Powerpoint template for scientific posters (Swarthmore College)

1
A Multi-Scale Tikhonov Regularization Scheme for
Implicit Surface Modelling Jianke Zhu, Steven
C.H. Hoi and Michael R. LyuDepartment of CSE,
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
Method Base Points Time
Proposed method 19.0K 1.1s
FastRBF 29.7K 70s
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
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

• 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