Wiener Subdivision Presented by Koray KAVUKCUOGLU Geometric Modeling Spring 2004

May 05, Introduction – Concepts Wiener Filtering – Theory Wiener Subdivision – Midpoint Subdivision – Application of Filter – Parameters Results Outline

May 05, aim – Derive and Implement a subdivison scheme Based on Marc Alexa’s Wiener Filtering of Meshes methodology – Midpoint Linear Subdivision – Create refined mesh – Wiener Filtering – Relocate vertices to obtain a smooth surface Introduction

May 05, – Filtering of Irregular Meshes using Wiener Filter – Recovering original smooth geometry from noisy data Wiener Filtering

May 05, Mesh – Triangular domain  (K,V) connectivity infovertices in R 3 –Topological Distance (  ) Wiener Filtering - Theory

May 05, – Neighborhood Definition m-ring neighborhood Collection of rings, with radius up to m – Expectation linear operator – Correlation Distance between two vertices Wiener Filtering - Theory

May 05, Representation of Vertex Locations vertex position in noisy mesh true vertex position random noise contribution Estimate each point as a linear sum of given noisy points Find coefficients that minimize square of discrepancy Wiener Filtering - Theory

May 05, Wiener Filtering - Theory Linear System Solution of this system gives, coefficients a ij Need to define distance and correlation functions i 1 2 d d d

May 05, Wiener Subdivision development environment – Language C++ – Mesh format GTS – Windows XP – Cygwin external libs / tools – TNT (template numerical toolkit) Supersedes Lapack++ – Jama/C++ (uses TNT - linear system solution) – Mesh Viewer for visualization

May 05, Wiener Subdivision mesh data structure – Tree  each triangle divided into 4 childs – Triangles – Edge Sharing

May 05, Wiener Subdivision mesh refinement – Linear midpoint subdivision

May 05, Wiener Subdivision filtering – computing Topology – compute m-ring neighborhood BFS over vertices – compute distance and correlation x is parameterized for smoothness control

May 05, filtering – solve linear system – LU decomposition method – Jama/C++ Wiener Subdivision

May 05, Wiener Subdivision parameters – size of m-ring neighborhood (1, 2, …) – smoothness parameter – fraction of old vertex location in new location

May 05, results -m1 / -n3 / -sp2

May 05, results -m1 / -n3 / -sp0-m2 / -n3 / -sp2

May 05, results -m1 / -n3 / -sp0 -m2 / -n3 / -sp2 -m1 / -n3 / -sp2

May 05, results -m1 / -n3 / -sp2-m1 / -n3 / -sp2 / -p0.3

May 05, results -m1 / -n3 / -sp2 / -p0.3 -m2 / -n3 / -sp2

May 05, results

May 05, results

May 05, Questions?