Seismic compression. François G. Meyer Department of Electrical Engineering University of Colorado at Boulder

Transcription

1 Seismic compression François G. Meyer Department of Electrical Engineering University of Colorado at Boulder fmeyer IPAM, MGA 2004 Seismic Compression 1

2 Collaborators Amir Averbuch Raphy Coifman Jan-Olov Strömberg Anthony Vassiliou IPAM, MGA 2004 Seismic Compression 2

3 Introduction seismic data: HUGE ( 100 Tbyte) datasets: large dynamic range, geometric features: important seismic data: highly oscillatory Wavelet compression of seismic data [Ergas et al., 1995, Reiter and Heller, 1994, Vassiliou and Wickerhauser, 1997] But: rapid variations of intensity many fine scale coefficients very diffuse representations in a standard wavelet basis IPAM, MGA 2004 Seismic Compression 3

4 Marine shot gathers original, range=[ , ], size = Compression by 20, SNR=35. Error, range =[ 10 6, 10 6 ]. IPAM, MGA 2004 Seismic Compression 4

5 Common depth gathers original, range=[ 4000, 4000], size= Compression by 20, SNR=46. Error, range =[ 10, 10]. IPAM, MGA 2004 Seismic Compression 5

6 Performance comparison compression wavelet LCT ratio (SNR) (SNR) Common depth gathers compression wavelet LCT ratio (SNR) (SNR) Marine shot gathers SNR = l2 ( f ) l 2 ( f f ) (1) Local cosine outperform wavelets for seismic datasets! IPAM, MGA 2004 Seismic Compression 6

7 Adaptive smooth local cosine transforms b n biorthogonal bases a α a a a a n - n n n + β n a - + β n+1 α n+1 n+1 n+1 n+1 R = n=+ n= [a n, a n+1 [ neighborhood around each point a n : [a n α n, a n + β n ] a n + β n < a n+1 α n+1. (2) b n : bell function lives over the interval [a n α n, a n+1 + β n+1 ]. IPAM, MGA 2004 Seismic Compression 7

8 Dual bells b n (x) = θ n (x)b n 1 (2a n x) if x [a n α n, a n + β n ] 1 b n (x) if x [a n + β n, a n+1 α n+1 ] θ n+1 (x)b n+1 (2a n+1 x) if x [a n+1 α n+1, a n+1 + β n+1 ] 0 otherwise (3) θ n (x) = 1 b n (x) b n 1 (2a n x) + b n (2a n x) b n 1 (x) (4) IPAM, MGA 2004 Seismic Compression 8

9 Biorthogonal local cosine bases DCT-IV: c n,k (x) = [ ] 2 x a n cos (k + 1/2) a n+1 a n a n+1 a n (5) We define the family and the dual family: w n,k = b n (x) c n,k (x) (6) w n,k = b n (x) c n,k (x) (7) IPAM, MGA 2004 Seismic Compression 9

10 Biorthogonal local cosine bases Lemma 1 w n,j and w n,j are Riesz biorthogonal bases: w n,j (x) w k,m (x) dx = δ j,k δ n,m (8) x L 2 (R), x(x) = n,j x(x) = n,j x n,j w n,j (x) with x n,j = x n,j w n,j (x) with x n,j = x(x) w n,j (x)dx (9) x(x)w n,j (x)dx (10) IPAM, MGA 2004 Seismic Compression 10

11 Choice of the bell function biorthogonal bases: more freedom to select the bells b n b n can be optimized seismic data : oscillatory [Matviyenko, 1996] other choices: reproduce constants [Jawerth and Sweldens, 1995], or polynomials [Bittner, 1999] IPAM, MGA 2004 Seismic Compression 11

12 Bells of Matviyenko class of signals: x = cos(ωk + ϕ) minimum number of coefficients x n,j to reconstruct x up to an error ε 0 b(x) 1 0 b(x) ( (11) (2) + 1)/2 IPAM, MGA 2004 Seismic Compression 12

13 Bells of Matviyenko b N (x) = ( ) N g n sin(n + 1/2)πx n=0 ) ( N n=0 ( 1) n g n cos(n + 1/2)πx coefficients g n are calculated numerically if x [ 1/2, 1/2] if x [1/2 : 3/2] 0 otherwise (12) IPAM, MGA 2004 Seismic Compression 13

14 N=1 N=2 N=3 N= Optimized bells. N = 1, 2, 3, 4 IPAM, MGA 2004 Seismic Compression 14

15 N=1 N=2 N=3 N= Optimized dual bells. N = 1, 2, 3, 4 IPAM, MGA 2004 Seismic Compression 15

16 Adaptive segmentation quadtree segmentation preserve the original aspect ratio of the data anisotropic segmentations [Bennett, 2000] depth first approach: minimize memory extension on the borders IPAM, MGA 2004 Seismic Compression 16

17 Choice of a cost function entropy [Coifman and Wickerhauser, 1992] h(x) = k x k 2 x 2 log x k 2 x 2 (13) Rate distortion [Ramchandran and Vetterli, 1993] { { max λ min T T node T min q Q { D node(q) + λr node (q)} λ R Q : set of all quantizers T : set of all bases T with quadtree structure very high computational complexity! first order entropy estimates only }} (14) IPAM, MGA 2004 Seismic Compression 17

18 Choice of a cost function cost function: estimate of the actual rate achieved by each node mimics the actual scalar quantization, and entropy coding much faster to compute composed of two complementary terms: c 1 (x): cost of coding the sign and the magnitude of the non zero output levels of the scalar quantizer, c 2 (x): cost of coding the locations of the non zero output levels (significance map), IPAM, MGA 2004 Seismic Compression 18

19 Choice of a cost function c 1 (x) = k/q(xk ) =0 max (log 2 Q(x k ), 0) fast implementation: representation of floating numbers c 2 (x) = N ( p log 2 (p) + (1 p) log 2 (1 p) ) first order entropy of a Bernoulli process: each coefficient x k is significant with a probability p IPAM, MGA 2004 Seismic Compression 19

20 Fast DCT-IV DCT-IV : FFT of half length DCT-IV coefficients, ˆx(j), j = 0,..., N 1 of the sequence x(n), n = 0,..., N 1 ˆx(j) =Re e ijπ 2N N/2 1 y(n)e 2iπjn N ˆx(N j 1) = Im n=0 e ijπ 2N N/2 1 n=0 y(n)e 2iπjn N (15) with + 1/4)π i(n y(n) = (x(2n) + i x(n 2n 1)) e N (16) IPAM, MGA 2004 Seismic Compression 20

21 Scanning the coefficients scanning coefficients by increasing frequency each LCT block divided into a fixed number of frequency subsets gather from all the LCT blocks all the coefficients that are in the same subset IPAM, MGA 2004 Seismic Compression 21

22 Scanning the coefficients IPAM, MGA 2004 Seismic Compression 22

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25 Laplacian based scalar quantization distribution of the cosine coefficients: Laplacian [Birney and Fischer, 1995] near optimal scalar quantizer [Sullivan, 1996] [ + δ, δ], the symmetric dead-zone,, the quantizer step size, δ, the reconstruction offset IPAM, MGA 2004 Seismic Compression 25

26 Laplacian based scalar quantization 2 δ δ -3 +δ - 2 +δ - + δ - δ - δ 2 - δ 3 - δ δ - 2 Scalar quantizer, with a dead zone. IPAM, MGA 2004 Seismic Compression 26

27 Entropy coding significance map: n C order arithmetic coder signs of the output levels: packed magnitude of the output levels: variable length encoded best basis geometry: adaptive arithmetic coder. IPAM, MGA 2004 Seismic Compression 27

28 Experiments Fast Local Cosine Transform (FLCT) coder and decoder actual bit stream: size equal to the targeted budget IPAM, MGA 2004 Seismic Compression 28

29 Comparison of the bells Standard bell: [ π ] b N (x) = sin 4 (1 + x N) x j = sin( π 2 x j 1) x 0 = x (17) Optimized bell: Matviyenko Gaussian b(x) = e α(x a n) 2 IPAM, MGA 2004 Seismic Compression 29

30 1 0.9 alpha=1 alpha= alpha=1 alpha= Gaussian bells. α = 1, 2 IPAM, MGA 2004 Seismic Compression 30

31 N=1 N=2 N=3 N= Standard bells. N = 1, 2, 3, 4 IPAM, MGA 2004 Seismic Compression 31

32 Test images Synthetic image: warped cosines, Seismic data: 2D slice uniform grid: blocks of size IPAM, MGA 2004 Seismic Compression 32

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35 Synthetic data : optimal N for optimized bell N=1 N=2 N=3 N=4 N=5 44 PSNR (db) Compression ratio IPAM, MGA 2004 Seismic Compression 35

36 Synthetic data : optimal overlap for optimized bell PSNR (db) Compression ratio alpha=0 alpha=2 alpha=4 alpha=6 alpha=8 alpha=10 alpha=12 alpha=12 alpha=16 IPAM, MGA 2004 Seismic Compression 36

37 Synthetic data : comparison of the bells Matviyenko (alpha=16, N =2) Standard (alpha =16) Gauss (alpha =16) No bell Wavelet packets PSNR (db) Compression ratio IPAM, MGA 2004 Seismic Compression 37

38 Seismic data : comparison of the bells Standard (alpha=16) Matviyenko (alpha=16, N =3) Wavelet packets No bell Gauss (alpha =16) PSNR (db) Compression ratio IPAM, MGA 2004 Seismic Compression 38

39 Compression 50, Matviyenko (N = 3), PSNR =20.56 db IPAM, MGA 2004 Seismic Compression 39

40 Compression 50, no bell, PSNR = db IPAM, MGA 2004 Seismic Compression 40

41 Conclusion compression of highly oscillatory signals: local cosine transforms optimal spatial tiling design of the window full 3-D compression: fast 3-D FFT The future: curvelets? IPAM, MGA 2004 Seismic Compression 41

42 References [Bennett, 2000] Bennett, N. (2000). Fast algorithm for best anisotropic Walsh bases and relatives. Applied and Computational Harmonic Analysis, 8(1): [Birney and Fischer, 1995] Birney, K. and Fischer, T. (1995). On the modeling of DCT and subbdand image data for compression. IEEE Trans. on Image Process., 4(2): [Bittner, 1999] Bittner, K. (1999). Error estimates and reproduction of polynomials for biorthogonal local trigonometric bases. Applied and Computational Harmonic Analysis, 6: [Coifman and Wickerhauser, 1992] Coifman, R. and Wickerhauser, M. (1992). Entropy-based algorithms for best basis selection. IEEE Trans. Information Theory, 38(2): [Ergas et al., 1995] Ergas, R., Donoho, P., and Villassenor, J. (1995). High-performance seismic trace compression. In Soc. Expl. Geophys., 65th Intern. Conv. Soc. Expl. Geophys. [Jawerth and Sweldens, 1995] Jawerth, B. and Sweldens, W. (1995). Biorthogonal local trigonometric bases. J. Fourier Anal. Appl., 2(2): [Matviyenko, 1996] Matviyenko, G. (1996). Optimized local trigonometric bases. Applied and Computational Harmonic Analysis, 3: [Ramchandran and Vetterli, 1993] Ramchandran, K. and Vetterli, M. (1993). Best wavelet packet bases in a ratedistortion sense. IEEE Trans. Image Processing, pages [Reiter and Heller, 1994] Reiter, E. and Heller, P. (1994). Wavelet transformation-based compression of nmocorrected cdp gathers. In Soc. Expl. Geophys., 64th Intern. Conv. Soc. Expl. Geophys. [Sullivan, 1996] Sullivan, G. (1996). Efficient scalar quantization of exponential and Laplacian random variables. IEEE Trans. Information Theory, 42(5): IPAM, MGA 2004 Seismic Compression 42

43 [Vassiliou and Wickerhauser, 1997] Vassiliou, A. and Wickerhauser, M. (1997). Comparison of wavelet image coding schemes for seismic data compression. In Soc. Expl. Geophys., 67th Intern. Conv. Soc. Expl. Geophys. IPAM, MGA 2004 Seismic Compression 43