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Blind Deconvolution of Backscattered Ultrasound Using Second-Order Statistics

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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 ¯rst 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 bene¯ts 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 convolu- tional 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 ul- trasound [1, 2, 3, 4, 5]. Lateral distortion in the (x; y) plane can be reduced by using a ¯nely 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 deconvo- lution 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 ¯lter-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 di±cult 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 transmit- ted pulse. The pulse could be measured o®-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 ap- proaches 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 ¯lter 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 di±cult 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 de¯nitive studies which suggest that reflectivity can be modeled as white noise. Moreover, higher-order methods re- quire 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 e®ect 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 ±h(z) g(z)= (3) 4z ±z 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 de- scribed. This approach o®ers 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 deconvo- lution 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 ¯rst 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 chan- nels [28, 29, 30]. However since the transmitted ultrasound has ¯nite 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: ² g(n): tissue reflectivity to be estimated, length K. ² p(n): transmitted ultrasound, length L, ² r(n): reflected pulse, length N ´ K + L ¡ 1. ² 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 2 3 6 g(0) 0 ¢¢¢ 0 7 6 7 6 7 6 7 6 g(1) g(0) ¢¢¢ 0 7 6 7 6 7 6 . 7 6 . .. 7 6 7 6 7 6 7 6 g(L ¡ 1) g(L ¡ 2) ¢¢¢ g(0) 7 6 7 6 7 6 . 7 G = 6 . .. 7 (8) 6 7 6 7 6 7 6 ¡ ¡ ¢¢¢ ¡ 7 6 g(K 1) g(K 2) g(K L) 7 6 7 6 7 6 ¡ ¢¢¢ ¡ 7 6 0 g(K 1) g(K L +1) 7 6 7 6 . 7 6 . ... 7 6 . 7 4 5 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].
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