B-MODE IMAGES. Jrgen Arendt Jensen, Jan Mathorne, Torben Gravesen and. Bjarne Stage. Electronics Institute, bldg Technical University of Denmark

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1 Paper published in Ultrasonic Imaging: DECONVOLUTION OF IN-VIVO ULTRASOUND B-MODE IMAGES Jrgen Arendt Jensen, Jan Mathorne, Torben Gravesen and Bjarne Stage Electronics Institute, bldg. 349 Technical University of Denmark DK-2800 Lyngby Denmark Published in Ultrasonic Imaging, Vol. 15, pp. 122{133,

2 DECONVOLUTION OF IN-VIVO ULTRASOUND B-MODE IMAGES Jrgen Arendt Jensen, Jan Mathorne, Torben Gravesen and Bjarne Stage Electronics Institute, Bldg. 349 Technical University of Denmark DK-2800 Lyngby, Denmark An algorithm for deconvolution of medical ultrasound images is presented. The procedure involves estimation of the basic one-dimensional ultrasound pulse, determining the ratio of the covariance of the noise to the covariance of the reection signal, and nally deconvolution of the rf signal from the transducer. Using pulse and covariance estimators makes the approach self-calibrating, as all parameters for the procedure are estimated from the patient under investigation. An example of use on a clinical, in-vivo image is given. A 2 2 cm region of the portal vein in a liver is deconvolved. An increase in axial resolution by a factor of 2.4 is obtained. The procedure can also be applied to whole images, when it is ensured that the rf signal is properly measured. A method for doing that is outlined. Key words: Deconvolution; estimation; image improvement; signal processing. 1 INTRODUCTION Medical ultrasound is today used routinely in nearly all hospitals for diagnosing soft tissue structures. The prime advantages of the technique are its real-time image formation, mobility, noninvasive nature, and that no ionizing radiation is employed. The widespread use can also be attributed to the relatively low cost of the equipment compared to, e.g., X-ray and magnetic resonance imaging. Some of the disadvantages are the relative diculty of mastering the investigation technique and the poor image quality. The resolution is low and the contrast between dierent tissues is low. This has led to the suggestion of a number of image enhancement procedures, one of which is deconvolution. Here the high-frequency signal from the transducer is deconvolved in order to increase resolution. Fairly convincing examples of deconvolved images of tissue mimicking phantoms have been made, but the techniques generally delivers less convincing results for in-vivo images [1{8]. A number of reasons can be given for this. First of all, the resolution enhancement obtainable is dependent on the precision with which the interrogating pulse is known and on the ratio between the covariances of the noise and of the reection signal. The interrogating pulse cannot be known ahead of time due to the dispersive attenuation of the tissue. Thus, it must be estimated from the actual signal returned from the tissue. This pulse determination problem has been investigated by a number of authors [9{13]. The best approach seems to be the prediction-error method investigated in [12, 13], where an ARMA parametrized pulse is estimated. This model is ecient for deconvolution purposes, and, using a number of A-lines, an accurate estimate of the pulse is obtained. 2

3 The determination of the covariance ratio is a very important, and often overlooked, problem in deconvolution of ultrasound images. The ratio ultimately determines the resolution enhancement and the amplication of the noise. It must be noted that the ratio varies substantially over the image with the reection strength being dependent on tissue type and the noise being determined by the gain of the time gain compensation (TGC) amplier. An algorithm for determining this ratio is given in section 3. With the pulse and covariance ratio determined, a deconvolution algorithm capable of handling time varying parameters is needed. Such an algorithm is the topic of the next section. Combining the algorithms for pulse estimation, covariance estimation, and deconvolution yields an ecient scheme for improving medical ultrasound images. An example from use of the combined algorithm is given in section 5. In-vivo clinical data is used and a quite satisfactory result is obtained for a small region of tissue. The algorithms used here all assume a linear measurement of the image data. This can be dicult to obtain by current standard measurement techniques, and section 6, therefore, details how to modify the acquisition process to obtain comparable or even improved results for whole images. The combined algorithm presented can be used for two-dimensional deconvolution, when suitably modied [8]. It is, however, important to note that the main limitation on axial resolution stems from the one-dimensional pulse estimated in this work. This pulse is also related to the one-dimensional pulse used for reconstruction of the full three-dimensional pressure eld [14, 15] and one can argue that the one-dimensional deconvolution is the rst step in performing the full two-dimensional deconvolution [8]. 2 THE DECONVOLUTION ALGORITHM The aim of deconvolution is to estimate the input sequence to a system described by: z(k) = p(k) w(k) + n(k) (1) where z is the measured signal, p the impulse response, w the input or reection signal and n is noise. Deconvolution can also be described as removing the pulse p from the output signal. In the case of no noise, a perfect reconstruction lter can be made, which is the "inverse of" p. In the frequency domain, this is stated as R(f) = 1=P (f), where R(f) is the Fourier transform of the inverse lter. When noise is present, gross errors will evolve in bands where the noise energy is large compared to the P (f). A compromise between perfect removal of the pulse and noise amplication is, therefore, needed. A common choice is to seek the minimum variance estimate of w. By this, the minimum value of: E[(w(k)? ^w(k)) 2 ] = E[(w(k)? p inv z(k)) 2 ] = E[(w(k)? p inv p w(k)? p inv n(k)) 2 ] (2) is found. E denotes the mean value, and ^w is the estimate of w. The lter found by solving 3

4 this equation is the celebrated Wiener lter [16, 17]. In the frequency domain, it is: D(f) = P (f) P 2 (f) + N 2 (f) W 2 (f) (3) where denotes complex conjugate, and N(f); W (f) are the spectral densities of the noise and reections, respectively. These can both, to a good approximation, be assumed to be white, so the lter can be expressed as: P (f) D(f) = P 2 (4) (f) + N 2 W 2 where N=W is the ratio between the covariance of the noise and the covariance of the reection sequence. This ratio gives the optimum balance between resolution and noise amplication, and can be considered a lter tuning parameter. In the case of no noise, the Wiener lter is equal to the ideal, inverse lter. The one-dimensional pulse is spatially varying due to the dispersive attenuation. This can be handled by a segmentation of the data, so the pulses can be considered quasistationary in the segments. A more elegant approach is, however, to use a more advanced algorithm. Using Mendel's xed-interval deconvolution algorithm [18, 19], which employs a Kalman lter, the nonstationarity of the pulses and of the reection and noise covariances can be handled optimally. Further, the estimate obtained is based on all samples in a single scan line, and the algorithm can handle the two-dimensional case. The algorithm consists of a Kalman lter and a reection estimator. First, a Kalman ltration is performed on the data, and then the reection estimation is performed backwards recursively in time. This enables the algorithm to calculate an estimate based on all the data in the A-line and to handle nonminimum phase pulses. Details of the algorithm's derivation and implementation will not be given, as this can be found in the literature. (e.g. [18{20]). The more advanced algorithm still minimizes the mean square error between the true and estimated reections, and is equivalent to the Wiener lter for a xed pulse and xed N=W ratio. Knowledge of the spectrum of pulse and the covariance ratio of the noise and reections is still needed. The solution to this problem will be elucidated in the next section. The equations for the Kalman lter and the backwards recursive estimation step can be found in [8]. 3 PULSE AND COVARIANCE ESTIMATION The nal result of the deconvolution process inherently relies on the knowledge of the pulse and of the covariances. As they all change from patient to patient and with position of the region of interest in the patient, it is necessary to estimate these parameters in-vivo. A prediction error algorithm [21] was used in [12] and [13] to estimate a set of ARMA (AutoRegressive Moving Average) parameters for the pulse. Using it on data from a tissue 4

5 mimicking phantom, on a calf's liver and on in-vivo data, it was shown that the basic attenuated pulse can be estimated. The change in pulse shape can be traced by estimating pulses at dierent depths and then interpolating between the estimates assuming slowly varying coecients. This is done by making a segmentation of the data into overlapping segments, and then using the estimate from the previous segment to initialize the next estimation step. The deconvolution algorithm also needs an estimate of the covariance ratio. An estimate of the noise covariance can be determined from prior measurement and the actual gain of the TGC amplier. The estimation of the reection signal covariance is more dicult, as this is spatially varying. An example of an image of a 13th week fetus is shown in gure 1. Large variations in scattering strength are seen when going from the placenta into the fetal water, and again at the transition from the water to the fetus. It is quite obvious that keeping a xed covariance ratio throughout the image would yield a lower than necessary image resolution enhancement for the tissue structures and result in a severe noise increase at the water surrounding the fetus. A xed covariance ratio would, in fact, render the deconvoluted image useless. To estimate the covariance ratio with sucient precision and still attain the dynamics of the ratio, smoothing in both the axial and lateral dimension must be used. Short axial segments are used combined with averaging over several A-lines. An estimate of the covariance in the individual segments can be obtained by two dierent techniques. One method is to use predictive deconvolution in which the rf A-lines are ltered by the inverse ARMA lter characterizing the pulse. This gives an estimate of ^w, which then can enter the covariance calculation. Another approach is to calculate the covariance by using the autocorrelation of the pulse. The measured signal is, without noise, generated by: z(k) = The autocorrelation of this signal is: Note that R zz (n) = lim NX 1X N!1 i=?n k 1 =?1 = R pp (n) R ww (n) R pp (n) = 1X k 0 =?1 p(k? k 0 )w(k 0 ) (5) p(i? k1)w(k1) 1X k=?1 1X k 2 =?1 p(i? k2 + n)w(k2 + n) (6) p(k)p(k + n) (7) Assuming w to be white, zero mean, and Gaussian distributed with a variance of 2 w, Eq. (7) reduces to: R zz (n) = R pp (n) 2 (n) = w 2 w R pp(n) (8) By knowing the lag zero autocorrelation of the pulse and calculating the variance of the received signal, the variance of the reection sequence can be found. This approach is used here, as it is more robust towards noise and measurement nonlinearities, and because it is considerably faster than predictive deconvolution. 5

6 The estimate at one location in the image is found by segmenting the data, calculating a Hann weighted covariance estimate, and then smoothing over a number of lines laterally weighted by a Hann window with its center at the line of interest. 4 IMPLEMENTATION DETAILS In this section, various aspects regarding the implementation of the algorithm is detailed. The deconvolution algorithm is based on the state-space model: x(k + 1) = (k + 1; k)x(k) +?(k + 1; k)w(k + 1) z(k) = H T (k)x(k) + n(k) (9) where x(k) is a state vector, and is matrix and? and H are vectors characterizing the ultrasound pulse. The correspondence between the ARMA model used for the pulse: (1 + a1q?1 + a2q?2 + a ns q?ns )z(k) = (1 + c1q?1 + + c ns?1q?(ns?2) )e(k) (10) and the state-space matrices and vectors is: =? = 0 B@ 0 B@ ?a ns?a ns?1?a2?a CA 1 CA (11) (12) H T = (c ns?1; c ns?2; c2; c1; 1) (13) using the controllable canonical form of the state-space model. As a new set of parameters are estimated for the pulse at each time instance, and new set of vectors and matrices are generated for each sample of data. Also a new set of covariance values enters the algorithm for each time instance, so the variation in pulse shape, noise, and reection strength is take into account by this processing scheme. The initial values for the Kalman gain, covariance matrix and state vector in the Kalman lter is found by computing the lter for 30 time steps. The set of pulse parameters and covariance values are kept xed for these 30 iterations, so the matrices and vectors converges to xed values, that are used in starting the deconvolution of the image data. 5 DECONVOLUTION OF IN-VIVO DATA In-vivo data were acquired in order to test the deconvolution algorithm. The data acquisition took place at Herlev University Hospital in Denmark. A diagram of the set-up is shown in 6

7 gure 2. An ultrasound scanner (Model 1846, Bruel & Kjr, Nrum, Denmark) was used with a Bruel & Kjr MHz mechanical sector scan probe. The scanner was connected to our dedicated sampling system [23], which acquires data at a rate of 20 MHz with a precision of 12 bits. The high frequency signal amplied by the TGC amplier was sampled and stored by a dedicated program system developed for clinical data acquisition [24, 25]. The images shown in this section were acquired from a 28 years old male with a normal liver function. The image used is a longitudinal scan of the right liver lobe showing also the right kidney. A view of the portal vein was taken from the image showing both the vessel and the speckle pattern surrounding it. The one-dimensional pulse was estimated by the multichannel algorithm derived in [13] using all the lines and all the samples in the image of the vessel and its surroundings. An ARMA(10,9) model was used. The pulse and its spectrum are shown in gure 3. The noise covariance was found from the gain of the TGC amplier. The spatial variation of the reection covariance was determined by the autocorrelation approach mentioned in section samples were used for the axial segments and averaging laterally was done over 7 lines. The B-scan image and the deconvolved image is shown in gure 4. The image covers an area of 2 2 cm and starts at 3.9 cm from the transducer surface. A clear increase in axial resolution is seen. To quantify this, the autocovariance of the envelope of the images was calculated and is shown in gure 5. The -3 db width of the unprocessed image's autocovariance is 0.52 mm, roughly 2 times the wavelength of the 3 MHz transducer center frequency. The width is 0.22 mm for the deconvolved image, so a factor of 2.4 increase in resolution was obtained. Another appealing feature of the deconvolved image is that the noise does not increase prohibitively. The vessel is still dark due to the use of a varying covariance ratio. This gives an optimal balance between resolution and noise amplication. Thus, in regions with a good signal-to-noise ratio, a good resolution is obtained, and in regions with more noise, the image is still preserved although with a lower resolution. 6 IMPROVED MEASUREMENT The images shown in gure 4 are from a small region of the liver, where no signals are so large that they overload the input amplier. This will not be the case for data from a normal scanning situation generating images like the one shown in gure 6. This is the full image from which the subimage shown in gure 4 was taken. In gure 6, clipping and nonlinear amplication takes place at the diaphragm, at various places on the surface of the kidney, behind the kidney, at vessel interfaces and at the transducer/skin interface. Normally this nonlinear amplication has no eect, as the regions always are displayed in bright white. It, however, has a profound inuence on the deconvolution and especially on the pulse estimation. Ensuring that no electrical nonlinearities degrade the image is no easy task when using a conventional system. Turning down the overall gain reduces the amplitude of the speckle signal 7

8 to a level where the analog-to-digital quantization noise becomes dominant. The problem is that the same TGC curve is used throughout the image, so local lateral variations cannot be handled. The solution is to apply one TGC curve for each A-line measured. The curve should be determined by a computer optimizing the gain, so the best signal-to-noise ratio is obtained within the linear range of the amplier taking into account both noise from the amplier and quantization noise. After the digital acquisition, the data must be compensated by the gain used and then multiplied by the gain set by the physician to preserve the relative gray levels in the image. The gains to use are determined from the previous acquisition. The postcompensation can be done exactly as the gains are known along with the gain time constant of the TGC amplier. The time constant of the gain regulation is determined by the lowest frequency component for the pulse spectrum and by the time constant of the TGC amplier. Keeping the regulation time constant below these two time constants should ensure a distortion free signal after postcompensation. Using the pre- and postcompensation scheme, it is possible to acquire linear signals for whole clinical images. The technique, thus, makes it possible to obtained full, deconvolved ultrasound images. ACKNOWLEDGMENT The research was funded by the Danish Technical Research Council, grant E, Bruel & Kjr A/S, Novo's Foundation, H.C. rsteds Foundation and Trane's Foundation and the Technical University of Denmark. M.D., Ph.D. Sren Torp Pedersen and M.D. Knud Erik Fredfelt, both at Herlev University Hospital, Denmark, acquired the images used in this paper. 8

9 References [1] Fatemi, M. and Kak, A.C., Ultrasonic B-scan imaging: Theory of image formation and a technique for restoration, Ultrasonic Imaging 2, 1-47 (1980). [2] Liu, C.N., Fatemi, M. and Waag, R.C., Digital processing for improvement of ultrasonic abdominal images, IEEE Trans. Med. Imag. MI-2, (1983). [3] Robinson, D.E. and Wing, M., Lateral deconvolution of ultrasonic beams, Ultrasonic Imaging 6, [4] Hundt, E.E. and Trautenberg, E.A., Digital processing of ultrasonic data by deconvolution, IEEE Trans. Sonics Ultrasonics SU-27, (1980). [5] Herment, A., Demoment, G. and Vaysse, M., Algorithm for On Line Deconvolution of Echographic Signals, in Acoustical Imaging, vol. 10, P. Alais and A.F. Metherell, eds., (Plenum Press, New York, 1980). [6] Demoment, G., Reynaud, R. and Herment, A., Range resolution improvement by a fast deconvolution method, Ultrasonic Imaging 6, (1984). [7] Kuc, R.B., Application of Kalman ltering techniques to diagnostic ultrasound, Ultrasonic Imaging 1, (1979). [8] Jensen, J.A., Deconvolution of ultrasound images, Ultrasonic Imaging 14, 1-15 (1992). [9] Hutchins, L. and Leeman, S., Pulse and impulse response in human tissues, in Acoustical Imaging, vol. 12, E.A. Ash and C.R. Hill, eds., pp , (Plenum Press New York, 1983). [10] Towg, F., Barnes, C.W., and Pisa, E.J., Tissue classication based on autoregressive models for ultrasound pulse echo data, Acta Electronica 26, (1984). [11] Jensen, J.A. and Leeman, S., Non-parametric estimation of ultrasound pulses, (submitted for publication.) [12] Jensen, J.A., Estimation of pulses in ultrasound B-scan images, IEEE Trans. Med. Imag. MI-10, (1991). [13] Jensen, J.A., Estimation of in-vivo pulses in medical ultrasound, (submitted for publication.) [14] Jensen, J.A. and Svendsen, N.B., Calculation of pressure elds from arbitrarily shaped, apodized, and excited ultrasound transducers, IEEE Trans. Ultrason., Ferroelec., Freq. Contr. 39, (1992). [15] Jensen, J.A., A model for the propagation and scattering of ultrasound in tissue, J. Acoust. Soc. Amer. 89, (1991). 9

10 [16] Wiener, N., Extrapolation, Interpolation and Smoothing of Stationary Time Series, with Engineering Applications, (Wiley & Sons, Inc., New York, 1949). [17] Rosenfeld, A. and Kak, A.C., Digital Picture Processing, (Academic Press, New York, NY, 1976). [18] Mendel, J.M. and Kormylo, J., New fast optimal white-noise estimators for deconvolution, IEEE Trans. Geo. Elec. GE-15, (1977). [19] Mendel, J. M., Optimal Seismic Deconvolution. An Estimation Based Approach (Academic Press, New York, 1983). [20] Jensen, J.A., Medical Ultrasound Imaging, An Estimation Based Approach, Ph.D. dissertation, (Electronic Institute, Technical University of Denmark, September, 1988). [21] Ljung, L., System Identication. Theory for the User, (Prentice-Hall Inc., 1987). [22] Anderson, B.D.O. and Moore, J.B., Optimal Filtering, (Prentice-Hall, Inc., New York, 1979). [23] Jensen, J.A. and Mathorne, J., A sampling system for clinical ultrasound images, Proc. Med. Imag. V Symposium, SPIE-1444, (1991). [24] Gravesen, T., Jensen, J.A., and Stage, B., Programs for the acquisition, storage, processing, and display of clinical ultrasound pictures, User's guide, Report D.1 (Electronics Laboratory, Technical University of Denmark, 1989). [25] Gravesen, T., Jensen, J.A., and Stage, B., Programs for the acquisition, storage, processing and display of clinical ultrasound pictures, Documentation, Report D.2A (Electronics Laboratory, Technical University of Denmark, 1990). 10

11 Figure 1: B-scan image of 13th week fetus. The markers indicate one centimeter. 11

12 Figure 2: Equipment used for acquiring in-vivo data. 12

13 Amplitude Time [s] x Normalized amplitude [db] Frequency [Hz] x10 6 Figure 3: Estimated pulse and its spectrum for the 3 MHz 8529 Bruel & Kjr transducer. 13

14 Figure 4: Normal (top) and deconvolved (bottom) 14 response for image measured by the Bruel & Kjr 8529 transducer. The images cover an area of 2 2 cm.

15 Normalized autocovariance Distance [mm] Figure 5: Autocovariance of the envelope of unprocessed ( ) and deconvolved (- - -) image in Fig

16 Figure 6: Image of the right liver lobe and right kidney. The markers indicate one centimeter. 16

31545 Medical Imaging systems

31545 Medical Imaging systems Simulation of ultrasound systems and non-linear imaging 545 Medical Imaging systems Lecture 9: Simulation of ultrasound systems and non-linear imaging Jørgen Arendt Jensen Department of Electrical Engineering