FID. Vol. 21 No. 4 Dec Chinese Journal of Magnetic Resonance (NMR) Gabor, NMR ( FID) NMR FID. Gabor. Gabor. , Gabor , NMR FID.

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1 Chinese Journal of Magnetic Resonance Vol 21 No 4 Dec 2004 : (2004) Gabor FID 1 3 2, (1, ; 2, ) : Gabor, (NMR FID) NMR FID, NMR FID,, NMR FID, Gabor Gabor NMR FID,,,, Gabor : NMR FID, Gabor, : O : A (NMR) NMR ( FID), NMR FID NMR, NMR FID,, [ 1 ] Gabor ( ), [1 4 ] : ; : : ( [ 2002 ]40 ) ; ( ) ; (2001kj020zd) ; : (19632),,,,,, : , , E2mail edu 3

2 Gabor NMR FID, NMR FID, NMR FID ( ), ( Gabor ), NMR FID,,, Gabor ( ), Gabor ( NMR FID ),, Gabor ( Gabor ) NMR FID, NMR FID, [1 ] NMR FID, Gabor, Gabor ;, [1 ],, ( ), Gabor,,, [5, ], Gabor, ( Gauss ),,,,, Gabor,,, Gabor [6 ] NMR FID Gabor 1 NMR FID Gabor [6 ] NMR FID, L x ( k), x ( k) Gabor : M - 1 x ( k) = a m, n gh m, n ( k) (1) m = 0 n = 0 N - 1 x ( k) Gabor (RD GT) : L - 1 a m, n = x ( k) k = 0 r m, n ( k) (2) a m, n x ( k) RD GT Gabor gh m, n ( k) = gh ( k - m ) gn ) cas(2 nk/ N ) (3) r m, n ( k) = r ( k - m) gn ) cas(2 nk/ N ) (4) cas( x) = cos ( x) + sin ( x) Hartley [7 ] Gabor Gabor L = gn M = N gm, Gabor, M N, gm gn, gn gm ( M N L ) ( M N < L ) ( M N > L ), M N / L,, Gabor ( M N ) gn gm = N M = L ( a m, n x ( k) ), M N / L = 1, Gabor

3 4 : Gabor FID 437 gh ( k) r ( k) h ( k) r ( k), gh ( k) = h ( k + il ) = gh ( k + L ) (5) i Gabor r ( k) = r ( k + il ) = r ( k + L ) (6) i a m, n, m n, a m + im, n + jn = a m, n i, j = 0, 1, 2, 3, (7) gh ( k) r ( k) ( [6 ] ) : L - 1 gh ( k + m N ) cas( 2 nk k = 0 gn ) r ( k) = L N M m n (8) 0 m gm - 1,0 n gn - 1, k Kronecker delta (8), H r = v (9) v = { L / ( M N ), 0, 0,, 0} T gm gn, r = { r (0), r (1),, r ( L - 1) } T = { r (0), r (1),, r ( L - 1) } T, H ( gm gn ) L, H ( m gn + n, k) = gh ( k + m N ) cas(2 nk/ N ) (10) r (9), M = gm N = gn ( L = gm gn ), H, r, gm gn < L, (9) r Gabor [5 ], ( minimum norm), r 0 (9), min r : H L - 1 r = k = 0 r ( k) 2 (11) r 0 = H T ( HH T ) - 1 v (12), L = 1024, : h ( k) = 0 5 exp { - [ ( k ) / 200} u ( k ) ( 1 (a) ), u ( k), M = 16 N = 64,, (12) r ( k) 1 (b), 1 (c) H ( f ) 1 (d) R ( f ) h ( k) r ( k) Fourier Gauss : h ( k) = 0 01 exp{ [ ( k ) 0 03 ] 2 } ( 2 (a) ), M = 256 N = 512,, N M / L = 128, Fourier 2, Gauss ( [5 ] ), 2 Gauss 1,,,, Gauss (the optimal joint time2frequency concentration) [8 ], Gauss NMR FID, Gabor NMR FID Gabor, Gabor,, ( ), NMR FID Gabor NMR FID Gabor,, Gabor NMR FID

4 (a) h ( k), L = 1024, (b) r ( k), M = 16, N = 64 ( ), (c) h ( k) Fourier H ( f ), ( d) r ( k) Fourier R ( f ) Fig 1 (a) Exponential synthesis window h ( k), L = 1024, (b) the corresponding analysis window r ( k), L = 1024, M = 16, N = 64, critical sampling, (c) H ( f ), the Fourier spectrum of h ( k), ( d) R ( f ), the Fourier spectrum of r ( k) r, Hartley (DHT) [7 ] (2) a m, n : L - 1 a m, n = k = 0 N - 1 x ( k) r ( k - m gn ) cas(2 nk/ N ) = j = 0 gm - 1 i = 0 R m ( in + j) cas(2 nj/ N ) R m ( k) = x ( k) r ( k - m gn ), k = in + j N DHT (DHT (DFT),, [7 ( FFT) ] ), a m, n M ( N 1 - D DHT) + M L + ( gm - 1) M N, [5 ] Gabor gm M ( N 1 - D DFT) + M L + ( gm - 1) M N, Gabor x ( k), (1) : M - 1 x 0 ( k) = m = 0 gh ( k - N - 1 m gn ) a m, n cas( 2 nk n = 0 N ) x 0 ( k) x ( k) k = qn + k 0, q = 0, 1,, gm - 1, k 0 = 0, 1,, N - 1, (13)

5 4 : Gabor FID 439 M - 1 x 0 ( qn + k 0 ) = gh ( qn + k 0 - m = 0, N DHT) N - 1 m gn ) a \ - m, ncas( 2 nk 0 n = 0 N ) (14) DHT M ( N 1 - D + M L + ( M - 1) L, [5 ] Gabor 2 M N L + ( M N - 1) L, Gabor 2 (a) Gauss h ( k), L = 1024, (b) r ( k), M = 256, N = 512 ( ), (c) h ( k) Fourier H ( f ), (d) r ( k) Fourier R ( f ) Fig 2 (a) a Gaussian synthesis window h ( k), L = 1024, (b) the corresponding analysis window r ( k), L = 1024, M = 256, N = 512, oversampling, (c) H ( f ), the Fourier spectrum of h ( k), ( d) R ( f ), the Fourier spectrum of r ( k), [ 9 ] Gabor Gabor [6 Gabor ], Gabor Gabor a m, n : C m, n = [ ( a m, n + a m, N - n ) / 2 ] 2 + [ ( a m, n - a m, N - n ) / 2 ] 2 m = 0, 1, 2,, M - 1 ; n = 0, 1, 2,, N / 2-1 Gabor 2 (15)

6 [2 ], NMR FID x ( k), j = x ( k) = s ( k) + n w ( k) = Q b q exp - q = 1 t k + j (2 f T q t k + q ) + n w ( k) (16) 2 q - 1, b q f q q T 2 q s ( k) q NMR FID, t, x ( k) L, 0 k L - 1, n w ( k), x ( k) Gabor [3,4 NMR FID ] L = 1024 ( 85 3 ms), t = 1/ s, (Reference) 5 FID ( P l PCr ), 1, NMR FID 3 = 4 000, SNR ( 0 k L - 1) ( Reference ), SNR = 20 log ( s / n w ) = db, 4 1 NMR FID Table 1 Parameters of the simulated phosphorus FID sequence Peaks f q ( Hz) 0 2 T 2 q (ms) b q q ( ) Reference P l PCr

7 4 : Gabor FID Gauss, Gabor 5 (a), NMR FID,, NMR FID,,, Gabor ( ), Gabor ( NMR FID ),, Gabor ( Gabor ) NMR FID x 0 ( k) X 0 ( f ) 6 (a), SNR = 20 log ( s / x 0 - s ) = 4 43 db ( Reference ), SNR 7 76 db,,, NMR FID 5 Gabor C m, n, (a) Gabor, : ms Hz ; (b) Gabor, : ms Hz Fig 5 (a) Gaborgram of the phosphorus FID sequence in the oversampling case using the Gaussian synthesis window, (b) Gaborgram of the phosphorus FID sequence in the critical2sampling case using the exponential synthesis window 6 NMR FID x 0 ( k) X 0 ( f ), (a) Gabor ; (b) Gabor Fig 6 (a) Fourier spectrum of the enhanced phosphorus FID sequence using the oversampled Gabor transform and the Gaussian synthesis window, ( b) Fourier spectrum of the enhanced phosphorus FID sequence using the critically2sampled Gabor transform and the exponential synthesis window

8 [1 ], 1, Gabor 5 (b), NMR FID x 0 ( k) X 0 ( f ) 6 ( b), SNR = 20 log ( s / x 0 - s ) = db ( Reference ), SNR 0 85 db, Gabor, 3 Gabor, NMR FID NMR FID, NMR FID, ( Gabor ), NMR FID, NMR FID, Gabor Gabor NMR FID,, ;, Gabor : [ 1 ] Lu Y, Joshi S, Morris J M Noise Reduction for NMR FID Signal via Gabor Transform [J ] IEEE T on Bio2med Eng, 1997, 44 (6) : [ 2 ] Joliot M, Mazoyer B M, Heusman R H In vivo NMR spectral parameter estimation : A comparison between time and frequency domain methods [J ] Magnet Reson Med, 1991, 18 : [ 3 ] Leclerc J H J Time2frequency representation of damped sinusoids [J ] J Magn Reson, 1991, 95 : [4 ] Angelidis P A, Sergiadis G D Time2frequency representation of damped sinusoids using the Zak transform [J ] Journal of Magnetic Resonance Series A, 1993, 103 : [ 5 ] Qian S, Chen D Discrete Gabor transforms [J ] IEEE T Signal Proces, 1993, 21 (7) : [6 ] Tao L, Cheng G J Real2valued discrete Gabor transforms for discrete signal and image representation [J ] Chinese Journal of Electronics, 2001, 10 (4) : [ 7 ] Bracewell R M The fast Hartley transform [J ] Proc IEEE, 1984 ; 72 (8) : [ 8 ] Qian S, Chen D Joint time2frequency analysis [J ] IEEE Signal Proc Mag, 1999, 6 (2) : [ 9 ] Friedlander B, Zeira A Oversampled Gabor Representation for Transient Signals [J ] IEEE T Signal Proces, 1995, 43 (9) :

9 4 : Gabor FID 443 NMR FID SIGNAL ENHANCEMENT VIA OVERSAMPL ING GABOR TRANSFORMATION TA O L iang 1 3, GU J uan2j uan 2 (1 Institute of Computer Science and Information Engineering, Anhui University, Hefei , China ; 2 Dept of Computer and Information Engineering, Hefei University, Hefei , China) Abstract : An efficient algorithm is proposed to reduce the noise in NMR FID signals based on t he real2value discrete Gabor t ransformation developed in our previous work As NMR FID signals in the time domain are short oscillating decay signals, the FID signals in Gabor t ransformation domain (i e, a joint time2f requency domain) are concent rated in very few number of Gabor transformation coefficients, while the noise is fairly distributed among all t he coefficient s Therefore, performing a t hreshold2limiting work in t he t ransform domain can significantly enhance t he NMR F ID signals The t heoretical and experimental analyses presented in t his paper show t hat oversampling Gabor t ransformation is more suitable for NMR F ID signal enhancement t han critically2sampling t ransformation, because t he synt hesis window and it s corresponding analysis window in t he oversampling case can have better localization in both time domain and frequency domain than that in the critically2sampling case Furt hermore, t he oversampling Gabor t ransform can lead to higher time and f requency resolution t han t he critically2sampling one Key words : NMR F ID signals, oversampling Gabor t ransformation, signal enhancement algorit hms 3 Correspondence author : Tao Liang, Tel : , , E2mail edu