Optimal Linear Receive Beamformer for Ultrasonic Imaging in Ndt

Phased arrays are relatively new tools that have been introduced to NDE in the recent decennium. Phased arrays have unquestionable advantages and open new possibilities of realtime ultrasonic imaging in NDE. However, the common way of designing focusing laws is based on the point source assumption, that is, it is assumed that the element size can be neglected. Since such assumption is valid in far field only, it may in many cases lead to degraded lateral resolution. Extended synthetic aperture focusing technique (ESAFT) that was proposed recently [1] compensates for the diffraction effects of a finite sized transducer both in transmission and reception. It has been shown that the lateral resolution offered by ESAFT is superior to the classical SAFT. Here, we propose a generalization of the ESAFT technique for phased arrays resulting in the Optimal Linear Receive Beamformer (OLIRB). The deconvolution concept used in ESAFT has been modified in OLIRB to suit phased arrays. A multidimensional linear filter optimized using the mean square error criterion is applied for compensating diffraction effects and delays introduced by the apertures used in transmission and reception. The filter replaces conventional delay and sum operation in ultrasonic system and, due to its parallel structure it can be used for processing ultrasonic signals acquired from array elements in real time. Performance of the OLIRB algorithm is demonstrated on simulated data and verified on data acquired from a real array system. Introduction: Conventional ultrasonic phased array (PA) systems perform focusing in reception by means of delay-and-sum (DAS) operations on the signals received by the individual array elements. DAS performs well in the receive mode provided that the array elements are small compared to the wavelength, otherwise, diffraction effects associated with the finite-sized elements will degrade the spatial resolution in the processed B-scans [1]. Generally, element size is a compromise between the resolution and the required signal-to-noise ratio (SNR) since the elements must be large enough to generate enough acoustic energy to obtain a sufficiently high SNR. Another important phenomenon that also limits size of the array element in DAS beamforming is the presence of grating lobes that can degrade images if the element spacing is larger than half of the wavelength. To avoid grating lobes the element spacing (and consequently element size) should be limited to less than half of the wavelength, which results in limiting the transmitted acoustic power. Focusing can be performed off-line after gathering signals received by all array elements (or transducer positions) using synthetic aperture focusing technique (SAFT), which also applies DAS operations, [2]. SAFT is based on the assumption that a point target will give rise to infinitely wide hyperbola in the B-scan, which is correct for a point transducer used for imaging a point scatterer only. The acquired signals are processed in SAFT with a kind of spatial matched filter, shaped like the expected hyperbola (its shape depends on the distance between the transducer and the scatterer). SAFT, which is essentially a post-processing technique, performs focusing in the reception only and, in fact, it does not use the information about the beampattern used in the transmission. Thus, both PA and SAFT that use DAS operations are built on an idealized model of the transducer and scatterers in the measurement setup. In practice due to the element’s (transducer’s) beam pattern normally only a small portion of the hyperbola corresponding to the main lobe is well pronounced in the B-scan and outside this portion the DAS operations process mainly noise. Moreover, the observed part of the hyperbola is modulated by the transducer’s electrical impulse and therefore takes the form of several parallel lines. This paper has two main objectives. Firstly, we will enlighten the above mentioned issues for the users of the PA systems who may believe that the DAS operations used in their systems are optimal for all measurement setups, and secondly, we will present the method of compensating the diffraction effects of finite-sized elements (transducers). Theory: In this section the theoretical model of ultrasonic imaging will be introduced that enables explaining how the proposed method operates and comparing it to DAS. The model applies to ultrasonic images that have been discretized in time and space, i.e. each pixel in the processed image has been expressed in a digital form and subsequently arranged in lexicographic order (i.e. expressed as a vector); details can be found in [1] and [3]. Then we can write the acquired image y as a result of a linear transformation of the real image o by the system propagation matrix P e Po y + = , (1) where e is the measurement noise vector and P is the transformation matrix. It can be shown that a discrete implementation of DAS can be expressed as linear operation on the acquired image y (see [1] for details) y D o T DAS = ˆ . (2) In the simplest DAS scheme matrix, D in eq. (2) contains zeroes and ones only, i.e. it performs sums of the image elements y with the delays corresponding to the respective hyperbolas. This operation is graphically illustrated in Fig. 1a where the delay operations are denoted as τi and the coefficients gi symbolize possible apodization used on the array elements (note that in general case there are M x N filter boxes in Fig. 1a). In the ESAFT algorithm, which we proposed recently [1], matrix D is replaced with the filter matrix K that represents the best linear filter in the minimum mean squared error (MMSE) sense. In other words, the matrix K is obtained by minimizing the minimum square error between the filter output ô and the real image o { } { } 2 2 ˆ Ky o o o − = − = E E J MSE , (3) where JMSE denotes the minimized MSE criterion, E{·} is the expectation operator, and ║·║ is the quadratic vector norm. The problem has an explicit solution given prior information about the image o and noise vector e in the form of their covariance matrices (see [1] for details). The operation of the ESAFT scheme is illustrated in Fig. 1b. Simple DAS operations in Fig. 1a have been replaced with linear discrete filters Fi that perform filtering of time shifted signals to compensate the effects of diffraction introduced by the finite sized elements. OLIRB technique: It should be noted that the ESAFT scheme, eq. (3), has been created for a monostatic case where the same transducer is used during transmission and reception. This technique can be generalized to the bistatic case when the apertures in transmission and reception are different. The result takes the form of the optimal linear filter used for all points in the regionof-interest (ROI) that includes diffraction characteristics of both apertures. This means that each row in the matrix K performs the optimal linear filtering operation required for compensating acoustic propagation effects encountered both in the transmit and receive modes for each image element oi. Even electrical characteristics of the individual elements can be compensated in this way. This technique will be referred to as optimal linear receive beamformer (OLIRB). Figure 1. Graphical illustration of the operation of parallel DAS (a), and the proposed OLIRB (b). Note that since the MMSE filter computes an estimate of the image o for all points simultaneously it is possible to process a full image from one emission only (if the signal from all array elements can be received simultaneously). The focusing operation can be viewed for all points as a bank of parallel optimal filters that are included in boxes with respective indices m,n in Fig. 1(b). Filter coefficients in Fi in each box m,n are constant for the setup used in imaging. In other words, filter coefficients are calculated only once for given imaging system and the desired ROI, based on the respective spatial impulse responses of the array used for imaging. Simulations: The purpose of this section is to evaluate the spatial resolution of the proposed method. Results presented below were obtained from the simulations of the phased array used for the experiments presented in the next section. The array, which is geometrically focused (concave) in the vertical plane has strip-like elements that can be electrically focused in the horizontal plane. Classical SAFT technique is compared with the above presented ESAFT and OLIRB techniques using synthetic B-scans obtained from the simulations performed with the Software Tool DREAM [4]. The simulated B-scans were created by a 3 MHz array with geometrical (vertical) focus at 190 mm in water, and with 1 mm wide elements. The simulations were performed for a point target located at the focal distance from the transducer in water. Processing results of the simulated B-scans are presented in Fig. 2. The results for the ultrasonic data received by a 8 mm (8 individual 1 mm array elements bridged electrically) aperture processed using SAFT and ESAFT are shown in Fig. 2a and 2b, respectively. It is apparent that even for a relatively small aperture a considerable resolution increase is observed in the images processed with ESAFT comparing to that obtained using an ordinary SAFT. −20 −10 0 10 20 257 0 2 4 x 10 −4