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. Laboratory of Human Motion Study & Virtual Reality - INB CNRHttp://www.inb.mi.cnr.it/Borghese.html Autoscan 3D digitiser Manual scanning (selective sampling) Motion Capture for scanning Portable & flexible

4. Laboratory of Human Motion Study & Virtual Reality - INB CNRHttp://www.inb.mi.cnr.it/Borghese.html 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. Laboratory of Human Motion Study & Virtual Reality - INB CNRHttp://www.inb.mi.cnr.it/Borghese.html a1(x)a1(x) a2(x)a2(x) s(x)s(x) r1(x)r1(x) r2(x)r2(x) aJ(x)aJ(x) rJ(x)rJ(x) s(x) aJ(x)aJ(x) a2(x)a2(x) a1(x)a1(x) x … a0(x)a0(x) r J-2 (x) r0(x)r0(x) … r J-1 (x) s(x) x a0(x)a0(x) r J-1 (x) r J-2 (x) s(x)s(x) a J-1 (x) a J-2 (x) r0(x)r0(x) MRA Analysis HRBF Analysis MRA Synthesis HRBF Synthesis

7. Laboratory of Human Motion Study & Virtual Reality - INB CNRHttp://www.inb.mi.cnr.it/Borghese.html The surface is therefore reconstructed as: HRBF Analysis HRBF Synthesis

8. Multi-scale Reconstruction with HRBF

10. Laboratory of Human Motion Study & Virtual Reality - INB CNRHttp://www.inb.mi.cnr.it/Borghese.html Multi-scale Reconstruction with HRBF

11. Laboratory of Human Motion Study & Virtual Reality - INB CNRHttp://www.inb.mi.cnr.it/Borghese.html Multi-scale Reconstruction with HRBF

12. Laboratory of Human Motion Study & Virtual Reality - INB CNRHttp://www.inb.mi.cnr.it/Borghese.html Position of the Gaussians

13. Laboratory of Human Motion Study & Virtual Reality - INB CNRHttp://www.inb.mi.cnr.it/Borghese.html 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