Digital Geometry Processing

Transcription

1 Digital Geometry Processing Spring 2011 physical model acquired point cloud reconstructed model 2 Digital Michelangelo Project Range Scanning Systems Passive: Stereo Matching Find and match features in both images 1G sample points 8M triangles Problem: Needs features to match 3 4 Range Scanning Systems Active: Structured Light Project special b/w patterns to identify pixels Range Scanning Systems Active: Laser Scanning Sweep laser, record where pixel intensity is max. 5 Problematic for materials / textures having strong color differences. 6 Problematic for difficult reflectance properties (highly specular, hairs) Page 1

2 Range Scanning Active systems are superior Ultimate Goal Multiple scans required for complex objects Scan path planning Scan registration Scans are incomplete and noisy Model repair, hole filling Smoothing for noise removal (later) set of raw scans reconstructed model 7 8 Input Data Set of irregular sample points Typically without normal vectors E.g., union of all range scan vertices Set of range scans Each scan is a regular quad- or tri-mesh Normal vectors are well defined Register scans into common coordinate system Explicit Reconstruction Methods Connect sample points by triangles Exact interpolation of sample points Bad for noisy or misaligned data Can lead to holes or non-manifold situations 9 10 Implicit Reconstruction Methods Estimate signed distance function (SDF) Extract zero isosurface by Marching Cubes Approximation of input points Watertight manifold results by construction Input Implicit Explicit 11 Page 2

3 Signed Distance Function Signed Distance Function < 0 0 > 0 < 0 0 > 0 Signed Distance Function Normal Estimation Construct SDF from point samples Distance to points is not enough Need inside/outside information Reconstruct normal vectors first + - To find normal ni for each sample point pi 1. Examine local neighborhood for each point Set of k nearest neighbors 2. Compute best approximating tangent plane Principle Components Analysis (PCA) 3. Determine normal orientation MST propagation pi Principal Components Analysis Normal Orientation Build graph connecting neighboring points Edge (ij) exists if pi knn(pj) or pj knn(pi) Propagate normal orientation through graph For neighbors pi,pj: Flip nj if ni T nj < 0 Fails at sharp edges/corners ni pi Center data at origin Compute 3x3 covariance matrix Normal direction is eigenvector with smallest eigenvalue Propagate along safe paths (parallel normals) Minimum spanning tree with angle-based edge weights wij = 1- ni T nj pj nj 18 Page 3

4 SDF from Point & Normals Signed distance from tangent planes Points + normals determine local tangent planes Use distance from closest point s tangent plane Linear approximation in Voronoi cell Simple and efficient, but SDF is only C -1 Smooth SDF Approximation Scattered data interpolation problem On-surface constraints dist(pi) = 0 Avoid trivial solution dist 0 Off-surface constraints dist(pi + ni) = 1 x Fx ( ) Smooth SDF Approximation Scattered data interpolation problem On-surface constraints dist(pi) = 0 Avoid trivial solution dist 0 Off-surface constraints dist(pi + ni) = 1 Radial basis functions Well suited for smooth interpolation Sum of shifted, weighted kernel functions RBF Interpolation Interpolate on- and off-surface constraints Choose centers ci = xi Solve symmetric linear system for weights wi RBF Basis Functions Other Radial Basis Functions Triharmonic basis functions Globally supported Leads to dense symmetric linear system Provably smooth Works well for highly irregular sampling 23 Polyharmonic spline k ( r) r log( r), k 2,4,6,... k ( r) r, k 1,3,5,... Multiquadratic () r r Gaussian () r e r B-Spline (compact support) ( r) piecewise-poly( r) Page 4

5 Image from: Digital Geometry Processing Off-surface Points Examples Without off-surface points With off-surface points Comparison Complexity Issues Solve the linear system for RBF weights Dense linear system Difficult to solve for large number of samples Compactly supported RBFs Sparse linear system Efficient solvers Distance to plane Compact RBF Global RBF Triharmonic Extracting the Surface Sample the SDF F(x) = 0 surface F(x) < 0 inside F(x) > 0 outside 30 Page 5

6 Sample the SDF 2D: Marching Squares? cases 15 cases Inversion 3D Marching Cubes Rotation Marching Cubes 33 Marching Cubes Problems Ambiguity Holes Generates HUGE meshes Millions of polygons Page 6

7 Ambiguity The Inversion Problem Reduction from 256 to 15 cases includes inversion Separate pink Separate blue The Inversion Problem Inversion Ambiguity Solution 256 cases 23 cases Rotation only Always separate same color Ambiguous faces triangulated consistently 8 new cases Mismatch Inversion Without Inversion Ambiguity Match demo No Ambiguity Page 7

8 Marching Cubes Issues Grid not adaptive Many polygons required to represent small features Ambiguity No Ambiguity Images from: Dual Marching Cubes: Primal Contouring of Dual Grids by Schaeffer et al. Page 8