Abstract
This paper describes a new method for resistant and robust alignment of sets of 2D shapes wrt. position, rotation, and iso-tropical scaling. Apart from robustness a major advantage of the method is that it is formulated as a linear programming (LP) problem, thus enabling the use of well known and thoroughly tested standard numerical software. The problem is formulated as the minimization of the norm of a linear vector function with a constraint of non-zero size. This is achieved by using the Manhattan distance between points in the plane. Unfortunately the Manhattan distance is dependent on the orientation of the coordinate system, i.e. it is not rotationally invariant. However, by simultaneously minimizing the Manhattan distances in a series of rotated coordinate systems we are able to approximate the circular equidistance curves of Euclidean distances with a regular polygonal equidistance curve to the precision needed. Using 3 coordinate systems rotated 30° we get a 12 sided regular polygon, with which we achieve deviations from Euclidean distances less than 2% over all directions. This new formulation allows for minimization in the L 1-norm using LP. We demonstrate that the use of the L 1-norm results in resistance towards object as well as landmark outliers. Examples that illustrate the properties of the robust norm are given on simulated as well as a biological data sets.
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References
Kendall, D.G.: The diffusion of shape. Adv. Appl. Probab. 9, 428–430 (1977)
Gower, J.C.: Generalized Procrustes analysis. Psychometrika 40, 33–50 (1975)
ten Berge, J.M.F.: Orthogonal Procrustes rotation for two or more matrices. Psychometrika 42, 267–276 (1977)
Goodall, C.: Procrustes methods in the statistical analysis of shape. J. R. Stat. Soc. Ser. B 53(2), 285–339 (1991)
Dryden, I.L., Mardia, K.: Statistical Shape Analysis. Wiley, Chichester (1998)
Siegel, A.F.: A robust comparison of biological shapes. Biometrics 38, 341–350 (1982)
Siegel, A.F.: Robust regression using repeated medians. Biometrika 69(1), 242–244 (1982)
Dryden, I.L., Walker, G.: Highly resistant regression and object matching. Biometrics 55, 820–825 (1999)
Huttenlocher, D.P., Klanderman, G.A., Rucklidge, W.J.: Comparing images using the Hausdorff distance. IEEE Trans. Pattern Anal. Mach. Intell. 15(9), 850–863 (1993)
Fischler, M.A., Bolles, R.C.: Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography. Commun. ACM 24(6), 381–395 (1981)
Er, F.: Robust methods in statistical shape analysis. PhD thesis, University of Leeds, Department of Statistics, Leeds, UK (1998)
Dempster, A., Laird, N., Rubin, D.: Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. Ser. B (1977)
Lima, V.M.C., Kent, J.T.: The non-uniqueness problem in l 1 ordinary Procrustes analysis. Student 4 (2003)
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Larsen, R. L 1 Generalized Procrustes 2D Shape Alignment. J Math Imaging Vis 31, 189–194 (2008). https://doi.org/10.1007/s10851-008-0077-2
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DOI: https://doi.org/10.1007/s10851-008-0077-2