A Multi-Scale Tikhonov Regularization Scheme for Implicit Surface Modelling

1. A Multi-Scale Tikhonov Regularization Scheme for Implicit Surface Modelling Jianke Zhu, Steven C.H. Hoi and Michael R. LyuDepartment of CS&E, The Chinese University of Hong Kong • Experimental Results • Experimental Setup • Stanford 3D Scanning Repository • Fast marching cube algorithm for rendering • CHOLMOD package for sparse factorization • FastRBF demo version • P4 3.0GHz with 1GB RAM • KFit Toolkit • Parameters setting Table 2. Compare with FastRBF toolbox 1. Multi-scale Fitting Problem and Overview • Relation Work • Regularization networks: • Points on the surface lie in the zero level set. • Slab SVM • SVR • Equivalent Eigenvalue problem • The additional regularization terms are usually to avoid the triviality issue. Table 1. Results of computational cost on various datasets • A fast solution for approximating implicit surfaces based on a multi-scale Tikhonov regularization scheme. • The optimization is formulated into a sparse linear equation system, which can be efficiently solved by factorization methods. • The approach does not employ auxiliary off-surface points, which not only saves the computational cost but also avoids the problem of injected noise. Fig.1 Illustration of a multi-scale fitting example by the Tikhonov regularization approach. Armadillo (170K points, 24.4 seconds) 2. Regularization 3. Interpolation of Incomplete Data • Main contributions: • A Tikhonov Regularization Approach • Representer theorem: • Compactly supported kernel functions pay off with respect to computational efficiency, and lead to a sparse system. • Object function: • K is sparse, the computational cost is determined by the number of base points and the total number of non-zero elements of K. • A fast nearest neighbor searching method is used to compute the • kernel expansion. Such an approach will usually decrease the complexity of computing K from to . • Cholesky and the factorization algorithms are employed to solve the optimization problem. • A Multi-scale Algorithm Fig 2. Eliminates the extra zero level-set that occurs on a complex topological object (28.7K, 0.9 second) Fig 4. An irregularly sampled Stanford Igea (73K points, 2.8 seconds). Fig 5. Incomplete data. The bunny (28K points, 1.1 seconds). 4. More results ~ Fig 3. Overfitting problem. Hand (39.2K, 1.7 second) Fig 6. Examples of large-scale implicit surface modelling. • Conclusion • We presented a novel and efficient solution for the implicit surface modelling using machine learning techniques. • Based on the regularization networks, a multi-scale Tikhonov regularization scheme is proposed. • Empirical evaluations on a number of datasets of different scales re presented. IEEE Conference on Computer Vision and Pattern Recognition 2007