Optimization of sparse synthetic transmit aperture imaging with coded excitation and frequency division

Ultrasonics 43 (2005) 777–788 www.elsevier.com/locate/ultras

Optimization of sparse synthetic transmit aperture imaging with coded excitation and frequency division Vera Behar a

a,*

, Dan Adam

b

Institute for Parallel Processing, Bulgarian Academy of Science, Acad. G. Bonchev Str., 25-A, Sofia 1113, Bulgaria b Department of Biomedical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel Received 28 February 2005; received in revised form 7 June 2005; accepted 15 June 2005 Available online 20 July 2005

Abstract An effective aperture approach is used for optimization of a sparse synthetic transmit aperture (STA) imaging system with coded excitation and frequency division. A new two-stage algorithm is proposed for optimization of both the positions of the transmit elements and the weights of the receive elements. In order to increase the signal-to-noise ratio in a synthetic aperture system, temporal encoding of the excitation signals is employed. When comparing the excitation by linear frequency modulation (LFM) signals and phase shift key modulation (PSKM) signals, the analysis shows that chirps are better for excitation, since at the output of a compression filter the sidelobes generated are much smaller than those produced by the binary PSKM signals. Here, an implementation of a fast STA imaging is studied by spatial encoding with frequency division of the LFM signals. The proposed system employs a 64-element array with only four active elements used during transmit. The two-dimensional point spread function (PSF) produced by such a sparse STA system is compared to the PSF produced by an equivalent phased array system, using the Field II simulation program. The analysis demonstrates the superiority of the new sparse STA imaging system while using coded excitation and frequency division. Compared to a conventional phased array imaging system, this system acquires images of equivalent quality 60 times faster, when the transmit elements are fired in pairs consecutively and the power level used during transmit is very low. The fastest acquisition time is achieved when all transmit elements are fired simultaneously, which improves detectability, but at the cost of a slight degradation of the axial resolution. In real-time implementation, however, it must be borne in mind that the frame rate of a STA imaging system depends not only on the acquisition time of the data but also on the processing time needed for image reconstruction. Comparing to phased array imaging, a significant increase in the frame rate of a STA imaging system is possible if and only if an equivalent time efficient algorithm is used for image reconstruction. Ó 2005 Elsevier B.V. All rights reserved. Keywords: Synthetic aperture; Stochastic optimization; Pulse compression; Orthogonal frequency division

1. Introduction Conventional phased array imaging systems employ all elements of the transducer during both transmit and receive during each excitation cycle, while employ-

*

Corresponding author. E-mail addresses: [email protected] (V. Behar), [email protected]. ac.il (D. Adam). 0041-624X/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2005.06.005

ing delays in order to steer the beam and scan a 2D plane. In the receive mode, dynamic (or composite) focus is used, by adjusting the delays of transducer elements as a function of the range (depth) at which the focusing is desired. In the transmit mode, usually the focus point is set in the middle of the depth being imaged. At the focus point, the lateral beamwidth is the smallest (and the best lateral resolution is therefore obtained) while away from the focus point, the lateral beamwidth increases. The spatial resolution of

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the ultrasound image can be improved by using several transmit beams during the interrogation of each sector, each of which is focused at a different range of depths [1]. It is done in modern ultrasound imaging systems at the cost of a decrease of the acquisition time, proportionally to the number of transmit foci [2]. An alternative way to obtain an appropriate spatial resolution, without a decrease in acquisition time, is to use the synthetic transmit aperture (STA) technique. This method makes it possible to generate images with dynamic focusing, during both transmit and receive, while maintaining or even drastically decreasing the acquisition time. In conventional STA imaging, a single transducer element is used to transmit, while all the elements receive the echo signals [3]. All the elements are excited sequentially one after the other, and the echoes received by all elements are recorded and stored in computer memory. Then all the echo data are focused synthetically by an appropriate algorithm. A disadvantage of an STA imaging system is the increased amount of the RF-data needed for image reconstruction. For an N-element array, N echo recordings are required to form a conventional phased array image, and, however, N Æ N echo recordings are required to synthesize a STA image. This disadvantage can be overcame to some extent, if only a few elements, M, act as transmitters. In that case M Æ N echo recordings are required to synthesize a STA image, where M < N [4]. This is equivalent to using of a sparse array in transmit. The sparse STA imaging offers image formation at higher frame rates, which makes this method very attractive for real-time 3D ultrasound imaging. The frame rate of an imaging system is defined as Frame rate ¼ 1=T processing ; and T processing ¼ T acquisition þ T reconstruction where Tprocessing is the time required for STA image formation, Tacquisition is the time required for RF-data acquisition, Treconstruction is the computational time for image reconstruction. The major advantage of a STA imaging system is the significant decrease in the acquisition time of the data. Comparing to phased array imaging, however, a significant increase in the frame rate of a STA imaging system is possible if and only if an equivalent time efficient algorithm is used for image reconstruction. A shortcoming of a sparse STA imaging is the low signal-to-noise ratio (SNR), caused by the use of a single transmit element [3]. Various approaches have been suggested to increase the SNR in the STA imaging. A popular approach is to use multiple elements in transmit, in order to create a spherical wave and, therefore, improve the SNR proportionally to the number of elements used for creating the transmit sub-aperture [5]. Another approach to improve the SNR is to use temporally encoded signals (e.g. linear frequency modulation (LFM)

or phase shift key modulation (PSKM)) for excitation of the transducer (temporal encoding) or to fire all M active transducer elements simultaneously (spatial encoding) [6]. The relation between the employed effective aperture function and the resultant radiation pattern of the imaging system can be used as a strategy for analysis and for optimization of a sparse STA imaging system. Actually, at the focal distance a focused beams directivity function is the Fourier transform of its aperture function. By reciprocity this applies to both transmit and receive, and when synthesizing, a focus the focal point of the synthesized beam is at the point of interest. Since the two-way radiation pattern of a system is the Fourier transform of the effective aperture function, the transmit and receive radiation patterns can be optimized by selecting the appropriate transmit and receive aperture functions, to produce the ‘‘desired’’ effective aperture of the imaging system. Thus, when the desired effective aperture of a system is defined, it also provides the two-way radiation pattern that should be used, with the appropriate width of the main-lobe and its sidelobes. In STA imaging, the transmit aperture function depends not only on the number of transmit elements, but also on their geometrical locations within the array. The receive aperture function depends on the length of a physical array and the apodization weights applied to the receiver elements. Thus, the shape of the effective aperture function and, therefore, the shape of the two, one-way radiation patterns of a system, can be optimized depending on the positions of the element in transmit and the weights of the element in receive. In this paper, a new two-stage algorithm is presented for a sparse STA imaging system that optimizes both the positions of the element and their weights. The first stage of the algorithm employs a new modification of the simulated annealing algorithm as a method for optimizing the positions of transmit elements, given a set of known apodization functions. In the second stage, the appropriate apodization function is selected. The new method is applied to a sparse STA imaging system that utilizes a linear array with four transmit elements and 64 receive elements. Since the STA imaging system suffers from low SNR and low frame rate—two studies are performed. In order to increase the SNR, temporal encoding is investigated for two types of coded signals—linear LFM and binary PSKM. In order to decrease the acquisition time as much as possible, spatial encoding with frequency division of LFM signals is studied, for the case of two and the case of four orthogonal signals. The sparse STA imaging system is evaluated by comparing its performance with that of an equivalent phased array system. This analysis is done by calculating, for both systems, a two-dimensional point spread function (PSF), using the Field II simulation program [10].

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2. Sparse synthetic transmit aperture imaging

RF-DATA ACQUISITION

Consider a sparse synthetic transmit aperture system with coded excitation. The system employs an array transducer with N elements and k/2 inter-element spacing. In the transmit mode, the code generator produces modulated signals (either LFM or PSKM) with a constant envelope. Only few number of array elements (M) are fired while all N elements receive the echoes (Fig. 1). The signal received at each element is acquired and stored in the computer memory. In order to decrease the acquisition time, part or all transmit elements can be fired simultaneously, i.e. the spatial coding scheme can be applied. The RF-signal received at the nth element, when mth element transmits, can be written as sm;n ðtÞ ¼ em ðtÞ  gm ðtÞ  gn ðtÞ  fscat ðtÞ  ha ðtÞ

ð1Þ

where em(t) is the coded signal applied to the mth transmit element, gm(t) and gn(t) are the impulse response of the mth transmit and the nth receive elements, respectively, fscat(t) is the scatterer strength, while ha(t) is the medium impulse response. After quadrature demodulation, the RF-signal sm,n(t) is transformed into the complex signal Sm,n(t), which is processed by a compression filter (Fig. 2). The signal at the output of a filter with an impulse response P(t) is Rm;n ðtÞ ¼ FFT1 fP ðf Þ  U m;n ðf Þg

Rm ðr; hÞ ¼

Rm;n ðt  sm;n Þ expð2pf0;m sm;n jÞ

n¼1

Transmit Aperture 1

Transmit Aperture 2

. . .

Transmit Aperture M

Receive Aperture 1

Receive Aperture 2

. . .

Receive Aperture M

Q-detector Filter1

Q-detector Filter2

. . .

Q-detector Filter N

QUDRATURE DEMODULATION & SIGNAL COMPRESSION

(r,θ)-image 1

(r,θ)-image 2

(r,θ)-image M

PARTIAL BEAMFORMING

Σ

ENVELOPE DETECTION LOG-COMPRESSION High resolution (r,θ)-image

Fig. 2. Processing steps of synthetic aperture imaging.

Fig. 3. Synthetic aperture focusing at a point (r, h).

ð3Þ where t = 2r/c, sm,n = (2r  rm  rn)/c, and where rm is the distance from mth transmit element to the current focal point, rn is the distance from the current focal point to the nth receiver element, f0,m is the carrier frequency of the RF signal that excites the mth transmit element and c is the velocity of sound. When no spatial encoding with frequency division is applied, f0,m = f0. The final high resolution image should have a number of lines (Nline) that is sufficiently high to satisfy the Nyquist sampling criteria for the full array of N elements: N line > sinðDb=2Þð2N  1Þ

Fig. 1. Synthetic aperture data acquisition.

. . .

ð2Þ

where P(f) = FFT{P(t)} and Um,n(f) = FFT{Sm,n(t)}. Since the wave emanating from a single transmit element is spherical, the return echo at each receiver element contains information from all directions, and scan lines can be formed in all directions synthetically by a computer. The principle of beamforming in a sparse STA imaging is shown in Fig. 3. The complex amplitude of the beamformed signal, which corresponds to a specific focal point at (r, h) within the region of view, is calculated by N X

779

ð4Þ

where Db is the angle sector of view. According to (4), the intensity of the final image is formed line by line as follows:

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2   X M   I k ðr; hk Þ ¼  Rm ðr; hk Þ ;   m¼1 where hk ¼ 

Given M; N and fBgK ; choose ði1 ; i2 ; . . . ; iM ÞK to minimize W subject to SL < Q

Db Db þ ðk  1Þ 2 N line  1

ð5Þ

for image lines k = 1, 2, 3, . . . , Nline and where hk is the direction of the kth image line.

3. Optimization of a sparse array For a sparse STA imaging system with an array with N-elements, the two-way radiation pattern is evaluated as the Fourier Transform of the effective aperture function eN, defined as M X am B; and eN ¼ m¼1

am ¼ ½0; 0; . . . ; im ; . . . ; 0 ;

where im ¼ 1

ð6Þ

where am is the transmit aperture during the mth firing, B is the appodization function applied to the receiver elements, and is the convolution operator. The speed of the image acquisition is determined by the number of transmit elements (M), M N. Since the geometrical locations of the transmit elements in a sparse array system impact the two-way radiation pattern of that system, the image quality parameters, the lateral resolution and contrast, all depend on the locations of the transmit elements within the sparse array (i1, i2, . . . , iM). Since the weighting applied to each receiver element also impacts the radiation pattern of the system, the image quality also depends on the type of the apodization function (B). Therefore, the optimization of a sparse STA imaging system can be formulated as an optimization problem of both the location of the elements of the sparse array in transmit and the weights assigned to the elements of the full array during receive [7]. Different algorithms have been proposed for optimization of the locations of the transmit elements in a sparse array—genetic, linear programming and simulated annealing algorithms [7]. For most cases the optimization criterion is minimal sidelobe peak of the radiation pattern. In this report, another optimization criterion is proposed. It is the minimal width of the mainlobe (W) combined with a condition on the maximum sidelobe level (SL < Q). It is suggested here to divide the optimization process into two stages. In the first stage, the optimal positions of transmit elements ((i1, i2, . . . , iM) are found, for a set of known apodization functions {Bk}, k = 1, 2, . . . , K. Such a set of apodization functions may include several well-known window-functions (Hamming, Hann, Kaiser, Chebyshev and etc.). At this stage, the optimization criterion can be written as follows:

ð7Þ

where Q is the threshold of acceptable level of the sidelobe peak. In the second stage, the final layout of transmit elements is chosen, which is a layout that corresponds to the most appropriate apodization function B = (b1, b2, . . . , bN). This choice is a compromise between minimal width of the mainlobe and the acceptable level of the peak of the sidelobes. Mathematically, it can be written as follows: Given M;N ; fBgK and fi1 ; i2 ;. . .; iM gK ; choose ðb1 ; b2 ;. . .; bN Þ to minimize W subject to SL < Q

ð8Þ

where {i1, i2, . . . , iM}K are the selected positions of transmit elements, as found at the first stage of the optimization. One way of selecting the positions {i1, i2, . . . , iM}K is by using a modification of the simulated annealing algorithm based on a Monte Carlo simulation. This approach was suggested initially for combinatorial optimization by Kirkpatrick et al. [8]. The simulated annealing algorithm realizes an iterative procedure that is determined by simulation of the arrays with variable transmit element positions. In order to speed up the simulation process it is assumed that two of the M transmit elements are always the two outer elements of the physical array; their positions are not changed and are assigned numbers 1 and N. The positions of the other transmit elements are shifted randomly, where a shift in position to the left or to the right has equal probability (of 0.5). Once the process is initiated, with an initial layout of transmit elements I 0 ¼ ði01 ; i02 ; . . . ; i0M Þ, a neighbor layout I 1 ¼ ði11 ; i12 ; . . . ; i1M Þ is generated, and the algorithm accepts or rejects this layout according to a certain criterion. The acceptance is decided stochastically and may be described in terms of probability as

P¼

8 1 > > > > > < expðDW =T k Þ

if DW < 0 & DSL < 0 if DW > 0 & DSL < 0

> if DW < 0 & DSL > 0 expðDSL=T k Þ > > > > : expðDW =T k Þ  expðDSL=T k Þ if DW > 0 & DSL > 0

ð9Þ

where P is the probability of acceptance, DW is the difference of width of the mainlobe, DSL is the difference of the height of the peak of the sidelobe between the current configuration of transmit elements and the best one obtained at preceding steps. Tk is the current value of temperature, where the current temperature is evaluated as Tk = 0.95Tk1, and the algorithm proceeds until the number of iterations reaches the final value. A pseudo-code of the proposed simulated annealing algorithm is given in Fig. 4.

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781

Fig. 5. LFM signal (chirp) and its spectrum.

Fig. 4. The simulated annealing algorithm.

4. Temporal encoding One major drawback of all STA imaging systems is the loss in SNR that is due to the small number of active transmit elements being used. A well-known approach that produces an increase of the SNR is to transmit long modulated signals, without an increase of the peak amplitude of the signals (temporal encoding). Two variants of temporal encoding will be considered here. The first variant of temporal encoding is a linear frequency modulated pulse with a constant envelope (LFM chirp): sðtÞ ¼ cosð2pf1 t þ pbt2 Þ;

where t  ½0; T S

ð10Þ

where f1 is the initial frequency, and b is the frequency sweep rate. The frequency bandwidth corresponding to the duration TS is Df = bTS and the central frequency of transmission is f0 = f1 + Df/2. The chirp amplitude spectrum has Fresnel ripples that are due to the sharp rise-time of the envelope. An effective way to reduce the Fresnel ripples is to apply amplitude tapering to the generated chirp [9]. In that case the signal applied to the mth transmit element can be written as em ðtÞ ¼ AðtÞ  sm ðtÞ;

where t  ½0; T S

ð11Þ

where A(t) is the tapering function. An example of a tapered chirp and its spectrum are shown in Fig. 5. The pulse duration is 25 ls, and the frequency bandwidth is 6 MHz. The corresponding time-bandwidth product is TB = 150. According to [9], the optimal mismatched compression scheme can be realized using a Dolph– Chebyshev filter that includes amplitude tapering of the chirp by a Lanczos window. An example of the LFM signal of Fig. 5, after being compressed as described above, is shown in Fig. 6. The other variant of temporal encoding is a binary phase shift key modulation (PSKM) signal, which is a

Fig. 6. LFM signal (chirp) at the output of a mismatched filter.

sinusoidal signal modulated by any pseudo-noise (PN) sequence. Thus, the signal applied to the mth transmit element can be written as em ðtÞ ¼ BL  sinð2pf0 tÞ; L ¼ T S =T chip

where t  ½0; T S ; and ð12Þ

where BL is the PN sequence of length L, and Tchip is the chip duration. One such PN sequence of length 16 and the corresponding PSKM signal are shown in Fig. 7. The duration of a signal is 25 ls, the frequency bandwidth is 6 MHz and the corresponding time-bandwidth product is TB = 150. The PSKM signal at the output of a matched filter is shown in Fig. 8. Comparison of the two outputs of the compression filters (Figs. 6 and 8) demonstrates the advantage of the LFM signal, since the level of the peak of the sidelobe of the compressed

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V. Behar, D. Adam / Ultrasonics 43 (2005) 777–788

Fig. 7. Binary PSKM signal.

elements, it is possible to reduce the acquisition time if some or all of the transmit elements are excited simultaneously by coded signals that can be distinguished from each other at the receiver elements. These coded signals should have good auto-correlation properties, if good axial resolution is required. But in order to obtain good separation of the signals at the output of the compression filters, however, the coded signals should have low cross-correlation properties, i.e. they should be orthogonal. Two approaches can be used to create such orthogonal signals [6]. One approach is to use binary PSKM signals, modulated by PN sequences that are either Golay pairs, Gold sequences or m-sequences. Unfortunately, the auto- and cross-correlation properties of PSKM signals are functions of the code length, which is a significant disadvantage for their applicability to medical ultrasound. Another approach to create orthogonal signals is to divide the available frequency band into K sub-bands and let each LFM signal sweep within its own frequency sub-band. In order to implement frequency division of the signals, a broadband receiver should be used. Such LFM signals can be written as ek ðtÞ ¼ AðtÞ  cosð2pf1;k t þ pbt2 Þ; t  ½0; T S ; k ¼ 1; . . . ; K

ð13Þ

where the initial frequency f1,k and the sweep rate b are determined by f1;k ¼ f1 þ Df ðk  1Þ=ðK  1Þ and b ¼ Df =ðK  1Þ

Fig. 8. Binary PSKM signal at the output of a matched filter.

LFM signal is much lower (80 dB) than that of the PSKM signal (12 dB). The level of the peak of the sidelobe of the PSKM signals is apfunction of the PN code ffiffiffi length and equal to 20 logð2= LÞ, where L is the code length. Since the PSKM signal in Fig. 7 was modulated by the PN sequence of length 16, the level of the peak of the sidelobe, at the output of a matched filter, is very high (12 dB) and not acceptable for temporal encoding (Fig. 8). The study shows that LFM signals are good candidates for temporal encoding and assure good axial resolution and contrast of the generated images.

5. Frequency division for fast imaging Since the acquisition time of a sparse STA imaging system is determined by the number of active transmit

ð14Þ

This approach of frequency division allows the data acquisition time to be decreased K times, when K orthogonal signals are transmitted simultaneously. However, this method causes deterioration of the axial resolution by K times, because the frequency sub-band of each LFM signal is divided K times, and consequently, the duration of each LFM signal at the output of a compression filter increases K times. The frequency division approach is demonstrated in Fig. 9, where the frequency band of 6 MHz is partitioned into four subbands and four orthogonal LFM signals are created using the frequency division approach. It is apparent from Figs. 6 and 9 that the frequency division causes deterioration of the axial resolution of images. Additionally, it causes deterioration of the contrast: when only one LFM signal is used to excite the transmit elements sequentially one after the other, the level of the peak of the sidelobe of the compressed signal is 80 dB; however, when four orthogonal LFM signals excite four transmit elements simultaneously, the level of the peak of the sidelobe of the compressed signal increases to 50 dB. Moreover, it should be kept in mind that the frequency division method can entail a great reduction of the SNR when the central frequency of

V. Behar, D. Adam / Ultrasonics 43 (2005) 777–788

783

Fig. 9. Frequency division (four chirps).

orthogonal LFM signals differ significantly from the central frequency of a transducer. Obviously, the appropriate number of orthogonal LFM signals should be chosen as a compromise between the following parameters of the imaging system—frame rate, spatial resolution, contrast and SNR.

6. Simulation results 6.1. Sparse array optimization Computer simulations were performed in order to optimize the design and performance of a sparse array probe, to be used for synthetic transmit aperture imaging. The example given here is of a 64 elements sparse array probe, where 64 active elements are used in receive and only four elements are used in transmit. The properties of the system are optimized using the two-stage algorithm described in Section 3. First, the optimal positions of transmit elements are found for three apodization functions—Boxcar (i.e. no apodization of the receiver elements), Hamming and the Blackman–Harris. For each apodization function, the positions of transmit

elements are shifted until optimal performance is obtained, as described earlier, using the simulated annealing algorithm presented in Fig. 4. In order to obtain a radiation pattern with a sharper mainlobe, the optimization criterion was formulated as the minimal width of the mainlobe at 20 dB ( instead of at 6 dB) below the maximum where the condition that the maximal level of the sidelobe peak is below 50 dB. The positions of transmit elements that were found to optimize the performance of the system, studied for a physical array with k/2 element spacing, together with the achieved widths of the mainlobe (at 6 dB, 20 dB and 40 dB) and the levels of the peaks of the sidelobe, are all presented in Table 1. Both optimized functions, the effective aperture function and the corresponding two-way radiation pattern, are plotted for each apodization function, for the Boxcar apodization—Fig. 10, for the Hamming apodization—Fig. 11 and for the Blackman–Harris apodization—Fig. 12. It may be seen that the apodization reduces the levels of the peaks of the sidelobes from 33 dB to 100 dB, but at the cost of widening the mainlobe of the radiation pattern. Since the dynamic range of a computer monitor is limited to about 50 dB, the comparison here is limited to two

Table 1 Numerical results obtained after employing the two stages of the optimization Optimized positions of transmit elements

Receiver apodization

1, 2, 63, 64 1, 26, 39, 64 1, 21, 44, 64

– Hamming Blackman–Harris

Mainlobe width (°) 6 dB

20 dB

40 dB

0.33 1.34 1.32

1.11 2.92 4.33

3.2 6.2 11.16

Sidelobe peak level (dB) 33 50 100

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Fig. 12. Optimized array with Blackman apodization.

Fig. 10. Optimized array without apodization.

imaging systems (synthetic transmit aperture and phased array), where the sparse array is used with the Hamming apodization and the locations of the transmit elements are set to be at positions 1, 26, 39 and 64. 6.2. Comparison analysis

Fig. 11. Optimized array with Hamming apodization.

As stated above, the STA imaging system with coded excitation is compared here to a conventional phased array system, using the following quality parameters: the system complexity, the transmitted power per image, the maximal intensity per transmission, the spatial resolution, the contrast and the SNR. Some of the quality parameters (the transmitted power, the maximal intensity, the SNR) are estimated analytically, while the other parameters (the spatial resolution and the contrast) are estimated by calculating the two-dimensional PSF using the Field II simulation program [10]. The parameters of the imaging systems that were optimized as described above, are given in Table 2.

Table 2 Parameters of the two imaging systems modeled by Field II Parameter name

Number of transmit elements Number of receiver elements Number of emissions Apodization Central frequency Focus in transmit Transducer fractional bandwidth Excitation pulse Max. amplitude of an excitation pulse Number of orthogonal pulses

Phased array

Synthetic transmit aperture

Notation

Value

Notation

N N Nline Hamming f0 F BW Sin, 2 cycle APA K

64 64 127

M 4 N 64 Nemis 4, 2, 1 Hamming f0 4 MHz – BW 100% Chirp, 25 ls, 6 MHz, TB = 150 ASTA 1 K 1, 2, 4

4 MHz 60 mm 60% 1 1

Value

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When analyzing the performance of the conventional phased array system, the time duration of image acquisition is linearly proportional to the number of scan lines (Nline) and the time required to acquire the echoes from one direction of view (T0), i.e. it is TPA = Nline Æ T0. In order to reduce this time duration, either Nline should be reduced at the cost of lowering the lateral resolution, or T0 should be reduced causing a decrease of the maximal depth of insonification. However, the STA imaging system performs differently, and better: the acquisition time of an image is linearly proportional to the number of emissions (M), i.e. it is TSTA = M Æ T0. Compared with the phased array system, the relative time of acquisition of an image in the STA system is T STA =T PA ¼ M=N line

ð15Þ

Using values from Table 2, the image formation by the STA system is 31 times faster than by the conventional phased array system. This result is obtained with the STA system using four sequential emissions during image formation. Furthermore, the acquisition time in the STA system can be decreased 60 times or even 127 times when, respectively, two or four orthogonal LFM signals are used for transducer excitation. In that case, two emissions, or just one emission, are used during image formation. Of course, the decrease in acquisition time by 60 or 127 times is achieved at the cost of a slight degradation of the axial resolution. The transmitted power per image of the conventional phased array imaging system is linearly proportional to the number of array elements (N), the number of scan lines (Nline) and the average power transmitted by each array element ðp0 Þ, i.e. it is P PA ¼ A2PA  N  N line . For the STA imaging system with coded excitation, the same parameter is P STA ¼ A2STA  M  TB, where TB is the timebandwidth product of the coded signal and M is the number of transmit elements. Compared with a phased array system, the relative transmitted power per image employed by the STA imaging system is P STA =P PA ¼ ðA2STA  M  TBÞ=ðA2PA  N  N line Þ

ð16Þ

Using the values from Table 2, it is found that the relative transmitted power per image by the STA imaging system is 0.064 and less. The maximal signal intensity that a phased array system can utilize during transmit is I PA ¼ A2PA  N 2 . The same parameter for a STA imaging system is I STA ¼ A2STA  K 2 , where K is the number of simultaneously transmitted orthogonal signals. Compared with a phased array system, the relative maximal intensity utilized by the STA imaging system during transmit is I STA =I PA ¼ ðA2STA  K 2 Þ=ðA2PA  N 2 Þ

ð17Þ

Using the values from Table 2, one may conclude that the relative maximal intensity utilized by a STA imaging system during transmit is very insignificant. It equals to

785

0.0039 only, when all transmit elements are fired simultaneously (K = 4), excited by orthogonal signals. The extremely low intensity of the signal used during transmit is an advantage of STA systems. This advantage is extremely important in special medical applications where there is an upper limit to the intensities allowed (e.g. for safe diagnosis of fetuses). The relative SNR of the echo signals achieved by a sparse STA system and by an equivalent phased array system can be estimated when the noise in a system is assumed to be primarily determined by uncorrelated electrical noise at the receiver elements. According to [4], the signal amplitude of a conventional phased array system is linearly proportional to the number of transmit elements (N), i.e. Signal  (APA Æ N), while the noise is inversely proportional to the number of receive elements pffiffiffiffi (N), i.e. Noise  1=ð N Þ. Therefore, the SNR of a phased array system (that uses the same number of transmit and receive elements) can be expressed as pffiffiffiffi SNRPA  APA  N  N ð18Þ The number of signals added together during beamforming is determined as a product of the number transmit elements (M) and the number of receive elements (N). Taking into account that the signal compression is done before beamforming, the SNR is proportional to the product of the number of simultaneously excited transmit elements (K), the square root of the number of transmit elements (M), the square root of the number of receive elements (N) and the square root of the timebandwidth product of the coded signals (TB): pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SNRSTA  ASTA  K  TB  M  N ð19Þ When compared to a phased array system, the relative SNR of a STA system with coded excitation can be written as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SNRSTA =SNRPA  ðASTA  K  TB  M Þ=ðAPA  N Þ ð20Þ Using the values from Table 2 and taking into account that TB = 150—for K = 1, TB = 75—for K = 2 and TB = 37.5—for K = 4, the calculated relative SNR of a sparse STA imaging system is proportional to 0.38— for K = 1, 0.55—for K = 2 and 0.77—for K = 4. It corresponds to 8.4 dB, 5.2 dB and 2.3 dB, respectively. It can be seen that the SNR is very low when the transmit elements are fired sequentially, i.e. for K = 1. This degradation is compensated by 3 dB when using two orthogonal LFM signals for the coded excitation. The SNR is improved further when using four orthogonal LFM signals to fire all transmit elements simultaneously. Therefore, the SNR of a sparse STA imaging system is within 6 dB of an equivalent phased array system only if orthogonal LFM signals are used for coded excitation.

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In order to compare the image quality (resolution and contrast) produced by both systems, a two-dimensional point spread function (PSF) of each system was simulated using Field II. The parameters of the two systems being modeled here, are given in Table 2. A sector scan of 90° is produced by shifts in steps of 0.71°, from 45° to 45°, which corresponds to 127 scan lines. A simulated point target is located at a distance of 60 mm corresponding to the transmit focus of the phased array system. Hamming apodization is applied to both imaging systems. For the sparse STA imaging system, the optimized positions of the transmit elements are used , i.e. (i1, i2, i3, i4) = (1, 26, 39, 64). In order to compare the radiation pattern of each system, all simulated images are displayed in the polar plane ‘‘distance—angle’’. But in order to compare the lateral resolutions, the radiation

pattern of each system is displayed by plotting the envelope received from a distance of 60 mm as a function of the scan angle. The axial resolutions are compared by displaying the envelope of the 0°-scan line as a function of the distance. The PSF of the conventional phased array system is presented in Fig. 13. The PSF of a sparse STA system is presented in Fig. 14, for the case when all transmit elements are fired sequentially one after the other (K = 1), and in Fig. 15 for the case of a doubled frame rate (K = 2). The latter is a case in which the 1st and 39th elements are fired at the first emission, while the 26th and 64th elements are fired at the second emission. The PSF of a sparse STA system in which all transmit elements are fired simultaneously, is presented in Fig. 16. The estimated spatial resolutions, axial and

Fig. 13. PSF of a phased array system.

Fig. 15. PSF of a STA system with two emissions.

Fig. 14. PSF of a STA system with four emissions.

Fig. 16. PSF of a STA system with one emission.

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Table 3 Image quality parameters of the 64-element imaging system performing as a phased array system Receiver apodization

(6 dB)-Spatial resolution

(20 dB)-Spatial resolution

Axial (mm)

Lateral (°)

Axial (mm)

Lateral (°)

– Hamming Blackman–Harris

0.40

1.63 2.44 3.59

0.66

3.11 4.82 7.16

Sidelobe peak level (dB) <70

Table 4 Image quality parameters of the 64-element imaging system performing as a sparse array using 4 transmit elements and employing either 1, 2 or 4 emissions Receiver apodization

Hamming

Number of emissions (Nemis)

4 2 1

(6 dB)-Spatial resolution

(20 dB)-Spatial resolution

Sidelobe peak level (dB)

Axial (mm)

Lateral (°)

Axial (mm)

Lateral (°)

0.36 0.67 1.33

1.55 2.13 2.74

0.62 1.18 2.33

2.97 6.26 10.1

Table 5 Comparison between a phased array system and a sparse STA imaging system Performance characteristics of the imaging system

Phased array

Sparse STA

Relative power per image Relative maximal intensity Relative SNR Relative image acquisition time Beam width (6 dB) Axial resolution (6 dB) Sidelobe level

1 1 0 dB 1 2.44° 0.40 mm <70 dB

<0.064 <0.0039 5.2 dB to 2.3 dB 1/61–1/127 2.13°–2.74° 0.67–1.33 mm 34 dB to 30 dB

lateral, are given in Table 3 for the conventional phased array system and in Table 4 for the sparse STA system. Table 5 summarizes the results of the comparison performed between the two systems, the sparse STA imaging system with coded excitation and an equivalent phased array system. It can be seen that the sparse STA system with coded excitation generates an image of the same quality as the phased array system, when using two emissions at most. Such a system produces images 60 times faster than a phased array system, while using only 6% of the power per image. The detectability and the frame rate of the sparse STA imaging system can be additionally increased using four orthogonal LFM signals for the excitation of the transmit elements. This improvement, however, can be achieved only at the cost of degradation of the axial resolution.

7. Conclusions Sparse synthetic transmit aperture (STA) imaging systems have been proposed as an alternative and supe-

40 34 30

rior approach to the phased array systems. Yet, the sparse STA imaging systems suffer from some deficiencies. It is demonstrated here that with proper design, these deficiencies can be overcome and the sparse STA imaging system can perform extremely well for specific applications. To do so, an effective aperture approach is used for optimization of the sparse STA imaging system, which is operated with coded excitation and frequency division. A two-stage algorithm is proposed for optimizing both the locations of transmit elements within the ultrasound probe and the weights of the receive element. The first stage of the optimization procedure employs a simulated annealing algorithm that optimizes the locations of the transmit elements for a set of apodization functions. At the second stage, an appropriate apodization function is selected. In order to increase the SNR of the synthetic aperture system, LFM signals are used for excitation of the transmit elements. A comparison analysis was performed between two different coded signals, the LFM signals and the binary PSKM signals. The comparison demonstrates that the chirps are the most appropriate kind of signals for the excitation of the transducer, since the filtered signals at the output of a compression filter have much lower sidelobes than those produced when the binary PSKM signals are used. In order to increase the frame rate of a sparse STA system, orthogonal LFM signals are proposed to be used for spatial encoding. The analysis of comparing the performance of two systems that are based on the two approaches, between the sparse STA system with coded excitation and an equivalent phased array system, was done by calculating the two-dimensional PSF of each system, using the Field II simulation program. Both systems employ 64-element linear arrays. The analysis shows that a sparse STA system that uses four transmit elements with two emissions,

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obtains images of equivalent quality to those produced by a conventional phased array system, but acquires data 60 times faster and transmits only at a small fraction of the power the conventional system transmits. The fastest acquisition time is achieved using four orthogonal LFM signals for the coded excitation. In that case, the detectability and the frame rate are increased at the cost of slight degradation of the axial resolution. In real-time implementation, however, the frame rate of a STA imaging system depends not only on the acquisition time of the data but also on the processing time needed for image reconstruction. It is shown in the paper that the STA method sufficiently reduces only the time required for RF-data acquisition. Comparing to a phased array imaging system, it should be kept in mind that the significant increase in the frame rate of a STA imaging system is possible if and only if an equivalent time efficient algorithm is used for image reconstruction.

Acknowledgements This work was partially supported by the Center of Excellence BIS21++ and the Bulgarian National Science Fund- grant MI-1506/05. This work was also partially supported by the Chief Scientist Magnet and

Magneton programs, of the Israel Ministry of Industry and Commerce.

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