Polynomial approximation on Lissajous curves on the d -cube 1 - PowerPoint PPT Presentation

Polynomial approximation on Lissajous curves on the d -cube 1 Stefano De Marchi 5´ emes Journe´ es Approximation, Universit´ e de Lille 1 Friday May 20, 2016 1 Joint work with Len Bos (Verona), Wolfgang Erb (Luebeck), Francesco Marchetti (Padova) and Marco Vianello (Padova),

Outline 1 Introduction and known results 2d Lissajous curves 3d Lissajous curves Hyperinterpolation Computational issues Interpolation The general approach 2 The tensor product case 3 4 Conclusion 2 of 48

Introduction and known results 3 of 48

Lissajous curves Properties and motivation Are parametric curves studied by Bowditch (1815) and Lissajous 1 (1857) of the form γ ( t ) = ( A x cos ( ω x t + α x ) , A y sin ( ω y t + α y )) . A x , A y are amplitudes, ω x , ω y are pulsations and α x , α y are phases. 4 of 48

Lissajous curves Properties and motivation Are parametric curves studied by Bowditch (1815) and Lissajous 1 (1857) of the form γ ( t ) = ( A x cos ( ω x t + α x ) , A y sin ( ω y t + α y )) . A x , A y are amplitudes, ω x , ω y are pulsations and α x , α y are phases. Chebyshev polynomials ( T k or U k ) are Lissajous curves (cf. J. C. 2 Merino 2003). In fact a parametrization of y = T n ( x ) , | x | ≤ 1 is � x = cos t � � nt − π y = − sin 0 ≤ t ≤ π 2 4 of 48

Lissajous curves Properties and motivation Are parametric curves studied by Bowditch (1815) and Lissajous 1 (1857) of the form γ ( t ) = ( A x cos ( ω x t + α x ) , A y sin ( ω y t + α y )) . A x , A y are amplitudes, ω x , ω y are pulsations and α x , α y are phases. Chebyshev polynomials ( T k or U k ) are Lissajous curves (cf. J. C. 2 Merino 2003). In fact a parametrization of y = T n ( x ) , | x | ≤ 1 is � x = cos t � � nt − π y = − sin 0 ≤ t ≤ π 2 Padua points (of the first family) [JAT 2006] lie on [ − 1 , 1 ] 2 on the 3 π -periodic Lissajous curve T n + 1 ( x ) = T n ( y ) called generating curve given in parametric form as γ n ( t ) = ( cos nt , cos ( n + 1 ) t ) , 0 ≤ t ≤ π , n ≥ 1 . 4 of 48

The generating curve of the Padua points ( n = 4) 1 0.5 0 -0.5 -1 -1 -0.5 0 0.5 1 Figure : Pad n = C O n + 1 × C E n + 2 ∪ C E n + 1 × C O n + 2 ⊂ C n + 1 × C n + 2 � ( j − 1 ) π � � � z n C n + 1 = j = cos , j = 1 , . . . , n + 1 : Chebsyhev-Lobatto points n on [ − 1 , 1 ] 5 of 48

The generating curve of the Padua points ( n = 4) 1 0.5 0 -0.5 -1 -1 -0.5 0 0.5 1 Figure : Pad n = C O n + 1 × C E n + 2 ∪ C E n + 1 × C O n + 2 ⊂ C n + 1 × C n + 2 � ( j − 1 ) π � � � z n C n + 1 = j = cos , j = 1 , . . . , n + 1 : Chebsyhev-Lobatto points n on [ − 1 , 1 ] Note: | Pad n | = ( n + 2 2 ) = dim ( P n ( R 2 ) ) 5 of 48

The generating curve and cubature Lemma (cf. JAT 2006) For all p ∈ P 2 n ( R 2 ) we have � π 1 � 1 1 − y 2 dxdy = 1 1 [ − 1 , 1 ] 2 p ( x , y ) p ( γ n ( t )) dt . √ π 2 � π 1 − x 2 0 6 of 48

The generating curve and cubature Lemma (cf. JAT 2006) For all p ∈ P 2 n ( R 2 ) we have � π 1 � 1 1 − y 2 dxdy = 1 1 [ − 1 , 1 ] 2 p ( x , y ) p ( γ n ( t )) dt . √ π 2 � π 1 − x 2 0 Proof. Check the property for all p ( x , y ) = T j ( x ) T k ( y ) , j + k ≤ 2 n . � 6 of 48

Lissajous points in 2D: non-degenerate case [Erb et al. NumerMath16 (to appear)] in the framework of Magnetic Particle Imaging applications, considered γ n , p ( t ) = ( sin nt , sin (( n + p ) t )) 0 ≤ t < 2 π , n , p ∈ N s.t. n and n + p are relative primes. 7 of 48

Lissajous points in 2D: non-degenerate case [Erb et al. NumerMath16 (to appear)] in the framework of Magnetic Particle Imaging applications, considered γ n , p ( t ) = ( sin nt , sin (( n + p ) t )) 0 ≤ t < 2 π , n , p ∈ N s.t. n and n + p are relative primes. γ n , p is non-degenerate iff p is odd. 7 of 48

Lissajous points in 2D: non-degenerate case [Erb et al. NumerMath16 (to appear)] in the framework of Magnetic Particle Imaging applications, considered γ n , p ( t ) = ( sin nt , sin (( n + p ) t )) 0 ≤ t < 2 π , n , p ∈ N s.t. n and n + p are relative primes. γ n , p is non-degenerate iff p is odd. Consider t k = 2 π k / ( 4 n ( n + p )) , k = 1 , ..., 4 n ( n + p ) . � � Lisa n , p := γ n , p ( t k ) , k = 1 , . . . , 4 n ( n + p ) , | Lisa n , p | = 2 n ( n + p )+ 2 n + p . Notice: | Lisa n , 1 | = 2 n 2 + 4 n + 1 while | Pad 2 n | = 2 n 2 + 3 n + 1 is obtained with p = 1 / 2. 7 of 48

Lissajous points: non-degenerate case Figure : From the paper by Erb et al. NM2016 (cf. arXiv 1411.7589) 8 of 48

Lissajous points: degenerate case [Erb AMC16 (to appear)] has then studied the degenerate 2 π -Lissajous curves 9 of 48

Lissajous points: degenerate case [Erb AMC16 (to appear)] has then studied the degenerate 2 π -Lissajous curves γ n , p ( t ) = ( cos nt , cos (( n + p ) t )) 0 ≤ t < 2 π , 9 of 48

Lissajous points: degenerate case [Erb AMC16 (to appear)] has then studied the degenerate 2 π -Lissajous curves γ n , p ( t ) = ( cos nt , cos (( n + p ) t )) 0 ≤ t < 2 π , Consider t k = π k / ( n ( n + p )) , k = 0 , 1 , ..., n ( n + p ) . , | LD n , p | = ( n + p + 1 )( n + 1 ) � � LD n , p := γ n , p ( t k ) , k = 0 , 1 , . . . , n ( n + p ) . 2 Notice: for p = 1, | LD n , 1 | = | Pad n | = dim ( P n ( R 2 )) and correspond to the Padua points themselves. 9 of 48

Lissajous points: degenerate case Figure : From the paper by Erb AMC16, (cf. arXiv 1503.00895) 10 of 48

An application Image reconstruction with adaptive filters −→ Work in progress with W. Erb and F. Marchetti. Ideas to avoid Gibbs phenomenon at discontinuites [Gottlieb&Shu SIAMRev97,Tadmor&Tanner IMAJN05] 11 of 48

An application Image reconstruction with adaptive filters −→ Work in progress with W. Erb and F. Marchetti. Ideas to avoid Gibbs phenomenon at discontinuites [Gottlieb&Shu SIAMRev97,Tadmor&Tanner IMAJN05] 1 Sampling with Lissajous for finding the interpolating polynomial 2 Initial non-adaptive filter e x α / ( x 2 − 1 ) � | x | ≤ 1 σ ( x ; α ) = 0 | x | > 1 ( α can vary with the point x in the adaptive case): this allow to avoid the Gibbs phenomenon 3 Detect the discontinuities by Canny edge-detector algorithm 4 Apply adptively the filter 11 of 48

Image reconstruction with adaptive filters: examples Figure : Original image: 115 × 115. Lissajous non degenerate curve with ( n , p ) = ( 32 , 33 ) ; Chebfun2 (modified) for the coefficients; α = 4 for the initial filter and α chosen “Ad hoc” for the remainig adapted filtering 12 of 48

Image reconstruction with adaptive filters: examples Figure : Original image: 115 × 115. Lissajous non degenerate curve with ( n , p ) = ( 32 , 33 ) ; Chebfun2 (modified) for the coefficients; α = 4 for the initial filter and α chosen “Ad hoc” for the remainig adapted filtering 13 of 48

3d case: notation Ω = [ − 1 , 1 ] 3 : the standard 3-cube The product Chebyshev measure w ( x ) = 1 1 d µ 3 ( x ) = w ( x ) d x , . π 3 � ( 1 − x 2 1 )( 1 − x 2 2 )( 1 − x 2 3 ) (1) P 3 k : space of trivariate polynomials of degree k in R 3 ( dim ( P 3 k ) = ( k + 1 )( k + 2 )( k + 3 ) / 6). 14 of 48

Fundamental theorem This results shows which are the admissible 3d Lissajous curves Theorem (cf. Bos, DeM, Vianello 2015, IMA J. NA to appear ) Let n ∈ N + and ( a n , b n , c n ) be the integer triple 4 n 2 + 1 4 n 2 + n , 3 4 n 2 + 3 � �  3 2 n , 3 2 n + 1 , n even     ( a n , b n , c n ) =  (2)  4 n 2 + 1 4 n 2 + 3 4 n 2 + 3  � � 3 4 , 3 2 n − 1 4 , 3 2 n + 3  , n odd.   4 Then, for every integer triple ( i , j , k ) , not all 0 , with i , j , k ≥ 0 and i + j + k ≤ m n = 2 n, we have the property that ia n � jb n + kc n , jb n � ia n + kc n , kc n � ia n + jb n . Moreover, m n = 2 n is maximal, in the sense that there exists a triple ( i ∗ , j ∗ , k ∗ ) , i ∗ + j ∗ + k ∗ = 2 n + 1 , that does not satisfy the property. 15 of 48

Consequence I Cubature along the curve On admissible curves follows Proposition Consider the Lissajous curves in [ − 1 , 1 ] 3 defined by ℓ n ( θ ) = ( cos ( a n θ ) , cos ( b n θ ) , cos ( c n θ )) , θ ∈ [ 0 , π ] , (3) where ( a n , b n , c n ) is the sequence of integer triples (2). Then, for every total-degree polynomial p ∈ P 3 2 n � π � [ − 1 , 1 ] 3 p ( x ) d µ 3 ( x ) = 1 p ( ℓ n ( θ )) d θ . (4) π 0 Proof. It suffices to prove the identity for a polynomial basis (ex: for the tensor product basis T α ( x ) , | α | ≤ 2 n ). � 16 of 48

Consequence II Exactness Corollary Consider p ∈ P 3 2 n , ℓ n ( θ ) and ν = n · max { a n , b n , c n } = n · c n . Then µ � � [ − 1 , 1 ] 3 p ( x ) w ( x ) d x = w s p ( ℓ n ( θ s )) , (5) s = 0 where w s = π 2 ω s , s = 0 , . . . , µ , (6) with µ = ν , θ s = ( 2 s + 1 ) π π , ω s ≡ µ + 1 , s = 0 , . . . , µ , (7) 2 µ + 2 or alternatively µ = ν + 1 , θ s = s π µ , s = 0 , . . . , µ , ω 0 = ω µ = π 2 µ , ω s ≡ π s = 1 , . . . , µ − 1 . µ , (8) 17 of 48

Remarks The points set � ℓ n ( θ s ) , s = 0 , . . . , µ � are a 3-dimensional rank-1 Chebyshev lattices (for cubature of degree of exactness 2 n ). Cools and Poppe [cf. CHEBINT, TOMS 2013] wrote a search algorithm for constructing heuristically such lattices. Formulae (2) (together with (6), (7), (8)) provide explicit formulas for any degree. 18 of 48