A High-Resolution and High-Contrast Beamforming Algorithm Based on Null Subtraction Imaging Applied to Synthetic Transmit Aperture

Abstract

In medical ultrasound imaging, achieving high-quality reconstructed images while avoiding a huge computational burden is an important challenge. The Null subtraction imaging (NSI) algorithm results in a high-resolution reconstructed image. However, this method is not successful in recovering the background speckle information. In this paper, a novel algorithm, known as NSI-based generalized coherence factor (GCF)-along with delay-and-sum (DAS), which is abbreviated as NSG-DAS, is developed to overcome this limitation. In the proposed method, by using a hybrid technique, the desired resolution and effective noise suppression of the NSI algorithm, as well as the background speckle information of the conventional DAS beamformer are recovered simultaneously. More precisely, by using the GCF method, a new weighing factor is introduced that enhances the coherent regions of the image and suppresses the off-axis signals. Evaluations prove the favorable performance of the suggested technique; in particular, by using the proposed NSG-DAS method, a resolution comparable to the NSI algorithm is achieved for the geabr0 dataset, which is improved by about 42% compared to DAS. Also, the contrast evaluation parameter of the suggested technique is comparable to the DAS algorithm and is improved by about 63% compared to the NSI method. This indicates the ability of the suggested technique to improve either resolution or contrast simultaneously.

Keywords

ultrasound synthetic transmit aperture high resolution speckle preservation null subtraction imaging

Introduction

Ultrasound imaging in medicine serves as a fundamental tools in the diagnosis and investigation of various diseases. This imaging modality is widely used due to its non-invasiveness and cost-effectiveness.¹ Conventional ultrasound imaging, primarily using delay-and-sum (DAS) beamforming in B-mode, is widely used in clinical diagnostics due to its simplicity and real-time performance. However, despite its widespread adoption, the conventional methods are limited by fixed focusing (which affects the image resolution) and a limited field of view. Also, while conventional techniques are widely used due to their simplicity and robustness, adaptive methods, such as minimum variance (MV) and null subtraction imaging (NSI) methods, which will be discussed in the following, aim to enhance resolution and contrast by tailoring the beamforming process to the signal and noise characteristics. These improvements, however, depend on scene complexity and implementation. As a complementary direction, the synthetic transmit aperture (STA) method has emerged as a promising alternative, offering the potential for high resolution due to the possibility of dynamic focusing.^2,3 In this imaging scenario, the received signals corresponding to multiple transmit elements are combined, allowing for improved image quality and increased resolution. The use of multiple transmit elements in the STA method enhances spatial resolution, minimizes noise, and improve image contrast. In addition, the processing of data obtained from multiple transmit elements allows for better reconstruction of tissue structures and increases the accuracy of diagnosis.⁴ One should note that the STA method is also associated with challenges such as reduced signal-to-noise ratio (in the case of using fewer emissions) and high computational complexity (in the case of using several emissions), which currently limit its application in commercial systems.

In recent years, several studies have been presented on different image reconstruction algorithms in the STA method to elevate the quality of the image output. Various studies have shown that the use of improved beamforming techniques can effectively contribute to improving image quality; in,^5,6 it was demonstrated that the adaptive MV algorithm effectively enhances image resolution in comparison to the conventional DAS method. It should be noted that the MV algorithm imposes a large computational burden on the system. This is mainly due to the calculation of the data covariance matrix in the procedure of obtaining the weighting coefficients. In,⁷ the well-known delay-multiplied-and-sum (DMAS) method was used in image reconstruction and it was shown that this method has a good performance in suppressing off-axis signals. The weighting coefficients of coherence factor (CF), generalized CF (GCF), sign CF (SCF), and other improved versions of it have also been used to improve image quality and suppress noise level more efficiently.^8
-13 In,¹⁴ an algorithm called signal eigenvalue factor was presented in which the eigen decomposed aperture and large eigenvalues are selected as weighting coefficients. This method improves the image resolution. In,¹⁵ by approximating the data matrix to the Hankel matrix, the two-dimensional matrix corresponding to the transmission and reception processes was converted into a one-dimensional virtual array. Then, the MV algorithm was applied to the resulting array, leading to an enhanced resolution of the final image. In,¹⁶ a weighting factor, called mean-to-standard deviation, was proposed, and it was shown that this method results in a better resolution compared to the CF method. In,¹⁷ two improved versions of DMAS algorithm were presented under the names filtered delay-weight-multiply-and-sum (F-DwMAS) and filtered Euclidian-weighted-multiply-and-sum (F-DewMAS); in general, in these algorithms, a predefined set of weighting factors is applied to the aperture data before multiplying the delayed signals mutually and calculating their correlation. In,¹⁸ a new algorithm called signal range-standard deviation factor-multiphase apodization with cross-correlation was presented which results in improved resolution. In the mentioned study, the presented method was implemented on FPGA to speed up the process. In,¹⁹ in order to remove clipping artifacts (i.e., the truncation of the signal value when it goes beyond the maximum limit of the system), a post-processing technique was presented. In,²⁰ two weighting factors, one aimed at suppressing the level of signal distortion inside the cysts, and the other aimed at improving resolution, were presented and applied to the DAS beamformer’s output to reconstruct the image. The study in²¹ was focused on sub-array beamforming, in which the array is divided into groups of neighboring elements, and the beamforming is performed in two stages. In the mentioned study, the limitations of the reduction of field of view and the deterioration of the resolution were overcome by using diverging waves as well as the steering of the received signals. In addition to the above-mentioned studies, various combinatorial techniques, in which the combination of two or more algorithms are used, have also been proposed so far which result in an overall improvement in the image quality (e.g.,^22,23).

Recently, a new algorithm called null subtraction imaging (NSI) has been proposed for plane wave imaging which leads to high-resolution improvement, as good as or even better than the MV algorithm, with a much lower computational burden.^24,25 This algorithm proposes a non-coherent combination of three beamformed images with pre-defined weighting coefficients, which has a computational complexity of about three times that of the DAS method. To enhance the efficiency of this method, different NSI-based techniques have been suggested so far.^26
-28 It should be noted that the NSI algorithm has been applied to plane wave imaging scenarios, and to the best of our knowledge, its application to the STA scenario has not been reported so far. Also, it is important to acknowledge that, despite enhancing resolution, the NSI algorithm is not very successful in recovering the background speckle. In general, there is a trade-off between preserving background speckle (and improving contrast) and resolution in the NSI algorithm. One should note that resolution is mainly defined as the ability to distinguish between point targets.²⁹ However, in practice, the lateral resolution is evaluated by measuring the Full Width Half Maximum (FWHM) of the lateral variations corresponding to the point target(s).

In this paper, the NSI algorithm in the STA imaging scenario is used as the basis of the proposed method with the aim of benefiting from its improved resolution. More precisely, with the goal of achieving high resolution with low computational complexity while preserving background speckle information, a new hybrid algorithm is proposed, which is named NSI-based GCF-along with DAS (NSG-DAS) beamformer. The results show that the proposed method achieves high resolution and preserves background speckle information, respectively, compared to the DAS and NSI algorithms. In summary, our contributions are as follows:

The NSI algorithm is implemented in the STA imaging scenario and used in the image reconstruction process.

A new two-part weighting coefficient is proposed using the NSI and GCF algorithms, in which an attempt is made to preserve the advantages of each algorithm.

The proposed new weighting coefficients are specifically employed in the outputs of the DAS and NSI methods, and the final image with improved quality is obtained.

The proposed NSG-DAS method has been evaluated on simulated and experimental data and its advantages over related methods has been concluded.

In the following, brief explanations of DAS, GCF, MV, and NSI algorithms are presented. Then, the suggested NSG-DAS technique is introduced and explained. Next, evaluation of the proposed NSG-DAS method using simulation as well as experimental data is reported. Ultimately, the paper concludes with a discussion and final remarks.

Background

Delay-and-Sum: In this study, a linear array including $N$ elements is considered. The time-delayed signal associated with $i^{t h}$ transmit element and $j^{t h}$ receive element is shown as $x_{i j} (n)$ , where $n$ demonstrates $n^{t h}$ imaging point. According to these considerations, the DAS beamformed image in the receive direction (i.e., for each transmit element) is formulated as below:

\begin{array}{r} z^{i} (n) = \sum_{j = 1}^{N} x_{i j} (n), \end{array}

(1)

where $z^{i} (n)$ denotes the DAS beamformed image of $i^{t h}$ transmit element. Note that the timed-delayed signal corresponding to $i^{t h}$ transmit and $j^{t h}$ receive elements depends on the calculated time delay $τ$ , which itself depends on the $(x, z)$ positions of the considered transmit and receive elements. These notations are omitted from the formulas for simple writing. Now, to achieve the high-quality image, the beamformed data corresponding to all the transmit elements are required to be coherently compounded. To this end, the DAS algorithm is applied in the transmit index (i.e., over the beamformed images of all the transmit elements) which is formulated as below:

\begin{array}{r} z_{D A S} (n) = \sum_{i = i}^{N} z^{i} (n) . \end{array}

(2)

The DAS algorithm can ideally preserves the information of the background speckle.

Generalized Coherence Factor: The GCF algorithm is characterized as the proportion of spectral energy within a low-frequency range to the overall spectral energy. More specifically, considering $M_{0}$ as a pre-defined cutoff frequency, and also, considering $z (n) = {[z^{1} (n), \cdot \cdot \cdot, z^{N} (n)]}^{T} \in C^{N \times 1}$ as a vector of DAS beamformed data in the receive direction, the GCF method is formulated as below:

\begin{array}{r} G C F (n) = \frac{\sum_{k = - M_{0}}^{M_{0}} {| z_{F} (k, n) |}^{2}}{\sum_{k = 0}^{N - 1} {| z_{F} (k, n) |}^{2}}, \end{array}

(3)

where $z_{F} (k, n)$ denotes the Fourier transform of $z (n)$ , and $k$ represents the frequency index. GCF suppresses unwanted signals, such as sidelobe artifacts, while retaining contrast-to-noise ratio (CNR), and has been observed to yield excellent CNR compared to conventional approaches.³⁰ These properties suggest that GCF preserves background speckle structure by avoiding over-suppression in diffuse areas. It should be mentioned that in the above equation, the GCF technique is executed along the transmit index.

Minimum Variance: The weighting coefficient for $n^{t h}$ imaging point using the adaptive MV algorithm is written as below:

\begin{array}{r} w (n) = \frac{R^{- 1} (n) a}{a^{H} R^{- 1} (n) a} . \end{array}

(4)

In the above equation, ${(.)}^{H}$ and ${(.)}^{- 1}$ represent the hermitian and inversion operators, respectively. Moreover, the steering vector, denoted as $a \in C^{L \times 1}$ , is assumed to be a unit vector consisting of ones. Also, $R (n) \in C^{L \times L}$ represents the covariance matrix which is estimated by partitioning the array into $N - L + 1$ distinct sub-arrays, each having a length of $L$ :

\begin{array}{r} R (n) = \frac{1}{N - L + 1} \sum_{l = 1}^{N - L + 1} z_{l} (n) {(z_{l} (n))}^{H}, \end{array}

(5)

where $z_{l} (n) = {[z^{l} (n), \cdot \cdot \cdot, z^{l + L - 1} (n)]}^{T} \in C^{L \times 1}$ denotes the $l^{t h}$ sub-array. Diagonal loading as well as temporal averaging can also be used to achieve a more robust approximated covariance matrix. The readers are recommended to refer to^5,31 to reach a more detailed explanation in this regard. Once the weighting coefficient is calculated according to,⁴ the MV beamformed image is obtained as below:

\begin{array}{r} z_{M V} (n) = \frac{1}{N - L + 1} \sum_{l = 1}^{N - L + 1} w^{H} (n) z_{l} (n) . \end{array}

(6)

Similar to the GCF method, the MV algorithm is also considered to be applied along the transmit index.

Null Subtraction Imaging: In the NSI method, the weighting process is applied to each sub-aperture the length of which is obtained based on a considered fixed F-number. In this technique, three coefficients are implemented in the sub-apertures and the results are compounded incoherently to achieve the high-resolution image. Here, the NSI algorithm is aimed to be applied along the receive direction. The first weighting coefficient (i.e., the zero-mean (ZM) weighting coefficient) which is assigned to $i^{t h}$ transmit element and $j^{t h}$ receive element is defined as below:

w_{Z M, i j} = {\begin{matrix} 1, & 1 \leq j < \frac{N_{a}}{2} \\ - 1, & \frac{N_{a}}{2} \leq j < N_{a}, \end{matrix}

(7)

where $N_{a}$ denotes the including elements for each receive sub-aperture. The second weighting coefficient is obtained by adding a DC bias to the first weighting coefficient:

\begin{array}{r} w_{D C 1, i j} = w_{Z M, i j} + ϵ, \end{array}

(8)

where $ϵ$ is the DC bias introduced to prevent complete signal cancelation in the subtraction operation. As shown in,³² the choice of $ϵ$ directly influences the quality of the NSI image; the lower bias value enhances image resolution but comes at the cost of degraded image contrast. Also, the third weighting coefficient is considered the inverted version of the second weighting coefficient, that is, $w_{D C 2, i j} = f l i p [w_{D C 1, i j}]$ . The weighted beamformed images of the first, the second, and the third weighting coefficients of the $i^{t h}$ transmit element are denoted as $z_{Z M}^{i} (n)$ , $z_{D C 1}^{i} (n)$ , and $z_{D C 2}^{i} (n)$ , correspondingly. Then, the output of the NSI method for $i^{t h}$ transmit element is expressed as below:

\begin{array}{l} z_{N S I}^{i} (n) = 0.5 (e n v {z_{D C 1}^{i} (n)} + e n v {z_{D C 2}^{i} (n)}) \\ - e n v {z_{Z M}^{i} (n)}, \end{array}

(9)

where $e n v {.}$ denotes the envelope operator. Note that the DC bias $ϵ$ is implicitly included in the above equation, as the second and third weighted beamformed images are computed using weighting coefficients (w_DC1,ij, w_DC2,ij) that are defined based on this bias. It can be seen from the above equation that the three weighted beamformed data are compounded incoherently. Once the NSI method is applied along the receive direction, the processed beamformed data are summed to obtain the final image which is denoted as $z_{N S I} (n)$ :

\begin{array}{r} z_{N S I} (n) = \sum_{i = 1}^{N} z_{N S I}^{i} (n) . \end{array}

(10)

Proposed Method

NSI achieves high resolution—often comparable to or better than that of the MV algorithm—yet requires substantially less computational effort. Also, by using the NSI algorithm, the off-axis signals are suppressed well. However, this algorithm is not able to preserve the information of background speckle well enough; although this limitation can be somehow tackled by adjusting the bias value, such an improvement will be at the expense of resolution degradation. On the other hand, the GCF method not only suppresses the noise level but also preserves the coherent areas of the imaging medium more successfully. In this paper, in order to benefit from the advantages of the NSI method and overcome its limitations, this algorithm is used in combination with the GCF and DAS beamformers. More precisely, by using the GCF method, the improved resolution of the NSI technique and the preserved background speckle of the DAS algorithm are identified and maintained in the final image.

The key stages of the suggested technique are outlined as follows:

1. The DAS beamformer is used in both transmit and receive directions to obtain the reference image from the perspective of background speckle preservation (denoted as $z_{D A S} (n)$ ).

2. The NSI algorithm is applied along the receive direction; that is, $N$ -NSI beamformed images are obtained (denoted as $z_{N S I}^{i} (n)$ , for $i = 1, \cdot \cdot \cdot, N$ ).

3. The NSI beamformed images are summed along the transmit index to obtain the final image (denoted as $z_{N S I} (n)$ ).

4. The GCF algorithm is applied to $N$ -NSI beamformed images to obtain the GCF-based weighting coefficients (denoted as $G C F (n)$ ).

5. The proposed combinatorial weighting coefficient is calculated as below (denoted as $w (n)$ ):

\begin{array}{l} w (n) = \frac{G C F^{α} (n)}{G C F^{α} (n) + {(1 - G C F (n))}^{α}} \\ \times {(1 - z_{N S I} (n))}^{β} \end{array}

(11)

where $α$ and $β$ are constants, the impact of which is discussed in the Results section.

6. Finally, the improved-quality reconstructed image is obtained according to the following equation (denoted as $z_{N S G - D A S}$ ):

\begin{array}{l} z_{N S G - D A S} (n) = w (n) \times z_{N S I} (n) \\ + (1 - w (n)) \times z_{D A S} (n) \end{array}

(12)

Pay attention to¹¹; the first term, where the impact of $G C F$ coefficient is more or less prominent by the constant $α$ , causes more emphasis on imaging points that are more coherent (on-axis targets). It also suppresses incoherent areas (such as noise and clutter), which improves detailed information of the imaging medium. The objective of the second term, the effect of which is increased or decreased by the constant $β$ , is to achieve more effective noise suppression within the cystic regions; this term prevents assigning a high weighting factor (erroneously) to areas where the NSI algorithm may be strong due to noise (not the desired signal). Then, the resulting weighting factor is applied according to,¹² and the final image is generated. Note that the DAS algorithm is used due to its favorable ability to preserve background speckle information. In general, the weighting function in¹¹ is designed to favor NSI in coherent, low-intensity regions (where it enhances resolution), and favor DAS in incoherent or artifact-prone areas (where DAS better preserves background texture):

(a) The first part is a generalized logistic function that emphasizes regions of high coherence, where speckle is more likely to be meaningful.

(b) The second part suppresses regions with strong NSI responses, which often correspond to high intensity but potentially incoherent artifacts.

(c) Together, the two terms form a weighting factor that emphasizes DAS (for speckle) in low GCF or high-intensity regions, and NSI (for resolution) in coherent, lower-intensity areas.

The use of $α$ and $β$ allows tuning the balance between resolution and speckle preservation. It is important to note that although GCF is typically applied to RF channel data, in this work, it is computed across the ensemble of NSI images obtained for different emissions. This adapted use reflects spatial variability across the emissions and serves as a coherence-based metric for fusion. Moreover, note that we employ the GCF method rather than a variance-based metric; the GCF method is widely used in ultrasound beamforming as a normalized and scale-independent measure of signal consistency, which remains efficient even for positive-valued inputs.

The general procedure of the suggested technique are according to the above-mentioned explanations. Note that the proposed method also includes some more detailed processing, which is elaborated upon in the subsequent explanations. It has been previously shown in²⁸ that applying spatial smoothing process to NSI improves the stability of the output and background speckle preservation. Since the primary objective of this paper is to overcome the limitation of the NSI algorithm (i.e., improve the background speckle preservation), it is decided to apply spatial smoothing to it; for each transmit element, the weighted beamformed data corresponding to the first, second and third weighting factors ( $z_{Z M}^{i} (n)$ , $z_{D C 1}^{i} (n)$ , and $z_{D C 2}^{i} (n)$ ) are partitioned into $N - L^{'} + 1$ sub-arrays, having a length of $L^{'}$ :

\begin{array}{r} {\begin{array}{l} z_{Z M, l} (n) = {[z_{Z M}^{l}, \cdot \cdot \cdot, z_{Z M}^{l + L^{'} - 1}]}^{T} \in C^{L^{'} \times 1}, \\ z_{D C 1, l} (n) = {[z_{D C 1}^{l}, \cdot \cdot \cdot, z_{D C 1}^{l + L^{'} - 1}]}^{T} \in C^{L^{'} \times 1}, \\ z_{D C 2, l} (n) = {[z_{D C 2}^{l}, \cdot \cdot \cdot, z_{D C 2}^{l + L^{'} - 1}]}^{T} \in C^{L^{'} \times 1} . \end{array} \end{array}

(13)

Then, spatial smoothing is performed as below:

\begin{array}{l} z_{S S N S I} (n) = \frac{1}{N - L^{'}' + 1} \\ \times [\frac{env {z_{D C 1, l} (n)} + env {z_{D C 2, l} (n)}}{2} - env {z_{Z M, l} (n)}], \end{array}

(14)

where $1 \in C^{1 \times L^{'}}$ , and the spatially-smoothed NSI image is obtained correspondingly ( $z_{S S N S I} (n)$ ). The obtained output is used instead of $z_{N S I} (n)$ in equations (11) and (12). Also, note that the outputs of SSNSI and DAS are required to be normalized, and their corresponding absolute values are considered in the formulas; that is, the combination process is performed incoherently. The final form of the proposed method is formulated as below:

\begin{array}{l} \begin{array}{r} \begin{array}{l} {\begin{cases} \hat{W} (n) = \frac{G C F^{α} (n)}{G C F^{α} (n) + {(1 - G C F (n))}^{α}} \times {(1 - N [| z_{S S N S I} (n) |])}^{β}, \\ z_{S S N S G - D A S} (n) = \hat{W} (n) \times N [| z_{S S N S I} (n) |] \\ + (1 - \hat{W} (n)) \times N [| z_{D A S} (n) |], \end{cases} \end{array} \end{array} \end{array}

(15)

where $N [.]$ denotes the normalization operator and $| . | represents$ the absolute value. The output of the suggested technique, with the consideration of the spatial smoothing process on NSI, is known as SSNSG-DAS. By using the suggested SSNSG-DAS technique, the high resolution of the NSI algorithm is maintained in coherent areas, the artifacts and noise levels are filtered out, and the background speckle information is prevented from being suppressed. Figure 1 demonstrates the schematic overview of the processing steps of the suggested technique.

Quantitative Performance Indicators

The quantitative evaluation indicators that are utilized in this paper are contrast ratio (CR), contrast-to-noise ratio (CNR), generalized CNR (gCNR), and speckle signal-to-noise ratio (sSNR) which are listed below:

C R = 20 \log_{10} (\frac{μ_{i}}{μ_{o}}),

(16)

C N R = 20 \log_{10} (\frac{| μ_{i} - μ_{o} |}{\sqrt{σ_{i}^{2} + σ_{o}^{2}}}),

(17)

g C N R = 1 - \int_{- \infty}^{\infty} \min {p_{i} (x), p_{o} (x)} d x,

(18)

s S N R = \frac{μ_{o}}{σ_{o}} .

(19)

In the above equations, the mean and variance corresponding to the interior/exterior regions of a cyst are denoted as µ_i/µo and $σ_{i}^{2} / σ_{o}^{2}$ , respectively. Also, the probability distribution corresponding to inside/outside the cystic region which is used in the gCNR metric, as a more robust evaluation metric compared to CNR,³³ is demonstrated as $p_{i} (x) / p_{o} (x)$ . The standard deviation value corresponding to exterior region of the cyst is also used in the sSNR metric which is denoted as $σ_{o}$ . In addition to the above-mentioned evaluation metrics that are used to evaluate the contrast and background speckle preservation of the suggested technique, the well-known full-width at half-maximum (FWHM) is also used to analyze the performance of the suggested technique in terms of resolution.

Simulation and Experimental Setup

In the simulation, which is executed utilizing the Field II Matlab toolbox,^34,35 a 96-element linear array is modeled, where sequential firing of single transmit elements, with no electronic steering employed. The central and sampling frequencies are considered as 4 and 20 $M H z$ , respectively. The spacing between elements is set to $λ / 2$ (where $λ$ denotes the wavenumber). To evaluate the suggested technique from the perspective of background speckle preservation, a phantom consisting of two cysts is simulated.

The ats-wires data corresponding to the University of Illinois at Urbana-Champaign is also utilized to analyze the performance of the suggested technique, in which a series of wire targets are distributed within a background scatter (here). A 64-element linear array is utilized in this experimental dataset. Moreover, the central frequency is considered as 2.6 $M H z$ , and also, the sampling frequency is equivalent to 25 $M H z$ . Moreover, the rat tumor public dataset is used to evaluate the performance of the proposed method in terms of preserving the structure of the tumor. Similar to the ats-wires, this dataset also corresponds to the University of Illinois (here). The imaging setup is the same as the one described for the ats-wires dataset. This dataset, as well as the ats-wires dataset, specifies 200× averaging, meaning that 200 repeated acquisitions were averaged to improve the raw signal-to-noise ratio before beamforming. This averaging improves data robustness but does not bias our comparisons, since all beamformers were applied to the same averaged dataset. Finally, it is worth evaluating the proposed method in a case where point targets and cysts appear simultaneously in a phantom. To this end, the public geab0 dataset from the biomedical ultrasonics laboratory website is used (here). Similar to the previously mentioned experimental datasets, a 64-element linear array is used here. The central frequency is 3.3 $M H z$ , while the sampling frequency is 17.76 $M H z$ in this dataset.

To leverage the high-resolution capability of the NSI method, the bias parameter was set to a small value (0.1) for both the simulation and experimental datasets. Moreover, the cutoff frequency corresponding to the GCF algorithm is considered to be $M_{0} = 1$ . Furthermore, the sub-array lengths in MV and the spatial smoothing process of NSI algorithms are considered as $L, L^{'} = N / 2$ . For the MV algorithm, temporal averaging over 22 samples is also considered. The value of the f-number is set as 1.5 and 0.75 for simulation and experimental datasets, respectively. Note that applying a fixed f-number during beamforming gives rise to a fan-shaped image geometry despite the underlying linear acquisition.

Results

Here, the reconstructed images of the suggested technique as well as other related algorithms are presented using simulation/experimental data. Quantitative evaluations are also conducted.

Simulation Results

The reconstructed images corresponding to the simulated cyst phantom are demonstrated in Figure 2. It is clear that the DAS algorithm successfully preserves the speckle information of the background. However, the reconstructed image of DAS contains some artifacts inside the cysts. The NSI algorithm better suppresses the noise inside the cyst, but it performs weakly in terms of background speckle preservation compared to DAS. In contrast, the proposed SSNSG-DAS method can successfully preserve the speckle information of the background compared to NSI, while the noise intensity within the cystic regions is better suppressed compared to DAS. Quantitative evaluations listed in Table 1 also prove the good performance of the suggested technique. In particular, the gCNR value of the suggested technique reaches the DAS algorithm, while the NSI and MV algorithms’ gCNR are degraded by about 0.14 and 0.62, respectively, in comparison with the suggested technique. Moreover, the sSNR metric shows that the suggested technique maintains the integrity of the background speckle well enough, as the sSNR of the proposed method is improved by about 0.16, 0.38, and 0.37 compared to DAS, NSI, and MV, respectively.

Table 1.

Quantitative Evaluations Corresponding to the Simulated Cyst Phantom Demonstrated in Figure 2.

Algorithm	Metric
Algorithm	$\| C R \|$ (dB)	CNR (dB)	gCNR	sSNR
DAS	30.74	5.75	0.98	1.57
NSI	17.42	−0.35	0.84	1.35
MV	17.48	−1.30	0.36	1.36
SSNSG-DAS	31.36	4.10	0.98	1.73

Experimental Results

In this section, evaluations of three experimental datasets are separately presented.

ats-wires Data: The resulting images related to the ats-wires dataset are shown in Figure 3. The superiority of the suggested SSNSG-DAS technique compared to the NSI and MV algorithms can be clearly seen; the suggested technique either enhances the resolution of the wire targets or preserves the information of the background speckle which results in a better reconstruction of the cysts on the right-hand side of the phantom. However, by using the NSI and MV algorithms, the cysts disappeared due to the suppression of the background speckle. To enhance the qualitative assessment of the suggested technique’s resolution, the lateral variations of a wire target at the depth of 35 mm are also depicted in Figure 4. Similar conclusions can also be made from the lateral variations.

Quantitative evaluations that are presented in Table 2 also confirm similar findings. For instance, the sSNR value of the suggested technique is comparable to that of the DAS algorithm which demonstrates that the SSNSG-DAS algorithm can successfully preserve the information of the background speckle. Also, the FWHM value of the suggested technique is enhanced by about 0.98 mm in comparison with DAS, and it is comparable to the high-resolution NSI and MV algorithms. It should be noted that according to the lateral variations shown in Figure 4, it seems that the proposed method couldn’t suppress the sidelobes well enough. However, note that the imaging medium is speckle-generating; that is, over-suppression of the energy around the targets is equivalent to the loss of background speckle information. If the goal is to only improve the resolution of the targets regardless of the background speckle preservation, using the NSI algorithm is sufficient and the proposed method is not applicable in such a case. The SSNSG-DAS algorithm is recommended in applications where resolution improvement and background speckle preservation are to be achieved simultaneously.

Table 2.

Quantitative Evaluations Corresponding to the Ats-Wires Dataset Demonstrated in Figure 3.

Algorithm	Metric
Algorithm	FWHM (mm)	$\| C R \|$ (dB)	CNR (dB)	sSNR
DAS	1.32	21.27	4.98	1.79
NSI	0.4	14.71	3.35	1.24
MV	0.33	14.48	1.37	1.08
SSNSG-DAS	0.38	21.02	4.86	1.73

Rat Tumor Data: The reconstructed images corresponding to the rat tumor dataset are illustrated in Figure 5. By using the proposed SSNSG-DAS method, the information of the background speckle is preserved comparable to the DAS algorithm. This is while the noise level inside the tumor is better degraded by using the suggested technique compared to DAS. These two factors result in better preservation of the tumor structure. Moreover, it can be seen that the NSI and MV algorithms don’t perform well enough compared to the two other algorithms. The gCNR values of the DAS, NSI, MV, and the suggested SSNSG-DAS algorithms are obtained as 1.58, 0.75, 0.72, and 1.51, respectively, which demonstrate the power of the suggested technique to well preserve the structure of the tumor. Also, the sSNR values of DAS, NSI, MV, and the proposed method are calculated as 1.39, 0.64, 0.61, and 1.52, respectively. It is important to note that the NSI algorithm is non-linear; this can cause skewed distributions and consequently influence gCNR computations. As reported in,³⁶ non-parametric gCNR estimation methods can provide a more robust estimation in such cases. We will benefit from such approaches in our future work to improve quantitative evaluations.

geabr0 Data: The resulting images related to the geabr-0 dataset are depicted in Figure 6(a1) to (d1). It can be observed that the DAS algorithm preserves the information of the background speckle well enough, as expected. However, it doesn’t perform well in resolution improvement. In contrast, the NSI method results in a better resolution than that of DAS. However, this method cannot preserve the background speckle successfully. Although this can be better by assigning a higher value to the bias parameter, the NSI output still doesn’t reach the DAS algorithm. As shown in Figure 6, the proposed SSNSG-DAS algorithm either reaches the improved resolution (similar to NSI) or speckle preservation (similar to DAS). Note that since the MV algorithm is incorporated into the N-NSI beamformed data in the transmit index, it cannot preserve the information of the background speckle. One can achieve the preserved background speckle using the MV algorithm if it is applied along the receive direction. However, it leads to a huge computational complexity which is not justified. The zoomed versions of the reconstructed images at the depth of 75 mm are also presented in Figure 6(a2) to (d2) to better compare the obtained results from the perspective of resolution. Furthermore, the lateral variations corresponding to the targets that are specified with yellow boxes in Figure 6(a1) are also illustrated in Figure 7. Similar conclusions are made according to the illustrated plots. In addition to the reconstructed images and the lateral variations, quantitative evaluations are also performed, and the calculated values are presented in Table 3. In general, it can be found from the table that the resolution of the suggested SSNSG-DAS technique is close to the NSI method, and its speckle preservation ability is close to the DAS algorithm.

Table 3.

Quantitative Evaluations Corresponding to the Geabr0 Dataset Demonstrated in Figure 6.

Algorithm	Metric
Algorithm	FWHM (mm)	$\| C R \|$ (dB)	CNR (dB)	gCNR	sSNR
DAS	1.75	14.49	4.57	0.88	1.99
NSI	1.08	5.70	−2.61	0.52	1.51
MV	0.94	4.91	−5.19	0.43	1.39
SSNSG-DAS	1.01	13.45	3.82	0.85	1.92

Impact of the Constants $α$ and $β$

The reconstructed images of the suggested technique for different values of the constant $α$ , and also, their corresponding lateral variations are presented in Figures 8 (first row) and 9, respectively. It can be observed that by reducing the value of the constant $α$ , less resolution improvement is obtained. In particular, at the depth of 35 mm, the FWHM values of 0.46, 0.5, 0.56, and 0.7 mm are obtained for $α = 2, 1.5, 1, 0.5$ , respectively. That is, to achieve a high-resolution image, a higher value should be assigned to this constant. On the other hand, better noise suppression is achieved for lower values of this constant, as illustrated in Figure 9. Therefore, depending on the attended application, a suitable value should be assigned to the constant $α$ . Usually, for mediums containing cysts, a small value of the constant $α$ is recommended.

The effect of the bias parameter $ϵ$ can also be qualitatively seen from Figure 8 (second row). As expected, the smaller the value of the parameter $ϵ$ , the better the image resolution. In particular, considering $ϵ = 0.1$ , the FWHM value is improved by about 0.51, 0.73, and 0.94 mm, compared to the cases where $ϵ = 0.3, 0.5, 1$ are considered, respectively. One should note that this improvement in the NSI algorithm comes at the expense of background speckle suppression. However, in the proposed method, as the information of the background speckle is inherited from the DAS algorithm, speckle degradation caused by a small value of the $ϵ$ is not noticeable. Therefore, the value of 0.1 is considered for this parameter to take advantage of the improved resolution of the NSI algorithm.

To better see the effect of the constant $β$ , it is better to perform the evaluation on a cyst phantom. In this regard, the simulated cyst phantom is considered, and the corresponding reconstructed images for different values of $β$ are presented in Figure 10. It can qualitatively be observed that by reducing the value of $β$ , the artifact inside the cysts is increased. However, this increment is not significant, as there exist 0.05 to 0.1 dB differences between the calculated CR values. In general, in cases where preserving the cystic regions is of interest, a high value for the constant $β$ is recommended.

In general, as $α$ increases, the first term of the proposed weighting factor in¹¹ increasingly favors values of GCF near 1 and suppresses those near 0; that is, it performs similarly to a step function. Of note, assigning a very high value to $α$ (tends to infinity) leads to numerical instability and undefined weighting coefficients. Also, as the value of the parameter $β$ increases, the second term of the proposed weighting factor in¹¹ behaves like a binary function; imaging points with high values are strongly suppressed (tend to 0), and the ones with small values are potentially retained (tend to 1). For a very large value of $β$ (tends to infinity), a binary behavior starts from the second part of,¹¹ which may not be stable. Therefore, it is important to initialize the parameters $α$ and $β$ from a limited interval.

Discussion and Conclusion

The primary objective of the work presented in this article is to preserve the background speckle information while achieving improved resolution with low computational complexity. To this end, the NSI algorithm is considered as the basis which results in an enhanced image resolution. The MV method can also prepare a high resolution, and in the case of applying temporal averaging, the information of the background speckle is successfully retained. However, the computational burden of this method is considerably high which clearly affects the performance of ultrasound imaging in real-time applications. This is why we considered the NSI algorithm to achieve an improved resolution. Here, the challenge is to simultaneously improve the resolution and contrast of the image, as the main drawback of the NSI algorithm is its poor contrast and its inability to preserve the background speckle information. Although the image contrast and the corresponding background speckle information can be changed by varying the bias value, two points should be noted: 1- the obtained image will be still degraded compared to the DAS algorithm, as the reference case in terms of background speckle preservation, and 2- this will be achieved at the cost of resolution degradation. To address this challenge, the new SSNSG-DAS algorithm is introduced in this paper in which the NSI, GCF, and DAS algorithms are used to obtain appropriate values for each imaging point. The GCF weighting factor, with its specific form in the proposed formula (i.e., the first part of the weighting coefficient $w (n)$ which is presented in¹⁵), helps to insist on coherent imaging points as well as suppressing the incoherent regions. This in turn enhances the image contrast and maintains the background speckle when combined with the NSI beamformed image. The effect of the described term is determined by using the constant $α$ . It is experimentally obtained that the value between the range of $[0.5, 2]$ is appropriate for this constant. Also, the results demonstrate that as the value of $α$ increases, the image resolution improves. However, better noise suppression is performed for smaller values of $α$ , as can be obtained from Figure 9. In general, for mediums containing point targets, the value of $1 - 1.5$ for the constant $α$ is recommended. Also, for mediums containing only cysts or tumor-like lesions, a value of around 0.5 is recommended for this constant.

The second part of the proposed weighting coefficient, that is, ${(1 - N [| z_{S S N S I} (n) |])}^{β}$ , is considered for further noise suppression process, especially inside the cystic regions. By assigning a suitable value to the constant $β$ , the effect of the NSI algorithm (or its spatially-smoothed version) becomes weaker for imaging points that have been incorrectly assigned high values, that is, for regions where noise is mistaken for signal. It is experimentally obtained that the constant $β$ is better to be initialized in the range of $[0.5, 4]$ . Although the second term of the proposed weighting coefficient helps further improve the image quality, its effect is not very significant, as shown in Figure 10. This ensures that the efficiency of the suggested technique does not change considerably for different values of the constant $β$ . In other words, the proposed SSNSG-DAS algorithm is not sensitive to the constant $β$ . Generally, a high value for the constant $β$ is recommended for phantoms including cysts and tumors.

In order to achieve the final image in the suggested SSNSG-DAS technique, the NSI (or its spatially-smoothed version) and the DAS algorithms are weighted and combined incoherently. Here, the DAS algorithm is also included to benefit from its speckle preservation ability. More precisely, the high-resolution NSI and high speckle-preserved DAS algorithms are weighted proportionally according to the proposed weighting coefficient such that the advantages of both of these two algorithms are achieved simultaneously.

In the proposed structure, the NSI operation is applied across all transmit events—that is, $N$ times—each requiring an incoherent sum of $N$ receive elements. This structure was chosen to fully leverage the high-resolution capability of NSI for each individual transmission. Moreover, the GCF map, computed across the NSI set, enables a spatially adaptive fusion with DAS, resulting in an improved balance between resolution and speckle preservation. While this approach does not minimize computational complexity, it prioritizes image quality and remains computationally feasible.

The proposed method may seem complex at first glance. To facilitate understanding of the processing steps of the proposed method, Figure 11 is provided. As shown schematically in the figure, each step plays a specific role; NSI is applied in the transmit index to improve the image resolution. The GCF method favors high-quality estimates and is used to design the proposed weighing factor (i.e., the next step). The proposed weighing function (i.e.,¹¹), which is shaped as a generalized softmax function, ensures a smooth and adaptive combination of NSI and DAS information based on the calculated GCF map. Furthermore, the spatial smoothing is also considered for better performance of the NSI algorithm. This structure is motivated to prevent sacrificing speckle fidelity of the NSI algorithm, which is crucial in the interpretation of medical ultrasound images.

The remarkable effectiveness of the suggested SSNSG-DAS technique is shown utilizing simulation and experimental datasets. In particular, the improved image resolution while keeping the background speckle by using the proposed method is demonstrated in Figure 3 that corresponds to the ats-wires dataset. Table 2 also confirms the qualitative findings, as the FWHM value of the suggested technique is enhanced by about 0.94 mm in comparison with DAS while it is comparable to that of NSI, and the sSNR value of the proposed method is improved by about 0.49 in comparison with NSI while it is comparable to that of DAS. Also, the ability of the proposed method only from the perspective of speckle preservation can be investigated in Figure 5. In addition to visual evaluation, quantitative evaluations also demonstrate a similar conclusion, as the sSNR value of the suggested technique is improved by about 0.13, 0.88, and 0.91 in comparison with DAS, NSI, and MV methods, respectively, for the rat tumor dataset. The geabr0 dataset, as a more complex dataset that contains either wire targets or anechoic and hyperechoic cysts, is also used to demonstrate that the suggested technique can successfully improve the resolution while the structures of the cystic regions are retained (Figure 6). In this dataset, the average value of the FWHMs over all the wire targets for the proposed method is improved by about 0.74 mm and 0.07 mm compared to the DAS and NSI methods, respectively. This demonstrates that the resolution of the proposed method is comparable with the NSI algorithm. This is while the CNR value of the suggested technique is enhanced by about 6.43 dB in comparison with NSI, indicating the superiority of the suggested technique from the perspective of contrast.

As mentioned earlier, the suggested technique improves the image quality while the computational complexity isn’t increased significantly. It is worth discussing the computational complexity of the related algorithms; the computational complexity of the DAS algorithm, as a high-speed beamforming method, is on the order of $O (N)$ . The NSI algorithm is similar to the case in which the DAS algorithm is applied three times with three different pre-defined weighting coefficients. Therefore, the NSI algorithm also exhibits computational complexity on the order of $O (N)$ . In the proposed method, the spatial smoothing process is used which has the computational complexity on the order of $O (L^{'})$ . In addition, the GCF, NSI, and DAS algorithms are also used, the computational complexity of which is on the order of $O (N)$ . Totally, the computational complexity of the proposed SSNSG-DAS algorithm is on the order of $N > L^{'}$ (where $O (N)$ ). Of course, compared to the high-speed DAS method, the proposed algorithm induces some additional computations, but it doesn’t push the order of the computational complexity to exceed O(N). In contrast, the MV algorithm has computational complexity on the order of $O (L^{3})$ . Comparing MV and the proposed method, one can conclude that the suggested SSNSG-DAS method improves the image quality with much less computational complexity.

Finally, it is worth mentioning that the drawback of the suggested technique is that its performance depends on two constants $α$ and $β$ . Although the appropriate values of these two constants are experimentally determined for different types of imaging mediums, adjusting them independently and adaptively will improve the performance of the proposed algorithm. One can consider an adaptive process to make the constants $α$ and $β$ user-independent. This could be studied in more detail in our future studies.

Footnotes

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is based upon research funded by Iran National Science Foundation (INSF) under project No. 4031015.

ORCID iDs

Roya Paridar

Babak Mohammadzadeh Asl

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A High-Resolution and High-Contrast Beamforming Algorithm Based on Null Subtraction Imaging Applied to Synthetic Transmit Aperture

Abstract

Keywords

Introduction

Background

Proposed Method

Quantitative Performance Indicators

Simulation and Experimental Setup

Results

Simulation Results

Experimental Results

Impact of the Constants α and β

Discussion and Conclusion

Footnotes

Declaration of Conflicting Interests

Funding

ORCID iDs

References

Impact of the Constants $α$ and $β$