Blind Deconvolution of Backscattered Ultrasound Using Second-Order Statistics

Blind Deconvolution of Backscattered Ultrasound using Second-Order Statistics

Carlos E. Davila and Tao He

Electrical Engineering Department, Southern Methodist University Dallas, Texas 75275-0338

E-mail: [email protected] Phone: (214) 768-3197 Fax: (214) 768-3573

Abstract

A method for blind deconvolution of ultrasound based on second-order statistics is proposed. This approach relaxes the requirement made by other blind deconvolution methods that the tissue reflectivity be statistically white or broad-band. Two methods are described, the first is a basic second-order deconvolution method, while the second incorporates an optimally weighted least squares solution. Experimental results using synthetic and actual ultrasound data demonstrate the potential benefits of this method.

Submitted to IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, July 23, 2001 1 Background

Ultrasound is used extensively in medicine as a relatively inexpensive approach for non- invasively imaging organ structures and measuring blood flow. The basic premise behind ultrasonic imaging is that an ultrasonic pulse transmitted through tissue will be reflected or backscattered, particularly at distinct tissue boundaries. The underlying tissue structures can then be imaged by measuring the reflections of the transmitted pulse at each tissue boundary. While models for ultrasonic backscattering can be quite complex, a simple model that is often used is based on a linear convolution

ZZZ µ ¶ e−2az 2z r(t)= g(x, y, z)s(x, y)p t − dxdydz (1) z c where s(x, y) is the lateral distribution of the transmitted ultrasound wave, c is the speed of sound through the tissue (assumed constant), a is the average attenuation constant of the tissue in the ultrasound propagation path, and g(x, y, z) is the tissue reflectivity. The variables x, y, and z are 3-dimensional spatial position variables, and t is time. This convolutional model is valid in a weakly scattering medium (no secondary or higher-order reflections) consisting of anisotropic scatterers whose size is small relative to the wavelength of the ultrasound [1, 2, 3, 4, 5]. Lateral distortion in the (x, y) plane can be reduced by using a finely focused beam generated by a phased array of transducers [6, 7, 8, 9]; moreover distance dependent attenuation can be compensated by multiplying the received reflected signal by cte2act, or more typically, by passing the received signal through a static log-like nonlinearity

[10]. Assuming that lateral distortion and attenuation have been compensated, the reﬂected

1 pulse can be expressed as a one-dimensional convolution

Z µ ¶ 2z r(t)= g(z)p t − dz (2) c

The resolution along the axial direction is limited by the spatial spread of the transmitted pulse; if the pulse were a true delta function, then the reflected pulse would correspond exactly to the tissue reflectivity. In practice the transmitted pulse deviates from an ideal impulse, and closely spaced tissue boundaries will not be resolved. For this reason deconvolution methods have been used to improve the axial resolution of ultrasound scans. If the transmitted pulse is known or can be easily measured, then a straight-forward least squares or Wiener filter-based estimation can be used to solve for the tissue reflectivity. A number of deconvolution approaches based on the assumption that the pulse is known have been published [1, 11, 12, 13, 14, 15]. In practice, the transmitted pulse is difficult to measure since this would require a second transducer. Even if a second transducer were present, it is not clear where this transducer could be placed in order to accurately measure the transmitted pulse. The pulse could be measured off-line using a target designed to behave like point reflector, however this would fail to incorporate temporal variations in the pulse shape, and the target may not adequately model a true point reflector. For this reason most recent approaches to ultrasound deconvolution have been based on so-called “blind” methods, wherein the tissue reflectivity is estimated without any knowledge of the ultrasonic pulse or probing signal. If the transmitted pulse is minimum phase, and the tissue reflectivity is modeled as white noise, then it is possible to use the second-order statistics of the reflected pulse to estimate the pulse signal [16, 5]. Given an estimate of the pulse, a Wiener filter or any one of the aforementioned deconvolution methods can subsequently be used to estimate the

2 tissue reﬂectivity. Unfortunately, the transmitted pulse is often not minimum phase [17, 18].

Several authors have used homomorphic deconvolution, wherein the complex cepstrum of

the reﬂected ultrasound is used to estimate the transmitted pulse [19, 17, 20, 21]. Again,

the pulse estimation step must be followed by a conventional deconvolution step. The com-

plex cepstrum used in homomorphic deconvolution is diﬃcult to measure if the transmitted

pulse has zeros close to the unit circle [22]. In this case, it is necessary to multiply the

received signal by an exponential function in order to migrate unstable zeros away from the unit circle [22]. Moreover, the homomorphic deconvolution approach also assumes that the tissue reflectivity can be modeled as wide-band (white) noise. Other investigators have used higher order statistics to estimate the transmitted ultrasound pulse [18, 23]. This approach assumes the tissue reflectivity can be modeled as white, non-Gaussian noise. In particular, the reflectivity kurtosis should be an impulse [24, 25, 26]. While it appears likely that tissue reflectivities are non-Gaussian [18], there do not appear to be any definitive studies which suggest that reflectivity can be modeled as white noise. Moreover, higher-order methods require a large amount of data to obtain reasonable estimates of the higher-order statistics and are computationally intensive. Both homomorphic and higher-order statistics deconvolution methods assume the tissue reflectivity can be modeled as white noise. Any deviation from this assumption is likely to have a detrimental effect on the quality of the deconvolution. The assumption that tissue reflectivity can be modeled as white noise appears to be questionable for a large class of tissue structures that undergo continuous (rather than abrupt) changes in acoustic impedance. The relationship between acoustic impedance h(z) and reflectivity has been studied in [27] and is given by

1 δh(z) g(z)= (3) 4z δz

3 This relationship takes into account higher-order reﬂections, which are usually ignored in convolutional models of backscattering. The relationship in (3) predicts that tissues which undergo a continuous variation in acoustic impedance with depth are unlikely to have white noise reﬂectivity. Another complicating factor with the use of homomorphic and higher order deconvolution methods is the assumption that the transmitted ultrasound pulse has a time invariant shape. It is well known that the pulse shape changes with depth due to frequency dependent attenuation [3].

In this paper, a method for ultrasound deconvolution based on second-order statistics is described. This approach offers several advantages over the two blind deconvolution methods mentioned above. First, no assumptions are made about the statistical nature of the tissue reflectivity. In theory, the reflectivity can be either broad-band or narrow-band without af- fecting the viability of the deconvolution. Second-order statistics can also be estimated more accurately and with fewer computations than higher order statistics. Second-order deconvolution methods are not sensitive to the location of the zeros of the transmitted ultrasound, so no exponential weighting or other kinds of preprocessing are necessary. Another appealing characteristic of this method is that it estimates the tissue reflectivity in a single step. In other words, it is not necessary to first estimate the transmitted ultrasound pulse prior to doing the deconvolution step. The deconvolution step is subject to errors, even when the pulse estimation is good due to the presence of noise in the reflected signal. The method to be described is closely related to second-order blind deconvolution of communications channels [28, 29, 30]. However since the transmitted ultrasound has finite duration, the equations which model the autocorrelation matrix of the reflected ultrasound do not possess multiple

Toeplitz matrices which can lead to sensitivity problems in blind second-order deconvolution of communications channels [31, 32, 33].

4 2 Second-Order Blind Deconvolution

If the spatial variable z and the time variable t in (2) are discretized, then the convolution integral can be expressed as a standard discrete-time convolution

r(n)=g(n) ?p(n)+v(n) (4)

where:

• g(n): tissue reﬂectivity to be estimated, length K.

• p(n): transmitted ultrasound, length L,

• r(n): reﬂected pulse, length N ≡ K + L − 1.

• v(n): measurement and modelling error, length N.

In matrix form, r(n) can be expressed as r = Gp where

· ¸T r = r(0) r(1) ··· r(N − 1) (5)

· ¸T p = p(0) p(1) ··· p(L − 1) (6)

· ¸T v = v(0) v(1) ··· v(N − 1) (7)

5 and    g(0) 0 ··· 0         g(1) g(0) ··· 0       . . . .   . . .. .         g(L − 1) g(L − 2) ··· g(0)       . . . .  G =  . . .. .  (8)        − − ··· −   g(K 1) g(K 2) g(K L)       − ··· −   0 g(K 1) g(K L +1)     . . .   ......   . . .    00··· g(K − 1)

The matrix G has dimension N × L. The transmitted ultrasound p(n) is assumed to be £ ¤ T persistently exciting, meaning that Rp ≡ E pp has full rank [34]. This implies that

the transmitted ultrasound should consist of a burst of wideband ultrasound rather than

2 the usual pulse wavelet. If v(n) is assumed to be white noise with variance σv , then the

autocorrelation matrix of r is given by

£ ¤ T T 2 Rr ≡ E rr = GRpG + σv IN (9)

T The matrix GRpG has rank L, and hence has an (N −L)-dimensional nullpsace. This makes

T it possible to use subspace methods to estimate g(n). The key to GRpG being singular is

T that p(n) has ﬁnite duration, if the stimulus signal has inﬁnite duration, then GRpG will

be nonsingular and second-order methods can not be used for estimating g(n) (unless p(n)

is minimum phase).

6 T Let the N × (K − 1) matrix VN be a basis for the nullspace (or “noise subspace”) of GRpG .

T Since the eigenvalues of GRpG are non-negative, the columns of VN can be taken to be

the eigenvectors of Rr associated with the N − L minimum repeated eigenvalues equal to

2 σv . The columns of G span the L-dimensional “signal subspace” which is the orthogonal complement of the noise subspace (a detailed description of subspace methods is found in

[16]). Clearly

T VN G =0(K−1)×L (10)

where the subscript indicates the dimension of the zero matrix. It is straight-forward to

T show that the matrix G is unique. If VN is partitioned as

· ¸ T T T T VN = VN,1 VN,2 ··· VN,L (11)

where each VN,i,i=1,...,L has dimension K × (K − 1). It follows that

T VN,ig =0K−1,i=1,...,L (12)

where · ¸T g = g(0) g(1) ··· g(K − 1) (13)

The equations in (12) can then be used to solve for g(n). In practice, Rr must be estimated

using measurement vectors given by

· ¸T

ri = ri(0) ri(1) ··· ri(N − 1) ,i=1,...,M (14)

7 where

ri(n)=g(n) ?pi(n)+vi(n),i=1,...,M (15)

The ri are response vectors corresponding to M independent transmitted pulse sequences, i.e., if pi is the ith transmitted pulse sequence:

· ¸T

pi = pi(0) pi(1) ··· pi(L − 1) (16)

£ ¤ T then E pipj =0(L×L),i=6 j. The matrix Rr can then be estimated as

XM 1 T Rˆr = riri (17) M i=1

Then g(n) can be estimated by solving in a least squares sense, the equations

T VˆN,igˆ ≈ 0K−1,i=1,...,L (18)

where the VˆN,i are partitions of the noise subspace eigenvector matrix VˆN derived from Rˆr

(see (11)) and · ¸T gˆ = gˆ(0)g ˆ(1) ··· gˆ(K − 1) (19)

One possibility is to constrain the norm ofg ˆ to be one, and solve for the eigenvector associated with the minimum eigenvalue of XL T RV = VˆN,iVˆN,i (20) k=1

The algorithm is summarized as follows:

1. Collect independent response vectors ri,i=1,...,M.

8 2. Compute XM 1 T Rˆr = riri M i=1

3. Find the eigendecomposition of Rˆr and set the columns of VˆN equal to the eigenvectors

associated with the smallest N − L eigenvalues of Rˆr.

4. Partition VˆN into VˆN,i,i=1,...,L as deﬁned in (11).

5. Compute XL T RV = VˆN,iVˆN,i k=1

6. Setg ˆ equal to the eigenvector associated with the minimum eigenvalue of RV .

This approach to ultrasound deconvolution is very similar to second-order methods which have been developed for blind deconvolution of communications channels [28, 29, 30]. The main difference between the two approaches is that for the communications channel case, the excitation signal p(n) (consisting of a stream of transmitted information symbols) has infinite duration. Therefore, second-order methods for blind deconvolution only work when the received signal is sampled at a rate higher than the symbol rate. This so-called “fractionally spaced sampling” leads to received signal autocorrelation matrices having a structure similar to Rr (see (9)), with the exception that the matrix G has block rows consisting of different Toeplitz matrices. The presence of more than one Toeplitz matrix in G for the fractionally sampled case can lead to robustness problems [31, 32, 33]. The method described here however is not subject to these robustness problems owing to the presence of only a single Toeplitz matrix in G. The accuracy of the proposed method ultimately depends on the accuracy of the estimated eigenvectors of Rr. This accuracy is linked to the condition number of the matrix Rp, the lower the condition number, the more accurate the eigenvector

9 estimates [35]. Hence, it is important that the excitation signal p(n) have an autocorrelation matrix which has a low condition number. This is best achieved by exciting the transducer with a white noise source. Since wideband ultrasound travelling through tissue will have a frequency-dependent attenuation, with higher frequency components attenuating more than the lower frequencies, the present method will give reﬂectivity estimates which are increas- ingly lowpass with depth. In other words, temporal nonstationarities in the transmitted ultrasound p(n) can be incorporated into rows of G. Estimated reﬂectivities can be compen-

sated by post ﬁltering with an appropriately designed time-varying highpass ﬁlter. Other

deconvolution approaches which account for time-varying pulse shape are given in [5, 36].

3 Weighted Least Squares

Let · ¸T

A = VˆN,1 VˆN,2 ··· VˆN,L (21)

The least squares problem solved to ﬁndg ˆ in the previous section can be expressed as

Agˆ ≈ 0L(K−1) (22)

The resulting response estimateg ˆ can be sensitive to the condition number in Rp. One way

of reducing this sensitivity is to solve the weighted least-squares problem

1/2 W Agˆ ≈ 0L(K−1) (23)

10 where W 1/2 is an L(K −1)×L(K −1) weight matrix which is chosen to minimize the variance £ ¤ of the estimate E (g − gˆ)(g − gˆ)T . It can be shown that if Agˆ is asymptotically Gaussian,

1/2 zero mean, and has covariance matrix Σ, then the optimal weight matrix, Wopt is given by

[37, 38]

1/2 −1/2 Wopt =Σ (24)

£ ¤ The ijth,(K − 1) × (K − 1) block entry of Σ = E AggT AT is given by

h i h i T T T T Σij = E VˆN,igg VˆN,j = E VˆN gigj VˆN ,i,j =1,...,L (25)

where gi is the ith column of G. Let the columns of VˆN be given by

qˆk = qk +∆k,k = L +1,...,N (26)

where qk are the actual noise subspace eigenvectors and ∆k are estimation error vectors.

Using the fact that the gi are orthogonal to the eigenvectors spanning the noise subspace, qk,k = L +1,...,N some straight-forward calculations give

  T T T T T T  g ∆L+1∆ gj g ∆L+1∆ gj ··· g ∆L+1∆ gj   i L+1 i L+2 i N     T T T T T T   g ∆L+2∆ gj g ∆L+2∆ gj ··· g ∆L+2∆ gj   i L+1 i L+2 i N  Σij = E   (27)  . . . .   . . .. .      T T T T T T gi ∆N ∆L+1gj gi ∆N ∆L+2gj ··· gi ∆N ∆N gj

11 The entire error covariance matrix is then  

 Σ11 Σ12 ··· Σ1L         Σ21 Σ22 ··· Σ2L    Σ=E   (28)  . . . .   . . .. .      ΣL1 ΣL2 ··· ΣLL

For large M, the ∆k are Gaussian, zero mean random vectors with covariance matrix [39]

£ ¤ XN T λ(i) λ(k) E i qkqkδ i − j ∆ ∆j = − 2 ( ) (29) M k=1 (λ(i) λ(k))

where q1,q2,...,qN and λ(1) ≥ λ(2) ≥ ... ≥ λ(N) are the eigenvectors and corresponding eigenvalues of Rr and δ(i−j) is the Dirac delta function . In practice, the qk and λ(k) are not available and hence computation of the weight matrix involves replacing these quantities with their estimates. In this case, if these estimates are asymptotically consistent, the resulting least squares solution is still optimal [40]. From (29), only the diagonal entries of Σij are nonzero. Nevertheless, the inversion of Σ can be computationally expensive. Good results

2 can still be obtained if only the diagonal entries of Σ are used. Moreover, if σv is estimated as the mean of the smallest N − L eigenvalues of Rˆr, the diagonal entries of Σij are equal and can be computed using

  £ ¤ XL T T T  2 λˆ(m) T  gk E ∆i∆i gk ≈ gˆk σˆv ³ ´2 qˆmqˆm gˆk,i = L +1,...,N (30) 2 m=1 λˆ(m) − σˆv k =1,...,L

12 whereq ˆ1, qˆ2,...,qˆN and λˆ(1) ≥ λˆ(2) ≥ ... ≥ λˆ(N) are the eigenvectors and corresponding

eigenvalues of Rˆr and XN 2 1 ˆ σˆv = − λ(k) N L k=L+1

The vectorg ˆi is the ith column of the matrix

   gˆ(0) 0 ··· 0         gˆ(1)g ˆ(0) ··· 0       . . . .   . . .. .         gˆ(L − 1)g ˆ(L − 2) ··· gˆ(0)       . . . .  Gˆ =  . . .. .  (31)        − − ··· −   gˆ(K 1)g ˆ(K 2) gˆ(K L)       − ··· −   0ˆg(K 1) gˆ(K L +1)     . . .   ......   . . .    00··· gˆ(K − 1)

The weighted second-order statistics algorithm for ultrasound deconvolution is given by:

1. Collect independent response vectors ri,i=1,...,M.

2. Compute XM 1 T Rˆr = riri M i=1

3. Find the eigendecomposition of Rˆr and set the columns of VˆN equal to the eigenvectors

associated with the smallest N − L eigenvalues of Rˆr.

4. Partition VˆN into VˆN,i,i=1,...,L as deﬁned in (11).

13 5. Compute XL T RV = VˆN,iVˆN,i k=1

6. Setg ˆ equal to the eigenvector associated with the minimum eigenvalue of RV .

7. Form the vectorsg ˆi,i=1,...,L using the deﬁnition in (31).

8. Compute   XL T  2 λˆ(m) T  ρ(k)=ˆgk σˆv ³ ´2 qˆmqˆm gˆk,k =1,...,L 2 m=1 λˆ(m) − σˆv

9. Compute XL T RV = VˆN,iVˆN,i/ρ(k) k=1

10. Setg ˆ equal to the eigenvector associated with the minimum eigenvalue of RV .

11. Go to step 6 and continue iterating until the change ing ˆ is suﬃciently small.

4 Experimental Results: Synthetic Data

The following deconvolution methods were compared using synthetic data:

• Higher-order statistics (HO)

• Homomorphic (HM)

• Second-order statistics (SO)

• Weighted second-order statistics (WSO)

14 The higher-order statistics deconvolution was implemented using the method in [25]. The

tissue reﬂectivity g(n) was synthesized using the following rule

   v(n), if u(n) > 0.4,n=0,...,499 g(n)= (32)   0, otherwise where u(n) and v(n) are sequences derived from a white noise generator that was uni- formly distributed on the interval [0, 1]. This insured that g(n) satisﬁed the requirements for higher-order blind deconvolution: that g(n) be white and have non-zero kurtosis. The pulse waveform was synthesized using

2 p(n)=e−(n−25) /100cos(0.7(n − 23)),n=0,...,48 (33)

In all, 1000 reﬂection sequences were generated by convolving p(n) with 1000 independently

generated g(n),

rk(n)=gk(n) ?p(n)+zk(n),k =1,...,1000 (34)

n =0,...,547

where zk(n) was taken from a zero-mean Gaussian random number generator. The rk(n)

were used to estimate the third-order moment matrix of the reﬂection data as described in

[25]. Third order moments were estimated for lags −48 ≤ m, n ≤ 48, for each of the rk and

then averaged. The estimated third-order moments were found to be in close agreement with

the theoretical moments derived from the known pulse sequence p(n). The optimal window

discussed in [41] was then applied to the estimated third-order moment matrix. Next, the

15 Fast Fourier Transform method was used to estimate the bicepstrum of the transmitted

ultrasound pulse, from which the complex cepstrum was subsequently derived. A detailed

description of this procedure is given in [25]. The complex cepstrum was then used to ﬁnd the

estimated pulse sequence. The actual deconvolution step was performed with a frequency

domain Wiener ﬁlter (see [20]). The known theoretical SNR was incorporated into the

Wiener ﬁlter as was the norm of the true pulse sequence, in practice, these quantities are

not directly available. The deconvolution was performed on r1(n). The reﬂectivity function

g1(n) is shown in Fig. 2(a). The experiment was repeated for signal to noise ratios ranging from 30 dB to 60 dB in 5 dB steps.

Next the homomorphic deconvolution method was applied to the same data matrix used for the higher-order statistics experiments. The complex cepstrum was estimated using the

Matlab routine “cceps” and averaging the cepstra over each of the 1000 independent rk(n).

A frequency invariant ﬁlter was then applied to the average cepstrum in order to estimate

the pulse. The ﬁlter zeroed the entries in the cepstrum from n =19ton = 542, and left

the other entries unchanged. These values were found to give reasonable estimates of the

pulse. The estimated pulse waveform was then applied to the same Wiener deconvolution

ﬁlter used for the higher order deconvolution experiments over the same range of signal to

noise ratios.

The second-order method was then tested using a single reﬂectivity function, g1(n). Note

that all of the methods compared require the estimation of a certain statistic. For the higher-

order statistics and homomorphic deconvolution, the corresponding statistics were averaged

over 1000 independent reﬂectivities. For the second-order statistics method, we used a single

reﬂectivity function and average over 1000 distinct, statistically independent transmitted

pulse sequences. The pulse sequences were chosen by passing Gaussian white noise through

16 a lowpass ﬁlter having the characteristic shown in Fig. 1. The frequency response of this

ﬁlter was chosen to closely correspond to the frequency response of the ultrasound transducer we have been using in our lab (5 MHz resonant frequency, -3dB points roughly at 3.75 MHz and 6.25 MHz and -10 dB points at 2 MHz and 8 MHz)). Since most of the energy in the simulated transmitted ultrasound is concentrated about 5 MHz, the condition number of the

6 matrix Rp was very high (7.29 × 10 ). This has a tendency to degrade the performance of the second-order method. The weighted second-order method was then applied to the same data. These experiments were repeated using the same range of SNR’s as before.

Fig. 2 shows the original reflectivity function g1(n) along with estimated reflectivities,g ˆ1(n) derived for a signal-to-noise ratio of 55 dB. The weighted second-order statistics method gives the best estimate followed by the second-order statistics method. For each of the methods tested, the squared error norm between the true reflectivity g1(n) and the estimated reflectivityg ˆ1(n) was computed for each SNR using:

NX−1 1 2 MSE = (g1(n)/α1 − gˆ1(n)/α2) (35) N n=0

Both the true reﬂectivity and the estimated reﬂectivity were normalized by the constants α1 and α2, respectively so as to have unit norm prior to doing the comparison. In other words,

v u uNX−1 t 2 α1 = g1(n) (36) n=0 and v u uNX−1 t 2 α2 = gˆ1(n) (37) n=0

17 For each SNR, the squared error norm was averaged over ﬁve independent trials. Since

cepstral methods estimate the pulse shape to within a scaling factor and a phase shift, the

squared error norm was computed over a range of phase shifts and the minimum error norm

over the phase shifts was retained. The results of these experiments are shown in Fig. 3.

The proposed second-order and weighted second-order deconvolution methods gave the best

results. At higher SNR’s, the weighted second-order method gave an averaged error norm

which was several orders of magnitude lower then the next lowest error norm.

Next the experiment was repeated using the second-order method on synthetic backscattered

data where the reﬂectivity function underwent continuous, rather than impulsive variations.

This may be a more realistic representation of many types of tissue structures. The weighted

second order method does not appear to provide any improvement over the unweighted

method, however both mean squared errors are low over the entire range of SNR’s as seen

in Fig. 4. We also note that this error can be made even lower by using a wider-band

isonification signal. Fig 5 shows the original reflectivity and the estimated reflectivities for

the second-order and weighted second-order method.

5 Experimental Results: Utrasound Data

A 5 MHz 0.375 in diameter transducer (Physical Acoustics Corp. part # IU5G1.5) was

connected to a Physical Acoustics Corp. PAC-IPR-100 pulser/receiver module. The pulser

unit can be triggered via an edge-sensitive input. To generate a persistently exciting isoni-

ﬁcation signal, a 12 KHz low duty cycle voltage (approximately 9.6 µs pulse width) was generated using a function generator (Leader LFG-1300S), this signal was connected to the control input of a second function generator (HP 8116A). By putting the HP generator in

18 “AM” mode, the control input was mixed with a 672 KHz 50% duty cycle square. Since the fundamental frequencies of the two signals being mixed were not integer multiples of each other, the high frequency square wave underwent a random phase shift each time it was mixed with the 9.6 µs pulse from the Leader function generator. The trigger signal generation set-up is shown in Fig. 6. The triggering signal was then applied to the trigger input of the pulser/receiver in order to generate a burst of ultrasound pulses. Each burst therefore had a random phase shift relative to the preceding bursts. This guaranteed a persistently exciting stimulus for the duration of the pulse burst (9.6 µs). In all, 1000 pulse bursts were generated by the immersed transducer which was aimed at a wire target. The wire target was approximately 1 mm in diameter and was positioned 5 mm in front of the transducer.

The reﬂected signal measured by the transducer was acquired using an HP 100 MHz digital oscilloscope controlled by a PC under a GPIB interface. Figure 7(a) shows the 1000 reﬂected signals as measured by the oscilloscope. Evidently, the pulses occurring during the latter portion of the burst had a lower amplitude than the pulses near the onset of the burst. This is an undesirable feature of this excitation method and future approaches will involve direct excitation of the transducer with a burst of white noise, appropriately modulated to insure a persistently exciting, stationary ultrasound burst. In theory the signal subspace dimension should be 100, however it was found that the amplitude of the pulse burst at latencies of

100 sampling intervals was exceedingly low. Therefore, we chose 25 as the signal subspace dimension, which corresponded to the approximate number of samples for which the pulse burst maintained a signiﬁcant amplitude as seen in Fig. 7a. Figure 7b shows the results of weighted second-order deconvolution after a single iteration. Subsequent iterations did not seem to improve the narrowness of the reﬂection from the wire target. As predicted by

(3), the reﬂection from the wire target has both positive and negative deﬂections since the

19 change in acoustic impedance is positive at the water-metal interface of the target and is negative at the metal-water interface.

6 Summary

A method for blind axial deconvolution of ultrasound signals was described. The method is blind in the sense that the transmitted ultrasound need not be measured. This approach, which makes use of second-order statistics, appears to oﬀer several advantages over other blind methods like homomorphic deconvolution and higher order statistics-based deconvolution since no assumptions are made about the statistical nature of the tissue reﬂectivity.

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23 0 10

−1 10

−2 10

−3 10 PSD

−4 10

−5 10

−6 10 0 10 20 30 40 50 frequency (MHz)

Figure 1: Frequency spectrum of simulated transmitted ultrasound used by the second-order and weighted second-order methods.

24 0.4 (a) 0.2 0 0 50 100 150 200 250 300 350 400 450 500 0.2 (b) 0

-0.2 0 50 100 150 200 250 300 350 400 450 500 0.2 (c) 0 -0.2 0 50 100 150 200 250 300 350 400 450 500 0.4 (d) 0.2 0 0 50 100 150 200 250 300 350 400 450 500 0.4 (e) 0.2 0 0 50 100 150 200 250 300 350 400 450 500 n

Figure 2: Estimated reﬂectivity functions for each of the methods tested at 55 dB SNR: (a) Actual g1(n), (b)g ˆ1(n) for HO, (c)g ˆ1(n) for HM, (d)g ˆ1(n) for SO, (e)g ˆ1(n) for WSO.

25 1 10

0 10

−1 10 MSE

HO −2 10 HM SO WSO −3 10 30 35 40 45 50 55 60 SNR (dB)

Figure 3: Averaged estimated reﬂectivity error norms for the four deconvolution methods tested: higher order statistics (HO), homomorphic (HM), second-order statistic (SO), and weighted second-order statistic. The best results were produced by the weighted second-order statistic method, followed by the second-order statistic method.

26 0.2 (a)

-0.2 0 100 200 300 400 500 n 0.2 (b)

0 SO WSO -0.2 0 100 200 300 400 500 n

Figure 4: (a) Actual reﬂectivity function g(n), (b) Estimated reﬂectivity functions for second- order statistic deconvolution (SO), and weighted second- order statistic-based deconvolution (WSO) at 55 dB SNR.

27 0 10 SO WSO −1 10

−2 10 MSE

−3 10

−4 10 30 35 40 45 50 55 60 SNR (dB)

Figure 5: Averaged error norms for the second-order statistic and weighted second-order statistic method for a smooth reﬂectivity function.

28 Leader LFG-1300S HP 8116A

out CTR in out

PAC-IPR-100 pulser/receiver transducer TRIG in

Figure 6: Generation of the trigger signal. The HP function generator’s high frequency square wave is modulated with a lower frequency, low duty cycle pulse train coming from a second function generator (Leader). This generates a burst of high frequency trigger pulses, each burst has a random phase shift.

29 4 (a) 2

-2

-4 0 100 200 300 400 500

0.5 (b)

-0.5 0 100 200 300 400 500 n

Figure 7: Results of second-order statistic-based deconvolution: (a) 1000 superimposed re- sponses from wire target. (b) Deconvolution result after two iterations.