PowellSabin splines on the sphere with applications in CAGD Jan - PowerPoint PPT Presentation

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis Powell–Sabin splines on the sphere with applications in CAGD Jan Maes Department of Computer Science Katholieke Universiteit Leuven Paris, November 17, 2006

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis Outline Powell–Sabin splines Section I Spherical Powell–Sabin splines Section II Multiresolution Analysis Section III

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis Powell–Sabin splines

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis Bernstein–Bézier representation = ⇒ Pierre Étienne Bézier (1910-1999)

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis Stitching together Bézier triangles = ⇒ No C 1 continuity at the red curve

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis C 1 continuity with Powell–Sabin splines Conformal triangulation ∆ PS 6-split ∆ PS S 1 2 (∆ PS ) = space of PS splines M.J.D. Powell M.A. Sabin

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis C 1 continuity with Powell–Sabin splines Conformal triangulation ∆ PS 6-split ∆ PS S 1 2 (∆ PS ) = space of PS splines M.J.D. Powell M.A. Sabin

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis C 1 continuity with Powell–Sabin splines Conformal triangulation ∆ PS 6-split ∆ PS S 1 2 (∆ PS ) = space of PS splines M.J.D. Powell M.A. Sabin

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis The dimension of S 1 2 (∆ PS ) ? There is exactly one solution s ∈ S 1 2 (∆ PS ) to the Hermite interpolation problem s ( V i ) = α i , ∀ V i ∈ ∆ , i = 1 , . . . , N . D x s ( V i ) = β i , D y s ( V i ) = γ i , The dimension of S 1 2 (∆ PS ) is 3 N . Therefore we need 3 N basis functions.

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis The dimension of S 1 2 (∆ PS ) ? There is exactly one solution s ∈ S 1 2 (∆ PS ) to the Hermite interpolation problem s ( V i ) = α i , ∀ V i ∈ ∆ , i = 1 , . . . , N . D x s ( V i ) = β i , D y s ( V i ) = γ i , The dimension of S 1 2 (∆ PS ) is 3 N . Therefore we need 3 N basis functions.

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis Powell–Sabin B-splines with control triangles 3 N � � s ( x , y ) = c ij B ij ( x , y ) i = 1 j = 1 B ij is the unique solution to [ B ij ( V k ) , D x B ij ( V k ) , D y B ij ( V k )] = [ 0 , 0 , 0 ] for all k � = i [ B ij ( V i ) , D x B ij ( V i ) , D y B ij ( V i )] = [ α ij , β ij , γ ij ] for j = 1 , 2 , 3 Partition of unity: � 3 � N j = 1 B ij ( x , y ) = 1, i = 1 B ij ( x , y ) ≥ 0 (Paul Dierckx, 1997)

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis Powell–Sabin B-splines with control triangles 3 N � � s ( x , y ) = c ij B ij ( x , y ) i = 1 j = 1 B ij is the unique solution to [ B ij ( V k ) , D x B ij ( V k ) , D y B ij ( V k )] = [ 0 , 0 , 0 ] for all k � = i [ B ij ( V i ) , D x B ij ( V i ) , D y B ij ( V i )] = [ α ij , β ij , γ ij ] for j = 1 , 2 , 3 Partition of unity: � 3 � N j = 1 B ij ( x , y ) = 1, i = 1 B ij ( x , y ) ≥ 0 (Paul Dierckx, 1997)

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis Powell–Sabin B-splines with control triangles 3 N � � s ( x , y ) = c ij B ij ( x , y ) i = 1 j = 1 B ij is the unique solution to [ B ij ( V k ) , D x B ij ( V k ) , D y B ij ( V k )] = [ 0 , 0 , 0 ] for all k � = i [ B ij ( V i ) , D x B ij ( V i ) , D y B ij ( V i )] = [ α ij , β ij , γ ij ] for j = 1 , 2 , 3 Partition of unity: � 3 � N j = 1 B ij ( x , y ) = 1, i = 1 B ij ( x , y ) ≥ 0 (Paul Dierckx, 1997)

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis Powell–Sabin B-splines with control triangles Three locally supported basis functions per vertex

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis Powell–Sabin B-splines with control triangles The control triangle is tangent to the PS spline surface.

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis Powell–Sabin B-splines with control triangles It ‘controls’ the local shape of the spline surface.

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis Spherical Powell–Sabin splines

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis Spherical spline spaces P . Alfeld, M. Neamtu, and L. L. Schumaker (1996) Homogeneous of degree d : f ( α v ) = α d f ( v ) H d := space of trivariate polynomials of degree d that are homogeneous of degree d Restriction of H d to a plane in R 3 \ { 0 } ⇒ we recover the space of bivariate polynomials ∆ := conforming spherical triangulation of the unit sphere S S r d (∆) := { s ∈ C r ( S ) | s | τ ∈ H d ( τ ) , τ ∈ ∆ }

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis Spherical Powell–Sabin splines s ( v i ) = f i , D g i s ( v i ) = f g i , D h i s ( v i ) = f h i , ∀ v i ∈ ∆ has a unique solution in S 1 2 (∆ PS )

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis 1 − 1 connection with bivariate PS splines ⇒ | v | 2 B ij ( v | v | ) ⇒ ← − Spherical PS B- piecewise trivari- Restriction to the spline B ij ( v ) ate polynomial of plane tangent to degree 2 that is S at v i ∈ ∆ homogeneous of degree 2

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis 1 − 1 connection with bivariate PS splines Let T i be the plane tangent to S at vertex v i Radial projection: R i v := v := v | v | ∈ S , v ∈ T i ⊂ ∆ PS be its Define ∆ i as the star of v i in ∆ , and let ∆ PS i PS 6-split. Theorem Let s ∈ S 1 ) . Let s be the restriction of | v | 2 s ( v / | v | ) to T i . 2 (∆ PS i 2 ( R − 1 Then s is in S 1 ) and ∆ PS i i s ( v i ) = s ( v i ) , D g i s ( v i ) = D g i s ( v i ) , D h i s ( v i ) = D h i s ( v i ) .

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis 1 − 1 connection with bivariate PS splines Let T i be the plane tangent to S at vertex v i Radial projection: R i v := v := v | v | ∈ S , v ∈ T i ⊂ ∆ PS be its Define ∆ i as the star of v i in ∆ , and let ∆ PS i PS 6-split. Theorem Let s ∈ S 1 ) . Let s be the restriction of | v | 2 s ( v / | v | ) to T i . 2 (∆ PS i 2 ( R − 1 Then s is in S 1 ) and ∆ PS i i s ( v i ) = s ( v i ) , D g i s ( v i ) = D g i s ( v i ) , D h i s ( v i ) = D h i s ( v i ) .

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis Spherical B-splines with control triangles

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis Applications on a spherical domain Approximation of a mesh: consider the triangles of the original triangular mesh as control triangles of a PS spline.

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis Applications on a spherical domain Compression by smoothing: Decimate a given triangular mesh and approximate the decimated mesh. triangular mesh reduced mesh (40000 triangles) (5000 triangles)

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis Applications on a spherical domain Compression by smoothing: Decimate a given triangular mesh and approximate the decimated mesh. triangular mesh Powell–Sabin spline (40000 triangles) (5000 control triangles)

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis Applications on a spherical domain triangular mesh decimated mesh spherical (40000 triangles) (5000 triangles) parameterization (5000 control triangles) PS spline surface

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis Multiresolution analysis

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis Multiresolution analysis (1989) Stéphane Mallat Yves Meyer A nested sequence of subspaces S 0 ⊂ S 1 ⊂ S 2 ⊂ · · · ⊂ S ℓ ⊂ · · ·

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis Multiresolution analysis (1989) Stéphane Mallat Yves Meyer Complement spaces W ℓ S ℓ + 1 = S ℓ ⊕ W ℓ

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis Multiresolution analysis (1989) Stéphane Mallat Yves Meyer A stable basis for the complement space W ℓ W ℓ = span { ψ ℓ, i }

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis Multiresolution analysis Refine the triangulation ∆ and its PS 6-split ∆ PS . V k V k R ki R ki R jk R jk Z ijk Z ijk V i V i R ij R ij V j V j dyadic refinement triadic refinement

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis Multiresolution analysis Refine the triangulation ∆ and its PS 6-split ∆ PS . V k V k V ki V ki R ki V kj V ik V jk R jk Z ijk V ijk V i V i V jk V ij V ij R ij V ji V j V j dyadic refinement triadic refinement