1 / 12

Affine Invariant Triangulation

Affine Invariant Triangulation. Ashish Myles. Overview. Goals/Motivation Delaunay Delaunay-based affine-invariant method Barycentric coordinates-based method. Goal and Motivation. Compute a “nice” triangulation of a point set P Connectivity should be affine invariant Application:

baruch
Télécharger la présentation

Affine Invariant Triangulation

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Affine Invariant Triangulation Ashish Myles

  2. Overview • Goals/Motivation • Delaunay • Delaunay-based affine-invariant method • Barycentric coordinates-based method

  3. Goal and Motivation • Compute a “nice” triangulation of a point set P • Connectivity should be affine invariant • Application: • 1-D b-spline knots (domain) – affine invariant • 2-D simplex spline knots need to be triangulated; desire affine invariance

  4. Delaunay – Strengths • Simple and intuitive criterion (circle-based) – makes proofs easy • Invariant under rotations, reflections and uniform scaling • Globally Local Delaunay  Global Delaunay • Implies uniqueness (up to degeneracy) • Guaranteed convergence of edge-flips • Objective function: maximize minimum angle • Convexity-check built-in

  5. Delaunay Weaknesses • Not general-affine invariant

  6. Affine-Invariant Delaunay • Apply principal components analysis • Rescale to make the principal components equal • Compute Delaunay triangulation • Rescale back to original data set • Demo

  7. Affine-Invariant Delaunay • C = • C = U D UT, U orthogonal, D diagonal • A = D½ UT • Run Delaunay on A-1(P–M), M = (x, x) to get T’ • T = A T’ + M is the final triangulation

  8. Affine-Invariant Delaunay • Advantages: • Same as regular Delaunay + affine-invariance • Disadvantages: • Point-insertion  global transformation change • Concept of locally “good” tainted by global transformation

  9. Barycentric-Based Approach ua + vb + wc = p (1/w)p – (u/w)a – (v/w)b = c –(v/u)b + (1/u)p – (w/u)c = a –(u/v)a + (1/v)p – (w/v)c = b • Goodness: • g(ab) = min(u/v, v/u)  1 • g(cd) = min(-w, -1/w)  1

  10. Barycentric-Based Approach • Geometric intuition • g(edge) = min(dist)/max(dist) = min(area)/max(area) • Choose edge with max goodness (good if closer to 1)

  11. Barycentric-Based Approach • Advantages • Point insertion does not automatically invalidate all edge choices • Automatic convexity check: g(ab) < g(cd)

  12. Barycentric-Based Approach • Disadvantages • No proofs for convergence for edge flips • Conjecture: converge to a local maximum • No proofs for uniqueness • Globally local good does not mean globally good • Conjecture: not always unique? • No obviously good choice for objective function • Max min goodness? • Max average goodness?

More Related