1 / 25

Adding charts anywhere Assume a cow is a sphere

Adding charts anywhere Assume a cow is a sphere. Cindy Grimm and John Hughes, “ Parameterizing n-holed tori ”, Mathematics of Surfaces X, 2003 Cindy Grimm, “ Parameterization using Manifolds”, International Journal of Shape Modelling, 2004. Motivation.

zacharyr
Download Presentation

Adding charts anywhere Assume a cow is a sphere

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. Adding charts anywhereAssume a cow is a sphere Cindy Grimm and John Hughes, “Parameterizing n-holed tori”, Mathematics of Surfaces X, 2003 Cindy Grimm, “Parameterization using Manifolds”, International Journal of Shape Modelling, 2004

  2. Motivation • Surface modeling: Adding more degrees of freedom • Don’t want to be restricted to existing charts/dof • Computers support vectors (points in Rn) • Functions on Rn • Splines, gradient descent algorithms • Define functions directly on sphere? • BRDFs, environment maps Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  3. Observation • We can define lots of meshes that are spheres • Which one’s the “right” one? • How do you compare two sphere meshes? Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  4. Approach • Find a computationally tractable way to represent some common domains • E.g., sphere, torus, plane • Doesn’t depend on any particular geometry • Questions to answer: • What is a point? (Convert from Rn to domain) • Movement in domain • Barycentric coordinates, linear interpolation Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  5. Once we have a domain… • Add charts. Anywhere. • Not restricted by existing atlas/construction • Questions to answer: • Chart specification? • Chart function form? Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  6. Applications • Surface modeling, texture mapping • Surface comparison, feature parameterization • Sphere: environment maps, BRDFs • Arbitrary chart placement • Port existing tools (linear embedding functions, gradient descent algorithms, etc.) to any domain transparently • Adaptive parameterization Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  7. Outline • Spherical domain • Defining • Adding charts • Building embedding function (surface • Torus, n-holed tori domains • Torus: Tiled plane • N-holed: Tiled plane, sort-of Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  8. Atlas from mesh • Embed mesh in domain • 1-1 correspondence • Specifies both chart locations and chart embeddings Sketch Embedded sketch Vertex charts Surface Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  9. In contrast… • Approaches presented earlier created abstract manifold from scratch • Specify transition functions/overlaps • Chart functions derived • Charts exactly match mesh connectivity • This approach • Define chart functions • Transition functions/overlaps are derived • Charts loosely tied to mesh connectivity • Both approaches fit geometry to subdivision surface Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  10. Representing a sphere • Representing points • x2+y2+z2 -1 = 0 • Average/blend points • Project back onto sphere (Gnomonic projection) • Valid in ½ hemisphere • Barycentric coordinates in spherical triangles • Interpolate in triangle, project • Line segments (arcs) Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  11. Charts • Stereographic projection • Sketch mesh charts • Circles to circles • Requires rotation to center at arbitrary point • Both C¥ Images: mathworld.wolfram.com Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  12. Charts, adjusting • Raw projections MD don’t provide much control • Map to consistent chart shape (e.g., unit disk) • Add additional “warp” function MW • Projective transform (scale, translate, rotate) • StillC¥ Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  13. Overlaps and transition functions • Overlaps • Chart inverse functions well-defined, C¥, over chart range/disk (but not over entire plane) • Coverage of chart determined by inverse map • Disk->sphere • Transition functions • Defined by composition a(a-1(s,t)) Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  14. Sketch atlas • Properties • Charts must cover sphere • Overlap substantially with neighbors • But not non-adjacent elements • Relate to sketch mesh • One chart for each vertex, edge, face • Vertex: centered, overlaps ½ adjacent edges/faces • Edge: covers end points, centroids of adjacent faces • Face: covers interior of face Vertex charts Edge charts Face charts Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  15. Creating embedding • Mesh in 1-1 correspondence with sphere • Subdivide original 3D mesh and mesh embedded in domain • Keeps correspondences • Fit chart embedding to corresponding part of subdivision surface Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  16. Blend and embedding functions • Blend: Uniform B-Spline basis function “spun” around origin • Produces smallest second derivative variations • Embed: C6 polynomial • No problems with end-conditions, skew • Continuity: Continuity of blend functions (everything else is C¥) Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  17. Advantages • Transition functions are “free”, C¥ • No difficulties with co-cycle condition • Can add additional charts anywhere • Embedding function provides a natural mechanism for “masking out” original surface • Mask out blend functions • Ck surface pasting with no constraints • Re-parameterization is easy • Take out old charts and add new ones Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  18. Disadvantages • Need to make/find canonical manifold for each genus • May prove difficult for higher dimensions • Requires “tweaking” of chart construction to get “nice” overlaps, guaranteed coverage • Regularization of embedded sketch mesh • Heuristics/optimizations for chart construction • Warping function Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  19. Torus • Similar to circle example • Repeat in both s and t: [0,2p) X [0,2p) • Chart is defined by projective transform • Care with wrapping (2p,2p ) X X (0,0) Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  20. Torus, associated edges • Cut torus open to make a square • Two loops (yellow one around, grey one through) • Each loop is 2 edges on square • Glue edges together • Loops meet at a point Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  21. b a a b d c c d N-Holed tori • Similar to torus – cut open to make a 4N-sided polygon • Two loops per hole (one around, one through) • Glue two polygon edges to make loop • Loops meet at a point • Polygon vertices glue to same point Front Back Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  22. Hyperbolic geometry • Why is my polygon that funny shape? • Need corners of polygon to each have 2p / 4N degrees (so they fill circle when glued together) • Tile hyperbolic disk with 4N-sided polygons Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  23. Hyperbolic geometry • Edges are circle arcs; circles meet boundary at right angles • Linear fractional transforms • Equivalent to matrix operations in Euclidean geometry, e.g., rotate, translate, scale • Invertible • Chart: Use a Linear fractional transform to map point(s) to origin, then apply warp function • Need to ensure we use correct copy in chart function Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  24. Summary • There are some manifolds we use often • Sphere, tori, circle, plane, S3 (quaternions) • Construct a general-purpose manifold + atlas + chart creation + transition functions • Now can use any tools that operate in Rn • Use same tools for all topologies • Can build charts at any scale, anywhere • Not dictated by initial construction/sketch Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  25. Open questions • Boundaries • Function discontinuities • Changing topologies • Establishing correspondences between existing surfaces and canonical manifold • Parameterization (next section) • Where and how to place charts Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

More Related