Intro PS r -splines Quasi-interpolation Conclusions On smooth spline spaces and quasi-interpolants over Powell-Sabin triangulations Hendrik Speleers Katholieke Universiteit Leuven Department of Computer Science MAIA Conference Erice, September 25–30, 2013 H. Speleers On smooth spline spaces and QIs over PS-triangulations
Intro PS r -splines Quasi-interpolation Conclusions Outline Introduction Smooth Powell-Sabin B-splines Spline space Normalized basis Quasi-interpolation Conclusions H. Speleers On smooth spline spaces and QIs over PS-triangulations
Intro PS r -splines Quasi-interpolation Conclusions Introduction Triangulation with Powell-Sabin split [Powell & Sabin, TOMS 1977] ◮ Every triangle is split into six subtriangles ◮ E.g., incenter as split point ⇒ H. Speleers On smooth spline spaces and QIs over PS-triangulations
Intro PS r -splines Quasi-interpolation Conclusions Introduction Univariate B-spline representation ◮ Basis: local support, convex partition of unity ◮ Control points (CAGD) ◮ Easy manipulation: stable evaluation [e.g. de Boor], differentiation, integration ◮ ... Bivariate B-spline representation ◮ Smooth splines on Powell-Sabin triangulations ◮ Basis with similar properties as univariate case ◮ Construction of quasi-interpolations (using blossoming) H. Speleers On smooth spline spaces and QIs over PS-triangulations
Intro PS r -splines Quasi-interpolation Conclusions Introduction Univariate B-spline representation ◮ Basis: local support, convex partition of unity ◮ Control points (CAGD) ◮ Easy manipulation: stable evaluation [e.g. de Boor], differentiation, integration ◮ ... Bivariate B-spline representation ◮ Smooth splines on Powell-Sabin triangulations ◮ Basis with similar properties as univariate case ◮ Construction of quasi-interpolations (using blossoming) H. Speleers On smooth spline spaces and QIs over PS-triangulations
Intro PS r -splines Quasi-interpolation Conclusions Spline space Basis Smooth Powell-Sabin (PS r ) splines PS r -spline space We consider piecewise polynomials of degree d with global C r -continuity and C ρ -supersmoothness at some points and edges, defined on a triangulation ∆ with PS-split ∆ ∗ d (∆ ∗ ) = { s ∈ C r (Ω) : s | T ∗ ∈ P d , T ∗ ∈ ∆ ∗ ; S r ,ρ s ∈ C ρ ( W ) , W ∈ ( V ∪ Z ∗ ); s ∈ C ρ ( e ) , e ∈ E ∗ } C ρ C ρ C ρ C ρ C ρ C ρ C ρ H. Speleers On smooth spline spaces and QIs over PS-triangulations
Intro PS r -splines Quasi-interpolation Conclusions Spline space Basis Smooth Powell-Sabin (PS r ) splines PS r -spline space We consider piecewise polynomials of degree d with global C r -continuity and C ρ -supersmoothness at some points and edges, defined on a triangulation ∆ with PS-split ∆ ∗ d (∆ ∗ ) = { s ∈ C r (Ω) : s | T ∗ ∈ P d , T ∗ ∈ ∆ ∗ ; S r ,ρ s ∈ C ρ ( W ) , W ∈ ( V ∪ Z ∗ ); s ∈ C ρ ( e ) , e ∈ E ∗ } C ρ PS r -splines: for a given r , d = 3 r − 1, ρ = 2 r − 1 C ρ C ρ C ρ r = 1: [Powell & Sabin, 1977, . . . ] C ρ r = 2: [Sablonni` C ρ ere, 1987, . . . ] C ρ r > 2: [S., 2013] H. Speleers On smooth spline spaces and QIs over PS-triangulations
Intro PS r -splines Quasi-interpolation Conclusions Spline space Basis Smooth Powell-Sabin (PS r ) splines PS r -spline space � 2 r +1 � � r � ◮ Let n v vertices and n t triangles in ∆; let N v = , N t = 2 2 ◮ Dimension equals N v n v + N t n t ◮ Interpolation problem: PS r -spline s is uniquely defined by D a x D b y s ( V l ) = f x a y b , l , l = 1 , . . . , n v , 0 ≤ a + b ≤ 2 r − 1 , D a x D b y s ( Z m ) = g x a y b , m , m = 1 , . . . , n t , 0 ≤ a + b ≤ r − 2 , for any given set of f x a y b , l -values and g x a y b , m -values. ◮ Basis? ◮ N t functions B t k , j ( x , y ) related to each triangle T k ◮ N v functions B v i , j ( x , y ) related to each vertex V i H. Speleers On smooth spline spaces and QIs over PS-triangulations
Intro PS r -splines Quasi-interpolation Conclusions Spline space Basis Smooth Powell-Sabin (PS r ) splines PS r -spline space � 2 r +1 � � r � ◮ Let n v vertices and n t triangles in ∆; let N v = , N t = 2 2 ◮ Dimension equals N v n v + N t n t ◮ Interpolation problem: PS r -spline s is uniquely defined by D a x D b y s ( V l ) = f x a y b , l , l = 1 , . . . , n v , 0 ≤ a + b ≤ 2 r − 1 , D a x D b y s ( Z m ) = g x a y b , m , m = 1 , . . . , n t , 0 ≤ a + b ≤ r − 2 , for any given set of f x a y b , l -values and g x a y b , m -values. ◮ Basis? ◮ N t functions B t k , j ( x , y ) related to each triangle T k ◮ N v functions B v i , j ( x , y ) related to each vertex V i H. Speleers On smooth spline spaces and QIs over PS-triangulations
Intro PS r -splines Quasi-interpolation Conclusions Spline space Basis Smooth Powell-Sabin (PS r ) splines PS r -spline space � 2 r +1 � � r � ◮ Let n v vertices and n t triangles in ∆; let N v = , N t = 2 2 ◮ Dimension equals N v n v + N t n t ◮ Interpolation problem: PS r -spline s is uniquely defined by D a x D b y s ( V l ) = f x a y b , l , l = 1 , . . . , n v , 0 ≤ a + b ≤ 2 r − 1 , D a x D b y s ( Z m ) = g x a y b , m , m = 1 , . . . , n t , 0 ≤ a + b ≤ r − 2 , for any given set of f x a y b , l -values and g x a y b , m -values. ◮ Basis? ◮ N t functions B t k , j ( x , y ) related to each triangle T k ◮ N v functions B v i , j ( x , y ) related to each vertex V i H. Speleers On smooth spline spaces and QIs over PS-triangulations
Intro PS r -splines Quasi-interpolation Conclusions Spline space Basis Smooth Powell-Sabin (PS r ) splines PS r -spline space � 2 r +1 � � r � ◮ Let n v vertices and n t triangles in ∆; let N v = , N t = 2 2 ◮ Dimension equals N v n v + N t n t ◮ Interpolation problem: PS r -spline s is uniquely defined by D a x D b y s ( V l ) = f x a y b , l , l = 1 , . . . , n v , 0 ≤ a + b ≤ 2 r − 1 , D a x D b y s ( Z m ) = g x a y b , m , m = 1 , . . . , n t , 0 ≤ a + b ≤ r − 2 , for any given set of f x a y b , l -values and g x a y b , m -values. ◮ Basis? ◮ B-spline-like basis [S., 2010, 2013] ◮ local support + nonnegativity + partition of unity H. Speleers On smooth spline spaces and QIs over PS-triangulations
Intro PS r -splines Quasi-interpolation Conclusions Spline space Basis Normalized basis B-spline related to a triangle T k ◮ The B-spline B t k , j ( x , y ) is the solution of the interpolation problem g x a y b , k = β ab k , j � = 0; g x a y b , m = 0 , m � = k ; f x a y b , l = 0 (local support) ◮ The values of β ab k , j are determined via Bernstein-B´ ezier representation of B-spline (nonnegativity) H. Speleers On smooth spline spaces and QIs over PS-triangulations
Intro PS r -splines Quasi-interpolation Conclusions Spline space Basis Normalized basis B-spline related to a triangle T k Example r = 2 , d = 5: • = zero BB-coefficient • = non-zero BB-coefficient polynomial of degree 2 r − 1 with single BB-coeff b 111 = 1 and other BB-coeffs b κµν = 0 H. Speleers On smooth spline spaces and QIs over PS-triangulations
Intro PS r -splines Quasi-interpolation Conclusions Spline space Basis Normalized basis B-spline related to a triangle T k Example r = 2 , d = 5: • = zero BB-coefficient • = non-zero BB-coefficient polynomial of degree 2 r − 1 with single BB-coeff b 111 = 1 and other BB-coeffs b κµν = 0 H. Speleers On smooth spline spaces and QIs over PS-triangulations
Intro PS r -splines Quasi-interpolation Conclusions Spline space Basis Normalized basis B-spline related a vertex V i ◮ Let M i be the molecule of vertex V i . The B-spline B v i , j ( x , y ) is the solution of the interpolation problem f x a y b , i = α ab i , j ; f x a y b , l = 0 , l � = i g x a y b , m = β ab i , j , T m ∈ M i ; g x a y b , m = 0 , T m / ∈ M i (Local support) � � ◮ Given α ab , the values of β ab i , j , 0 ≤ a + b ≤ 2 r − 1 i , j are determined via Bernstein-B´ ezier representation of B-spline H. Speleers On smooth spline spaces and QIs over PS-triangulations
Intro PS r -splines Quasi-interpolation Conclusions Spline space Basis Normalized basis B-spline related a vertex V i Example r = 2 , d = 5: • = zero BB-coefficient • = non-zero BB-coefficient polynomial of degree 2 r − 1 given by α ab i , j H. Speleers On smooth spline spaces and QIs over PS-triangulations
Intro PS r -splines Quasi-interpolation Conclusions Spline space Basis Normalized basis B-spline related a vertex V i Example r = 2 , d = 5: • = zero BB-coefficient • = non-zero BB-coefficient polynomial of degree 2 r − 1 given by α ab i , j H. Speleers On smooth spline spaces and QIs over PS-triangulations
Intro PS r -splines Quasi-interpolation Conclusions Spline space Basis Normalized basis B-spline related a vertex V i Example r = 2 , d = 5: • = zero BB-coefficient • = non-zero BB-coefficient polynomial of degree 2 r − 1 given by α ab i , j H. Speleers On smooth spline spaces and QIs over PS-triangulations
Intro PS r -splines Quasi-interpolation Conclusions Spline space Basis Normalized basis B-spline related a vertex V i ◮ For each V i , choose a PS-triangle t i ◮ Choose � 3 r − 1 � � ( θ i ) a + b D a a + b α ab x D b y B 2 r − 1 i , j = � 2 r − 1 κµν ( V i ) , a + b with B 2 r − 1 κµν ( x , y ) a Bernstein polynomial defined on t i , for some κ + µ + ν = 2 r − 1 (partition of unity) H. Speleers On smooth spline spaces and QIs over PS-triangulations
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