A Multi-Scale Tikhonov Regularization Scheme for Implicit

Surface ModellingJianke Zhu, Steven C.H. Hoi and Michael R. Lyu

Department of CS&E, The Chinese University of Hong KongProblem 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.

Dataset Points Scale

#Base1 SVR #Base2 Our

Hand 39.2K 4 37.0K 28.1s 17.4K 1.7s

Amadillo

173.0K 6 234.4K 131.2s

121.9K 24.4s

Bunny 28.0K 5 25.0K 17.3s 19.0K 1.1s

Squirrel

76.3K 6 133.1K 120.7s

70.1K 17.0s

Igea 72.5K 6 63.9K 22.3s 42.1K 2.8s

Knot 28.7K 4 38.0K 37.7s 12.3K 0.9s

Dino 56.2K 5 71.1K 33.4s 42.9K 2.8s

Feline 199.5 6 202.8K 114.3s

99.9K 11.5s

Dragon 437.6K 7 346.3K 365.9s

201.9K 77.9s

~

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 2007IEEE 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 Points Time

Proposed method 19.0K 1.1s

FastRBF 29.7K 70s

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.

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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 thekernel 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.

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•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.