Pseudoinverse Decoding Process in Delay-Encoded ... - IEEE Xplore

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Pseudoinverse Decoding Process in Delay-Encoded Synthetic Transmit Aperture Imaging Ping Gong, Michael C. Kolios, and Yuan Xu

Abstract— Recently, we proposed a new method to improve the signal-to-noise ratio of the prebeamformed radio-frequency data in synthetic transmit aperture (STA) imaging: the delay-encoded STA (DE-STA) imaging. In the decoding process of DE-STA, the equivalent STA data were obtained by directly inverting the coding matrix. This is usually regarded as an ill-posed problem, especially under high noise levels. Pseudoinverse (PI) is usually used instead for seeking a more stable inversion process. In this paper, we apply singular value decomposition to the coding matrix to conduct the PI. Our numerical studies demonstrate that the singular values of the coding matrix have a special distribution, i.e., all the values are the same except for the first and last ones. We compare the PI in two cases: complete PI (CPI), where all the singular values are kept, and truncated PI (TPI), where the last and smallest singular value is ignored. The PI (both CPI and TPI) DE-STA processes are tested against noise with both numerical simulations and experiments. The CPI and TPI can restore the signals stably, and the noise mainly affects the prebeamformed signals corresponding to the first transmit channel. The difference in the overall enveloped beamformed image qualities between the CPI and TPI is negligible. Thus, it demonstrates that DE-STA is a relatively stable encoding and decoding technique. Also, according to the special distribution of the singular values of the coding matrix, we propose a new efficient decoding formula that is based on the conjugate transpose of the coding matrix. We also compare the computational complexity of the direct inverse and the new formula. Index Terms— Delay-encoded synthetic transmit aperture imaging, inverse problem, pseudoinverse (PI), singular value decomposition (SVD), ultrasound imaging.

I. I NTRODUCTION

S

YNTHETIC transmit aperture (STA) imaging is a novel imaging technique that has been widely used in different fields such as radar and sonar systems [1], nondestructive testing [2], [3], seismic monitoring [4], and medical ultrasound imaging [5], [6]. It offers dynamic focusing in both transmit and receive, leading to high resolution and detectability over the entire imaging area [7]–[11]. The major limitation of STA ultrasound imaging is the low signal-to-noise ratio (SNR) of the radio-frequency (RF) signals compared with conventional ultrasound (B-mode) imaging. This is due to the fact that only one or a small number elements are excited in each

Manuscript received July 28, 2015; accepted June 7, 2016. Date of publication June 9, 2016; date of current version September 12, 2016. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada, in part by the Canada Foundation for Innovation, and in part by Ryerson University. (Corresponding author: Yuan Xu.) P. Gong is with the Department of Radiology, Mayo Clinic College of Medicine, Rochester, MN 55905 USA (e-mail: [email protected]). M. C. Kolios and Y. Xu are with Ryerson University, Toronto, ON M5B 2K3, Canada (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TUFFC.2016.2578952 .

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transmission under STA mode, resulting in low transmitted power into imaging area [11]. Whereas in B-mode imaging, the RF signals are acquired using focused beams with much greater transmitted power. Various approaches have been proposed to overcome this low SNR limitation in STA imaging involving encoded transmission schemes [12]–[18], which can be generally grouped into two categories: temporal encoding and spatial encoding. Temporal encoding has been applied to STA imaging using linear frequency-modulated pulse (chirp) to increase the transmitted energy and, therefore, to increase the SNR while still retaining axial resolution [12], [13]. Spatial encoding techniques generally encode the excitation pattern of the transmitting elements using binary codes such as [−1, 1] or [0, 1]. Hadamard encoding method encodes the transmission schemes with [−1, 1] codes, which represent the negative and positive transmit √ pulses. The SNR of the RF signals can be increased by I times (I is the total number of active transmitters) compared with the traditional STA [14], [15]. S-sequence encoding uses [0, 1] codes to avoid the negative/inverted pulses as used in Hadamard encoding [16], [17]. Temporal and spatial encoding techniques can also be combined into spatiotemporal encoding such as Hadamard–Golay encoding to improve the SNR further [18]. Gong et al. [19], [20] proposed a delay-encoded STA (DE-STA) imaging method based on the Hadamard encoding technique. The transmission scheme in DE-STA imaging is encoded with half-period delay (t = 1/2 f 0 , where f 0 is the central frequency of the ultrasound wave), rather than with reverting the polarity as in Hadamard encoding. This enables the implementation of DE-STA in commercial scanners. The coding matrix A in the DE-STA method is generated from a Hadamard matrix with [−1, 1] codes. In this case, however, one represents that no delay is added to the transmitted pulse and negative one represents a half-period delay to the transmitted pulse from the transducer element. The coding matrix A is then calculated by replacing all the −1 elements in a Hadamard matrix with e− j 2π f t = e− j π( f / f 0) , where f is an arbitrary frequency in the spectrum. In the decoding process, a decoding matrix D (i.e., A−1 ) is applied to the received RF data. The SNR improvement of the equivalent STA RF data restored from the DE-STA raw data has been shown to be comparable with that in the Hadamard encoding technique. The decoding step in DE-STA was obtained by directly inverting the coding matrix A in [19] and [20]. However, when the signals in DE-STA are degraded by noise,

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the decoding process to obtain the equivalent STA signals is a discrete ill-posed problem for the frequencies close to 0 or 2 f0 [21], i.e., if an arbitrarily small perturbation of the DE-STA data can cause a large perturbation of the restored STA data. When the frequency is close to 0 or 2 f 0 , the coding matrix becomes a matrix with all elements being equal to one, a severely ill-conditioned matrix. As a result, the direct inversion of the coding matrix may be instable in the DE-STA decoding process. A bandpass filter may narrow down the signal spectrum and eliminate part of the unstable frequency components. However, it may also lead to undesired axial resolution loss. Consequently, in such cases, a pseudoinverse (PI) is commonly used instead of direct inverse of matrix A. In this paper, we study the PI decoding process in the DE-STA method using different noise levels. We also apply singular value decomposition (SVD) to the encoding matrix to study the properties of singular values and singular vectors of the matrix. We find that the noise mainly affects the prebeamformed signals from the first transmit channel. The difference in the overall image qualities of the enveloped beamformed images between the direct inverse and PI is negligible. Based on the SVD results, an efficient decoding method is derived to reduce the computational complexity of the decoding process. In Section II, the theory of DE-STA technique in [19] and [20] is first reviewed. The process of SVD implementation is explained. The properties of singular values and singular vectors of the coding matrix are analyzed, and a more efficient decoding process is then derived. The setup for simulation and experiments are introduced finally. Section III shows all the simulation and experiment results. Section IV discusses the influence of PI on the first transmitting element and its minor effect on the overall image quality. The reason for that is also given in this section. The computational complexity of direct inverse and PI is compared. The conclusions are drawn in Section V. II. M ETHODS A. Delay-Encoded Synthetic Transmit Aperture Imaging In DE-STA imaging, I transmitting elements are encoded in L transmission events with various delays tli (l = 1 : L and i = 1 : I ) to form one high-resolution frame (the same I elements for all the L transmissions). tli denotes the delay that is applied to the i th transmitting element in the lth transmission event compared with a reference time (t = 0). We define an L-by-I delay matrix T whose element is tli . When there is no delay, tli is 0; when there is delay, tli is half period (1/2 f 0 , where f0 is the central frequency of the ultrasound wave). When multiple elements are excited together, the received signal equals to the summation of the equivalent received signals when the same multiple elements are excited individually with the same delay. Therefore, for the lth transmission, we have I  i=1

pik (t − tli ) = m lk (t)

(1)

where m lk (t) is the RF signal in DE-STA, which is received by the kth (k = 1 : K ) receiving element in the lth transmission when multiple elements are excited together; pik (t) is the equivalent traditional STA signal, which is received by the kth receiving element when only the i th (i = 1 : I ) transmitting element is activated (while all the other imaging conditions are the same as in DE-STA). According to the translation properties of Fourier transform, applying a time delay tli to the received signal pik (t) is equivalent to multiplying the signal spectrum Pik ( f ) by a factor of Ali ( f ) = e− j 2π f tli

(2)

where f is any frequency in the spectrum. Therefore, after applying Fourier transform to both sides of (1), it can be transformed into the frequency domain at each frequency as I 

Ali ( f )Pik ( f ) = Mlk ( f )

(3)

i=1

or AP = M

(4)

where Mlk ( f ) is the Fourier transform of m lk (t) and A, P, and M are the matrices with elements of Ali ( f ), Pik ( f ), and Mlk ( f ), respectively. A is called the coding matrix that is constructed from the delay matrix T. Thus, the column index of A corresponds to a particular transmission element position, and each row includes the delays applied to all transmitting elements in one transmission event. Note that the coding matrix Ali ( f ) depends on frequency. Equation (4) can be solved to yield P, the spectrum of the traditional STA data. After obtaining the Pik ( f ) for each frequency, an inverse Fourier transform is used to obtain pik (t), the traditional STA data. Finally, low-resolution images can be formed using delay and sum from the restored STA data pik (t) and they can be combined together to yield a highresolution image as in STA imaging. It is worth noting that when f equals to 0 or 2 f 0 , the coding matrix will be a square matrix with all the elements equaling to 1, which cannot be inversed stably. To deal with this, the signal is processed by a bandpass filter to cut off the frequencies of 0 or 2 f 0 . B. Pseudoinverse in DE-STA Imaging The goal of decoding is to recover P from M in a stable manner, since M is usually contaminated by noise. To obtain P, one can multiply both sides of (4) with the decoding matrix D for each frequency within the frequency bandwidth of the RF signals P = DM.

(5)

In principle, D = A−1 . However, because of the measurement noise in M, the direct inverse may not be stable. A PI method can be used instead to solve (5). PI is implemented by first applying SVD to the coding matrix A, then we have A = USV∗ =

R  i=1

u i σi v∗i

(6)

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where U = (u 1 , u 2 , . . . u R ) and V = (v1 , v2 , . . . v R ) are the two orthonormal matrices and u i and vi are the i th column of U and V, which are defined as the left and right singular vectors of A; R is the rank of A; and S is a diagonal matrix that contains all the singular values (σi ) of A, distributing in a nonincreasing order (such that σ1 ≥ σ2 ≥ · · · ≥ σ R ≥ 0). Since both U and V are orthonormal matrices, we also have D = A−1 = VS−1 U∗

(7)

A∗ = VS∗ U∗

(8)

and

or D = A−1 = VS−1 U∗ =

R  vi u ∗ i

i=1 ∗

A =

R 

σi

vi u ∗i σi∗

Fig. 1. Singular values of coding matrix A in DE-STA method at three different frequencies (i.e., 2, 3.5, and 5 MHz). The singular values are all the same (σ0 ) except for the first one (σ1 ) and the last one (σ R ) at 2 and 3.5 MHz. At the central frequency 5 MHz (dotted line), the singular values are constant.

(9)

(10)

Then the terms associated with σ1 and σ R can be extracted and the decoding matrix D can be written as

i=1

where A∗ means the complex conjugate transpose of A. One or more small σi (i = 1 . . . R) indicate that A is nearly rank deficient and ill conditioned. Consequently, to stabilize the solution, the decoding matrix D is obtained by deleting the terms that are associated with the small singular values in (9) in the truncated PI (TPI) method in this paper. Moreover, from (9) and (10), it is clear that if the singular values are constant or almost constant, the decoding matrix D can be replaced by A∗ by some modifications (as will be shown later in this paper). Therefore, (5) can be solved exactly or approximately to yield P, which can be used through inverse Fourier transform to produce the signals pik (t). In both simulations and experiments, we decode the prebeamformed RF data from each receiver separately since it is independent from the others. C. Singular Values and the Singular Vectors Study Applying SVD to A in MATLAB illustrated that the singular values are all the same (σ0 ) for all the frequencies we have studied except for the first one (σ1 ) and the last one (σ R ) as shown in Fig. 1. The singular values are displayed for the frequency-dependent coding matrix A for three different frequency components as examples: 2, 3.5, and 5 MHz. At 5 MHz (the central frequency of the transducer), e− j π( f / f 0 ) is −1 and A is exactly a Hadamard coding matrix with binary codes [−1, 1]. Then, A is a stable matrix with condition number σ1 /σ R (the ratio between the largest singular value to the smallest one) equaling to 1. The singular value distribution is a straight line. As the frequency component used in the calculation diverges from the central frequency, the coding matrix becomes more and more ill conditioned with an increasing condition number as demonstrated by the curves corresponding to 2 and 3.5 MHz in Fig. 1.

D=

R−1 v R u ∗R v1 u ∗1 1  + + vi u ∗i σ1 σR σ0 i=2

=

R v1 u ∗1 v R u ∗R 1  ∗ ∗ + + vi u i σi σ1 σR σ0 σ0∗ i=1

−

 1  ∗ ∗ v1 u 1 σ1 + v R u ∗R σ R∗ σ0 σ0∗

(11)

so that     v R u ∗R σ R σ R∗ σ1 σ1∗ v1 u ∗1 1 1− + 1− + A∗ . D= ∗ ∗ σ1 σ0 σ0 σR σ0 σ0 σ0 σ0∗ (12) Then the calculation of A−1 has been replaced by A∗ , which enables a more efficient DE-STA decoding algorithm. The computational complexity has been reduced. If the terms associated with all the singular values are kept [as in (12)], it is equivalent to directly inverting A. Such a process is referred to as complete PI (CPI) in the following text. On the other hand, due to the special property of singular value distributions of A (Fig. 1), the term associated with the last singular value (σ R ) can be deleted in order to stabilize the inversion and this decoding process is referred to as TPI [as in the following] (the consequences of TPI will be described in Sections III and IV):     R−1  vi u ∗ σ1 σ1∗ v R u ∗R σ R σ R∗ v1 u ∗1 i 1− + − = D= σi σ1 σ0 σ0∗ σR σ0 σ0∗ i=1

+

1 A∗ . σ0 σ0∗

(13)

D. Application to Ultrasound Imaging—Proposed Approach The PI method was first tested with simulation data and then with experimental data acquired by the Ultrasonix RP research platform (Ultrasonix, CA).

GONG et al.: PI DECODING PROCESS IN DE-STA IMAGING

1) Simulation Setup: The PI was tested with Field II simulation program [22], [23]. The probe was a 4-cm-wide 128-element linear array with 0.28-mm width, 0.02-mm kerf, and 1540-m/s speed of sound, the characteristics of the array used in experiments. Standard STA and DE-STA data were acquired from a 4 cm × 1 cm × 5.5 cm (lateral × azimuthal × axial) phantom that contained seven wire targets located at 5-mm intervals from 15 to 45 mm in depth. The diameter of the wire targets was 200 μm. Two noise levels (SNR equaled to 0 and −10 dB, respectively, in the case of standard STA imaging) were added to both the standard STA and DE-STA data. Afterward, PI steps were applied to restore the traditional STA data. Finally, the restored prebeamformed RF signals from all receiving channels were examined to investigate the performance of SNR improvement undergoing the two inversions (CPI and TPI) with different noise levels. 2) Experimental Acquisition Setup: The experimental RF data were acquired using an Ultrasonix RP research platform equipped with the parallel channel acquisition system SonixDaq (Ultrasonix, CA), using probe L14-5. The central frequency of the transducer was 5 MHz with 40-MHz sampling frequency. RF data were acquired in DE-STA mode from a 4 cm × 4 cm tissue-mimicking phantom that contained a hyper- (on the right side) and a hypo-echoic (on the left) inclusion with a diameter of 1.2 cm as well as three wire inclusions of 0.5-mm diameter (a detailed description of the phantom can be found in [20]). The DE-STA data were processed by both CPI and TPI. The beamformed signals were processed by Hilbert transform followed by the logarithm compression and were then displayed as log-enveloped images. III. R ESULTS A. Simulation Results In simulation, we applied the PI method to investigate the decoding process and the image reconstruction at various noise levels. In Field II simulation results, the decoded RF signals from an arbitrary receive channel were analyzed (e.g., 30th receiving channel as an example) to compare the performance of TPI and CPI at two SNR levels of the prebeamformed RF signals: 0 and −10 dB before decoding. The noise was filtered by a fourth-order Butterworth bandpass filter to simulate the filtering effect of the transducer. No other filtering process was applied to decode DE-STA data except for a rectangle window to cut the frequencies close to 0 and 2 f 0 . Decoding steps were applied to the frequency range from 2 to 9 MHz (with nonzero amplitude). It has been found that the restored RF signals from all the combination of transmitting and receiving elements decoded from CPI almost completely overlapped with those from TPI with similar SNR values except for the case when the first element was used as the transmitter. Figs. 2 and 3 show the simulation results corresponding to the signals received by the 30th element at the transmission of the 30th and the 1st element, respectively, with 0-dB additive noise. Fig. 2(a) displays the line plot of a typical noiseless prebeamformed RF signal in the standard STA mode as a reference. Fig. 2(b) illustrates the same signal after

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Fig. 2. Simulated prebeamformed RF signals received by the 30th receiving element at the transmission of the 30th element from four different cases. (a) Standard STA signal without noise. (b) Standard STA signal with 0-dB additive bandpassed noise. Restored DE-STA signal after applying (c) CPI and (d) TPI to the RF data with the same amount of noise as in (b).

Fig. 3. Simulated prebeamformed RF signals received by the 30th receiving element at the transmission of the first element from four different cases. (a) Standard STA signal without noise. (b) Standard STA signal with 0-dB additive bandpassed noise. Restored DE-STA signal after applying (c) CPI and (d) TPI to the RF data with the same amount of noise as in (b).

adding 0-dB bandpassed noise. We added the same level of noise to the simulated DE-STA signals. Fig. 2(c) and (d) shows the restored signals by decoding the noisy raw DE-STA signals using the CPI and TPI processes, respectively. The signals in Fig. 2(c) and (d) are similar to the noiseless STA signal as shown in Fig. 2(a). They both have a better SNR (improved by 22.62 and 21.68 dB, respectively) than the noisy traditional STA signal in Fig. 2(b). The enhancements match the theoretical value as well (10 log10 I = 21 dB). Fig. 3 displays the prebeamformed RF signals received by the 30th element at the transmission of the first element in the same order as those in Fig. 2: the noiseless STA signal [Fig. 3(a)], STA signal with 0 dB noise [Fig. 3(b)], restored DE-STA signal after CPI [Fig. 3(c)] and TPI [Fig. 3(d)]. The restored DE-STA signal in Fig. 3(c) is similar to the reference in Fig. 3(a) while the signal in Fig. 3(d) is quite different. Also, the SNR enhancement was reduced compared with the theoretical value, especially for TPI decoded signal [Fig. 2(d)]. The signal amplitude in Fig. 2(d) was reduced compared with that in Fig. 2(c), leading to a lower SNR in TPI than that in CPI (Table I). Figs. 2 and 3 demonstrate that the PI process mostly affected the signals related to the first transmitter. The detailed RF SNR quantifications are shown in Table I. As a comparison, the RF signals in these four different modes were also presented while using the same setup, but

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TABLE I D ETAILED Q UANTIFICATIONS OF STA, CPI, AND TPI D ECODED RF S IGNALS W ITH 0-dB A DDITIVE N OISE AT THE T RANSMISSION OF 30 TH AND 1 ST E LEMENT, R ESPECTIVELY

Fig. 6. Reconstructed images using (a) CPI and (b) TPI decoding processes with 0-dB additive noise.

Fig. 4. Simulated prebeamformed RF signals received by the 30th receiving element at the transmission of the 30th element from four different cases. (a) Standard STA signal without noise. (b) Standard STA signal with −10-dB additive bandpassed noise. Restored DE-STA signal after applying (c) CPI and (d) TPI to the RF data with the same amount of noise as in (b).

Fig. 7. Reconstructed images using (a) CPI and (b) TPI decoding processes with −10-dB additive noise.

Fig. 5. Simulated prebeamformed RF signals received by the 30th receiving element at the transmission of the first element from four different cases. (a) Standard STA signal without noise. (b) Standard STA signal with −10-dB additive bandpassed noise. Restored DE-STA signal after applying (c) CPI and (d) TPI to the RF data with the same amount of noise as in (b). TABLE II D ETAILED Q UANTIFICATIONS OF STA, CPI, AND TPI D ECODED RF S IGNALS W ITH −10-dB A DDITIVE N OISE AT THE T RANSMISSION OF 30 TH AND 1 ST E LEMENT, R ESPECTIVELY

with a higher level noise (SNR = −10 dB) as shown in Figs. 4 (30th transmitter and 30th receiver) and 5 (1st transmitter and 30th receiver). The detailed RF SNR quantifications of Figs. 4 and 5 are shown in Table II. Similar general trends have been observed while using −10-dB noise. The SNR improvements after both CPI (22.67 dB) and TPI (22.57 dB) agreed well with the theoretical value at the transmission of the

30th transmitter, whereas the SNR improvement was decreased at the transmission of the first element. However, under −10-dB noise, TPI provided a better SNR value while using first transmitter. The detailed quantification values are shown in Table II. Other SNR levels such as −20 or −30 dB have also been investigated. The same trend was observed: at the 30th transmitter, the SNR improvement for both CPI and TPI decoded data followed the theoretical value, whereas at the first transmitter, as the noise increased, the superiority of TPI became more and more significant. The SNR improvement comparison between Tables I and II illustrated that the decoding process, by either the CPI or the TPI, was stable using the DE-STA technique. For low noise levels, the CPI was better than the TPI. The superiority of the TPI was evident at higher noise levels, and this was mainly reflected by the signals from the first transmitting element. The log-enveloped beamformed images obtained from CPI and TPI with 0- and −10-dB additive noise are presented in Figs. 6(a) and (b) and 7(a) and (b), respectively. CPI and TPI provided similar reconstructed images in both Figs. 6 and 7, which can also be verified by the representative lateral and axial line plot comparison through the wire target located at 3-cm depth with −10-dB additive noise (Fig. 8). The TPI lateral and axial plots almost exactly overlapped with the CPI plots. This is due to fact that the effect of deleting the smallest singular value is minor (mainly affecting the first transmitting element). We will return to this point in Section IV.

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noise, P, is P = Pn − P =

R  i=1

Fig. 8. Representative (a) lateral and (b) axial line plots through the wire target located at 3-cm depth with −10-dB additive noise. Dashed lines: CPI decoded signal plot. Solid lines: TPI decoded signal plot. Dotted lines: noiseless standard STA plots are shown as reference. TABLE III D ETAILED Q UANTIFICATIONS OF CPI AND TPI D ECODED I MAGES OF THE E XPERIMENTAL P HANTOM

B. Experimental Results Fig. 9(a) and (b) shows DE-STA images of the experimental tissue-mimicking phantom as described in Section II with CPI and TPI, respectively. Fig. 9(c) shows the lateral line plots through the wire targets in Fig. 9(a) and (b) as indicated by the white line, respectively. Image qualities obtained from CPI and TPI were quantified by peak SNR (as assessed by the wire target), spatial resolution (as assessed by the full-width at half-maximum of the wire target), and contrast-to-noise ratio of the hyper inclusion. The detailed values are shown in Table III. Based on the results presented in Fig. 9 and Table III, we concluded that the performances of CPI and TPI were similar, which also agreed well with the simulation results. IV. D ISCUSSION The SNR improvement for both CPI and TPI processes matches the theoretical value as in Hadamard encoding and is independent of the additive noise level. This demonstrates that the decoding process in the DE-STA technique is stable. Truncating the smallest singular value mainly affects the signals corresponding to the first transmit channel (Tables I and II). Considering that the measured signals M are contaminated by noise e, then from (5) and (9), the decoded RF signal under noise Pn can be solved as Pn = D(M + e) = DM + De =

R  i=1

 u∗ u ∗i M+ vi i e. σi σi R

vi

i=1

(14) The difference between the solutions with and without

vi

u ∗i e σi

(15)

where u i and vi are the singular vectors associated with the output and input spaces, respectively. From (15), it can be clearly seen that the term corresponding to the small singular values will be more sensitive to the measurement noise e. The coding matrix A has a special distribution of singular values (Fig. 1), which is a straight line (σ0 ) except for σ1 and σ R (R = 128 in both simulations and experiments). Therefore, we need to check only the term associated with the smallest singular value σ128 (i.e., v128(u ∗128 /σ128 )e). Fig. 10 shows the plots of vector v128 and v64 (as a reference). The index of the horizontal axis, i , is the index of transmitting elements in vectors v128 and v64 . For example, v128 (i ) and v64 (i ) indicate the weight used to restore the signals corresponding to the i th transmitting element. Fig. 10 shows that when i equals 1, v128 has a much greater value compared with the other points (others are all the same with a value around 0). Consequently, the P in (15) mostly has an impact on the first transmitting element as indicated in Fig. 10. On the other hand, other singular vectors, such as v64 , will affect signals related to majority of transmitters. The fact that only the signals related to one transmit channel are severely affected by noise shows that DE-STA is a relatively stable encoding and decoding technique in terms of the overall image quality of the enveloped beamformed images since deleting the signals corresponding to the first transmitter will not compromise the overall image qualities significantly. The signal amplitudes for TPI decoded data [Figs. 3(d) and 5(d)] have been decreased compared with the corresponding CPI decoded data [Figs. 3(c) and 5(c)] at the transmission of the first element. This is because deleting the terms associated with the last singular value σ R results in loss of part of the signal information. If the noise level is low, the information loss will compromise the performance of TPI, leading to a lower SNR compared with CPI decoded data (Fig. 3 and Table I). However, if the noise level is high, the amplification of noise e in the term of v128 (u ∗128/σ128 )e by σ128 in the CPI process becomes dominant. Then TPI shows superiority as illustrated in Fig. 5 and Table II: the reduction of noise amplitude compensates for the information loss, leading to a higher SNR in TPI decoded data than the CPI case. This term corresponding to σ128 mainly affects the first transmitting element, which explains the similarity of the image quality in the reconstructed images from CPI and TPI decoded DE-STA images (Figs. 6–9). Another novelty of this paper is the derivation of a new inversion algorithm. Comparing CPI and TPI decoding processes (12), (13) with the original direct inversion as shown by (9), the calculation of coding matrix inversion (A−1 ) has been replaced by the multiplication with the complex conjugate transpose of coding matrix (A∗ ) with some modifications. The efficiency improvement in PI includes two aspects: storage memory requirement and computational complexity. We will discuss each inverse process in their optimal situation.

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Fig. 9. Experimental DE-STA log-enveloped beamformed images decoded from (a) CPI and (b) TPI. (c) Lateral line plots through the wire target as indicated by the white line in (a). Dashed lines: CPI decoded signal plot. Solid lines: TPI decoded signal plot.

Pk =

R  vi u ∗ Mk i

i=1

σi

⇔ O(R × 2R + R 2 ) = O(3R 2 )

(16)

where Pk is the kth column in matrix P (restored STA signal in frequency domain). For the PI process, the calculation of each step and the corresponding complexity are v1 u ∗1 Mk σ1 v R u ∗R Mk σR A∗ Mk 1 A∗ Mk σ0 σ0∗ Fig. 10. Plots of vectors v128 and v64 (reference) of coding matrix A at 8 MHz.

For both direct inverse and PI (including CPI and TPI), we assume that all the singular values and singular vectors (u i , vi , σi ) are precalculated and preloaded in the memory. Then the memory space needed for direct inverse must be sufficient for all the (u i , vi , i = 1 : R) plus (σ1 , σ R , and σ0 ). This requirement for memory is approximately R/2 times compared with that in CPI or TPI in which only (u 1 , v1 , σ1 , u R , v R , σ R , and σ0 ) are needed to be stored. To estimate the computational complexity, we estimate the order of the number of the multiplication and summation in the decoding processes. For simplicity, we assume that the number of transmitters and receivers is the same order as R, (I = K = R), the rank of the encoding matrix. Then for the direct inverse process, the calculation of each step and the corresponding complexity to decode the kth column in matrix M, Mk (k = 1 : K , receiver index), are shown as follows: u ∗i Mk ⇔ O(R) σi ∗ vi u i Mk ⇔ O(R + R) = O(2R) σi

⇔ O(2R) ⇔ O(2R) ⇔ O(R 2 ) ⇔ O(R 2 + R).

In total      ∗ v R u ∗R σ1 σ1∗ σ1 σ1∗ v1 u 1 1 ∗ 1− + 1− + Mk Pk = A σ1 σ0 σ0∗ σR σ0 σ0∗ σ0 σ0∗ ⇔ O(R 2 + 5R).

(17)

From (16) and (17), the computational complexity has been reduced to 1/3 after implementing (12) or (13) compared with the direct inverse equation (9). In addition, the simplicity of (12) or (13) may also bring the opportunity to implement the decoding through hardware, which will further improve the speed of the decoding process. V. C ONCLUSION We demonstrated the stability of the DE-STA method by testing the decoding process at various SNR levels using both CPI and TPI. We studied the property of encoding matrix A with SVD and found that all the singular values were all the same except for the first one and the last one. The last singular value had very minor effect on the overall image quality from the simulation and experimental results. We also derived a more efficient decoding process by replacing the calculation of A−1 by A∗ with some modifications. The computational complexity was reduced from O(3R 2 ) to O(R 2 ). ACKNOWLEDGMENT The authors would like to thank Dr. R. Cobbold, Dr. J. Tavakkoli, Y. Li, and D. Young for their valuable discussion and suggestions.

GONG et al.: PI DECODING PROCESS IN DE-STA IMAGING

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Ping Gong was born in Changchun, China, in 1987. She received the B.Sc. degree in biomedical engineering from Tianjin University, Tianjin, China, in 2010, the M.Sc. degree from the Department of Electrical and Computer Engineering, Lakehead University, Thunder Bay, ON, Canada, in 2012, and the Ph.D. degree in biomedical physics from Ryerson University, Toronto, ON, Canada, in 2016. She is currently with the Department of Radiology, Mayo Clinic College of Medicine, Rochester, MN, USA. Her current research interests include developing ultrasound transmission and beamforming algorithms to improve the ultrasound image quality.

Michael C. Kolios was born in Toronto, ON, Canada. He received the B.Sc. degree (Hons.) in physics, with a minor in computer science, from the University of Waterloo, Waterloo, ON, Canada, in 1991, and the M.Sc. and Ph.D. degrees from the Department of Medical Biophysics, University of Toronto, Toronto, in 1994 and 1998, respectively. He joined the Department of Physics, Ryerson University, Toronto, in 1997, where he is currently a Full Professor and the Associate Dean of Research and Graduate Studies with the Faculty of Science. He held a Canada Research Chair in Biomedical Applications of Ultrasound from 2004 to 2014. His current research interests include ultrasound/optical imaging and characterization of tissues and cells, high-frequency ultrasound imaging and spectroscopy, acoustic microscopy, ultrasound- and laser-based therapy, optical coherence tomography, and photoacoustic imaging. Dr. Kolios has been a recipient of numerous awards for his research and teaching.

Yuan Xu was born in China in 1971. He received the Ph.D. degree in physics from the Institute of Physics, Chinese Academy of Sciences, Beijing, China, in 1999, and the Ph.D. degree in biomedical engineering from Texas A&M University, College Station, TX, USA, in 2003. He has been with Ryerson University, Toronto, ON, Canada, since 2005, where he is currently an Associate Professor with the Department of Physics. His current research interests include the development of novel algorithms to improve the image qualities of ultrasound images and to reduce the cost of ultrasound imaging systems.