On multivariate Multi-Resolution Analysis, using generalized (non - PowerPoint PPT Presentation

MAIA conference Erice (Italy), September 26, 2013 On multivariate Multi-Resolution Analysis, using generalized (non homogeneous) polyharmonic splines or: A way for deriving RBF and associated MRA Christophe Rabut ∗ , Mira Bozzini and Milvia Rossini University of Toulouse (INSA ; IMT, IREM, MAIAA), France 1. Some known tools... using Fourier Transform 2. Extension of polyharmonic splines, and associated MRA 1

Part 1: Some known tools... using Fourier Transform 2

A word on (odd degree) polynomial splines Definition: � R( f ( m ) ( x )) 2 dx σ m = Argmin ∀ i ∈ [1: n ] , f ( x i )= y i I i =1: n λ i | x − x i | 2 m − 1 + p m − 1 ( x ) � = ⇒ σ m ( x ) = � with ∀ q ∈ I P m − 1 , i =1: n λ i q ( x i ) = 0 and p m − 1 ∈ I P m − 1 1 2( m !) | x | 2 m − 1 “Radial basis functions”: writing u m ( x ) = � σ m ( x ) = i =1: n µ i u m ( x − x i ) + q m − 1 ( x ) (same constraints on µ and q m − 1 ) Derivative and Fourier Transform: 1 E ( u m ) := ( − 1) m u (2 m ) = Dirac u m ( ω ) = � m ω 2 m 3

... and on associated functions ϕ m ( ω ) = | sin ω | 2 m ϕ m = ( − 1) m δ 2 m u m “Basic spline”, or “B-spline”: � | ω | 2 m “Lagrangian spline”, or “L-spline”: L m (0) = 1 ; ∀ j ∈ Z Z \{ 0 } , L m ( j ) = 0 � � L m = Z L m ( j/ 2) L m (2 • − j ) ; ϕ m = Z ϕ m ( j ) L m ( • − j ) j ∈ Z j ∈ Z ω − 2 m ϕ m ( ω ) ϕ m ( ω ) � � � L m ( ω ) = ϕ m ( ω − 2 πℓ ) = Z ϕ m ( j ) exp( − i j ω ) = � � � Z ( ω − 2 πℓ ) − 2 m � ℓ ∈ Z Z j ∈ Z ℓ ∈ Z ψ m = ( − 1) m D 2 m L 2 m Semi-orthogonal wavelet: (to be normalized) ω − 2 m ψ m ( ω ) = ω 2 m � � L 2 m ( ω ) = � Z ( ω − 2 πℓ ) − 4 m ℓ ∈ Z ω − 2 m � ψ ( ω ) � ψ ⊥ ( ω ) = Orthogonal wavelet: ψ ( | ω − 2 πℓ | )) 2 = � � � � � Z ( ω − 2 πℓ ) − 4 m � � � � Z ( � � � ℓ ∈ Z ℓ ∈ Z 4

Linear case B-spline L-spline psi-spline function B ; norm(B) = 0.408 function L ; norm(L) = 0.41 function psi ; norm(psi) = 1 1 3 1 0.9 2.5 0.8 0.8 2 0.7 1.5 0.6 0.6 1 0.5 0.4 0.5 0.4 0 0.2 0.3 −0.5 0.2 0 −1 0.1 −1.5 0 −0.2 −3 −2 −1 0 1 2 3 −5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 κ = (0 ) ; γ = (1 ) ; κ = (0 ) ; γ = (1 ) ; κ = (0 ) ; γ = (1 ) ; psi-ortho psi-hat psi-ortho-hat function psi ortho function psi ortho ; norm(psi ortho ) = 1 function psi hat hat 12 1 3.5 0.9 3 10 0.8 2.5 0.7 8 2 0.6 6 0.5 1.5 0.4 1 4 0.3 0.5 0.2 2 0 0.1 0 0 −5 −4 −3 −2 −1 0 1 2 3 4 5 −30 −20 −10 0 10 20 30 −30 −20 −10 0 10 20 30 κ = (0 ) ; γ = (1 ) ; κ = (0 ) ; γ = (1 ) ; κ = (0 ) ; γ = (1 ) ; 5

6

Cubic case B-spline L-spline psi-spline function B ; norm(B) = 0.346 function L ; norm(L) = 0.467 function psi ; norm(psi) = 1 1 2 0.6 1.5 0.8 0.5 1 0.6 0.4 0.5 0.4 0.3 0 −0.5 0.2 0.2 −1 0.1 0 −1.5 0 −0.2 −3 −2 −1 0 1 2 3 −5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 κ = (0 0 ) ; γ = (1 1 ) ; κ = (0 0 ) ; γ = (1 1 ) ; κ = (0 0 ) ; γ = (1 1 ) ; psi-ortho psi-hat psi-ortho-hat function psi ortho function psi ortho ; norm(psi ortho ) = 1 function psi hat hat 600 1 3 0.9 2.5 500 0.8 2 0.7 400 0.6 1.5 300 0.5 1 0.4 200 0.3 0.5 0.2 100 0 0.1 −0.5 0 0 −5 −4 −3 −2 −1 0 1 2 3 4 5 −30 −20 −10 0 10 20 30 −30 −20 −10 0 10 20 30 κ = (0 0 ) ; γ = (1 1 ) ; κ = (0 0 ) ; γ = (1 1 ) ; κ = (0 0 ) ; γ = (1 1 ) ; 7

8

Use of the associated functions (translation invariant spaces) � y is defined by � y ( ω ) = Z y j exp( − i j ω ). � j ∈ Z B-spline approximation of vector y : (or of points P j for B-spline curve) � σ m = Z y j ϕ m ( • − j ) ⇐ ⇒ σ = � y � ϕ m . � j ∈ Z Interpolating spline of vector y : (or P j instead of y j for interpolating spline curve) � σ m = Z y j L m ( • − j ) j ∈ Z � σ = � y L . � Wavelet decomposition of some f ∈ L 2 ( I R ) : Z ( f , ψ ⊥ (2 ℓ • − j )) ψ ⊥ (2 ℓ • − j ) � � f = j ∈ Z Z ℓ ∈ Z ( f , ψ ⊥ (2 ℓ • − j )) = 2 − ℓ exp( i 2 − ℓ j ) ( ψ ⊥ (2 − ℓ • ) � � f , 9

Spline under tension Definition and Fourier transform � R( f ( m ) ( x )) 2 dx + ρ 2 � R( f ( k ) ( x )) 2 dx σ m,k = Argmin ∀ i ∈ Z (say k < m ) I I Z , f ( x i )= y i 1 E ρ ( u ) := ( − 1) m D 2 m u + ( − 1) k ρ 2 D 2 k u = Dirac ; u ρ ( ω ) = � ω 2 m + ρ 2 ω 2 k ϕ ρ ( ω ) = sin 2 m ω + ρ 2 sin 2 k ω ( ω 2 m + ρ 2 ω 2 k ) − 1 � L ρ ( ω ) = � ω 2 m + ρ 2 ω 2 k � � − 1 ( ω − 2 πℓ ) 2 m + ρ 2 ( ω − 2 πℓ ) 2 k � ℓ ∈ Z Z � � − 1 ω 2 m + ρ 2 ω 2 k ψ ρ ( ω ) = ω 2 m � ρ ( ω ) + ρ 2 ω 2 k � � L 2 L 2 ρ ( ω ) = � � − 2 ( ω − 2 πℓ ) 2 m + ρ 2 ( ω − 2 πℓ ) 2 k � ℓ ∈ Z Z � � − 1 ω 2 m + ρ 2 ω 2 k � ψ ⊥ ρ ( ω ) = � � Z (( ω − 2 πℓ ) 2 m + ρ 2 ( ω − 2 πℓ ) 2 k ) − 2 � � � � ℓ ∈ Z 10

Tension splines (linear-cubic) B-spline L-spline psi-spline function B ; norm(B) = 0.346 0.375 0.398 0.407 function L ; norm(L) = 0.467 0.461 0.442 0.422 function psi ; norm(psi) = 1 1 1 1 3 0.9 1 2.5 0.8 2 0.8 0.7 1.5 0.6 0.6 1 0.5 0.5 0.4 0.4 0 0.3 0.2 −0.5 0.2 −1 0 0.1 −1.5 0 −0.2 −3 −2 −1 0 1 2 3 −5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 κ = (0 0 ) ; (0 1 ) ; (0 3 ) ; (0 10 ) ; γ = (1 1 ) ; (1 1 ) ; (1 1 ) ; (1 1 ) ; κ = (0 0 ) ; (0 1 ) ; (0 3 ) ; (0 10 ) ; γ = (1 1 ) ; (1 1 ) ; (1 1 ) ; (1 1 ) ; κ = (0 0 ) ; (0 1 ) ; (0 3 ) ; (0 10 ) ; γ = (1 1 ) ; (1 1 ) ; (1 1 ) ; (1 1 ) ; psi-ortho psi-hat psi-ortho-hat function psi ortho function psi hat function psi ortho ; norm(psi ortho ) = 1 1 1 1 hat 1800 1 3 0.9 1600 0.8 2.5 1400 0.7 1200 2 0.6 1000 1.5 0.5 800 0.4 1 600 0.3 0.5 400 0.2 0 200 0.1 −0.5 0 0 −5 −4 −3 −2 −1 0 1 2 3 4 5 −30 −20 −10 0 10 20 30 −30 −20 −10 0 10 20 30 κ = (0 0 ) ; (0 1 ) ; (0 3 ) ; (0 10 ) ; γ = (1 1 ) ; (1 1 ) ; (1 1 ) ; (1 1 ) ; κ = (0 0 ) ; (0 1 ) ; (0 3 ) ; (0 10 ) ; γ = (1 1 ) ; (1 1 ) ; (1 1 ) ; (1 1 ) ; κ = (0 0 ) ; (0 1 ) ; (0 3 ) ; (0 10 ) ; γ = (1 1 ) ; (1 1 ) ; (1 1 ) ; (1 1 ) ; 11

12

Tension splines (cubic-quintic) B-spline L-spline psi-spline function B ; norm(B) = 0.314 0.332 0.343 0.346 function L ; norm(L) = 0.479 0.477 0.471 0.468 function psi ; norm(psi) = 1 1 1 1 2 1 0.6 1.5 0.8 0.5 1 0.6 0.4 0.5 0.4 0.3 0 −0.5 0.2 0.2 −1 0.1 0 −1.5 0 −0.2 −3 −2 −1 0 1 2 3 −5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 κ = (0 0 0 ) ; (0 0 1 ) ; (0 0 3 ) ; (0 0 10 ) ; γ = (1 1 1 ) ; (1 1 1 ) ; (1 1 1 ) ; (1 1 1 ) ; κ = (0 0 0 ) ; (0 0 1 ) ; (0 0 3 ) ; (0 0 10 ) ; γ = (1 1 1 ) ; (1 1 1 ) ; (1 1 1 ) ; (1 1 1 ) ; κ = (0 0 0 ) ; (0 0 1 ) ; (0 0 3 ) ; (0 0 10 ) ; γ = (1 1 1 ) ; (1 1 1 ) ; (1 1 1 ) ; (1 1 1 ) ; psi-ortho psi-hat psi-ortho-hat function psi ortho function psi hat function psi ortho ; norm(psi ortho ) = 1 1 1 1 4 hat x 10 7 1.4 3 6 1.2 2.5 5 1 2 1.5 4 0.8 1 3 0.6 0.5 2 0.4 0 1 0.2 −0.5 0 0 −5 −4 −3 −2 −1 0 1 2 3 4 5 −30 −20 −10 0 10 20 30 −30 −20 −10 0 10 20 30 κ = (0 0 0 ) ; (0 0 1 ) ; (0 0 3 ) ; (0 0 10 ) ; γ = (1 1 1 ) ; (1 1 1 ) ; (1 1 1 ) ; (1 1 1 ) ; κ = (0 0 0 ) ; (0 0 1 ) ; (0 0 3 ) ; (0 0 10 ) ; γ = (1 1 1 ) ; (1 1 1 ) ; (1 1 1 ) ; (1 1 1 ) ; κ = (0 0 0 ) ; (0 0 1 ) ; (0 0 3 ) ; (0 0 10 ) ; γ = (1 1 1 ) ; (1 1 1 ) ; (1 1 1 ) ; (1 1 1 ) ; 13

14

Fractional splines Definition and Fourier transform Let s ∈ [0 .. 1) and m ∈ I N such that α := m + s > d/ 2 � R( f ( α ) ( x )) 2 dx σ α = Argmin ∀ i ∈ [1: n ] , f ( x i )= y i I � 2 dω � � R ( ω 2 ) s F ( f ( m ) ( ω ) = Argmin ∀ i ∈ [1: n ] , f ( x i )= y i I 1 E α ( u α ) := ( − 1) ⌊ α ⌋ D 2 α u α = Dirac ; u α ( ω ) = � ( ω 2 ) α u α ( x ) = c α | x | 2 α − 1 if 2 α − 1 is not an even integer number. u α ( x ) = c α | x | 2 α − 1 ln x 2 if 2 α − 1 is an even integer number. ( c α is some known real valued constant) They too are in some place between order 1 (linear) and order 2 (cubic) splines, but are different from splines under tension.  sin 2 ω   α   B-spline : ϕ α ( ω ) =   �  ω 2 15