Sparse and Fast Fourier Transforms in Biomedical Imaging

Transcription

1 Sparse and Fast Fourier Transforms in Biomedical Imaging Stefan Kunis 1 1 joint work with Ines Melzer, TU Chemnitz and Helmholtz Center Munich, supported by DFG and Helmholtz association

2 Outline 1 Trigonometric polynomials and FFTs 2 Specific sparse FFTs 3 Butterfly approximation scheme 4 Numerical experiments 5 Applications & Summary

3 Trigonometric polynomials and FFTs torus T d = R d /Z d = [ 1 2, 1 2) d, index set trigonometric polynomials [ I N = Z d N 2, N ) d 2 f (x) = k I N ˆf k e 2πikx discrete Fourier transform (DFT) f = Fˆf, f j = k I N ˆf k e 2πikj/N, j I N FFT (Gauß; Cooley, Tukey; Frigo, Johnson) O ( N d log N )

4 Trigonometric polynomials and FFTs motivation, randomly drawn X I N, sensing matrix F X IN = (e 2πikj/N ) j X,k IN Fourier coefficient support T I N (Candes, Tao; Rauhut; Needell, Vershynin) recovery scheme C T log 4 N X N d 1 identify support T I N - hard part (Gilbert, Tropp; Iwen) 2 solve for the nonzero Fourier coefficients F X Tˆf T = f X well conditioned F X T F X T matrix vector multiplications with F X T in less than O ( X T )?

5 Specific sparse FFTs sparse DFT, T, X I N, S = T = X N d f j = k T ˆf k e 2πikj/N, j X O ( S 2) divide and conquer, compute on nonzeros - pruning ˆf 0 ˆf 1 ˆf 2 ˆf 3 ˆf 4 ˆf 3 ˆf 2 ˆf 1 -i w 8 -i -i w 3 8 f 0 f 4 f 2 f 2 f 1 f 3 f 3 f 1 ( ) O N(1 + log S2 N ) for S N

6 Specific sparse FFTs T HCFFT X HCFFT, S = T = X = O(N log d 1 N) (Baszenski, Delvos; Hallatschek) O(S log N) for arbitrary X T d, S = T = X (Döhler, Fenn, Kämmerer, Potts, K.) O(S log d 1 N log ε d )

7 Specific sparse FFTs general sparsity pattern T I N, S = T = m d, m N x j = j/m, j I m, coarse grid O(S log S) x j = Gj, G R d d, coarse lattice O(S(log S + log ε d )) reduction to O ( S log α N ) ε only for very specific situations

8 Butterfly approximation scheme Lexing Ying. Sparse Fourier transform via butterfly algorithm, SIAM J. Sci. Comput. 31 (2009) smooth and sparse T [ N 2, N 2 )d, X [ N 2, N 2 )d T = X = O(N d 1 ) nodes in frequency/spatial domain lth dyadic subdivision of [ N 2, N 2 )d and [ 1 2, 1 2 )d into 2 dl boxes have only O(2 (d 1)l ) nonempty ones O(N d 1 log N p d+1 ) like in FMM, H-matrices, nonequispaced FFT, local expansion degree p = log ε, independent of N?

9 Butterfly approximation scheme considered model problem, nonsparse, univariate, given T = {k l [0, N] : l = 0,..., N} X = {x j [0, N] : j = 0,..., N} ˆf = (ˆf k ) k T C N evaluate almost periodic function for x X f (x) = k T ˆf k e 2πikx/N FFT for nonequispaced nodes in time and frequency domain (type-3 nufft, Greengard, Lee; nnfft, Keiner, Potts, K.)

10 Butterfly approximation scheme well known low rank property, p max(2eπ, log 2 ε ) e2πikx/n p 1 s=0 (2πi) s N s s! ks x s ε, kx N 2 admissible partitions of T X = [0, N] 2 diam(t )diam(x ) N dyadic decompositions of T and X, examples for N = 8 T T T T X X X X (Edelman; Überhuber,...)

11 dyadic decompositions of T and X, examples for N = 4 T 20 T 21 T 22 T 23 X 00 X 10 X 11 T 10 T 11 X 20 X 21 X 22 X 23 T 00 butterfly graph, nodes are admissible pairs X 00 T 20 X 00 T 21 X 00 T 22 X 00 T 23 X 10 T 10 X 10 T 11 X 11 T 10 X 11 T 11 X 20 T 00 X 21 T 00 X 22 T 00 X 23 T 00

12 Butterfly approximation scheme local in T, global in X : start with f T 30 (x) = ˆf k e 2πikx k T 30 T approximate f T 30 on X 00 f X00T30 approximate f X 00T 30 + f X 00T 31 on X go on local in X, global in T : finally f X 10T 20 f X 30T 00 is an approximation to f T 00 = f on X 30 (..., Rokhlin, O Neill,...)

13 Butterfly approximation scheme approximations local in T, global in X f X 00 T 20 f X 00 T 21 f X 00 T 22 f X 00 T 23 f X 10 T 10 f X 10 T 11 f X 11 T 10 f X 11 T 11 f X 20 T 00 f X 21 T 00 f X 22 T 00 f X 23 T 00 local in X, global in T

14 Butterfly approximation scheme frequency band T = [k min, k max ], admissible X = [x min, x max ] almost periodic function g Π T, T T, g : X C, g(x) = k T ĝ k e 2πikx/N T p = k min + t p 1 (k max k min ), t = 0,...p 1 x s = x min+x max 2 + xmax x min interpolation operator Jp XT 2 cos 2s+1 2p π, s = 0,...p 1 : Π T Π Tp J XT p g(x s ) = g(x s ), s = 0,..., p 1

15 Butterfly approximation scheme Theorem (Melzer, K. 2010) For admissible T, X, g Π T, and p 3, we have with C p = g J XT p g C(X ) C p g C(X ) 2π p 4 p 1 p! π p c 0 c p 1. Sketch of proof. shift invariance Jp XT e 2πikx = e 2πik0x J X (T k 0) p e 2πi(k k 0)x polynomial interpolation of Jp XT g and g coincide Bernstein inequality for g Π T

16 Butterfly approximation scheme Corollary (Melzer, K. 2010) For N = 2 L, T, X = [0, N] d, f Π T, ε > 0, and ( ) 1 2c0 d(l + 1) p = log 3 log c 1 ε we have f f C(X ) ε ˆf 1. Sketch of proof. J XT p = d r=1 J X (r) T (r) p J X (r) T (r) p 1 + C p

17 Numerical experiments T, X [0, N], T = X = N f (x) = ˆf k e 2πikx/N k T computation time, naive O(N 2 ), 1d-butterfly nnfft O(N log N ( log ε + log log N) 2 )

18 Numerical experiments 1d-butterfly nnfft, accuracy p log ε + log log N ε = c N c p 1 ε = c p log N

19 Numerical experiments smooth and sparse T, X [0, N] d, T = X = N d 1 f (x) = ˆf k e 2πikx/N k T computation time naive O(N 2d 2 ) butterfly O(N d 1 log N ( log ε + log log N) d+1 ) d = 2 d = 3

20 Applications & Summary 2d spherical mean values, forward simulation N 2 image data N acoustic sensors, N times/radii spherical mean value operator M : C(R d ) C(R d+1 ) (Mf ) (y, r) = 1 f (y + rξ)dσ(ξ) ω d 1 S d 1 let e k (x) = e 2πikx then (Me k )(y, r) = 2 d 2 2 Γ ( ) J d 2 (2π k r ) d 2 2 (2π k r ) d 2 2 e 2πiky

21 Applications & Summary special case d = 3 J 1 (2π k r) = 2 2 sin(2π k r) π 2π k r 3d photoacoustic imaging, forward simulation N 3 volume data N 2 acoustic sensors, N times/radii number of floating point operations naive O(N 5 ) 4d butterfly sparse FFT O(N 3 log N ( log ε + log log N) 5 )

22 Applications & Summary Thank you!

23 Stability local approximation scheme computes expansion coefficients of Jp XT in a monomial basis matrix G C p p, g s,t = e 2πiktxs, s, t = 0,..., p 1 x s = 1 2 (2s + 1)π cos, k t = 1 2p 2 + t p 1 condition number cond(g) ( p 1 p π ) p 1 d = 2, N = 2 10, limits to p 9 and ε 10 8