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Spherical manifolds for hierarchical surface modeling

Spherical manifolds for hierarchical surface modeling. Cindy Grimm. Goal. Organic, free-form Adaptive Can add arms anywhere Hierarchical Move arm, hand moves C k analytic surface Build in pieces. Overview. Building an atlas for a spherical manifold Embedding the atlas

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Spherical manifolds for hierarchical surface modeling

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  1. Spherical manifolds for hierarchical surface modeling Cindy Grimm

  2. Goal • Organic, free-form • Adaptive • Can add arms anywhere • Hierarchical • Move arm, hand moves • Ck analytic surface • Build in pieces Graphite 2005, 12/1/2005

  3. Overview • Building an atlas for a spherical manifold • Embedding the atlas • Surface modeling • Sketch mesh • Adaptive editing Graphite 2005, 12/1/2005

  4. M Traditional atlas definition • Given: Sphere (manifold) M • Construct: Atlas A • Chart • Region Uc in M (open disk) • Region c in R2 (open disk) • Function ac taking Uc to c • Inverse • Atlas is collection of charts • Every point in M in at least one chart • Overlap regions Graphite 2005, 12/1/2005

  5. Operations on the sphere • How to represent points, lines and triangles on a sphere? • Point is (x,y,z) • Given two points p, q, what is (1-t) p + t q? • Solution: Gnomonic projection • Project back onto sphere • Valid in ½ hemisphere • Line segments (arcs) • Barycentric coordinates in spherical triangles • Interpolate in triangle, project All points such that… (1-t)p + tq q p Graphite 2005, 12/1/2005

  6. -1 c Chart on a sphere • Chart specification: • Center and radius on sphere Uc • Range c = unit disk • Simplest form for ac • Project from sphere to plane • (stereographic) • Adjust with projective map • Affine Uc 1 Graphite 2005, 12/1/2005

  7. Defining an atlas • Define overlaps computationally • Point in chart: evaluate ac • Coverage on sphere (Uc domain of chart) • Define in reverse as ac-1=MD-1(MW-1(D)) • D becomes ellipse after warp, ellipsoidal on sphere • Can bound with cone normal Graphite 2005, 12/1/2005

  8. Embedding the manifold • Write embed function per chart (polynomial) • Write blend function per chart (B-spline basis function) • k derivatives must go to zero by boundary of chart • Guaranteeing continuity • Normalize to get partition of unity Normalized blend function Proto blend function Graphite 2005, 12/1/2005

  9. Final embedding function • Embedding is weighted sum of chart embeddings • Generalization of splines • Given point p on sphere • Map p into each chart • Blend function is zero if chart does not cover p • Continuity is minimum continuity of constituent parts Map each chart Embed Blend Graphite 2005, 12/1/2005

  10. Surface editing • User sketches shape (sketch mesh) • Create charts • Embed mesh on sphere • One chart for each vertex, edge, and face • Determine geometry for each chart (locally) Graphite 2005, 12/1/2005

  11. Charts • Optimization • Cover corresponding element on sphere • Don’t extend over non-neighboring elements • Projection center: center of element • Map neighboring elements via projection • Solve for affine map • Face: big as possible, inside polygon • Use square domain, projective transform for 4-sided Face Face charts Graphite 2005, 12/1/2005

  12. Edge and vertex • Edge: cover edge, extend to mid-point of adjacent faces • Vertex: Cover adjacent edge mid points, face centers Vertex charts Edge Edge charts Vertex Graphite 2005, 12/1/2005

  13. Defining geometry • Fit to original mesh (?) • 1-1 correspondence between surface and sphere • Run subdivision on sketch mesh embedded on sphere (no geometry smoothing) • Fit each chart embedding to subdivision surface • Least-squares Ax = b Graphite 2005, 12/1/2005

  14. Summary • CK analytic surface approximating subdivision surface • Real time editing • Works for other closed topologies • Parameterization using manifolds, Cindy Grimm, International Journal of Shape modeling 2004 Graphite 2005, 12/1/2005

  15. Hierarchical editing • Override surface in an area • Add arms, legs • User draws on surface • Smooth blend • No geometry constraints Graphite 2005, 12/1/2005

  16. Adding more charts • User draws new subdivision mesh on surface • Only in edit area • Simultaneously specifies region on sphere • Add charts as before • Problem: need to mask out old surface Graphite 2005, 12/1/2005

  17. Masking function • Alter blend functions of current surface • Zero inside of patch region • Alter blend functions of new chart functions • Zero outside of blend area • Define mask function h on sphere, • Set to one in blend region, zero outside 1 1 0 0 Graphite 2005, 12/1/2005

  18. Defining mask function • Map region of interest to plane • Same as chart mapping • Define polygon P from user sketch in chart • Define falloff function f(d) -> [0,1] • d is min distance to polygon • Implicit surface • Note: Can do disjoint regions • Mask blend functions 0 1 d Graphite 2005, 12/1/2005

  19. Patches all the way down • Can define mask functions at multiple levels • Charts at level i are masked by all j>i mask functions • Charts at level i zeroed outside of mask region Graphite 2005, 12/1/2005

  20. Summary • Flexible modeling paradigm • No knot lines, geometry constraints • Not limited to subdivision surfaces • Alternative editing techniques? Graphite 2005, 12/1/2005

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