ANALYSIS OF MRS SIGNALS WITH METABOLITE-BASED AUTOCORRELATION WAVELETS

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1 ANALYSIS OF MRS SIGNALS WITH METABOLITE-BASED AUTOCORRELATION WAVELETS A. Schuck Jr. A. Suvichakorn C. Lemke J.-P. Antoine Unité de Physique Théorique et Mathematique, FYMA Université Catholique de Louvain, UCL Louvain-la-Neuve, BELGIUM Wavelets and Fractals, Esneux, April 26-28, 2 BELGIUM Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets /37

The traditional analysis The Autocorrelation Function Objectives and outline THE ANALYSIS SO FAR Time-domain and frequency-domain are traditional methods to quantify metabolites; Also Time-Frequency

2 The traditional analysis The Autocorrelation Function Objectives and outline THE ANALYSIS SO FAR Time-domain and frequency-domain are traditional methods to quantify metabolites; Also Time-Frequency (Scale) methods have been used to analyzing MRS signals; The CWT using Morlet was presented in [4]. An example of pure Creatine using the Morlet wavelet: Fig. Freq. response of CRE at 4.7 T and its Morlet CWT (w = rad/s, σ =, Fs = 46, 4 s ). Extracted from A.-Suvichakorn and J.-P. Antoine, The Continuous Wavelet Transform in MRS, e-book FAST website, 29 Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 2/37

3 The traditional analysis The Autocorrelation Function Objectives and outline QUESTIONS MADE BY THE AUTHORS Questions made by the authors: Can one make a Wavelet function with its spectrum "Matched" with some MRS metabolite signal? 2 Can one estimate this metabolite s parameters using the CWT with such a wavelet function? Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 3/37

4 The traditional analysis The Autocorrelation Function Objectives and outline THE AUTOCORRELATION FUNCTION One answer can be the use of the concept of Autocorrelation Functions. The autocorrelation function estimator R xx (τ) of an ergodic process time series x(t) is ([3]): R xx (t) = x(τ)x(τ t) dt = where x(t) means the complex conjugate of x(t). x(τ)x(τ + t) dτ, () In the frequency domain, the Fourier transform of R xx, called S xx (ω) can be evaluated by the Wiener-Khintchine relation: S xx (ω) = F {R xx (τ)} = X(ω) 2. (2) Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 4/37

5 OBJECTIVES AND OUTLINE The traditional analysis The Autocorrelation Function Objectives and outline Objectives: Create a wavelet from a MRS pure metabolite signal autocorrelation function; 2 Perform the CWT of a complex MRS signal using this wavelet; 3 Estimate the metabolite parameters. Outline: Find an Analytical Solution for CWT using classic MRS models and autocorrelation wavelets form this model; 2 Create the discrete versions of signals and wavelets presented in the analytical part and analyze them with YAWtb Matlab Toolbox(Lorentzian Models); 3 Create discrete versions of signals and wavelets now based on the metabolite database and analyze them with Matlab (Metabolite-based Models); Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 5/37

6 One peak model Generalizing for N-peaks What happens with more realistic signals? LORENTZIAN LINESHAPE A simple classical FID MRS signal model x(t): The Lorentzian lineshape. x(t) = A e Dt e i(ωs t+φ) A >, D >. (3) In the frequency domain this signal is defined by X(ω) = 2πA e iφ δ(ω (ω s + id )), (4) where δ is the Dirac delta function. Fig. Real part (blue line) and imaginary part (red dash) of x(t) for A =, D = s, ω s = 32 rad/s, φ = rad. What is the autocorrelation function of this signal? NOT a CLUE!!! Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 6/37

7 MODIFIED LORENTZIAN LINESHAPE One peak model Generalizing for N-peaks What happens with more realistic signals? A slightly modified FID MRS signal model x (t) can be: x (t) = A e D t e iωs t θ(t), D >, (5) where θ(t) is the Heaviside function (or step function). In the frequency domain this signal is defined by X (ω) = A [D + i(ω ω s )]. (6) Now, the autocorrelation function (on freq. domain) is easily computed: S xx (ω) = A D + i(ω ω s ) 2 = A 2 D 2 + (ω ω s) 2. (7) Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 7/37

8 One peak model Generalizing for N-peaks What happens with more realistic signals? THE ADMISSIBLE WAVELET FUNCTION Is this function admissible? PROBABLY NOT. What to do, then? As with Morlet, use a correction term. Then the admissible wavelet function Ψ adm (ω) becomes: Ψ adm (ω) = Is this term really necessary? A 2 D 2 + (ω ω s ) 2 A 2 D 2 + (ω2 + ω 2 s ). (8) As with Morlet, in practice the appropriate choice of D and ω s make its value numerically negligible, so that the correcting term can indeed be omitted. Ψ(ω) = A 2 D 2 + (ω ω s ) 2. (9) Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 8/37

9 THE WAVELET FUNCTION: One peak model Generalizing for N-peaks What happens with more realistic signals? Fig. Ψ adm (ω) (blue line) and Ψ(ω) (red dot) for A =, D = and ωs = 32 rad/s. Fig. ψ(t), for A =, D = and ωs = 32 rad/s: real part (blue line) and imaginary part (red dash). Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 9/37

10 One peak model Generalizing for N-peaks What happens with more realistic signals? THE CWT OF A LORENTZIAN MRS SIGNAL Then, the CWT of the the Lorentzian lineshape signal x(t) using Ψ(ω) is: S(b, a) = a 2πA e i φ A 2 δ(ω (ω s + i D )) 2π D 2 + (aω ω s) 2 eiωb dω = a x(b) A 2 D 2 + [ωs (a ) + i a D ] 2. () S(b, a) diverges for a = cause signal and wavelet have the same damping factors" D. It cannot be used to estimate parameters, as in the Morlet case, but... Using different damping factors" D for the wavelet function and D for the signal, then S(b, a), for a = will become: S(b, ) = x(b) A 2 D 2 D2 = A e D b So the signal s damping factor can be calculated by: A 2 D 2 D2. () D = d ln S(b, ) (2) db Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets /37

S(b, a) for A =, D =, ω s = 32 rad/s,b = [, 2] s and a = [.5,.5], a. Fig.

11 THE (BEAUTIFUL) GRAPHICS One peak model Generalizing for N-peaks What happens with more realistic signals? Fig. S(b, a) for A =, D =, ω s = 32 rad/s,b = [, 2] s and a = [.5,.5], a. Fig. arg S(b, a) for A =, D =, ω s = 32 rad/s,b = [, 2] s and a = [.5,.5], a. Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets /37

12 One peak model Generalizing for N-peaks What happens with more realistic signals? GENERALIZING FOR N-PEAKS SIGNALS As many metabolites are multipeaked, we will generalize for N-peaks Lorentzian signals. The signal x N (t), a weighted sum of N Lorentzian components: N x N (t) = A ne Dnt e i(ωnt+φn) ; D n >. (3) The Fourier transform of x N (t): n= X N (ω) = 2π N A ne iφn δ(ω (ω sn + id n)). (4) n= The left truncated version of x N (t): x N (t) = n= where θ(t) is the Heaviside function. Now the Fourier transform of x N (t) N A ne Dnt e i(ωsnt) θ(t), D n >, (5) X N (ω) = N n= A n [D n + i(ω ω sn)]. (6) Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 2/37

13 One peak model Generalizing for N-peaks What happens with more realistic signals? THE N-PEAK WAVELET FUNCTION AND CWT The autocorrelation of (5) in the frequency domain is: S x N x N 2 N A n N (ω) = D n + i(ω ω n= sn) n= A 2 n D 2 n + (ω ω sn) 2. (7) Under the considerations made, the autocorrelation wavelet function is: Ψ N (ω) = N n= A 2 n D 2 n + (ω ω sn) 2 (8) The CWT of the signal (3) using (8) as wavelet function is given by: S N (b, a) = a N N A k e iφ k δ(ω (ω sk + id k )) k= n= A 2 n D 2 n + (aω ω sn ) 2 eiωb dω = N N A 2 n a x k (b) D 2 (9) k= n= n + [a(ω sk + id k ) ω sn] 2 Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 3/37

S 2 (b, a) for A = A 2 =, D = D2 =, ω s = 32 and ω s2 = 64 rad/s, b = [, 2] s and a = [., 3], a.; Fig.

14 One peak model Generalizing for N-peaks What happens with more realistic signals? (ALSO BEAUTIFUL)GRAPHS OF 2 AND 3 PEAKS CWT Fig. S 2 (b, a) for A = A 2 =, D = D2 =, ω s = 32 and ω s2 = 64 rad/s, b = [, 2] s and a = [., 3], a.; Fig. S 3 (b, a) for A = A 2 = A 3 =, D = D 2 = D 3 =, ω s = 3, ω s2 = 6 and ω s2 = 9 rad/s, b = [, 2] s and a = [., 4], a.; Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 4/37

15 One peak model Generalizing for N-peaks What happens with more realistic signals? THE N-PEAK DAMPING FACTORS ESTIMATION An oscillating" local maxima at a = ; Also, S N (b, a) have horizontal ridges at a = ω sn /ω sk, k n; k, n =, 2,..., N. Estimation of the damping factors: Consider the wavelet function have damping factors D n and the signal have D nn, D n D nn for any n. 2 Choosing one of the other local maxima a = ω sn /ω sk, and considering that the other factors are small enough in this scale, so only one term of (9) will be significative and we can estimate the damping factor as: d db ln S(b, ωsn ω sk ) d db ln e D kk b + d db ln A k A 2 n d db ln D2 n + [(ωs k + id kk ) ω sn ] 2 D kk. (2) Summarizing: if (a) peaks are far enough from each other and;(b) damping factors provide sharp peaks, the damping factors can estimated by D kk d ωsn ln S(b, db ω sk ). Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 5/37

16 One peak model Generalizing for N-peaks What happens with more realistic signals? WHAT HAPPENS WITH MORE REALISTIC SIGNALS? What happens when one analyzes limited, discretized, noisy and more realistic signals? Two procedures proposed: Create Discrete versions of signals and wavelets presented before and analyze them with Matlab and YAWtb Toolbox (Lorentzian Models); 2 Finally, create Discrete versions of signals and wavelets based on the metabolite database and analyze them with Matlab (Metabolite-based Models); Next, in the Numerical Analysis section. Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 6/37

17 NUMERICAL ANALYSIS WITH PEAK LORENTZIAN FUNCTION A MATLAB c function which implements Ψ(ω) = N A n n= D n+i(ω ω sn ) 2 were created (called LorentzNd.m") and added to YAWtb toolbox ([5]); A N component discrete exponential signal was defined by:, n ( N 4 ), x[n] = N n= A n e D (n N 4 )ts e iωsn (n N 4 )ts, ( N 4 ) n ( 3N 4 ),, ( 3N 4 ) n N, where t s = /f s is the sampling period in seconds. The CWT of this signal using the LorentzNd" function, with the same frequency and damping factors as the signals, for n =, 2, 3 was performed; Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 7/37

THE CWT RESULTS FOR COMPONENT Fig. CWT x [n] for A =, D =, ω s = 32 rad/s, n = [, 496] and a = [.5,.5] using the Lorentzd" wavelet. Fig. Skeleton of the CWT, for A =, D =, ω s = 32 rad/s, n = [, 496] and a = [.

18 THE CWT RESULTS FOR COMPONENT Fig. CWT x [n] for A =, D =, ω s = 32 rad/s, n = [, 496] and a = [.5,.5] using the Lorentzd" wavelet. Fig. Skeleton of the CWT, for A =, D =, ω s = 32 rad/s, n = [, 496] and a = [.5,.5] using the Lorentzd" wavelet. Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 8/37

19 THE CWT RESULTS FOR 2 COMPONENTS Fig. CWT x 2 [n] for A = A 2 =, D = D 2 = /s, ω s = 32 and ω s2 = 64 rad/s, t s = /256 s and N = 496 for Lorentz2Pkd" wavelet. Fig. Skeleton of CWT for A = A 2 =, D = D 2 = /s, ω s = 32 and ω s2 = 64 rad/s, t s = /256 s and N = 496 for Lorentz2Pkd" wavelet. Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 9/37

20 ...AND THE CWT RESULTS FOR 3 COMPONENTS: Fig. CWT x 3 [n] for A = A 2 = A 3 =, D = D 2 = D 3 = /s, ω s = 3, ω s2 = 6 and ω s2 = 9 rad/s, t s = /256 s, N = 496 and a = [.5, 5] for Lorentz3Pkd" wavelet. Fig. Skeleton of CWT for A = A 2 = A 3 =, D = D 2 = D 3 = /s, ω s = 3, ω s2 = 6 and ω s2 = 9 rad/s, t s = /256 s, N = 496 and a = [.5, 5] for Lorentz3Pkd" wavelet. Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 2/37

21 NUMERICAL ANALYSIS FOR SIGNALS WITH NOISE Here the goal was to find the limit of signal detection in presence of noise by means of autocorrelation wavelets ; White gaussian noise was added to signals at different SNR levels; 2 The CWT was performed (N peak was analyzed by its related N autocorrelation wavelet; 3 The results are in the limit of signal detection by its wavelet (still have the horizontal ridge); 4 The SNR of this limit were in average 2dB Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 2/37

22 SOME NOISE RESULTS Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 22/37

23 CREATING AND USING METABOLITE-BASED WAVELETS Numerical analysis with Metabolite-based wavelets: Take signals from one in vitro metabolite database; 2 Create its related Autocorrelation Wavelets; 3 Analyze a mixture of metabolites by one related autocorrelation wavelet with Matlab; Metabolite profile (Cre) 3 Cre The Algorithm: 25 Calculate autocorrelation function R of metabolite profile φ: Φ Cre (f) 2 5 R[n] = +N k= N φ[k] φ[n k] (2) f [khz].5 using Matlab function xcorr; 4 2 Subtract its mean value: ψ[n] = R[n] E{R[n]} (22) Re{φ Cre (t)} t [ms] Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 23/37

24 WAVELET CONSTRUCTION RESULT Metabolite profile (Cre) 3 Cre Cre-based wavelet x 4 Cre 25 8 Φ Cre (f) Ψ Cre (f) f [khz] f [khz] Re{φ Cre (t)} 2 Re{ψ Cre (t)} t [ms] t [ms] Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 24/37

25 HOW TO DILATE DISCRETE WAVELETS? PROBLEM: Mother wavelet is Discrete, no analytical expression; SOLUTION: An discrete upsampler/downsampler system was used. Fig. Block diagram of Upsampler/downsampler system. L, M (Z ). Wavelet will be expanded by an integer factor of L and contracted by an factor of M. a = L/M. Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 25/37

26 EXAMPLE OF DILATED WAVELET Cre-based wavelet (real part) at different :.5 scale a=.5 scale a=.2 Re{ψ Cre (t)}.5 Re{ψ Cre (t)} t [ms] 5 5 t [ms] Fig. scale a = Fig. scale a =.2 Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 26/37

27 SIGNAL : COMBINATION NAA + CRE WITHOUT LAC Sum of pure Naa and Cre signals, no noise: 2 x 7 Naa 2 x 7 Cre 2 x Y Naa (f) 6 Y Cre (f) 6 S(f) f [khz].5.5 f [khz].5.5 f [khz].5 Fig. Naa part Fig. Cre part Fig. Naa + Cre Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 27/37

28 SIGNAL 2: COMBINATION NAA + CRE + LAC Sum of pure Naa and Cre signals PLUS Lac, no noise: 2 x 7 2 x 7 Lac 2 x S(f) 6 Y Lac (f) 6 S(f) f [khz].5.5 f [khz].5.5 f [khz].5 Fig. Naa + Cre Fig. Lac part Fig. Naa + Cre + Lac Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 28/37

29 CWT ANALYSIS OF SIGNAL WITH NAA-BASED WAVELET CWT Naa: Max= 75.2 at scale=.99 and τ=47ms CWT Naa: Max= at scale= and τ=27ms horizontal ridges, thresh= horizontal ridges, thresh= Fig. Naa-wavelet CWT of Naa reference signal Fig. Naa-wavelet CWT of Naa+Cre composed signal Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 29/37

30 CWT ANALYSIS OF SIGNAL 2 WITH NAA-BASED WAVELET CWT Naa: Max= 75.2 at scale=.99 and τ=47ms CWT Naa: Max= at scale= and τ=27ms horizontal ridges, thresh= horizontal ridges, thresh= Fig. Naa-wavelet CWT of Naa reference signal Fig. Naa-wavelet CWT of Naa+Cre+Lac composed signal Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 3/37

31 CWT ANALYSIS OF SIGNAL WITH CRE-BASED WAVELET CWT Cre: Max= 63.4 at scale=.98 and τ=52ms CWT Cre: Max= at scale= and τ=6ms horizontal ridges, thresh= horizontal ridges, thresh= Fig. Cre-wavelet CWT of Cre reference signal Fig. Cre-wavelet CWT of Naa+Cre composed signal Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 3/37

32 CWT ANALYSIS OF SIGNAL 2 WITH CRE-BASED WAVELET CWT Cre: Max= 63.4 at scale=.98 and τ=52ms CWT Cre: Max= at scale= and τ=6ms horizontal ridges, thresh= horizontal ridges, thresh= Fig. Cre-wavelet CWT of Cre reference signal Fig. Cre-wavelet CWT of Naa+Cre+Lac composed signal Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 32/37

33 CWT ANALYSIS OF SIGNAL WITH LAC-BASED WAVELET CWT Lac: Max= 64.7 at scale=.99 and τ=5ms CWT Lac: Max= at scale=.2 and τ=6ms horizontal ridges, thresh= horizontal ridges, thresh= Fig. Lac-wavelet CWT of Lac reference signal Fig. Lac-wavelet CWT of Naa+Cre composed signal Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 33/37

34 CWT ANALYSIS OF SIGNAL 2 WITH LAC-BASED WAVELET CWT Lac: Max= 64.7 at scale=.99 and τ=5ms CWT Lac: Max= at scale=.2 and τ=3ms horizontal ridges, thresh= horizontal ridges, thresh= Fig. Lac-wavelet CWT of Lac reference signal Fig. Lac-wavelet CWT of Naa+Cre+Lac composed signal Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 34/37

35 THE CONCLUSIONS SO FAR Analytical analysis: Analytical expressions for Wavelet function and CWT, using Lorentzian models were made; 2 Horizontal ridges at a = means Presence of Metabolite which generated the autocorr.wavelet; 3 For peak, the Damping factor can be found. For more than one peak, the Damping factors can be approximated from horizontal ridges at a ; Numerical analysis with Lorentzian signals: Discretization and time limitation of signals changed a little the results ( lateral" Ridges are not smooth); 2 With multi-peaked signal/autocorr. wavelet, the damping estimation is harder; 3 Single signals could be detected at 2dB SNR; Numerical analysis with metabolite-based signals: Algorithms for autocorrelation wavelet creation, dilation and CWT were made; 2 The metabolites presence could be detected in the mixture by its related wavelet; 3 Parameters estimation is almost done (so wait a little bit more). Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 35/37

36 ACKNOWLEDGEMENTS The authors would like to thank Diana Sima, Maria Isabel Osorio Garcia and Sabine van Huffel from KUL; Adalberto Schuck Jr. would like to thank Conselho Nacional de Pesquisa e Desenvolvimento - CNPq, Brasil, for its financial support. Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 36/37

37 REFERENCES Introduction J-P. Antoine and R. Murenzi and P. Vandergheynst and S.T. Ali. Two-Dimensional Wavelets and their Relatives. Cambridge University Press, 24. A. Oppenheim and R.W. Schaffer with J. R. Buck. Discrete-time signal processing, 2nd Ed.. Prentice-Hall, Inc., 999. A. Papoulis and S. U. Pillai. Probability, random variables and stochastic processes, 4th ed.. McGraw-Hill, Inc., 22. A. Suvichakorn and H. Ratiney and A. Bucur and S. Cavassila and J-P. Antoine. Toward a quantitative analysis of in vivo proton magnetic resonance spectroscopic signals using the continuous Morlet wavelet transform. Meas. Sci. Technol., 2:429 44, 29. L. Jacques and A. Coron and P. Vandergheynst and A. Rivoldini. YAWTb toolbox : Yet Another Wavelet Toolbox Adalberto Schuck Jr. Analysis of MRS Signals with Autocorr. Wavelets 37/37

Joint ICTP-TWAS School on Coherent State Transforms, Time- Frequency and Time-Scale Analysis, Applications.

585- Joint ICTP-TWAS School on Coherent State Transforms, Time- Frequency and Time-Scale Analysis, Applications - June Wavelets and multiresolution : from NMR spectroscopy to motion analysis J-P. Antoine