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Objectives

Hierarchical Radial Basis Function Networks for 3-D Surface Reconstruction ( Borghese and Ferrari , Neurocomputing , 1998 ). A constructive hierarchical RBF network for 3-D surface reconstruction from irregularly sampled data. Objectives.

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Objectives

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  1. Hierarchical Radial Basis Function Networks for 3-D Surface Reconstruction (Borghese and Ferrari, Neurocomputing, 1998) A constructive hierarchical RBF network for 3-D surface reconstruction from irregularly sampled data

  2. Objectives • Reconstruction of 3-D surfaces from irregularly sampled data About this presentation • Presentation of the problem • HRBF and wavelet MRA approaches • Examples • Conclusions

  3. Autoscan 3D digitiser • Manual scanning (selective sampling) • Motion Capture for scanning • Portable & flexible

  4. How to go from points to meshes Problem: noise Solution: regularised reconstruction • Human body parts are smooth • Noise has higher frequencies than the surface • Surface has been oversampled

  5. Properties of HRBF • Requires no regularly sampled 3-D data • Multiresolution, coarse-to-fine (bottom-up), regularly grid, dyadic RBF network • The number of layers is not determined on a a priori basis: the network grows until a convergence criterion is met • RBF parameters (, ) are constrained through sampling theory • RBFs are selectively located on a regular grid based on the local error

  6. rJ(x) aJ-2(x) aJ-1(x) r1(x) r2(x) s(x) s(x) a1(x) rJ-1(x) a2(x) aJ(x) aJ(x) rJ-2(x) a0(x) r0(x) x x … a2(x) r0(x) a0(x) a1(x) rJ-1(x) rJ-2(x) … s(x) s(x) MRA Analysis HRBF Analysis MRA Synthesis HRBF Synthesis

  7. HRBF Analysis HRBF Synthesis The surface is therefore reconstructed as:

  8. Multi-scale Reconstruction with HRBF

  9. Multi-scale Reconstruction with HRBF

  10. Multi-scale Reconstruction with HRBF

  11. Multi-scale Reconstruction with HRBF

  12. Position of the Gaussians

  13. Multi-scale Reconstruction with Wavelet MRA

  14. Advantages of HRBF vs wavelet MRA • Coarse-to-fine (bottom-up) vs fine-to-coarse (top-down) approximation. The number of layers is not determined on a a priori basis (the network grows until a convergence criterion is met) • The approximation process can be stopped at a certain level of detail having the outline of the surface with few coefficients • Coefficent elimination is carried out during learning on the basis of the local error. There is no analogous mechanism in MRA. • Requires no regular sampling • Like MRA, HRBF features a firm foundation in data sampling theory Disadvantages of HRBF vs wavelet MRA • Computational cost

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