensing, pending publication.
[22] M. Sonka, V. Hlavac, and R. Boyle, Image Processing, Analysis, and Machine Vision. Chapman and Hall, 1993.
[23] A.D. Ventura, A. Rampini, and R. Schettini, Image Registration by Recognition of Corresponding Structures IEEE Trans. Geoscience and Remote Sensing, vol. 28, no. 3, pp. 305-314, 1990.
[24] J. Willats, Art and Representation: New Principles in the Analysis of Pictures. Princeton Univ. Press, 1997.
[25] D.J. Williams and M. Shah,“A fast algorithm for active contours and curvature estimation,” Computer Vision, Graphics, Image Processing, vol. 55, pp. 14-26, 1992.
[26] A. Witkin, Shape from Contour Technical Report 589, MIT, 1980.

Index Terms:
Shape, polygons, rectilinearity, measurement.
Jovi?a ?unic, Paul L. Rosin, "Rectilinearity Measurements for Polygons," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 25, no. 9, pp. 1193-1200, Sept. 2003, doi:10.1109/TPAMI.2003.1227997
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