Relaxing the 3L algorithm for an accurate implicit polynomial fitting

This paper presents a novel method to increase the accuracy of linear fitting of implicit polynomials. The proposed method is based on the 3L algorithm philosophy. The novelty lies on the relaxation of the additional constraints, already imposed by the 3L algorithm. Hence, the accuracy of the final solution is increased due to the proper adjustment of the expected values in the aforementioned additional constraints. Although iterative, the proposed approach solves the fitting problem within a linear framework, which is independent of the threshold tuning. Experimental results, both in 2D and 3D, showing improvements in the accuracy of the fitting are presented. Comparisons with both state of the art algorithms and a geometric based one (non-linear fitting), which is used as a ground truth, are provided.

[1]  Cem Ünsalan.  A model based approach for pose estimation and rotation invariant object matching , 2007, Pattern Recognit. Lett..

[2]  Hans-Jürgen Warnecke,et al.  Orthogonal Distance Fitting of Implicit Curves and Surfaces , 2002, IEEE Trans. Pattern Anal. Mach. Intell..

[3]  Cem ínsalan.  A model based approach for pose estimation and rotation invariant object matching , 2007 .

[4]  David Malah,et al.  Stable fitting of 2D curves and 3D surfaces by implicit polynomials , 2004, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[5]  Giovanni Marola.  A technique for finding the symmetry axes of implicit polynomial curves under perspective projection , 2005, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[6]  Aytül Erçil,et al.  Combining implicit polynomials and geometric features for hand recognition , 2003, Pattern Recognit. Lett..

[7]  Martin Aigner,et al.  Gauss-Newton-type techniques for robustly fitting implicitly defined curves and surfaces to unorganized data points , 2008, Shape Modeling International Conference.

[8]  Petko Faber,et al.  Pros and Cons of Euclidean Fitting , 2001, DAGM-Symposium.

[9]  David B. Cooper,et al.  Improving the stability of algebraic curves for applications , 2000, IEEE Trans. Image Process..

[10]  Katsushi Ikeuchi,et al.  3D Model Segmentation and Representation with Implicit Polynomials , 2008, IEICE Trans. Inf. Syst..

[11]  Mustafa Unel,et al.  Fitting globally stabilized algebraic surfaces to range data , 2005, Tenth IEEE International Conference on Computer Vision (ICCV'05) Volume 1.

[12]  Farzin Mokhtarian,et al.  A Theory of Multiscale, Curvature-Based Shape Representation for Planar Curves , 1992, IEEE Trans. Pattern Anal. Mach. Intell..

[13]  David B. Cooper,et al.  The 3L Algorithm for Fitting Implicit Polynomial Curves and Surfaces to Data , 2000, IEEE Trans. Pattern Anal. Mach. Intell..

[14]  Takeshi Oishi,et al.  A Fast Registration Method Using IP and Its Application to Ultrasound Image Registration , 2009, IPSJ Trans. Comput. Vis. Appl..

[15]  Gabriel Taubin,et al.  Estimation of Planar Curves, Surfaces, and Nonplanar Space Curves Defined by Implicit Equations with Applications to Edge and Range Image Segmentation , 1991, IEEE Trans. Pattern Anal. Mach. Intell..