ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 1, january

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1 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 5, no. 1, january Coherent-Array Imaging Using Phased Subarrays. Part I: Basic Principles Jeremy A. Johnson, Student Member, IEEE, Mustafa Karaman, Member, IEEE, and Butrus T. Khuri-Yakub, Fellow, IEEE Abstract The front-end hardware complexity of a coherent array imaging system scales with the number of active array elements that are simultaneously used for transmission or reception of signals. Different imaging methods use different numbers of active channels and data collection strategies. Conventional full phased array (FPA) imaging produces the best image quality using all elements for both transmission and reception, and it has high front-end hardware complexity. In contrast, classical synthetic aperture (CSA) imaging only transmits on and receives from a single element at a time, minimizing the hardware complexity but achieving poor image quality. We propose a new coherent array imaging method phased subarray (PSA) imaging that performs partial transmit and receive beamforming using a subset of adjacent elements at each firing step. This method reduces the number of active channels to the number of subarray elements; these channels are multiplexed across the full array and a reduced number of beams are acquired from each subarray. The low-resolution subarray images are laterally upsampled, interpolated, weighted, and coherently summed to form the final high-resolution PSA image. The PSA imaging reduces the complexity of the front-end hardware while achieving image quality approaching that of FPA imaging. I. Introduction Array-based imaging systems have been used for various applications, including radar, sonar, and medical ultrasound [1], []. In general, a collection of transmitters emits electromagnetic or mechanical waves into the medium to be imaged. The waves scattered or reflected by different parts of the medium then are received by a set of sensors. The received signals are processed to form an image with intensities that represent some physical parameter of the medium. Although applicable to a range of array-based imaging system, the work presented in this paper will be presented from the perspective of pulse-echo ultrasound imaging using a phased array. A typical phased-array ultrasound imaging system consists of a one-dimensional (1-D) linear array of transducers that are capable of both transmitting and receiving acoustic energy [3]. A short modulated pulse is transmitted from each of the elements. The timing of the trans- Manuscript received July 6, 004; accepted August 11, 004. J. A. Johnson and B. T. Khuri-Yakub are with Stanford University, Image Guidance Laboratory, Stanford, CA ( public@ drjjo.com). J. A. Johnson is also with Stanford University, Image Guidance Laboratory, Stanford, CA. M. Karaman is with Işik University, Department of Electrical Engineering, Istanbul, Turkey. mission from each element can be adjusted to steer the transmitted waveform to a desired direction. Additionally, the timing can be set to cause the transmitted pulses to converge, forming a beam that is narrowest at the desired focal depth. When the wavefront crosses boundaries between materials with different acoustic impedances, a portion of the pulse is reflected or scattered back toward the transducer array. These reflected signals are received by the elements of the array. In the same way that the transmitted pulses can be timed to achieve beam steering and focusing, the received signals can be delayed such that a reflection from a single point along the beam will add constructively. The process of steering and focusing the array signals is called beamforming [4]. A simplified diagram illustrating the concept is shown in Fig. 1. Current ultrasound systems use a highly parallel hardware architecture for signal conditioning between the transducer array and the beamformer. For every transducer element that is active for a given firing event, an independent front-end electronic processing channel performs transmit/receiveswitching, amplification, filtering, time-gain compensation, and digital-to-analog conversion. A high-level block diagram of a phased array imaging system is shown in Fig., and Fig. 3 illustrates the frontend hardware components. The cost and complexity of the front-end hardware scales with the number of active channels [5] [7]. The beamforming architecture determines the number of active channels for a particular system design. The image quality produced by an ultrasound system generally improves with the number of active channels, meaning that better images correspond to the systems with larger arrays and thus of greater cost and size. One of the most space- and power-consuming parts of a typical ultrasound imaging system is the front-end hardware [5], [7]. This is especially true since the advent of digital beamforming, which has greatly reduced the back-end hardware requirements [8], [9]. Early beamforming hardware was primarily analog. The increase in performance and decrease in size of digital and mixed-signal electronics eventually enabled the implementation of digital beamformers. Unfortunately, the analog nature of the front-end hardware has not experienced an equal reduction in cost and size. High-end commercial ultrasound machines still house the analog and mixed-signal, front-end electronics within the base unit, requiring costly and bulky probe cables that contain dedicated coaxial transmission lines for each transducer element. Although acceptable for today s 1-D arrays containing on the order of 100 elements, al /$0.00 c 005 IEEE

2 38 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 5, no. 1, january 005 Fig. 1. Delay-and-sum beamforming in a phased array ultrasound imaging system. The set of beamforming delays defines the steering angle and focal distance of the transmit and receive beams. Fig.. Block diagram of a phased array ultrasound imaging system. ternate configurations will be needed for tomorrow s -D arrays consisting of thousands of elements. Classical full phased array (FPA) imaging sweeps out a sector image by electronically steering beams across image space [10] [14]. All N transducer elements are active during both transmit and receive formation of each beam, requiring N dedicated front-end processing channels as shown in Fig. 4(a). In addition to high front-end hardware complexity, the large number of received signals required to form each beam causes a significant increase in beamformer complexity. One benefit of using all N elements is that high signal-to-noise ratio (SNR) is achieved. A variety of beamforming methods have been developed, each requiring a different number of active channels [15], [16]. Thomenius [17] has provided an overview and history of beamformers. Classical synthetic aperture (CSA) techniques using a single channel for transmit and receive minimize the hardware complexity [18]. Synthetic aperture techniques were first developed for radar [19] and later adapted to ultrasound imaging [0]. Synthetic aperture was first used with linear arrays with reconstruction in the spatial domain [1], but it since has been modified for use with circular arrays [], [3], and frequency-domain reconstruction methods also have been developed [4]. For the standard linear array method, a single processing channel is time multiplexed across all N transducer elements [Fig. 4(b)]. Because only a single element is used for both transmit and receive, the transmitted power and receive sensitivity are minimal and lead to a low SNR. Array imaging techniques have continued to strike compromises between CSA and FPA, aiming to improve the SNR of CSA methods and reduce the number of chan- Fig. 3. Front-end hardware for a phased array imaging system. T/R: transmit/receive switch; TX: transmit driver; LNA: low-noise amplifier; TGC: time-gain compensation; LPF: low-pass anti-alias filter; ADC: analog-to-digital converter. nels required for FPA imaging. Karaman et al. [5] have suggested transmitting from multiple elements to emulate a more powerful transmit element for synthetic aperture imaging, and later described how to correct for motion and phase aberration [6]. Lockwood et al. [7] used a similar method by transmitting from five virtual elements and using the full aperture in receive in order to achieve higher frame rates needed for 3-D imaging with a 1-D transducer array. Frazier and O Brien [8] proposed a synthetic aperture method for increasing the lateral resolution by considering virtual elements located at the focal point.nikolov and Jensen [9] combined the previous two techniques to achieve increased frame rate and improved resolution in elevation for 3-D imaging with a mechanically scanned 1- D transducer array. Nikolov et al. [30] introduced recursive ultrasound imaging as a method for increasing frame rate. Synthesizing an effective aperture for 3-D imaging using -D transducer arrays using only the outermost elements was described in [31]. The use of coded excitation is a current area of study that has benefits of improved frame rates, increased SNR, and improved depth penetration [3] [35]. Multielement synthetic aperture using a frequency-modulated signal also has been proposed as a method for increasing the SNR [36]. Early proposals for reducing the number of active channels in phased-array imaging systems did so by transmitting on a single central portion of the array and receiving on a number of overlapping [37] or adjacent [38] subarrays. Later developments

3 johnson et al.: proposed phased subarray imaging method 39 array. This allows the realization of a significantly wider effective aperture compared to the earlier methods using a fixed-transmit subarray, resulting in improved lateral resolution. Second, a new reconstruction method is presented that uses a set of subarray-dependent, -D filters for wideband imaging. This extends the capabilities of previous reconstruction methods that used 1-D lateral filters suitable for narrowband imaging. This paper presents the basic principles and theory of phased subarray (PSA) imaging. The companion paper presents the performance of the PSA method on simulated and experimental data [4]. The paper is organized as follows. In Section II basic terminology is described that will provide the framework for the presentation of the PSA imaging technique. Section III describes the data acquisition and image reconstruction methods for PSA imaging in beamspace. Sections IV and V present the same process described in the frequency domain along with reconstruction filtering. Section VI includes a generalization of the PSA method. The theoretical performance of PSA imaging is provided in Section VII and compared to FPA imaging. II. Basic Definitions A. Spatial and Spatial Frequency Responses The response of a phased-array imaging system at the far-field or focal depth can be approximated using linear systems theory. The performance of the system then is characterized by its two-way point spread function (PSF) in the spatial domain, or equivalently, its transfer function in the spatial frequency domain [43] [45]. There is a oneto-one relationship between these two functions given by the -D discrete Fourier transform (DFT): Fig. 4. Front-end hardware configuration for (a) phased array, (b) synthetic aperture, and (c) proposed phased subarray imaging methods. improved the frame rate of subarray imaging by acquiring a subset of the beam lines and interpolating the others [39] [41]. These methods are limited in use to narrowband imaging systems and did not achieve FPA-equivalent image quality due to a smaller effective aperture. This paper presents a new method that combines the principles of phased-array and synthetic-aperture imaging methods to reduce the system cost and size by decreasing the number of active channels while maintaining highimage quality. The method presented is most useful for cost- or size-constrained real-time acoustic imaging systems. The phased-subarray technique described here extends the capabilities of the earlier subarray imaging approach [39] [41] by contributing in two ways. First, the transmit subarray is not fixed in the center of the array but is placed at a number of positions across the entire u[m, n] =I D {U[p, q]}, (1) where U[p, q] isthepsf,andp and q are the axial and lateral indices, respectively; u[m, n] is the transfer function, and m and n are the axial and lateral spatial frequency indices, respectively; and I D { } is the -D DFT operator. The 1-D lateral spatial frequency response at a given axial spatial frequency is referred to as the coarray or the effective aperture, and can be computed as the convolution of the transmit and receive aperture functions [46], [47]: u[n] =a T [n] a R [n]. () The axial PSF of a phased-array imaging system is determined by the transmit pulse, s(t), centered at time t =0: (( U[p] =s p P ) ) r, (3) where P is the number of axial samples and r is the axial distance between beam samples. The -Dspatial frequency

4 40 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 5, no. 1, january 005 response can be expressed as the product of the lateral and axial spatial frequency responses: u[m, n] =u[m] u[n], (4) where u[m] is the 1-D axial spatial frequency response given by the 1-D DFT of the axial PSF. Note that the temporal frequency response of the excitation pulse determines the axial spatial frequency response of the system, and are related by a scaling factor. B. Comatrix Representation A useful technique for visualizing the relationship between the aperture function and the coarray is by means of the comatrix. Comatrix representation was introduced in [5] and formalized as the transmit-receive apodization matrix in [48]. Each transmit/receive element pair contributes to a specific bin of the coarray representing a specific lateral spatial frequency. Each entry in the comatrix represents a specific pair of single transmit element and receive element. For example, the topmost comatrix entry represents transmitting on array element 16 and receiving from array element 1. The transmit and receive arrays shown above each comatrix in Fig. 5 are the same physical array, and are used to index into the comatrix with the given element numbers. The numbers in the comatrix indicate the corresponding coarray bin. All entries in a comatrix column contribute to the same coarray bin. For the FPA example in Fig. 5(a), all TX/RX pairs are acquired at once as all array elements are used during each firing event. This results in a comatrix with all elements equal to unity, and a coarray equal to a triangle function. The comatrix and coarray are also useful for evaluating methods in which individual TX/RX pairs are obtained through separate firings. In contrast to FPA imaging, CSA imaging transmits and receives to/from a single element at a time. As shown in Fig. 5(b), each of these firing events contributes to a single comatrix and coarray entry. The corresponding comatrix structure is in the form of the identity matrix, resulting in a coarray equal to the shah [49] or comb function. The resulting coarray is zero valued at every other bin, which results in grating lobes. In the previous subaperture imaging work done by Karaman and O Donnell [40], a single transmit subaperture located at the center of the array fires multiple times, and a receive subaperture is scanned across the full array. The comatrix and coarray representation of this modality is shown in Fig. 5(c). The width of the coarray is less than that of both FPA and CSA imaging, and thus the lateral resolution is poorer. The PSA imaging method advances this approach by translating both the transmit and receive subapertures to obtain an FPA-equivalent effective aperture. III. Phased Subarray Image Formation A 1-D,N-element array is subdivided into K overlapping subarrays, each consisting of M elements. The subarray pitch is J elements, thus the amount of subarray overlap is M J. A general condition that is satisfied for a valid subarray configuration is: J(K 1) = N M. (5) The first M-element subarray can be used as a reference subarray and expressed by its aperture function: { 1, n [0,M 1] a 0 [n] =. (6) 0, otherwise The aperture functions of all K subarrays then can be expressed in terms of this reference subarray: a k [n] =a 0 [n kj], k [0,K 1], (7) where k is the subarray index. For the sake of simplicity, the examples in this paper and results in the companion paper [4] assume that the subarray pitch is equal to half the width of a subarray (J = M/). In this case, the full array response a triangle function is equal to a linear combination of the subarray responses, each a narrower triangle function with its base aligned with the peak of the adjacent triangle function. As a result, the interpolation filters only need to suppress aliases and do not need to reshape the amplitude response of each subarray function. The interpolation filters therefore simplify to bandpass filters. When this condition is not met, the filters also must modify the amplitude response in order to reconstruct an FPA-equivalent image. Methods for designing these filters are beyond the scope of this paper and are presented in [50]. As shown earlier in Fig. 4(c), the number of front-end transmit and receive processing channels is equal to the number of elements in each subarray. The hardware may be designed to allow the active transmit subarray to differ from the active receive subarray as is the case in Fig. 4(c) or may be designed such that the same subarray is used for transmit and receive. Both cases are discussed, but the discussion primarily focuses on the latter case due to simplicity of presentation. Fig. 6 illustrates all acquisition and processing steps involved in PSA image formation as compared to FPA. The object of PSA imaging is to generate an FPA-equivalent set of Q FPA beams by transmitting and receiving on only M elements at a time. This full set of Q FPA beams forms a high-resolution image. Although each of the K subarrays could acquire a high-resolution image, the frame rate is increased by acquiring only a subset of the Q FPA beams over the same sector angle. Each of the K subarrays acquires a total of Q PSA subarray beams, forming a low-resolution image. With the exception of limiting the transmit and receive elements to those of the transmit and receive subarrays, the beamforming used to generate each subarray image is identical to that used for FPA imaging. The beamformer delays are calculated in the same manner as for FPA imaging; for example, the delays shown in Fig. 1 required to

5 johnson et al.: proposed phased subarray imaging method 41 Fig. 5. Comatrices and coarrays for different imaging methods. Shaded comatrix elements correspond to TX/RX element or subarray pairs that contribute to the image. The diagonal sum of the comatrix produces the coarray that represents the lateral spatial frequency response of the imaging system. All figures correspond to a 16-element array with half-wavelength element pitch. (a) Phased array imaging uses all elements in TX and RX. (b) Synthetic aperture imaging transmits and receives on the same single element, resulting in an undersampled coarray. (c) Synthetic receive aperture imaging uses a single transmit subarray and several receive subarrays. (d f) Three representative phased subarray methods. (d) Seven overlapping subarrays are used for both transmit and receive (configuration used by examples in text). (e) Four adjacent subarrays acquire some redundant spatial frequency information. (f) Four adjacent subarrays use all TX/RX subarray combinations.

6 4 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 5, no. 1, january 005 Fig. 6. Image formation and reconstruction in the spatial domain. Right: FPA imaging directly acquires Q FPA beams using the full array. Left: K phased subarrays sequentially acquire a subset (Q PSA )ofthebeams,formingk subarray images. These images are laterally upsampled and interpolated to reconstruct the unacquired beams, then weighted and summed to form and FPA-equivalent image. steer and focus the beam to the desire point would be the same for FPA and for all of the subarray images. The subarray beam origins are located at the center of the full array, not at the center of each subarray, as illustrated in Fig. 7. The next step is to form high-resolution subarray images from each of the low-resolution subarray images. Each of the subarray images is first upsampled laterally by a factor of L by inserting L 1 zero-valued beams between the acquired beams. An interpolation filter is applied to this set of upsampled beams, reconstructing the empty beams and leaving the originals intact. Two methods for design of the reconstruction filter are presented in the next two sections. The resulting K high-resolution subarray images are then coherently weighted and summed to form the final PSA image. At this point, the final PSA image is equivalent to the FPA image. Note that the equivalence of PSA imaging to FPA imaging holds true in the far-field or at the focal distance as it depends on the Fourier transform relationship between the aperture function and the beam pattern. The extent to which the equivalence holds away from the focal distance is shown experimentally in the companion paper [4]. The beam profiles for each subarray differ from each other and from the FPA beam profile; however, the equivalent beam profile representing the PSA system at the focal distance are equivalent to that of FPA imaging as illustrated in Fig. 7. Although not obvious in the spatial

7 johnson et al.: proposed phased subarray imaging method 43 Fig. 7. Approximate beam profiles for PSA and FPA imaging when acquiring the center beam. (a) (c) The beam profiles for a single subarray are wider at the focal point than the FPA beam profile. (d) The combined response of PSA is equivalent to a beam profile equal to that of FPA imaging. domain, this observation is clearer when considering that the coarrays of PSA systems are equivalent to those of FPA systems [compare Fig. 5(a) to Figs. 5(d) (f)]. IV. Narrowband PSA Processing Due to the minimal support in the temporal frequency domain of a narrowband system, its array response can be approximated by the 1-D cross section through the coarray at the center operating frequency. We refer to the lateral spatial frequency response of a single subarray as a cosubarray. In order to understand the function of the reconstruction filter, we outline the PSA procedure by illustrating the resulting cosubarrays at each step in Fig. 8. The drawings represent a system with a 16-element array, 7 subarrays, each with 4 elements, and an upsampling rate of 4 (N = 16, K =7,M =4,L = 4). The array in these drawings is sampled with an element pitch of d = λ/ to avoid grating lobes, and Q PSA =Msin(Θ/) beams are acquired to avoid aliasing in the lateral spatial frequency domain. The subarray aperture functions are shown in Fig. 8(a). The corresponding cosubarrays equal to the convolution of the transmit and receive aperture functions are triangle functions of width M as shown in Fig. 8(b). These cosubarrays correspond to acquiring all Q FPA beams with each subarray. Because the subarray pitch, J, isequalto half the subarray size, M/, then these triangle-function cosubarrays overlap one another by half the width of the triangle, and an FPA-equivalent subarray can be reconstructed as a linear combination of the cosubarrays. The cosubarrays corresponding to the acquisition of only Q PSA beams by each subarray are shown in Fig. 8(c). Due to the band-limited nature of the ideal cosubarrays, no information is lost when the beam sampling rate is sufficiently large (equivalently the subsampling ratio, L, sufficiently small): ( ) Θ Q PSA > M sin. (8) The upsampling and filtering steps performed in the spatial domain correspond to reconstruction of the ideal cosubarrays. The resulting cosubarrays after upsampling are shown in Fig. 8(d). At this point, the cosubarrays contain the lateral spatial frequency components of the original cosubarrays in addition to periodic replicates of these components shifted by multiples of N/L. The purpose of the interpolation filter is to suppress the aliases while preserving the original cosubarray response. The lateral spatial frequency response of the ideal interpolation filters is shown in black behind the original cosubarray components in Fig. 8(d). The spatial frequency domain representation of the ideal filter for each subarray can be written as: (k 1)M 1, +1 n H k [n] = 0 otherwise (k +1)M Because this is a bandpass filter, h k [q] is sinc-like and thus the energy decreases further from the filter center. The size of the filter is reduced by truncation. The resultant rippling in the spatial frequency response is minimized by applying a window function. The reconstructed cosubarrays are found by applying the filter transfer function to the upsampled cosubarrays, and are shown in Fig. 8(e). These upsampled cosubarrays are equivalent to the high beam rate cosubarrays shown in Fig. 8(b). The final PSA coarray [Fig. 8(h)] is formed by linearly combining the upsampled coarrays [Fig. 8(e)] using a triangular weighting function: w k = K +1. (9) k K +1. (10) The final PSA coarray is equivalent to the FPA coarray [Fig. 8(g)]. The closeness of this approximation is dependent on the performance of the reconstruction filter. V. Wideband PSA Processing The PSA imaging for a wideband system differs from narrowband imaging in that the -D spatial frequency response must be taken into account. The -D spatial frequency response is the sum of the coarrays over all tem-

8 44 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 5, no. 1, january 005 Fig. 8. PSA image formation and reconstruction in the lateral spatial freqency (coarray) domain. (a) Subarray aperture functions in array space. (b) Ideal cosubarrays if full set (Q FPA ) of beams were acquired. (c) Cosubarrays when a subset (Q PSA )ofthebeamsisacquiredby each subarray. (d) Upsampling leads to periodic replication of the cosubarrays in (c). Reconstruction filters are designed as bandpass filters to suppress the unwanted aliases. (e) Cosubarrays of the reconstructed subarray system should approximate ideal cosubarrays shown in (b). By properly weighting and summing these responses, the final coarray (e) is equivalent to the FPA coarray (g) corresponding to phased array imaging with the full aperture (f). poral frequencies contained in the transmit pulse. The lateral spatial frequency width of the response varies linearly with the temporal frequency of operation; the axial spatial frequency extent of the -D spatial frequency response is determined by the minimum and maximum temporal frequencies present, f min and f max. The nonzero portion of the -D spatial frequency response for an N-element array therefore is limited to a trapezoidal region. For example, consider the wideband cosubarrays illustrated in Fig. 9. The -D spatial frequency response of the center M-element subarray acquiring Q FPA beams (high beam rate) is shown in Fig. 9(a). A horizontal cross section through any part of the nonzero spatial frequency response yields a triangular function whose peak location is indicated in Fig. 9(a) by the dashed line. For off-center subarrays, the spatial frequency response is sheared proportional to the distance of the subarray from the center of the full array. Fig. 9(b) illustrates the spatial frequency response of the end subarray. The -D spatial frequency responses corresponding to each stage of the PSA imaging system are outlined in Figs. 9(c) (h). Because the beams are treated as analytic signals, the negative temporal frequency response is zero; therefore, the illustrations only show the response for positive temporal frequencies. The illustrations are similar to those given in Fig. 8, except that only the responses for the center and end subarrays are shown. With the exception of using a -D rather than a 1-D filter kernel, the processing steps are identical. The drawings are based on the same system as in Figs. 7 and 8 (N = 16, K =7,M =4, L = 4). The center frequency, f 0, is identical to that used in the narrowband example. However, the temporal frequencies now range from f min to f max. This increase in temporal frequency causes a corresponding increase in the lateral spatial frequency bandwidth. Assuming that the element pitch is d = λmin, the number of beams that must be acquired to avoid aliasing of the -D spatial frequency response is: ( ) Θ Q PSA > M sin. (11) The drawings in Fig. 9 correspond to a system that is slightly oversampled. The spatial frequency responses of the 1-D filters used in the previous section correspond to lateral frequency bands that are constant over all temporal frequencies. Although the 1-D filter is suitable for reconstructing the center subarray image, a -D filter kernel is needed for all other subarrays because the lateral spatial frequency passband varies with temporal frequency. The ideal passbands for the center and end subarrays are indicated as dark gray regions behind the original subarray spatial frequency responses in Figs. 9(e) and (f). After the appropriate -D filters are applied to each subarray image, the result approximates the high beam rate spatial frequency response [Figs. 9(a) and (b)]. The reconstructed subarray images are then coherently weighted and summed, using the same weights as given in (10). The nonzero portions of the subarray spatial frequency responses are all shown overlapping one another in Fig. 9(g). A lateral cross section through these subarray spatial frequency responses would reveal that adjacent cosubarrays overlap one another by half their

9 johnson et al.: proposed phased subarray imaging method 45 Fig. 9. PSA image formation and restoration in the -D spatial frequency domain for wideband system consideration. Coarrays shown in Fig. 8 correspond to horizontal cross-sections through these figures at the center frequency, f 0. Shaded areas correspond to regions of nonzero response. (a) Response of center subarray when full set (Q FPA ) beams are acquired. (b) Subarray shift along the array causes a sheared response relative to (a). (c) Critical beamsampling (Q PSA beams) leads to a laterally compressed frequency domain. (d) The wrapping of the response in (b) is evident due to the subsampling of beamspace. (e), (f) Upsampling in beamspace leads to periodic replication of the response in (c), (d), and requires reconstruction filtering to suppress the aliases. (g) The overlapping individual subarray responses are weighted and summed to form (h) the overall response of the system, equivalent to FPA imaging.

10 46 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 5, no. 1, january 005 width independent of the temporal frequency. Fig. 9(h) shows the -D spatial frequency response of the overall PSA system, equivalent to that of FPA imaging. The -D interpolation filter is applied in both the lateral and axial dimensions by -D convolution. The subarray filters illustrated are best described in the spatial frequency domain as shown in (1) (see next page). This expression describes the filters shown in Figs. 9(e) and (f). The filter for the center subarray is independent of temporal frequency and, thus, is a sinc function in the spatial domain. The spatial-frequency-domain representations of the other subarray filters are sheared versions of the centerarrayfilter.justasforthe1-dcase,thespatialfilter must be truncated and windowed. In this case, the filter is truncated to a finite -D kernel over the axial and lateral dimensions. A window function is applied to the ideal filter to obtain the final filter. VI. Alternate Acquisition Methods The PSA acquisition method presented earlier assumed that the same subarray is used in transmit and receive. Although this is the simplest method to describe and illustrate, the reconstruction filters presented are equally useful when reconstructing subarray images acquired by other pairs of transmit and receive subarrays. The similarities between these different methods of acquisition are best visualized using the comatrix representation described in Section II-B. Figs. 5(d) (f) illustrate the comatrices and resultant coarrays for three different PSA acquisition methods. The comatrix for the example used throughout this paper (N = 16, K =7,M =4,L =4)is shown in Fig. 5(d). Each of the small, gray diamonds corresponds to a firing event from one of the seven subarrays. The dashed line indicates the overall coarray without any weighting; applying the weights shown in Fig. 5(d) restores an FPA-equivalent response. The same overall coarray response can be acquired with fewer subarrays, although with additional subarray image acquisitions; Fig. 5(e) illustrates a configuration that uses four subarrays (K = 4) to acquire 10 subarray images. As an example, the cosubarray formed by transmitting and receiving from the second subarray in PSA-1 is identical to that formed by PSA- by transmitting on the first subarray and receiving on the second. In order to avoid distortion due to phase asymmetry, the opposite subarray pair also must be used that is, transmitting on the second and receiving on the first. Because some of the cosubarrays have been acquired twice using this method, the weights for achieving an FPA-equivalent response are different. A third PSA example with only four subarrays uses all TX/RX subarray pairs and is shown in Fig. 5(f). No weighting is necessary for this method. The subsampling and reconstruction are not illustrated using the comatrix representation. However, the same principles apply regardless of the subarray pair used for acquisition. The subsampling rate, the beam sampling requirement, and the reconstruction filters both 1-D and -D remain the same for these alternate acquisition methods. The only part of reconstruction that differs are the weights used prior to summation. The primary differences between these alternate methods are due to the difference in the total number of firings. The two alternate PSA schemes (PSA- and PSA- 3) require an increased number of firings. The frame rate is inversely proportional to the number of firings. While the frame rate decreases, the SNR increases. VII. Results The reduction of front-end hardware complexity can be represented by the ratio of the number of front-end hardware channels required for PSA versus FPA imaging: Complexity of PSA Complexity of FPA = M N. (13) Other theoretical performance metrics of FPA, CSA, and PSA imaging are summarized in Table I and described here in detail. The number of firings per frame required for FPA and PSA imaging is equal to the total number of beams formed: ( ) Θ B FPA =Nsin, and (14) B PSA =KM sin ( Θ ). (15) CSA imaging does not form beams directly; rather, the image is formed from pulse-echo scans acquired from individual array elements. Therefore, the number of firings is simply equal to the number of elements: B CSA = N. (16) The ratio of the number of firings required for the two phased array methods is: ( ) N B PSA = KM B FPA N = M 1 M = N M, N N (17) where the substitution for K is based on (5) with J = M/. Note that this ratio is bounded by: B PSA <, (18) B FPA meaning that the number of firings required for any PSA configuration will never require more than twice the number of firings required for FPA imaging. The frame rate of an ultrasound imaging system is determined by the number of scans required per frame (B), the velocity of sound in the medium (c), and the desired imaging depth (R): F c RB. (19)

11 johnson et al.: proposed phased subarray imaging method 47 ( )( ) ( )( ) 1, K +1 K +1 1 H k [m, n] = k P m 1 N n 1 < 1. (1) 0 otherwise Number of front-end hardware channels Number of firings per frame (B) Frame Rate (F ) TABLE I FPA CSA PSA Exact O( ) Example 1 Exact O( ) Example Exact O( ) Example N N M M 3 ( ) N sin( Θ ) N 181 N N 18 KM sin Θ c 1 4NR sin( Θ ) 39 N c RN 1 N 58 KM 317 c 1 4KMR sin( Θ ) 3 KM Normalized SNR (db) 0 log 10 (N N) log 10 ( N) log in (M KM) The numerical results are given for the following example setup: number of array elements (N), 18; number of subarray elements (M), 3; number of subarrays (K), 7; scan angle (Θ), 90 ; velocity of sound (c), 1430 m/s; imaging depth (R), 10 cm; and half-wavelength array spacing (d = λ min /). This is the basis for the results presented in Table I. The SNR of a single point reconstructed from multiple pulse-echo scans generally is accepted to be: ( ) SNR = 0 log 10 N T NR +SNR 0, (0) where N T is the number of active transmit elements for each scan, N R is the number of receive scans used to construct each image pixel, and SNR 0 is the SNR of the pulseecho signal generated by transmitting and receiving on a single array element. The basis for the equation is that the total signal power is proportional to the number of elements firing simultaneously (N T ), and uncorrelated additive noise on receive, reducing the noise by the root of the number of independent measurements ( N R ). For a given firing event in PSA imaging, there are M transmit elements and M receive elements, resulting in an SNR gain of 0 log 10 (M M) for each subarray image. An additional gain is realized by combining the K subarray images. If all K subarray images are weighted uniformly, as they are in PSA-3 [Fig. 5(f)], then the SNR becomes: SNR PSA =0log 10 (M KM)+SNR 0. (1) However, in cases in which the subarray images are nonuniformly weighted when combined to form the final image, the SNR becomes: ( SNR PSA =0log 10 M ) M Σw k + SNR Σw 0. k () When uniform weights are used, () simplifies to (1). One of the primary advantages of PSA imaging over FPA imaging is the reduced cost and size of the frontend hardware. For the example illustrated in Table I, the FPA front-end requires 18 buffers, time-gain compensation circuits, low-pass filters, and analog-to-digital converters (ADCs). In contrast, the PSA front-end adds M multiplexers (one K : 1 multiplexer for each subarray channel) and reduces the number of buffers, time-gain compensation (TGC) circuits, low-pass filters, and ADCs to 3. In addition to the cost savings this represents, the reduction in size increases the feasibility of incorporating the ADCs into the transducer head or integrating them with the beamformer on a single application specific integrated circuit (ASIC). Compared to the previous work by Karaman and O Donnell [40] the method described here has improved lateral spatial resolution and more closely approximates phased array imaging using the full array. This can be observed by comparing the coarray of the previous method shown in Fig. 5(c) to any of the PSA method in Figs. 5(d) (f). By translating both the transmit and receive subapertures across the full array as in PSA, the FPA response can be achieved. There are multiple factors that influence the motion artifacts present in a PSA image. First, the order in which the subarray beams are acquired will affect motion artifacts. As illustrated in Fig. 6, all subarrays acquire the first beam in the subarray image before acquiring the second, and so on. Although it also is possible to acquire all beams from each subarray consecutively, the order illustrated minimizes the time between acquisitions of the same beam by different subarrays, thus reducing motion artifacts. Second, the total acquisition time for acquiring the beams used to form a single PSA image is longer than that required by FPA imaging. The size of this artifact depends on the total number of beams acquired to form a single image, and, therefore, will be less than twice the size of the FPA artifact (18). Third, application of the in-

12 48 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 5, no. 1, january 005 terpolation filter will influence motion artifacts. Moving objects detected by any of the acquired beams used in the interpolation will effect the reconstructed beam. The impact on the reconstructed beam will depend on both the number of beams used in the interpolation determined by the interpolation filter length and the filter weight of each beam. In general, beams further from the reconstructed beam will have less impact due to the sinc-like nature of the interpolation filter. VIII. Discussion and Conclusions The proposed PSA imaging method provides a balance between conventional FPA imaging having a high frame rate and good SNR but high hardware complexity and classical synthetic aperture imaging having a high frame rate and low hardware complexity, but with poor SNR. The PSA imaging significantly reduces the front-end hardware complexity compared to FPA imaging at the expense of a slight decrease in frame rate and SNR, in which the SNR of the proposed method is significantly higher than CSA imaging. The aperture functions for both the subarrays and the full array were assumed to be uniform with no apodization. In addition, the subarray spacing was assumed to be half thesubarraysize(j = M/). These two requirements allow the full FPA coarray to be a linear combination of the PSA cosubarrays. Using arbitrary apodization or different subarray spacing requires a more generalized restoration filter. Methods for calculating such generalized subarray filters are presented in [50]. One of the most exciting potential applications of the synthetic aperture and subarray imaging methods is their application to 3-D imaging using -D transducer arrays [51] [53]. 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Ultrason., Ferroelect., Freq. Contr., vol. 50, no. 11, pp , 003. Jeremy A. Johnson (S 9) received his B.S. in electrical engineering and a minor in mathematics with honors from Walla Walla College, College Place, Washington, in He received his M.S. and Ph.D. in 1999 and 003, respectively, in electrical engineering from Stanford University, Stanford, California. He completed the Biodesign Innovation Fellowship at Stanford University in 004. Jeremy is a Senior R&D Engineer at Medtronic Vascular, Santa Rosa, CA. He has several summers of industry experience. He worked as a software engineer at Interactive Northwest, Inc., Tualatin, OR, during the summer of 1995; worked as an ASIC design engineer at Intel, Inc., Hillsboro, OR, during the summers of 1996 and 1997; performed research in color science at Sony Research Laboratories, San Jose, CA, during the summer of 1998; performed research in computer vision at Hughes Research Laboratories, Malibu, CA, during the summer of 1999; and developed an endoscopic calibration routine for image-enhanced endoscopy at Cbyon, Inc., Palo Alto, CA, during the summer of 000. His research interests include medical imaging, computer-aided diagnosis, and surgical navigation systems. Mustafa Karaman (S 88 S 89 M 89 M 93 M 97) was born in Balıkesir, Turkey, in He received the B.Sc. degree from the Middle East Technical University, Ankara, Turkey, and the M.Sc. and Ph.D. degrees from Bilkent University, Ankara, Turkey, in 1986, 1988, and 199, respectively, all in electrical and electronics engineering. From 1993 to 1994, he was a post-doctoral fellow in the Biomedical Ultrasonics Laboratory in the Bioengineering Department, University of Michigan, Ann Arbor. From 1995 to 1996, he was on the faculty with the Electrical and Electronics Engineering Department of Kırıkkale University, Turkey, first as Assistant Professor and later as Associate Professor. In 1996, he joined Başkent University, Ankara, Turkey, as the Chairman of Electrical and Electronics Engineering and Acting Chairman of the Computer Engineering Department and served in founding these departments. He was a visiting scholar in the Biomedical Ultrasonics Laboratory at the University of Michigan, Ann Arbor, and in the E. L. Ginzton Laboratory at Stanford University, Stanford, California, in the summer terms of and 1999, respectively. Between , he was with the E. L. Ginzton Laboratory at Stanford University, Stanford, California, as a visiting faculty in electrical engineering. In 00, he joined Işık University, Istanbul, Turkey, where he is currently working as a faculty member in Electronics Engineering. In 1996, he was awarded H. Tuĝaç Foundation Research Award of Turkish Scientific and Technical Research Council for his contributions to ultrasonic imaging. His research interests include signal and image processing, ultrasonic imaging and integrated circuit design. Dr. Karaman is a member of the IEEE. Butrus T. Khuri-Yakub (S 70 S 73 M 76 SM 87 F 95) was born in Beirut, Lebanon. He received the B.S. degree in 1970 from the American University of Beirut, Beirut, Lebanon, the M.S. degree in 197 from Dartmouth College, Hanover, NH, and the Ph.D. degree in 1975 from Stanford University, Stanford, CA, all in elecrical engineering. He joined the research staff at the E. L. Ginzton Laboratory of Stanford University in 1976 as a research associate. He was promoted to a senior research associate in 1978 and to a professor of electrical engineering (research) in 198. He has served on many university committees in the School of Engineering and the Department of Electrical Engineering at Stanford University. Presently, he is the Deputy Director of the E. L. Ginzton Laboratory.