# Spatial Coherence of Backscattered Signals in Multi-Line Transmit Ultrasound Imaging and Its Effect on Short-Lag Filtered-Delay Multiply and Sum Beamforming

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Spatial Coherence

^{2}for a uniformly weighted aperture. Away from the focus, the beam widens and spatial coherence decreases. The same effect is observed in all those cases in which a decorrelation occurs, e.g., because of the presence of acoustic and/or electronics noise, beam sidelobes, and aberrations, etc. [9,20,23].

_{n}(t) are the backscattered radio-frequency (RF) signals received by each n-th transducer of the N-element receive aperture and focused (delayed). l is an integer number representing the spatial lag, i.e., the number of elements between the couple of multiplied signals s

_{n}(t) and s

_{n+l}(t) (l = 0 … N − 1), whose product is integrated over a short time interval t = [t

_{1}; t

_{2}]. C(l,t) is the spatial covariance at the l-th lag; the normalization factor (denominator of Equation (1)) is given by the zero-lag covariance C(0,t), as in [3].

#### 2.2. Multi-Line Transmission

_{TX}beams simultaneously in different focusing directions. After each transmission (TX), the backscattered signals are collected by the array elements in reception (RX) and beamformed in parallel along the considered steering directions. Then, the beams are moved by an angular step θ

_{STEP}= θ

_{SECT}/N

_{LINES}in order to cover the full θ

_{SECT}-wide image field of view with N

_{LINES}scan lines. Therefore, the simultaneous TX beams are separated by an angular distance θ

_{TX}= θ

_{SECT}/N

_{TX}[22].

_{TX}) and RX (h

_{RX}) ultrasound beams at the focal depth can be generally expressed as follows:

^{TX}refers to the focusing direction of the TX beam (θ

^{TX}) and u

^{RX}to the focusing direction of the RX beam (θ

^{RX}). N is the number of array elements, p

_{x}is the array pitch, and λ is the wavelength. Then, in the classic SLT case (i.e., each time one beam is transmitted and received), the pulse-echo (h

_{PE}) response is given by:

_{i}

^{TX}refers to the focusing direction of the i-th TX beam (θ

_{i}

^{TX}). Hence, in this case the pulse-echo beam in (5) is made of N

_{TX}terms, where one term is the one generated when θ

_{i}

^{TX}= θ

^{RX}, i.e., the TX and RX responses are equal (as in a classic SLT scan), and the other terms with θ

_{i}

^{TX}≠ θ

^{RX}are the so called cross-talk contributions that originate from interferences among the TX and RX beams. Consequently, in the SLT case, when N

_{TX}= 1, the pulse-echo beam has a sinc

^{2}shape, while in MLT with N

_{TX}> 1, the beam has a more complex shape, with N

_{TX}peaks in the simultaneous TX focusing directions, as shown in Figure 1 for MLT with four or 12 TX beams (i.e., 4-MLT and 12-MLT).

#### 2.3. Short-Lag Filtered-Delay Multiply and Sum Beamforming

_{n}(t) are amplitude-rescaled, by means of the signed square-root operator, and then combinatorially coupled and multiplied. The beamformed output is thus computed as:

_{FDMAS}(t) is subsequently obtained by band-pass (BP) filtering y

_{DMAS}(t), in order to pass the second-harmonics that originates after signal cross-multiplications, and to attenuate as much as possible the baseband and higher frequency components which originate from such non-linear operations. After beamforming, the obtained RF image lines are demodulated using the Hilbert transform, normalized, and logarithmically compressed to produce the final B-mode image.

#### 2.4. Simulation Setup and Study Organization

^{3}volume centered around (x, y, z) = (0, 0, 65) mm (i.e., >10 scatterers per resolution cell).

_{N}(l,t) and total spatial correlation SC(t) were evaluated using Equations (1) and (2), and averaging the values obtained in a small 5 × 5 mm

^{2}area centered at (x, z) = (0, 70) mm, i.e., over the TX focus. Three different scenarios were analyzed (Table 1):

- the number of MLT beams varies (i.e., N
_{TX}= 1/4/6/8/12), but the total image sector is fixed (θ_{SECT}= 90°), as well as the number of lines (192); consequently, the angle among the TX beams (θ_{TX}= θ_{SECT}/N_{TX}) changes together with the number of beams (usually, this is the classic MLT implementation); - the number of MLT beams varies (i.e., N
_{TX}= 1/4/6/8/12), but the same angle (θ_{TX}) among the beams is used in all configurations; in particular, this angle was set to be equal to the one that would be obtained applying 12-MLT to scan a 90° sector (i.e., θ_{12}). Thus, in this case, the total image sector also changes in the different MLT configurations; - the number of beams is fixed (i.e., N
_{TX}= 4), while the angle among them changes, as it would do in 4/6/8/12-MLT when a 90° sector is acquired. Thus, also here, the total image sector changes in each case.

^{3}uniform tissue background starting at z = 30 mm. Measurements were performed considering two 6-mm-diameter circular areas inside and outside the cyst.

## 3. Results and Discussion

#### 3.1. Spatial Coherence Trends in MLT Images of a Homogeneous Phantom

_{TX}only. The plots clearly show that this time the SC oscillatory trend becomes similar for all MLT configurations (except for SLT), but the lobes have different peak amplitudes and widths, which still makes the total SC decrease when increasing the number of TX beams (Figure 3b). On the other hand, when the number of MLT beams N

_{TX}is fixed, while the angle among them varies, we see in Figure 2c that the SC trend shows a different pitch between secondary lobes. Instead, total SC becomes similar in all cases (Figure 3c).

^{2}, while a sort of damped oscillatory behavior is shown in MLT imaging, as expected because of the more complex beam shape obtained when multiple beams are transmitted simultaneously (cf. Figure 1). Furthermore, Figure 2b,c and Figure 3b,c demonstrate that it is the angular distance between the multiple TX beams (i.e., θ

_{TX}) that mainly influences the SC trend over lags in MLT imaging, while the total SC value depends on the number of TX beams. When N

_{TX}is fixed to four, for example, as in Figure 2c and Figure 3c, total SC remains almost constant at ~8, while the lobes of the SC trend gradually move at higher lags with increasing values of N

_{TX}. Conversely, when θ

_{TX}is fixed (Figure 2b and Figure 3b), we have a similar trend in all the MLT cases, with a peak at about lag 19, but the total SC decreases as N

_{TX}becomes higher. Both behaviors can be observed in Figure 2a and Figure 3a, where N

_{TX}and θ

_{TX}change at the same time in the different MLT configurations analyzed.

#### 3.2. Simulated Images with MLT and Short-Lag F-DMAS

_{TX}.

_{TX}is the one that would be used in 4- or 12-MLT, respectively, cf. Figure 7CR-c). In scenario 2, this threshold is similar for all plots (i.e., lag = 39/41/42/42 for 4/6/8/12 TX beams, cf. Figure 7CR-b), since θ

_{TX}is the same (i.e., θ

_{12}). Moreover, panels CR-a and CR-c show that there is a dependence of the CR trend on the angular distance among TX beams, but no significant relation exists with the number of MLT beams (in panel CR-b in fact, all curves are almost overlapped). In particular, the CR gets worse as θ

_{TX}decreases from θ

_{4}to θ

_{12}.

_{TX}values; on the other hand, when θ

_{TX}is fixed (Figure 7CNR-b, scenario 2), the curves are almost overlapped. Thus, the CNR trend also seems to be related to θ

_{TX}, even if in a less pronounced way as compared to the CR one.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Examples of theoretical beampatterns in 4-MLT and 12-MLT: in the panels on the

**left**, the TX (gray) and RX (black) beampatterns are plotted in each MLT case, when the RX beam is steered in one of the TX beams’ direction (e.g., at 0°); on the

**right**, the resulting pulse-echo beam shapes are shown.

**Figure 2.**Normalized spatial covariance trend vs. lags, measured on the simulated uniform phantom data, and averaged over a 5 × 5 mm

^{2}area around (x, z) = (0, 70) mm in the different scenarios of Table 1: (

**a**) both the number of TX beams and the angle among them varies (as in SLT or 4/6/8/12-MLT); (

**b**) only the number of TX beams varies (N

_{TX}= 1/4/6/8/12), while the angular distance among them is fixed to the one of 12-MLT; (

**c**) the number of TX beams is set to four, while the angular distance among them is the one of 4/6/8/12-MLT (in this last case, the legend refers to the number of TX beams which would determine the angular distance).

**Figure 3.**Total average spatial coherence, measured in a 5 × 5 mm

^{2}area around (x, z) = (0, 70) mm in the different scenarios of Table 1: (

**a**) both the number of TX beams and the angle among them varies (as in SLT or 4/6/8/12-MLT); (

**b**) only the number of TX beams varies (N

_{TX}= 1/4/6/8/12), while the angular distance among them is fixed to the one of 12-MLT; (

**c**) the number of TX beams is set to four, while the angular distance among them is the one of 4/6/8/12-MLT (in this last case, the legend refers to the number of TX beams which would determine the angular distance).

**Figure 4.**Lateral resolution (at −6 dB) trends, measured on the PSF images with varying maximum-lag, in the different scenarios of Table 1: (

**a**) both the number of TX beams and the angle among them varies (as in SLT or 4/6/8/12-MLT); (

**b**) only the number of TX beams varies (N

_{TX}= 1/4/6/8/12), while the angular distance among them is fixed to the one of 12-MLT; (

**c**) the number of TX beams is set to four, while the angular distance among them is the one of 4/6/8/12-MLT (in this last case, the legend refers to the number of TX beams which would determine the angular distance).

**Figure 5.**4-MLT PSF images obtained in scenario 1 by applying (

**a**) DAS with Tukey apodization in TX/RX, or F-DMAS with (

**b**) maximum lag = 63 (standard version), (

**c**) maximum lag = 10, (

**d**) maximum lag = 40, and Tukey apodization in TX only. Figures are displayed over an 80 dB dynamic range, in order to better highlight the possible presence of small cross-talk artifacts.

**Figure 6.**Lateral (

**a**) and axial (

**b**) profiles of the PSF at 70 mm (i.e., the TX focal depth) shown in Figure 5 for DAS, F-DMAS (standard formulation with maximum lag = 63), short-lag F-DMAS with maximum lag = 10 (SL F-DMAS (10)) and maximum lag = 40 (SL F-DMAS (40)).

**Figure 7.**CR (

**top**row), CNR (

**middle**row), and sSNR (

**bottom**row) trends, measured on the cyst-phantom images with varying maximum lag, in the different scenarios of Table 1: (

**a**) both the number of TX beams and the angle among them varies (as in SLT or 4/6/8/12-MLT); (

**b**) only the number of TX beams varies (N

_{TX}= 1/4/6/8/12), while the angular distance among them is fixed to the one of 12-MLT; (

**c**) the number of TX beams is set to four, while the angular distance among them is the one of 4/6/8/12-MLT (in this last case, the legend refers to the number of TX beams which would determine the angular distance).

**Figure 8.**4-MLT cyst-phantom images obtained in scenario 1 by applying (

**a**) DAS with Tukey apodization in TX/RX, or F-DMAS with (

**b**) maximum lag = 63 (full-version), (

**c**) maximum lag = 10, (

**d**) maximum lag = 40, and Tukey apodization in TX only. Figures are displayed over a 60 dB dynamic range.

**Figure 9.**Cross-section of the anechoic cyst in Figure 8, for DAS, F-DMAS (standard formulation with maximum lag = 63), short-lag F-DMAS with maximum lag = 10 (SL F-DMAS (10)) and maximum lag = 40 (SL F-DMAS (40)).

Scenario 1 | Scenario 2 | Scenario 3 | |||
---|---|---|---|---|---|

N_{TX} | θ_{TX} | N_{TX} | θ_{TX} | N_{TX} | θ_{TX} |

1 | θ_{1} = 90° | 1 | θ_{12} | - | - |

4 | θ_{4} = θ_{1}/4 | 4 | θ_{12} | 4 | θ_{4} |

6 | θ_{6} = θ_{1}/6 | 6 | θ_{12} | 4 | θ_{6} |

8 | θ_{8} = θ_{1}/8 | 8 | θ_{12} | 4 | θ_{8} |

12 | θ_{12} = θ_{1}/12 | 12 | θ_{12} | 4 | θ_{12} |

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**MDPI and ACS Style**

Matrone, G.; Ramalli, A. Spatial Coherence of Backscattered Signals in Multi-Line Transmit Ultrasound Imaging and Its Effect on Short-Lag Filtered-Delay Multiply and Sum Beamforming. *Appl. Sci.* **2018**, *8*, 486.
https://doi.org/10.3390/app8040486

**AMA Style**

Matrone G, Ramalli A. Spatial Coherence of Backscattered Signals in Multi-Line Transmit Ultrasound Imaging and Its Effect on Short-Lag Filtered-Delay Multiply and Sum Beamforming. *Applied Sciences*. 2018; 8(4):486.
https://doi.org/10.3390/app8040486

**Chicago/Turabian Style**

Matrone, Giulia, and Alessandro Ramalli. 2018. "Spatial Coherence of Backscattered Signals in Multi-Line Transmit Ultrasound Imaging and Its Effect on Short-Lag Filtered-Delay Multiply and Sum Beamforming" *Applied Sciences* 8, no. 4: 486.
https://doi.org/10.3390/app8040486