Abstract
In this paper, we employ low-rank matrix approximation to solve a general parameter estimation problem: where a non-linear system is linearized by treating the carrier terms as separate variables, thereby introducing heteroscedastic noise. We extend the bilinear approach to handle cases with heteroscedastic noise, in the framework of low-rank approximation.
The ellipse fitting problem is investigated as a specific example of the general theory. Despite the impression given in the literature, the ellipse fitting problem is still unsolved when the data comes from a small section of the ellipse. Although there are already some good approaches to the problem of ellipse fitting, such as FNS and HEIV, convergence in these iterative approaches is not ensured, as pointed out in the literature. Another limitation of these approaches is that they cannot model the correlations among different rows of the “general measurement matrix”. Our method, of employing the bilinear approach to solve the general heteroscedastic parameter estimation problem, overcomes these limitations: it is convergent, at least to a local optimum, and can cope with a general heteroscedastic problem. Experiments show that the proposed bilinear approach performs better than other competing approaches: although it is still far short of a solution when the data comes from a very small arc of the ellipse.
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Aguiar, P.M.Q., Moura, J.M.F.: Factorization as a rank 1 problem. In: CVPR (1999)
Aguiar, P.M.Q., Moura, J.M.F.: Weighted factorization. In: ICIP (2000)
Aguiar, P.M.Q., Moura, J.M.F.: Rank 1 weighted factorization for 3D structure recovery: algorithms and performance analysis. IEEE Trans. Pattern Anal. Mach. Intell. 25(9), 1134–1149 (2003)
Anandan, P., Irani, M.: Factorization with uncertainty. Int. J. Comput. Vis. 49(2–3), 101–116 (2002)
Chen, P., Suter, D.: Recovering the missing components in a large noisy low-rank matrix: Application to SFM. IEEE Trans. Pattern Anal. Mach. Intell. 26(8), 1051–1063 (2004)
Chojnacki, W., et al.: On the fitting of surfaces to data with covariances. IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1294–1303 (2000)
Chojnacki, W., et al.: Revisiting Hartley’s normalized eight-point algorithm. IEEE Trans. Pattern Anal. Mach. Intell. 25(9), 1172–1177 (2003)
Chojnacki, W., et al.: From fns to heiv: a link between two vision parameter estimation methods. IEEE Trans. Pattern Anal. Mach. Intell. 26(2), 264–268 (2004)
Fitzgibbon, A., Pilu, M., Fisher, R.B.: Direct least square fitting of ellipses. IEEE Trans. Pattern Anal. Mach. Intell. 21(5), 476–480 (1999)
Golub, G.H., Loan, C.F.V.: Matrix Computations, 3rd edn. John Hopkins University Press, Baltimore (1996)
Guerreiro, R.F.C., Aguiar, P.M.Q.: Estimation of rank deficient matrices from partial observations: two-step iterative algorithms. In: Energy Minimization Methods in Computer Vision and Pattern Recognition (2003)
Hartley, R., Zisserman, A.: Multiple view geometry in computer vision. Cambridge University Press, Cambridge (2000)
Huffel, S.V., Vandewalle, J.: The total least squares problem: computational aspects and analysis. SIAM (1991)
Irani, M., Anandan, P.: Factorization with uncertainty. In: Proc. European Conf. Computer Vision (2000)
Jacobs, D.: Linear fitting with missing data: Applications to structure-from-motion and to characterizing intensity images. In: CVPR (1997)
Jacobs, D.: Linear fitting with missing data for structure-from-motion. Comput. Vis. Image Underst. 82, 57–81 (2001)
Kahl, F., Heyden, A.: Structure and motion from points, lines and conics with affine cameras. In: Proc. European Conf. Computer Vision (1998)
Kahl, F., Heyden, A.: Affine structure and motion from points, lines and conics. Int. J. Comput. Vis. 33(3), 163–180 (1999)
Kanatani, K.: Unbiased estimation and statistical analysis of 3-D rigid motion from two views. IEEE Trans. Pattern Anal. Mach. Intell. 15(1), 37–50 (1993)
Kanatani, K.: Statistical bias of conic fitting and renormalization. IEEE Trans. Pattern Anal. Mach. Intell. 16(3), 320–326 (1994)
Kanatani, K.: Statistical Optimization for Geometric Computation: Theory and Practice. Amsterdam, Elsevier (1996)
Leedan, Y., Meer, P.: Estimation with bilinear constraints in computer vision. In: Proc. International Conf. Computer Vision (1999)
Leedan, Y., Meer, P.: Heteroscedastic regression in computer vision: problems with bilinear constraint. Int. J. Comput. Vis. 37(2), 127–150 (2000)
Mahamud, S., et al.: Provably-convergent iterative methods for projective structure from motion. In: Proc. Conf. Computer Vision and Pattern Recognition (2001)
Manton, J.H., Mahony, R., Hua, Y.: The geometry of weighted low-rank approximations. IEEE Trans. Signal Process. 51(2), 500–514 (2003)
Matei, B., Meer, P.: A general method for Errors-in-Variables problems in computer vision. In: Proc. Conf. Computer Vision and Pattern Recognition (2000)
Matei, B., Meer, P.: Reduction of bias in maximum likelihood ellipse fitting. In: ICPR (2000)
Morris, D.D., Kanade, T.: A unified factorization algorithm for points, line segments and planes with uncertainty models. In: Proc. International Conf. Computer Vision (1998)
Press, W.H., et al.: Numerical Recipes in C, 2nd edn. Cambridge University Press, Cambridge (1992)
Reid, I.D., Murray, D.W.: Active tracking of foveated feature clusters using affine structure. Int. J. Comput. Vis. 18(1), 41–60 (1996)
Shum, H., Ikeuchi, K., Reddy, R.: Principal component analysis with missing data and its applications to polyhedral object modeling. IEEE Trans. Pattern Anal. Mach. Intell. 17(9), 854–867 (1995)
Stewart, G.W., Sun, J.G.: Matrix Perturbation Theory. Academic Press, San Diego (1990)
Vidal, R., Hartley, R.: Motion segmentation with missing data using PowerFactorization and GPCA. In: Proc. Conf. Computer Vision and Pattern Recognition (2004)
Zhang, Z.: Parameter estimation techniques: A tutorial with application to conic fitting. Image Vis. Comput. 15, 59–76 (1997)
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Chen, P., Suter, D. A Bilinear Approach to the Parameter Estimation of a General Heteroscedastic Linear System, with Application to Conic Fitting. J Math Imaging Vis 28, 191–208 (2007). https://doi.org/10.1007/s10851-007-0003-z
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DOI: https://doi.org/10.1007/s10851-007-0003-z