Efficient reconstruction of functions on the sphere from scattered - PowerPoint PPT Presentation

Efficient reconstruction of functions on the sphere from scattered data Daniel Potts Department of Mathematics Chemnitz Universitiy of Technology email: potts@mathematik.tu-chemnitz.de http://www.tu-chemnitz.de/ ∼ potts

Content • NFFT • NFSFT • Iterative reconstruction on S 2 • Probabilistic arguments numerical examples

NFFT (Dutt,Rokhlin;Beylkin; Steidl; P .,Steidl,Tasche) fast computation of of the sums N/ 2 − 1 N/ 2 − 1 � � f k e − 2 π i kv j f ( v j ) = . . . ( j = − M/ 2 , . . . , M/ 2 − 1) k 1 = − N/ 2 k d = − N/ 2 M/ 2 − 1 ( − N 2 ≤ k < N � f j e 2 π i kv j h ( k ) = 2 ) j = − M/ 2 for equispaced nodes v j := j N ( M = N d ) FFT ( fast Fourier transform ) in O ( N d log N ) for arbitrary nodes v j ∈ [ − 1 / 2 , 1 / 2) d NFFT ( nonequispaced FFT ) in O ( N d log N + m d M )

Fourier algorithms on the sphere Problem: fast computation of N k � � a n k Y n f ( θ, φ ) = k ( θ, φ ) n = − k k =0 at arbitrary nodes ( θ d , φ d ) ∈ S 2 ( d = 0 , . . . , M − 1) • discrete spherical Fourier transform (FFT on S 2 ) ( θ d 1 , φ d 2 ) := ( d 1 π D 1 , 2 d 2 π d 1 = 0 , . . . , D 1 − 1 , d 2 , . . . , D 2 − 1 D 2 − 1 ) Driscoll, Healy (1994, 2003, ...); Potts, Steidl, Tasche (1998); Mohlenkamp (1999); Suda, Takami (2002); Rokhlin, Tygert (2006) • nonequispaced discrete spherical Fourier transform (NFFT on S 2 ) ( θ d , φ d ) ∈ S 2 , d = 0 , . . . , M − 1 Kunis, P . (2003); Keiner, P . (2008)

Inverse NFFT on the sphere i N k � � ˆ f n k Y n ∈ Π N ( S 2 ) f = k k =0 n = − k ”inverse” problem, f ∈ C M given in � � ˆ k =0 ,...,N, | n |≤ k ∈ C ( N +1) 2 , f ∈ C M Y ˆ ˆ f n f ≈ f , f = k � � f ( ξ j ) j =0 ,...,M − 1 ≈ f spherical Fourier matrix j =0 ,...,M − 1; k =0 ,...,N, | n |≤ k ∈ C M × ( N +1) 2 . Y n � � �� Y := ξ j k

The geodetic distance of ξ , η ∈ S 2 is given by dist ( ξ , η ) := arccos ( η · ξ ) . We measure the “nonuniformity” of a sampling set X := { ξ j ∈ S 2 : j = 0 , . . . , M − 1 } , M ∈ N , by the mesh norm δ X and the separation distance q X , defined by δ X := 2 max j =0 ,...,M − 1 dist ( ξ j , ξ ) , min ξ ∈ S 2 q X := 0 ≤ j<l<M dist ( ξ j , ξ l ) . min The sampling set X is called • δ -dense for some 0 < δ ≤ 2 π , if δ X ≤ δ , and • q -separated for some 0 < q ≤ 2 π , if q X ≥ q .

0 10 −5 10 −10 10 −15 10 0 10 20 30 40 Distribution of the singular va- lues of the spherical Fourier ma- trix Y ∈ C M × ( N +1) 2 with respect Generalised spiral nodes to the polynomial degrees N = 0 , . . . , 40 for M = 400 generali- sed spiral nodes

Least squares approximation M > ( N + 1) 2 over-determined M − 1 ˆ f � � f − Y ˆ f � 2 w j | f j − f ( ξ j ) | 2 W = → min j =0 W := diag ( w j ) j =0 ,...,M − 1 ∈ R M × M , weights w j > 0 The least squares problem is equi- valent to the normal equation of first kind Y ⊢ ⊣ W Y ˆ f = Y ⊢ ⊣ W f .

Theorem: (Filbir ,Themistoclakis, 2008) Let a δ -dense sampling set X ⊂ S 2 of cardinality M ∈ N be given. Mo- reover let for N ∈ N with 154 Nδ < 1 and W = diag ( w j ) j =0 ,...,M − 1 , with Voronoi weights w j be given. Then we have for arbitrary spherical polyno- mials f ∈ Π N ( S 2 ) , for the vector f = � � f ( ξ j ) j =0 ,...,M − 1 the weighted norm estimate (1 − 154 Nδ ) � f � 2 L 2 ≤ � f � 2 W ≤ (1 + 154 Nδ ) � f � 2 L 2 . Proof: based on spherical Marcinkiewicz-Zygmund inequalities (Mhaskar, Narcowich and Ward, 01; Filbir and Themistoclakis, 06). Corollary: Y ⊢ ⊣ W Y Y ⊢ ⊣ W Y � � � � 1 − 154 Nδ ≤ λ min ≤ 1 ≤ λ max ≤ 1 + 154 Nδ i.e. a constant number of iterations in CGNR method is suffices to decrease the residual to a certain fraction

Optimal interpolation M < ( N + 1) 2 under-determined • given sample values f j ∈ C , j = 0 , . . . , M − 1 and weights ˆ w k > 0 2 � � � ˆ f n N k N k � � k � � � � � ˆ f n k Y n � � min = f j subject to ξ j k w k ˆ f ∈ C ( N +1)2 ˆ k =0 n = − k k =0 n = − k The optimal interpolation problem is equivalent to the normal equations of second kind ⊣ ˜ ⊣ ˜ Y ˆ W Y ⊢ f = ˆ ˆ W Y ⊢ f = f , f , where ˆ w n w k , k = 0 , . . . , N, | n | ≤ k . W := diag ( ˜ w ) with ˜ k = ˆ • polynomial kernel K N : [ − 1 , 1] → C and its associated matrix N 2 k + 1 � � � �� ξ j · ξ l K N ( t ) := w k P k ( t ) , ˆ K := K N j,l =0 ,...,M − 1 4 π k =0 K = Y ˆ W Y ⊢ ⊣

Theorem: (Kunis, 2005; Keiner, Kunis, P ., 2006) Let a q -separated sampling set X ⊂ S 2 of cardinality M ∈ N and with q ≤ π be given. Then for N, β ∈ N , N ≥ β − 1 ≥ 2 , the kernel matrix K = ( K j,l ) j,l =0 ,...,M − 1 , K j,l = B β,N ( ξ j · ξ l ) , has bounded eigenvalues | λ ( K ) − 1 | ≤ 25 c β ζ ( β − 1) (( N + 1) q ) β . Corollary: Let a q -separated sampling set X ⊂ S 2 of cardinality M ∈ N and with q ≤ π be given. Moreover, let N ∈ N , ( N + 1) q > 11 . 2 , and weights be given by the sampled cubic B-Spline. Then we have � 4 � 4 � 11 . 2 � 11 . 2 ≤ λ min ( Y ˆ ⊣ ) ≤ λ max ( Y ˆ W Y ⊢ W Y ⊢ ⊣ ) ≤ 1+ 1 − . ( N + 1) q ( N + 1) q i.e. a constant number of iterations in CGNE method is suffices to decrease the error to a certain fraction

Numerical example 90 300 Latitude 250 0 200 150 -90 -180 -90 0.0 90 180 Longitude The original atmospheric temperature of the earth from 5 November 2006 measured by a satellite in Kelvin.

90 300 Latitude 250 0 200 150 -90 -180 -90 0.0 90 180 Longitude Least squares approximation to the global temperature data with N = 32 .

90 300 Latitude 250 0 200 150 -90 -180 -90 0.0 90 180 Longitude Least squares approximation to the global temperature data with N = 128 .

Probabilistic Marcinkiewicz-Zygmund Inequalities (B¨ ottcher, Kunis, P . 08) Observation: in practice theoretically ill-conditioned systems often behave better than one would expect Idea: probabilistic arguments • Bass, Gr¨ ochenig, Rauhut (03,07): results for randomly chosen sampling nodes • deterministic MZ inequality (Filbir ,Themistoclakis) (1 − 154 Nδ ) � f � 2 L 2 ≤ � f � 2 W ≤ (1 + 154 Nδ ) � f � 2 L 2 • Now: randomly chosen polynomials (1 − ǫ ) � f � 2 L 2 ≤ � f � 2 W ≤ (1 + ǫ ) � f � 2 � � ≥ 1 − η P L 2

Example on S 2 : • to ensure the inequality 1 2 � f � 2 ≤ � f � W , 2 ≤ 3 2 � f � 2 (1) for N ≤ 13 , one has to require that R ≤ 1 / (2 · 13 · 84) ≈ 0 . 000 46 (Filbir, Themistoclakis 06, case q=3, p=2); • ˆ f n k taken at random from the uniform distribution, then � 1 2 � f � 2 ≤ � f � W , 2 ≤ 3 � 2 � f � 2 ≥ 0 . 95 P whenever R ≤ 0 . 000 46 and N ≤ 2 184 , on the earth l = 2 . 03 km , using (1) for N ≤ 2184 we have to take l = 12 m . R is partition norm R = max diam R j := max ξ , η ∈ R j d ( ξ , η ) max j j

By A. B¨ ottcher, S. Grudsky (EJP ,03) it was shown that if x is randomly drawn from the uniform distribution and A ∈ C m × n � � 2 � � AA ⊢ ⊣ � 2 � Ax � 2 � x � 2 − � A � 2 � � A � 2 �� � � ≥ 1 − 2 � F � F F � ≤ ε − P � � n nε 2 n n � Corollary: Let A ∈ C m × n and suppose � A � 2 F = n . If x is taken at random from the uniform distribution on B n , then ≥ 1 − 2 � AA ⊢ ⊣ � 2 � 1 − ǫ ≤ � Ax � 2 � F ≤ 1 + ǫ P � x � 2 n 2 ǫ 2 (2 − ǫ ) 2 for every ǫ ∈ (0 , 1) .

Theorem: (B¨ ottcher, Kunis, P ., 08) If NR ≤ 1 then � 1 − ǫ ≤ � f � W , 2 � 2(1 + B d NR ) ≤ 1 + ǫ ≥ 1 − P � f � 2 N d ( N ) ǫ 2 (2 − ǫ ) 2 for each ǫ ∈ (0 , 1) . B d = 3 d/ 2 is Filbir/Themistoclakis constant depending only on d N d ( N ) is the dimension of Π d N R is partition norm R = max diam R j := max ξ , η ∈ R j d ( ξ , η ) max j j

Corollary: Let NR ≤ 1 . If 0 < α < d/ 2 , then a N := 2(1 + B d NR ) N 2 α ∼ Γ( d + 1)(1 + B d NR ) N d ( N ) N d − 2 α and � 1 − 1 N α ≤ � f � W , 2 ≤ 1 + 1 � ≥ 1 − a N . P � f � 2 N α If 0 < β < d , then � � Γ( d + 1)(1 + b d NR ) 2 N β (1 + B d NR ) b N := ∼ N d ( N ) N ( d − β ) / 2 and � 1 − b N ≤ � f � W , 2 � ≥ 1 − 1 ≤ 1 + b N N β . P � f � 2

Theorem: Let d ≥ 2 , ǫ ∈ (0 , 1) , η ∈ (0 , 1) , L ∈ (1 , ∞ ) , and suppose the set X has partition norm R and separation distance q . Then there exists a positive number ̺ 0 = ̺ 0 ( d, ǫ, η, L ) > 0 such that � 1 − ǫ ≤ � f � W , 2 � ≤ 1 + ǫ ≥ 1 − η P � f � 2 for every polynomial degree N ≥ 0 whenever the uniformity condition R/q < L and the density condition R < ̺ 0 hold.

Conclusions • NFFT and inverse NFFT • NFFT on the sphere • Iterative reconstruction on S 2 • Probabilistic arguments http://www.tu-chemnitz.de/ ∼ potts