Eigenvector-Based Spectral Enhancement of Nuclear Magnetic Resonance Profiles of Small Volumes from Human Brain Tissue

G. P. A B O U S L E M A N , R. J O R D A N , * and R. H. G R I F F E Y Department of Electrical and Computer Engineering (G.P.A., R.J.) and Center for Non-invasive Diagnosis (R.H.G.), University of New Mexico, Albuquerque, New Mexico 87131

Nuclear Magnetic Resonance (NMR) spectroscopy is a low-energy tech- nique which suffers from poor inherent signal-to-noise ratio (SNR). In a clinical setting, it is often desirable to study small regions of tissue in patients to aid in the detection and diagnosis of disease states. Analysis of the smaller regions, however, degrades the SNR further and renders conventional spectral estimation techniques such as the discrete Fourier transform useless. We demonstrate the utility of two complex eigenvec- tor-based algorithms, Multiple Signal Classification (MUSIC) and Min- imum Norm, in the detection of resonances within small sample volumes. The results indicate that these methods are clearly superior to Fourier transform-based techniques currently available on clinical N M R scan- ners. Index Headings: Nuclear magnetic resonance spectroscopy; Computer, applications; Data analysis.

INTRODUCTION

Proton spectroscopic evaluation of brain metabolism has demonstrated sensitivity in the identification of dis- ease states, including those for tumor, multiple sclerosis, and stroke. 1-s A major limitation to the clinical imple- mentation of the technique is the disparity between the small size of most lesions and the larger size of the sample volume (voxel) under study. Current single voxel and chemical shift imaging methods performed with a mag- netic field strength of 1.5 Tesla (T) utilize voxels of 4-8 cubic centimeters (cc) to ensure that adequate signal-to- noise ratios are obtained in approximately 15 rain of data acquisition. 4,5 Many cerebral lesions or areas of interest are an order of magnitude smaller than 8 cc. Hence, spectroscopic information from the lesion is averaged with the signal from surrounding normal tissue, which can compromise the information content of the exam.

Studies of smaller voxels now entail a significant time penalty even at higher magnetic field strengths2 Analysis of a 1-cc region would require 2-4 h of signal averaging to produce frequency spectra with a signal-to-noise ratio (SNR) comparable to the results now obtained from 4-cc voxels. This is impractical, given that the SNR in spectra from patients does not increase with longer ac- quisition times, because of motion and intolerance. The identification of spectral peaks from the smaller voxel may provide a differential diagnosis in many disease con- ditions. 7

One approach to determining discrete frequency com- ponents in NMR data is to model the frequency spectrum with the use of a set of parameters. Autoregressive (AR) parametric models have been applied to NMR spectros-

Received 15 Augus t 1990. * Author to whom correspondence should be sent.

copy. Their ability to provide good results with relatively small data samples and their freedom from windowing effects characterize them as superior to Fourier trans- form techniques. Multiple peaks can be represented with resolution that is theoretically infinite, making AR mod- eling an attractive alternative to other methods. ~14

The estimated parameters of the model determine the "goodness" of the resulting power spectrum. Techniques such as the Maximum Entropy Method (MEM or Burg's method), Yule-Walker methods, Bartlett 's method, and a host of others, as well as adaptive algorithms such as LMS/MLMS transversal (tapped delay) and LMS/ MLMS lattice, all make unique assumptions on the data record. While these methods have proven relatively suc- cessful in the analysis of NMR data, as compared to Fourier transform methods, they suffer from the inability to accurately track a power spectrum with a low SNR. Moreover, in vivo (living tissue) proton NMR spectro- scopic data have several defects which limit the utility of these methods. These include complex time-domain data, the large residual signal from unsuppressed water magnetization, the presence of multiplet structures from strong and weak scalar couplings, and variations in the linewidths of the peaks.

We discuss the practical implementation of two para- metric methods, Multiple Signal Classification (MUSIC) and Minimum Norm, for analysis of in vivo proton NMR spectra. Their potential utility and limitations are shown in studies of spectra from normal human brain tissue and a spectrum from a pituitary adenoma. The results suggest that these alternative processing methods can be used under extremely noisy conditions. Consequently, specific peaks may be identified in spectra from voxels smaller than 1 cc with data acquired in only 15 rain.

PRELIMINARIES

Classical methods of spectral estimation using Fourier transform or periodic operations on windowed data un- realistically assume that data outside the window are zero, resulting in a smeared spectral estimate.

Use of a priori information may permit the selection of an exact model or a close approximation for the process that generated the data. Thus, parametric modeling de- fines a three-step procedure: model selection, estimation of the assumed model from the available data samples, and generation of a spectral estimate by substitution of the estimated parameters into the theoretical power spectral density implied by the model. In this way, more realistic assumptions can be made concerning the nature of the measured process outside the measurement in- terval. --16

202 Volume 45, Number 2, 1991 0003-7028/91/4502-020252.00/0 © 1991 Society for Applied Spectroscopy

APPLIED SPECTROSCOPY

The general autoregressive moving average (ARMA) or pole-zero model is represented by Eq. 1, where un is a white noise driving sequence with zero mean and vari- ance ~ ; xn is the output sequence; n is the time index; k is the delay; and ah and bh are the model parameters:

P q

X n = - - ~ akXn_ h -4- ~ bkUn--k. (1) kffil k ~ 0

Taking the z-transform and rearranging terms yields the transfer function

B(z) H(z ) = - - (2)

A ( z ) "

The assumption that A(z ) has all its zeros within the unit circle on the complex plane will guarantee a stable H(z). 17

The power spectral density of the output sequence is defined as

where * denotes the complex conjugate operator. S~( z ) is then evaluated along the unit circle to obtain

B(o~) 2. S(o~) = S=(z) l~=¢~ = a~ A~w) (4)

Hence, specification of the parameters ah, bh, and a~ is equivalent to specifying the PSD of the process x,. The autoregressive moving average model is of the pole-zero variety and is denoted ARMA(p, q).

If all ah coefficients except a0 = 1 vanish for ARMA parameters, the moving average or all-zero model is ob- tained as shown below:

q

x , = ~ bkUn-h. (5) hffi0

The moving average model is denoted MA(q). The power spectral density of the MA process is

SMA(~0) = ~b IB(w) 12. (6)

If all bk coefficients except bo = 1 vanish for ARMA parameters, then the autoregressive model is obtained

P

xn = - ~ akX,-h + Un. (7) kffil

Equation 7 is termed all-pole or autoregressive since the sequence x, is a linear regression on itself with u, rep- resenting the error. The present value of the process is expressed as a weighted sum of past values plus a noise term. The model is denoted AR(p). The power spectral density of the AR process is

~b S~(w) = I A(w) 12 (8)

Model selection is critical in any parametric modeling application. A model must be chosen to represent spec- tral peaks, valleys, and roll-offs. For spectra with sharp peaks it is necessary to employ a model which has poles,

e.g., AR or ARMA. For spectra with sharp valleys a model with zeros may be used, e.g., MA or ARMA.

In proton spectroscopic studies it is known that the power spectrum will consist of sharp peaks representing distinct chemicals within a tissue sample. These spectral features would seem to dictate the use of an all-pole model. Also, since the noise variance a~ of the data set may not be known and our interest lies in the relative magnitudes of the peaks in the resulting spectrum, this term may equal one. Our interest will then lie in the determination of an all-pole power spectrum estimate written as

~(w) = 1

I A(z ) I ~ I ~=¢~. (9)

In the following sections, the parameters ak constituting the power spectrum estimate will be determined with the use of two complex eigenvector-based techniques, Mul- tiple Signal Classification (MUSIC) and Minimum Norm. It will be shown that these algorithms are capable of producing unbiased spectrum estimates with infinite res- olution regardless of the signal-to-noise ratio.

E I G E N D E C O M P O S I T I O N

Multiple Signal Classification and Minimum Norm are two parametric modeling techniques used to detect si- nusoidal components in noisy environments. These methods are based on eigendecomposition of the auto- correlation matrix of sampled random process. 11,18-21 Con- sider L complex sinusoids buried in white additive noise vn with zero mean and variance a~ such that

L

Yn = 1)n -4- ~ Aiexp( jwin + J¢i). (10) i=1

The ¢i are uniformly distributed and independent of the noise and of each other. Thus

E[v~*Vm] = O'~2~nm E[q~iPn] = 0. (11)

The autocorrelation of y . is L

r(k) = E[yn+kY~*] = a~2~(k) + ~ Piexp( jwik) (12) i=1

where the P~ = ]Ai ] 2 denote the power levels. The objective is to find a polynomial A(z ) = ao + alz -1

+ . . . + aMZ TM of order M _> L with zeros lying close to or on the unit circle corresponding to the L sinusoids. It will be shown that A(z ) may be determined on the basis of the autocorrelation matrix Ryy. Let the phase vector and weight vector be defined as follows:

| | [ elj~ ] I a l

S,, = |e2Jo,[ a = al . (13)

keJ~Mj a

The sinusoidal and diagonal power matrices are written respectively as

S = [S~lS~2 • • • SoL] P = diag{P1P2 . . . PL}. (14)

The polynomial corresponding to the ith sinusoid is de- noted as

APPLIED SPECTROSCOPY 203

where

A(zi ) = A(o~i) = O, i = 1, 2 . . . . . L

A(o~) = A(z) Iz=¢w (15)

r(k, m) = r(k - m) = a~6(k - m) (16) L

+ ~ P iexp( jw i (k - m) ) i = l

which in matrix form becomes L

Ryy = a~I + ~ Pis,~,s,~i + (17) i=1

o r

Rry = a~I + S P S + (18)

where S ÷ is the conjugate transpose of S. Rry is Hermitian and Toeplitz since we assume the data to be stationary; i.e., the autocorrelation may be dependent only on the time difference. Since A(00) = s~-a, Eq. 15 may be writ- ten as

[-A(,o,) l

Multiplying both sides of Eq. 18 by a yields

Rrya = a~a + SPS+a = Zv2a. (20)

Thus, Ryya = a~ a implies that a~ is an eigenvalue of Ryy with a being the corresponding eigenvector. It can be shown that a~ is the smallest eigenvalue. Consider any other eigenvector b of Rry and normalize it to unit norm. That is,

Ryyb = ~b with b + b = 1. (21)

Multiplying Eq. 21 on the left by b ÷ gives

b+Ryyb = b+~b = ~b+b = h. (22)

Substituting for R~ as given in Eq. 18 yields

X = b+Ryyb = a~2b+b + b+SPS+b = a~ 2 + b+SPS+b (23)

which can be written as

[a(O l)l h = ~ + [B*(o~t)B*(w2)...B*(~OL)]PIB(°~2)[" " (24)

LS(!=aj or

L

X = a~ + ~ Pi lB(w,) l 2. (25) i=1

Any X is equal to the noise variance shifted by a non- negative amount. If the eigenvector satisfies A(w~), i.e., X = a~, then the shift in h vanishes. Hence, there are M + 1 - L degenerate eigenvalues and L eigenvalues great- er than a~.

Thus, the eigenspace of Ryy consists of the (M -t- 1 - L)-dimensional noise subspace spanned by the eigen-

vectors corresponding to the minimum eigenvalue a~, and the L-dimensional signal subspace spanned by the re- maining L eigenvectors having eigenvalues greater than a~. Also, the first (M -t- 1 - L) eigenvectors form an orthonormal basis for the noise subspace, and the last L eigenvectors form a basis for the signal subspace. That is,

E = [EN, Es] = [e0, e l , . . . , eh_l, ek . . . . , eM] (26)

where e~, i = 0, 1 . . . . , M are the eigenvectors of Ryr in increasing eigenvalue order, and K = M + 1 - L is the dimension of the noise subspace.

Several observations are noted. Since a signal block of analysis data is available, ensemble statistics are re- placed with time averages. Thus, a biased estimate of the autocorrelation matrix Ryy is calculated by letting

1 N - 1 - K

f ( k ) = ~ ~ yn+hY,*, k = O, l , . . . , M (27) n=0

and forming the (M .1. 1) × (M -t- 1) sample autocor- relation matrix Ryy with the assumptions of Toeplitz and Hermitian characteristics. Also, since Ryy is an estimate, the (M -t- 1 - L) eigenvalues will not necessarily be equal but will be approximately so relative to the larger eigen- values of the signal subspace. Finally, choosing the ei- genvector A(z~) corresponding to the minimum eigen- value and forming the spectrum estimate

= 1 (28) [A(zi) 12

is simply Pisarenko's method of harmonic retrieval. We now discuss two eigenvector-based methods for deter- mining the polynomial A(z ) .

MULTIPLE SIGNAL CLASSIFICATION (MUSIC)

Multiple Signal Classification (MUSIC) considers all eigenvectors of the noise subspace to form the PSD es- timate without effects of spurious zeros. 11,18,19,21 Consider the eigenfilters El(z ) and the corresponding coefficient vector el of the noise subspace

Ei(z), ei, i = 0, 1 . . . . . k - 1. (29)

We can write El(Z) = A i ( z )F i ( z ) where Ai(z ) is the re- duced-order polynomial whose zeros occur at the desired frequency values. Fi(z) contains spurious zeros. From the previous discussion, each Ei(z ) has a common set of L zeros at the desired spectral locations. However, each has a different set of k - 1 spurious zeros. The problem is the possibility of the spurious zeros lying close to or on the unit circle. If only one eigenfilter is used, it may not be possible to distinguish between the spurious and de- sired zeros.

MUSIC attempts to smear the effects of spurious zeros by averaging the magnitude responses of the k eigenfil- ters. We write

1 k--1 2 1

h--1

~ IE,(z)l 2= IA(z) ~ I F,(z) 12 (30) "= i = 0

and note that, since the polynomials F~(z) are dissimilar, the averaging operation will tend to smear spectral con- tributions of spurious seros of any individual term. The MUSIC spectrum is defined as

204 Volume 45, Number 2, 1991

1 SMUSIC(O0) = 1 k - - 1 (31)

~ ] Ei(6o) [ 2 i=o

M I N I M U M NORM

The Minimum Norm method attempts to eliminate the effects of spurious zeros by pushing them inside the unit circle. ",1s,2°,2~ A vector d = [dodl • • • riM] r representing the coefficients of a polynomial D(z) must be found such that its zeros represent the desired spectral values. The first coefficient do is constrained to be unity and the sum

M

I dh 12 -- d+d (32) k=0

is minimized. The M - L extraneous zeros will be forced inside the unit circle. The polynomial D(z) can be fac- tored into two polynomials, A(z) and F(z):

D(z) = A(z)F(z) (33)

where

L

A(z) = ~ akz -h ao = 1 k=O M - - L

f ( z ) = ~ h z -h bo = 1. k=O

The signal and spurious zeros are contained in A(z) and F(z), respectively. Minimizing d÷d is equivalent to min- imizing

f lD[exp(j~o)]12d~o. (34)

This can be seen as a problem of finding F(z) to minimize the above quantity, given A(z) , which is simply the au- tocorrelation method of linear prediction. 22,23 It is shown by Kumaresan 24 that F(z) has zeros strictly inside the unit circle independent of the zeros of A(z) . Further, it is also shown that the spurious zeros are approximately uniformly distributed about the unit circle in regions separated from the signal zeros. Consequently, the spu- rious zeros are less likely to produce false peaks in the angular spectrum.

Let A(z) and F(z) be defined, respectively, as

A(z) = 1 + alz -1 + . . . + aLz -L (35) F(z) = /o + / l Z -1+ . . . + f x - l z -¢g-~). (36)

If the K delayed polynomials are

B~(z) = z-~A(z), i = 0, 1, . . . , K - 1, (37)

Eq. 33 may be written as

D(z) = [oBo(z) + [lB,(z) + . . . + f~:-~BK_i(z) (38)

or in coefficient form

e = ~ f i b i = [ b o b l . . . b h _ ~ ] fl = B f = d (39) i=O

--1

where B is a nonorthogonal basis of the noise subspace since each polynomial Bi(z) has L desired zeros, and the corresponding vector bi will lie in the noise subspace. The coefficients f of any noise subspace vector e are the coefficients of the polynomial F(z) . The (M + 1)-dimen- sional basis vectors bi are delayed versions of the coef- ficient vector a. The basis B must be linearly related to the orthonormal basis EN by B = ENC, where C is a (K x K) invertible matrix. Since f are the coefficients of F(z), the constraint equation becomes u~-Bf = 1 where u~ = [1, 0 . . . . . 0]. Thus,

uJ-B = [u~ b0, U~bl . . . . . Uo+bk_l] = [1 , 0 . . . . . 0 ] = U +. (40)

In the B basis, Eq. 32 becomes

d+d = f+Raaf = m i n u+f = 1 (41)

where Raa = B+B. This is a Toeplitz matrix of autocor- relations of the filter a. Equation 41 represents an or- dinary linear prediction problem and its solution f will be a filter with all its zeros within the unit circle, e.g., minimum phase. The solution is

f = Raa-lu = (B+B)-lu. (42)

If u = B+uo then

f = (B+B)-IB+uo. (43)

It follows that

d = B f = B(B+B)-IB+u o = ENEN+Uo . (44)

Noting that EN+U0 is the complex conjugate of the top row of EN, we obtain

k - - 1

d = ~ * Eoiez, (45) i~0

which is used to form the Minimum Norm spectrum estimate:

1 S M I N O R M - - - - (46)

isZdl 2"

EXPERIMENTAL METHOD

A simulation is first presented to demonstrate the re- solving properties of the MUSIC and Minimum Norm algorithms in environments having an extremely low sig- nal-to-noise ratio. The file contains 512 data points with a sampling frequency of 128 Hz. Additive white noise is present in sufficient quantity to ensure that the SNR is -15 dB.

Proton NMR spectra of normal human brain and tu- mor tissue were acquired on a General Electric Medical Systems Signa scanner with a magnetic field strength of 1.5 Tesla. All in vivo spectra were obtained with the use of the VAPOR technique based on acquisition of a lo- calized stimulation echo. 4 A repetition time of 1.5 s was used in all cases. A total of 1024 complex points were digitized with a sweep width of 1000 Hz. NMR spectra were multiplied by an exponential function equivalent to 2 Hz of line broadening prior to discrete Fourier trans- formation. MUSIC and Minimum Norm frequency pro- files were generated in under three minutes on a DEC

APPLIED SPECTROSCOPY 205

3100 workstation. The square root of the calculated pow- er spectrum has been displayed for each profile. 24

RESULTS AND DISCUSSION

Figure 1A represents the DFT of 2 equal-powered si- nusoidal components at 20 and 21 Hz. The sinusoids can be identified at their appropriate frequencies but with little confidence due to the many false noise components.

The Minimum Norm profile of the same data is shown in Fig. lB. The sinusoids are reproduced at the correct frequencies, and the spectral contributions from the noise have been reduced dramatically.

The frequency spectrum produced by the MUSIC al- gorithm is shown in Fig. 1C. Spectral components of noise have been nullified entirely by this method, while the peaks corresponding to the two sinusoids are present at the correct frequencies.

The DFT of the data record from a 1.7-cc region of grey matter in human brain tissue is shown in Fig. 2A. A total of 512 averages were summed over 12 min, and the resulting finite induction decay (FID) was treated with 2 Hz of exponential broadening prior to transfor- mation. Small signals from, possibly, choline (Cho) meth- yls and phosphocreatine/creatine (PCr/Cr) are seen at - 9 1 Hz (3.25 ppm) and -104 Hz (3.03 ppm). A large peak from N-acetylaspartate (NAA) is observed at - 169 Hz (2.06 ppm). Several other spectral features are sug- gestive of signals, but are not resolved clearly above the noise.

Application of the Minimum Norm method to the data record using 67 weights and 512 points produces the frequency distribution shown in Fig. 2B. Resolved signals are observed at - 40, - 63, - 78, - 91, - 104, - 129, - 146, -169, -226, and -252 Hz (4.01, 3.84, 3.61, 3.25, 3.03, 2.75, 2.5, 2.06, 1.35, and 0.9 ppm). These resonances have been observed previously in spectra from 8-64 cc regions of human brain tissue. The degree of distortion in the relative amplitudes of the observed peaks is difficult to assess, since the metabolic contents of such a "pure" region of grey matter are not well characterized.

The MUSIC technique generates the spectrum shown in Fig. 2C. The frequency components are identical to those of Minimum Norm, with some improvement in the resolution of the peak from glutamate at -147 Hz.

The DFT of data from a 24-cc region of pathology- proven pituitary adenoma (tumor) is shown in Fig. 3A. A total of 256 averages were summed and treated with 2 Hz of exponential broadening prior to Fourier transfor- mation. The spectrum contains signals from lactate ( - 4 0 Hz, 4.08 ppm), C-alpha protons of amino acids ( - 6 5 Hz and - 7 4 Hz, 3.63 ppm and 3.59 ppm), choline ( - 9 6 Hz, 3.26 ppm), and PCr and/or Cr ( -110 Hz, 3.06 ppm). In addition, there is a large peak from glutamate ( - 160 Hz, 2.14 ppm). Other resonances are lactate at - 226 Hz (1.3- 1.6 ppm) and possibly methyl groups of lipid or choles- terol esters at - 252 Hz (0.9 ppm). The SNR for the Cho resonance is ~25. Since SNR is not an issue, the DFT should yield an accurate spectrum. The intent is to show that, with good SNR, the MUSIC and Minimum Norm algorithms produce excellent approximations of the DFT spectrum.

The spectral pattern of the same frequency region ob-

A

Frequency (Hz)

Frequency (Hz)

64

64

0 64 Frequency (Hz)

FIG. 1. Frequency profile of two sinusoids buried in additive white Gaussian noise. (A) Discrete Fourier transform. (B) Minimum Norm profile using 99 weights and 512 points. (C) MUSIC profile using 99 weights and 512 points.

tained by the Minimum Norm technique is shown in Fig. 3B. A total of 99 weights were employed, and 512 points were calculated. All of the spectral features are repro- duced accurately by this method, including the relative amplitudes. The MUSIC analysis of the same data record with the use of 99 weights generates a similar spectral pattern, as seen in Fig. 3C. Note that the two spectra are identical, which suggests that, with increasing SNR, the two algorithms converge to the same result.

206 Volume 45, Number 2, 1991

A

' ' I ' ' ' ' I ' ' ' ' I . . . . I . . . .

- 1 0 0 - - 2 0 0 - - 3 0 0 - 4 0 0 - 5 0 0 H z

' ' ' ' I ' ' ' ' I ' ' ' ' I ' ' ' '

0 - - 1 0 0 - - 2 0 0 - 3 0 0

I i i i i

- 4 0 0 - 5 0 0 H z

' ' ' ' I ' ' ' ' I ' ' ' ' I ' ' ' ' I ' ' ' '

- 1 0 0 - 2 0 0 - 3 0 0 - 4 0 0 - 5 0 0 H z

FIG. 2. Frequency profile of a 1.7-cc voxel from normal human grey matter. (A) Discrete Fourier transform of 512 averages with exponential filtering. (B) Minimum norm profile using 67 weights and 512 points. (C) MUSIC profile using 67 weights and 512 points.

C O N C L U S I O N S

The clinical appl ica t ion of p ro ton M R spect roscopy cont inues to be l imited by the large size of the voxel required to produce a spec t rum with adequa te signal-to- noise. 21,25 Wi th a field s t rength of 1.5 T, t issue volumes of 4 cc are the smal les t t ha t can be s tudied because of decreased SNR. Studies of l -cc volumes would require a lmos t 4 h of signal averaging to obtain an S N R com- parable to t ha t of 4-cc regions. In a clinical setting, a spectroscopic s tudy m u s t be comple ted in 15-25 min to be pract ical as an " a d d - o n " to an imaging exam. This l imi ta t ion can be overcome by analyzing the complex t ime -doma in da ta with an e igenvector-based technique.

. . . . I ' ' ' ' 1 ' ' ' ' 1 ' ' ' ' 1 ' ' ' '

0 - 1 0 0 - 2 0 0 - 3 0 0 - 4 0 0 - 5 0 0 H z

. . . . I ' ' ' ' I ' ' ' ' I ' ' ' ' I ' ' ' '

0 - 1 0 0 - 2 0 0 - 3 0 0 - 4 0 0 - 5 0 0 H z

I . . . . I . . . . I . . . . I . . . . B ' ' ' '

0 - 1 0 0 - 2 0 0 - 3 0 0 - 4 0 0 - 5 0 0 H z

FIG. 3. Frequency profile of a 24-cc voxel from a pituitary adenoma. (A) Discrete Fourier transform of 256 averages with exponential fil- tering. (B) Minimum Norm profile using 99 weights and 512 points. (C) MUSIC profile using 99 weights and 512 points.

T h e eigenvector me thods accurate ly describe the am- pl i tudes and frequencies of the signals in complex da ta records with a signal-to-noise of 10 dB. In addit ion, the major f requency componen t s within a da ta set can be identified even when the inherent S N R falls below 0 dB. As shown in the profile f rom the 1.7-cc voxel of h u m a n brain tissue, spectra l componen t s other t han the large resonance f rom NAA are lost in the noise. In compar ison, the eigenvector me thods genera te a spec t rum in which signals f rom other cerebral metabol i tes such as choline esters, g lu tamate , and sugars can be identified. The MU- SIC and M i n i m u m N o r m algor i thms theoret ical ly pro-

APPLIED SPECTROSCOPY 207

duce a frequency record from infinitely low SNR envi- ronments.

These characteristics were dependent on the assump- tion that the true autocorrelation matrix Ryy was avail- able. However, since ensemble statistics were unavail- able, the sample autocorrelation was calculated with the use of a single data set, and an autocorrelation matrix estimate Ryy was formed, under the assumption of Her- mitian and Toeplitz characteristics.

When the autocorrelation matrix is estimated from a finite number of independent data samples, the eigen- vector methods exhibit deviations from the true signal zeros, resulting in a loss of resolution. This is due to the statistical sampling perturbation of the signal and noise subspaces. The perturbation may be dependent upon signal-to-noise ratios, signal characteristics, data set length, and model order. Combined, these parameters determine the resolving capability of either eigenvector methodJ la8,26-28 The "practical" SNR is determined by the appearance of extraneous peaks in the frequency record. We have not observed such peaks using optimized eigenvector methods, although the amplitudes of some resonances may be distorted. Some broad spectral com- ponents may be lost, however, when the signal-to-noise is very low.

Model order selection is an important step to suc- cessful application of the eigenvector techniques. As the number of frequency components increases in low SNR environments, the model order will tend to increase. This is intuitively necessary, as the polynomial A(z) men- tioned previously must be of sufficient order to represent the zeros of the signal and noise components. The num- ber of weights can roughly be determined as lying be- tween N/8 and N/4 where N is the number of data sam- ples. This range was found exper imenta l ly to accommodate a wide range of NMR spectra and may therefore provide a starting point for other studies.

Two practical problems with implementing the eigen- vector methods should be noted. First, the ability of the technique to find small signals is diminished by the pres- ence of a large resonance from water. These round-off errors can be avoided through suppression of the water peak by a factor of 750 or more with the use of a spec- troscopic pulse technique such as VAPOR? Second, the resonances displayed from the eigenvector analysis do not contain phase information. Thus, care must be taken in the interpretation of the frequency data, since phase anomalies which reflect poor spatial localization of im- perfections in system hardware may be missed.

Finally, it is stressed that no preprocessing of the data was performed prior to the application of the MUSIC and Minimum Norm algorithms. Cascading of noise can- cellation techniques with the eigenvector methods may enhance detection of discrete frequency components in lower SNR environments (smaller voxel sizes).

Application of eigenvector analysis to 3,p, 13C, and 19F NMR data should provide a similar improvement to these spectroscopic procedures.

ACKNOWLEDGMENTS

We wish to thank the State of New Mexico for supporting the Center for Non-invasive Diagnosis. The assistance of Jeremy Worley and Don- na Koechner is gratefully acknowledged. A special thanks goes to Dr. N. A. Matwiyoff, Director of the Center for Non-invasive Diagnosis.

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