Generalized spline subdivision Jorg Peters Surf Lab Purdue

Generalized spline subdivision Jorg Peters Surf. Lab (Purdue, UFL) • Polynomial Heritage • Computing Moments • Shape and Eigenvalues Jorg Peters, Surf. Lab

Polynomial heritage of generalized spline subdivision • Doo-Sabin Catmull. Clark Jorg Peters, Surf. Lab

Polynomial heritage of generalized spline subdivision • Increasing regions are regular: points and faces have standard valence Jorg Peters, Surf. Lab

Polynomial heritage of generalized spline subdivision • Doo-Sabin • Catmull-Clark • Midedge • Loop bi-2 B-spline bi-3 B-spline Zwart-Powell C^1 box-spline C^2 box-spline = generalization of B-spline to shift-invariant partitions book: [de Boor, Hollig, Riemenschneider 94] Jorg Peters, Surf. Lab

Polynomial heritage of generalized spline subdivision • Subdivision of the Zwart-Powell C^1 quadratic box-spline basis function Subdiv ision a Subdiv ision c Subdivision Rule d Subdiv ision Jorg Peters, Surf. Lab

Polynomial heritage of generalized spline subdivision Mid-edge Rule (“simplest rule”) Zwart-Powell subdivision = 2 steps of Midedge subdivision 2 steps regular: 4 -valence, quadrilaterals 4 steps 1 1 2 Jorg Peters, Surf. Lab

Polynomial heritage of generalized spline subdivision • Increasing regions are regular (polynomial) • Union of surface layers at an extraordinary point Jorg Peters, Surf. Lab

Polynomial heritage of generalized spline subdivision • Uses: Representation as Bezier patches Evaluation at non-binary points Fast moment computation Jorg Peters, Surf. Lab

Generalized spline subdivision Jorg Peters Surf. Lab (Purdue, UFL) • Polynomial Heritage • Computing Moments • Shape and Eigenvalues Jorg Peters, Surf. Lab

Moments of objects enclosed by generalized subdivision surfaces Inertia Frame • Challenge: Exponential increase in the number of facets! Volume Center of mass Jorg Peters, Surf. Lab

Moments of objects enclosed by generalized subdivision surfaces Theory: Gauss’ Divergence Theorem: The integral of the divergence over the volume equals the integral of the normal component over the surface S ò Ñ × f d. V V ò f × n/ n d. S S f d. V Ñ × = × f n/ n d. S f ò ò ò n d. U V S U Jorg Peters, Surf. Lab

Moments of objects enclosed by generalized subdivision surfaces Theory: Change of variables The area of the surface element S equals the integral of the Jacobian |n| of the surface parametrization (x, y, z) over the domain U ò d. S S ò n d. U U f d. V Ñ × = × f n/ n d. S f ò ò ò n d. U V S U Jorg Peters, Surf. Lab

Moments of objects enclosed by generalized subdivision surfaces f=[0, 0, z] n = x u y v - x v y u For example, f n = z xu yv - x v yu Volume = is piecewise polynomial in regular regions ò 1 d. V = ò z [x V u y v - x v y u ] du dv U = å p p ò z (x y x v y u ) du dv p p u p v patch p Up Jorg Peters, Surf. Lab

Moments of objects enclosed by generalized subdivision surfaces “Volume” patch p = p p p z ( x y x ò u v v y u ) du dv Up Schema for bi-3 Bezier patch Jorg Peters, Surf. Lab

Moments of objects enclosed by generalized subdivision surfaces Volume patch p = p p p z ( x y x ò u v v y u ) du dv Up Jorg Peters, Surf. Lab

Moments of objects enclosed by generalized subdivision surfaces Work: at each subdivision step linear for each extraordinary point add volume contribution of 3 n patches Doo-Sabin Jorg Peters, Surf. Lab

Moments of objects enclosed by generalized subdivision surfaces Vi = å p p p z ( x y x ò u v v y u ) du dv p in layer i Up m V i+1 Wm Vi Volume = å Vi + Wm i =0 Jorg Peters, Surf. Lab

Moments of objects enclosed by generalized subdivision surfaces • Error estimation: bounding boxes Jorg Peters, Surf. Lab

Moments of objects enclosed by generalized subdivision surfaces • Geometric decay of error volume 1, 1/8, 1/64, . . . Jorg Peters, Surf. Lab

Moments of objects enclosed by generalized subdivision surfaces • Computing geometry given a fixed volume Bisection Jorg Peters, Surf. Lab

Moments of objects enclosed by generalized subdivision surfaces • Higher moments and the inertia frame center of mass: ò x d. V, ò y d. V, ò z d. V V inertia tensor: V V . . . , ò xy d. V, . . . V eigenvector frame Jorg Peters, Surf. Lab

Moments of objects enclosed by generalized subdivision surfaces • Higher moments and the inertia frame center of mass Jorg Peters, Surf. Lab

Moments of objects enclosed by generalized subdivision surfaces • Physics-based animation Center of mass support Jorg Peters, Surf. Lab

Moments of objects enclosed by generalized subdivision surfaces • Simple registration, comparison matching frames = computing a 3 x 3 matrix Q: IP Q = IS Jorg Peters, Surf. Lab

Moments of objects enclosed by generalized subdivision surfaces Inertia Frame Volume • Solution: Moments efficiently and exactly computed via Gauss’ theorem and polynomial Center of mass heritage Jorg Peters, Surf. Lab

Jorg Peters, Surf. Lab

Shape and eigenvalues • Union of surface layers at an extraordinary point • Control points transformed by the subdivision matrix Jorg Peters, Surf. Lab

Shape and eigenvalues eigenvector expansion i Jorg Peters, Surf. Lab

Shape and eigenvalues • If all l < 1, then collapse • If some l > 1, then unbounded growth • Good sequence: 1, , , … where | | < 1 • Eigenvectors of determine the tangent plane ll l l Jorg Peters, Surf. Lab

Shape and eigenvalues • Fast contraction of 3 -sided facets l = (1+cos(2 pi/ 3))/2 =. 25 • Slow contraction of large facets midedge subdivision l = (1+cos(2 pi/16))/2 =. 962. . . Jorg Peters, Surf. Lab

Shape and eigenvalues • adjust subdominant eigenvalues midedge subdivision) <=> (modified l =. 5 Jorg Peters, Surf. Lab

Shape and eigenvalues Jorg Peters, Surf. Lab

Generalized spline subdivision Summary • Polynomial Heritage regular regions • Computing Moments Gauss’ theorem • Shape and Eigenvalues subdominant values Jorg Peters, Surf. Lab