1. A Multi-Scale Tikhonov Regularization Scheme for Implicit Surface Modelling Jianke Zhu, Steven C.H. Hoi and Michael R. Lyu Department of CS&E, The Chinese University of Hong Kong 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. DatasetPointsScale#Base1SVR#Base2Our Hand39.2K437.0K28.1s17.4K1.7s Amadillo173.0K6234.4K131.2s121.9K24.4s Bunny28.0K525.0K17.3s19.0K1.1s Squirrel76.3K6133.1K120.7s70.1K17.0s Igea72.5K663.9K22.3s42.1K2.8s Knot28.7K438.0K37.7s12.3K0.9s Dino56.2K571.1K33.4s42.9K2.8s Feline199.56202.8K114.3s99.9K11.5s Dragon437.6K7346.3K365.9s201.9K77.9s ~ Fig 2. Eliminates the extra zero level-set that occurs on a complex topological object (28.7K, 0.9 second) IEEE Conference on Computer Vision and Pattern Recognition 2007 Fig.1 Illustration of a multi-scale fitting example by the Tikhonov regularization approach. Armadillo (170K points, 24.4 seconds) Method# Base PointsTime Proposed method19.0K1.1s FastRBF29.7K70s Table 1. Results of computational cost on various datasets 1. Multi-scale Fitting 2. Regularization 4. More results 3. Interpolation of Incomplete Data Fig 3. Overfitting problem. Hand (39.2K, 1.7 second) Fig 4. An irregularly sampled Stanford Igea (73K points, 2.8 seconds). Fig 5. Incomplete data. The bunny (28K points, 1.1 seconds). Fig 6. Examples of large-scale implicit surface modelling. Table 2. Compare with FastRBF toolbox 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 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. 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 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.