Combining Windowed Enveloping and the Delay and Sum Algorithm for Photoacoustic Image Reconstruction

Abstract

Delay and sum (DAS) is one of the most common beamforming algorithms for photoacoustic image reconstruction. Owing to its high computational efficiency and ease of implementation, this method is particularly well-suited for real-time photoacoustic imaging. However, its shortcomings, such as that the algorithm can make high sidelobes and strong artifacts, are as prominent as its advantages. Some improved algorithms based on spatial coherence theory, such as DMAS, have significantly enhanced imaging quality. In this paper, we analyzed the beamforming principle and propose a photoacoustic imaging method by combining windowed enveloping and the delay and sum beamforming algorithm. The delay and sum beamforming algorithm is used for ensuring high computational efficiency, and windowed enveloping for the suppression of sidelobes and artifacts. Tests were performed for a simple circular source model and a multiple-source model. The results show that our method can effectively improve the quality of reconstructed images compared with DAS and some improved methods. In addition, this method also retains the advantage of the high parallelism of the DAS algorithm and is suitable for real-time imaging systems.

1. Introduction

Photoacoustic imaging (PAI) is an emerging biomedical imaging method which integrates the principles of optical and ultrasonic imaging [1]. It combines the high-resolution characteristic of optical imaging with the deep-penetration capability of ultrasonic imaging [2]. PAI has garnered significant attention and become a research hotspot in the field of biomedical imaging. It has great application potential in many fields such as tumor detection [3,4,5], vascular imaging [6,7,8], skin disease analysis [9], and drug delivery monitoring [10]. The principles of PAI are based on the photoacoustic effect [11]. When tissue absorbs light energy, the temperature of the tissues increases. This results in transient thermal expansion and the contraction of the tissues. The thermal expansion and contraction propagate outward in the form of ultrasonic waves, which are known as photoacoustic signals (PA signals) [12]. Photoacoustic signals are detected by ultrasonic transducers and subsequently processed through various reconstruction algorithms, enabling the generation of images that reflect the spatial distribution of optical absorption [13].

The delay and sum (DAS) algorithm is one of the most commonly used beamforming algorithms in photoacoustic image reconstruction [14,15]. Known for its simple principles and ease of implementation, DAS is frequently used in real-time imaging systems. However, its simplicity also results in a higher level of side lobes and more pronounced artifacts in the reconstructed images. In recent years, a great deal of research has been conducted to address these issues, leading to the development of several improved algorithms. These enhancements have shown promising results in reducing side lobes and artifacts, contributing to the ongoing evolution of PAI techniques. For instance, Matron and colleagues proposed the Delay Multiply and Sum (DMAS) algorithm, which improves the reconstructed image in terms of contrast and resolution [16,17]. Moein Mozaffarzadeh and colleagues introduced a coherence factor (CF), which integrates the coherence factor (CF) and modified coherence factor (MCF) into the traditional DAS beamforming technique, known as DAS-CF [18,19,20]. This combined approach is applied in PAI to mitigate the impact of unwanted off-axis signals on beamforming. Subsequently, the team proposed an improved version of the DMAS algorithm, known as the double-stage DMAS algorithm, which enhances image resolution and side-lobe levels [21,22]. Compared to the DMAS algorithm, the side-lobe level is reduced by approximately 10 dB. Several other optimization algorithms, which are derived from the DMAS method, have also contributed to improving the quality of the reconstructed images to some extent [23,24]. There are also some other methods that have made significant contributions to improving the quality of the reconstructed images [25]. Although the aforementioned algorithms have improved the imaging quality, their effectiveness in suppressing the aliasing issue between adjacent microstructures remains limited. However, these methods, like the DAS method, analyze the photoacoustic signal as a single-valued signal. In reality, the acoustic pressure signal is a short pulse lasting for a certain period, composed of both positive and negative amplitudes, forming an N-shaped signal. The duration of this N-shaped signal is related to the size of the source [26]. For a particular pixel within the imaging area, the complete PA signal for that point over the entire imaging time can be obtained through multiple DAS calculations. This signal should also be an N-shaped signal. Furthermore, the N-shaped signals acquired from different pixel points exhibit variations in amplitude and in the timing of their peak values. Thus, the real source points in the imaging region and the pixels with artifacts can be distinguished.

Ma Xiang and colleagues proposed the Multiple Delay and Sum with Enveloping (multi-DASE) algorithm, which aims to suppress side lobes and artifacts [27]. This method not only computes the initial acoustic pressure signal but also computes the complete signal for each point. The complete signal is then enveloped. By comparing the differences between the envelope signals, the method distinguishes between signal sources and artifacts. This method has shown strong performance in suppressing artifacts. However, when imaging a slightly larger signal source, the interior of the signal source is also suppressed, leading to a reconstructed image that exhibits missing internal structures, thereby causing significant distortion in the final reconstructed result. Furthermore, the larger the volume of the source, the more pronounced this phenomenon becomes.

By analyzing this problem, we find that the original N-shaped signal contains abrupt changes. Directly obtaining the envelope signal through the Hilbert transform results in overshoot. Overshoot makes it difficult to distinguish between some pixels inside the real source and the pixels in the artifact regions, resulting in the suppression of both. This is determined by the properties of the Hilbert transformation. When the signal contains an abrupt change, the Hilbert transform produces the Gibbs phenomenon. Specifically, the Hilbert transform involves convolution of the signal using a kernel function of ${\displaystyle \frac{1}{\pi t}}$, which is singular and particularly sensitive to abrupt points, resulting in overshoot. The simplest and most effective way to solve this problem is the smooth preprocessing of the original signal. We chose to use the window function to smooth the original signal. This paper builds upon the recovery of the complete signal, and a window function is multiplied before taking the envelope to weaken the abrupt changes in the N-shaped signal, thereby suppressing the overshoot in the envelope signal obtained via the Hilbert transform. The simulation of a simple circular source model and an irregularly shaped source model has verified that this method can effectively avoid distortions caused by overshooting when directly using envelope signals for imaging.

2. Methods

2.1. Multiple Delay and Sum with Enveloping

The basic principle of PAI technology is the photoacoustic effect. When a pulsed laser irradiates a biological tissue, the chromophore in the tissue absorbs the optical energy, causing temperature changes. This, in turn, leads to local thermoelastic expansion and vibration of the tissue, generating sound waves that propagate outward, termed PA signals. These signals travel from the interior of the biological tissue to the surface, where they are captured by an array of transducers surrounding the tissue and converted into electrical signals. Through signal processing and reconstruction algorithms, these electrical signals are utilized to generate images that depict the internal structure and functional characteristics of the tissue.

The delay and sum (DAS) algorithm is a commonly used image reconstruction method in PAI. The fundamental principle of DAS is to account for the different times it takes for a PA signal to propagate from the acoustic source to each transducer. By applying appropriate delays to the signals received by each transducer, the signals originating from the same source can be aligned in time. These aligned signals are then summed, thereby enhancing the signal strength from specific source locations while suppressing noise and non-coherent signals. The advantages of DAS are its straightforward principles and ease of implementation, making it suitable for real-time imaging systems.

In the DAS algorithm, the output signal is the sum of the received signals from each sensor after applying appropriate delays. This process can be mathematically represented. Suppose there are $N$ ultrasonic transducers, each receiving a signal $S\left(i;t\right)$, where $i$ denotes the sensor number and $t$ denotes time, indicating that at time t, the i-th sensor receives a signal of $S\left(i;t\right)$. For a specific source location $\left(x,y\right)$, each sensor receives the signal with a corresponding delay time $\tau \left(x,y,i\right)$. The output signal $Y_{DAS}\left(x,y\right)$ of the DAS algorithm at location $\left(x,y\right)$ can be defined as:

$$Y_{DAS}\left(x,y\right)={\sum }_{i=1}^{N}S\left(i,\tau \left(x,y;i\right)\right),$$

(1)

the output $Y_{DAS}\left(x,y\right)$ is the sum of the signals received by each transducer at the corresponding delayed time $\tau \left(x,y,i\right)$; it can be understood as returning the data received by each sensor at the corresponding delay time back to the corresponding position, which is calculated by the following formula:

$$\tau \left(x,y,i\right)={\displaystyle \frac{l\left(x,y,i\right)}{c}},$$

(2)

here, $l\left(x,y,i\right)$ in the formula represents the distance from the place $\left(x,y\right)$ to the i-th transducer, and $c$ is the propagation speed of the acoustic signal.

Figure 1 illustrates the imaging principle of DAS. As shown in Figure 1, when a transducer receives the PA signal from a certain position, points surrounding the certain position that are equidistant to this transducer will influence the data of this point. Conversely, in the reconstruction process, the reconstructed data received by the surrounding pixels will also be affected by the data from this point. This phenomenon is the fundamental cause of the arc-shaped artifacts observed in reconstructed images using DAS.

During photoacoustic imaging, when a pulsed laser irradiates a spherical absorber, first of all, the incident laser is selectively captured by the absorber in the form of photons. This process strictly follows the Beer–Lambert law, and the absorption intensity depends on the optical properties (such as the absorption coefficient) and the geometric size of the absorber. The absorbed light energy is rapidly converted into heat energy through non-radiative relaxation processes. Due to the extremely short duration of the laser pulse (usually in the nanosecond-to-microsecond range), the absorber undergoes an adiabatic heating process, and its temperature significantly increases within a very short period of time. According to the theory of thermal elasticity, this rapid temperature rise causes the absorber to undergo transient volume expansion. Notably, due to the characteristic time of thermal diffusion being much longer than the laser pulse width, the heat is almost completely confined within the absorber during the expansion stage. This confined thermal elastic expansion generates a significant initial pressure rise within the absorber, and the resulting pressure disturbance propagates to the surrounding medium in the form of ultrasound. As time evolves (usually in the nanosecond-to-millisecond range), the heat begins to transfer to the surrounding medium through the thermal diffusion mechanism, which follows Fourier’s law of heat conduction. As heat is lost, the absorber temperature gradually decreases and triggers a contraction effect, thereby generating a negative pressure signal with opposite polarity to the initial expansion stage. This dynamic process of first expansion and then contraction ultimately produces characteristic N-shaped bipolar pressure pulses. Moreover, the distance between the signal peak and the signal valley is proportional to the size of the absorber. Shown in Figure 2 are the theoretical data of the photoacoustic signal of a spherical absorber. Based on the aforementioned descriptions, it is evident that the photoacoustic signal generated by an absorber is not a single-valued signal. If one focuses only on the instantaneous data received by the sensor at a certain moment during the imaging process, this is obviously not comprehensive enough. Given the impact of factors such as the sampling frequency of the sensor data, treating them as a single-valued signal during analysis may potentially introduce certain errors. Therefore, we need to analyze the complete signal of each imaging point throughout the whole period.

By incorporating a time index into Equation (1), we derive Equation (3). It is defined as follows:

$$Y_{multi-DAS}\left(x,y;t\right)={\sum }_{i=1}^{N}S\left(i,t+\tau \left(x,y;i\right)\right)$$

(3)

where $Y_{multi-DAS}\left(x,y;t\right)$ represents the sum of the data returned by each sensor to the pixel $\left(x,y\right)$ at time t. Equation (3) demonstrates that for any position $\left(x,y\right)$ within the imaging region, the complete beamforming signal across the entire imaging period can be reconstructed through multiple DAS operations. $S\left(i;t\right)$ is the signal received by the i-th sensor at time t, $\tau \left(x,y;i\right)$ is defined in the same way as in Equation (2). From Equation (3), we can see that when $t=0$, the output of $Y_{multi-DAS}\left(x,y;t\right)$ is the same as that of $Y_{DAS}\left(x,y\right)$. This also indicates that the DAS algorithm only computes the signal at time $t=0$.

$$Y_{multi-DAS}\left(x,y;0\right)={\sum }_{i=1}^{N}S\left(i,\tau \left(x,y;i\right)\right)=Y_{DAS}\left(x,y\right).$$

(4)

According to Equation (3), the complete signal for each pixel during the whole period can be acquired. This signal corresponds to the photoacoustic signal emitted by the pixel point throughout the imaging period, and the photoacoustic signals generated at different positions are different. We set the following experimental conditions within the imaging area: Firstly, at the center position of the 21.6 mm × 21.6 mm square imaging area, a uniform circular sound source model with a radius of 0.2 mm was placed, as shown in Figure 3a. We selected two pixel points for comparison analysis—the first point was located at the center of the sound source (the true source point), and the second point was located 1.2 mm to the right of the center, outside the sound source (a non-source point). Through Equation (3), the complete signals of the two points were obtained, respectively, as shown by the black line in Figure 3b,c.

The complete photoacoustic signals acquired from the two pixel points both exhibit nearly N-shaped bipolar characteristics. However, there are also some obvious differences in signal amplitude and phase between the two. To further investigate these differences, we computed the envelopes of the complete signals. The envelope can effectively extract the main features of the signal and also eliminate the negative components present in the PA signal. PA imaging is used to reveal information about the optical absorption characteristics within tissues. Consequently, it was established that the final imaging result should, in theory, contain no negative components. This is because the photoacoustic effect results from the conversion of absorbed optical energy into acoustic waves, which inherently does not produce negative values. If negative values are involved in actual imaging, they will cause discontinuities in the imaging results. By extracting the envelope, we can effectively avoid this issue. The Hilbert transform is used to compute the envelope, as it is a standard method for determining the amplitude envelope of a signal. It is defined as [28]:

$${Env}_{multi-DAS}\left(x,y;t\right)=\left|H\left(Y_{multi-DAS}\left(x,y;t\right)\right)\right|,$$

(5)

$$H\left(f\left(t\right)\right)=f\left(t\right)\ast {\displaystyle \frac{1}{\pi t}}=F^{-1}\left(F\left(f\left(t\right)\right)\times F\left({\displaystyle \frac{1}{\pi t}}\right)\right),$$

(6)

$$F\left({\displaystyle \frac{1}{\pi t}}\right)=-j\mathrm{s}\mathrm{g}\mathrm{n}\left(\omega \right),$$

(7)

where $H\left(f\left(t\right)\right)$ is the Hilbert transform of $f\left(t\right)$, $F\left(f\left(t\right)\right)$ is the Fourier transform of $f\left(t\right)$, and $\mathrm{sgn}\left(\omega \right)$ is the sign function the value of which is 1 if $\omega$ is greater than 0, otherwise, it is −1.

By applying the Hilbert transform to an N-shaped signal, its envelope signal can be extracted, which exhibits a pulse-like characteristic. There exists a peak in the envelope signal, which is the maximum value of the envelope signal, and the corresponding moment of the peak is $t_{peak}$, which is defined as:

$$t_{peak}\left(x,y\right)=\arg \max \left(En{v}_{multi-DAS}\left(x,y;t\right)\right).$$

(8)

The black lines in Figure 3b,c represent the envelopes obtained, respectively. It is easy to find that when $\left(x,y\right)$ is the pixel at the source, the $t_{peak}$ of the envelope signal is the initial time $t_0=0$. When $\left(x,y\right)$ is not the location of the source, the $t_{peak}$ corresponding to the envelope signal is no longer the initial time. $\left|t_{peak}\right|$ is introduced here to represent the difference between $t_{peak}$ and $t_0$. It was found through experimentation that when the distance between $\left(x,y\right)$ and the source is greater, $\left|t_{peak}\right|$ will be of a larger value.

From this, the source and the artifact can be distinguished during imaging, thus the artifact can be suppressed. Here, a suppressor factor needs to be introduced, which is defined as

$$inh\left(x,y;t\right)={\displaystyle \frac{En{v}_{multi-DAS}\left(x,y;t\right)}{En{v}_{multi-DAS}\left(x,y;t_{peak}\right)}}$$

(9)

${\mathit{E}\mathit{n}\mathit{v}}_{\mathit{m}\mathit{u}\mathit{l}\mathit{t}\mathit{i}-\mathit{D}\mathit{A}\mathit{S}}\left(\mathit{x},\mathit{y};{\mathit{t}}_{\mathit{p}\mathit{e}\mathit{a}\mathit{k}}\right)$ is the peak value or the maximum value in the envelope signal. Given that the envelope signal is a pulse wave as is shown in Figure 3b,c, then we can see from Equation (9) that if the value of $\left|t_{peak}-t\right|$ is larger, we will obtain a smaller value of $inh\left(x,y;t\right)$, and the envelope signal will be suppressed by the following expression

$$En{v}_{inh}\left(x,y;t\right)=En{v}_{multi-DAS}\left(x,y;t\right)\times in{h}^k\left(x,y;t\right)$$

(10)

where $k$ represents the level of inhibition. The larger $k$ is, the more obvious the inhibition effect is. After obtaining the suppressed signal, use the initial value of the suppressed signal as the pixel value to image, which can be defined as

$$Y_{multi-DAS}\left(x,y\right)={Env}_{inv}\left(x,y;0\right)={Env}_{multi-DAS}\left(x,y;0\right)\times {inh}^k\left(x,y;0\right)$$

(11)

${\mathit{Y}}_{\mathit{m}\mathit{u}\mathit{l}\mathit{t}\mathit{i}-\mathit{D}\mathit{A}\mathit{S}}\left(\mathit{x},\mathit{y}\right)$ represents the data that is ultimately used for reconstruction at the position $\left(\mathit{x},\mathit{y}\right)$.

2.2. Signal Overshoot and Smoothing Signal with Window Function

When there are abrupt changes in the signal, directly applying the Hilbert transform to extract the envelope results in overshoot phenomena in the envelope signal. As shown in Figure 4, this is a function with abrupt changes, as well as the envelope signal obtained after applying the Hilbert transform. The envelope signal shows overshoot at two positions, which correspond exactly to the locations where the data in the original signal undergoes abrupt changes.

The reasons for this phenomenon are as follows. First of all, Hilbert transform is actually a convolution operation; that is, the convolution of the original function and ${\displaystyle \frac{1}{\pi t}}$. It has a strong long-tail effect, meaning that the energy distribution at the points of signal abrupt changes cannot be fully smoothed by the filter, leading to oscillations near the abrupt changes. Secondly, the edge effect caused by these abrupt changes leads to an excessive concentration of signal energy at the points of abrupt changes, further exacerbating the overshoot phenomenon in the transformed signal. In addition, the Gibbs phenomenon also plays a crucial role, as it describes that the ideal filter cannot completely suppress the oscillation when dealing with the sudden change signal, which is consistent with the wideband filtering property of the Hilbert transform. This results in the spectral energy near the abrupt changes not being evenly distributed, causing significant spectral leakage. Finally, the existence of spectrum leakage makes part of the energy overflow into the adjacent frequency components, which shows the oscillatory behavior of the transformed signal in the time domain.

Combining the above reasons, we believe that smoothing the original signal is an effective way to solve this problem. In our study, we consider using the method of window functions to smooth the original signal. The principle of window functions is to apply weighting to the signal, causing it to decay smoothly at the edges, thereby avoiding discontinuities in the signal. This smoothing process effectively reduces abrupt changes in the signal, thereby lowering high-frequency components in the spectrum and mitigating spectral leakage issues. Therefore, in signal processing, selecting an appropriate window function can significantly enhance the quality and accuracy of spectral analysis. Commonly used window functions include the Hanning window, Hamming window, Blackman window, and Gaussian window. In our experiment, to facilitate the adjustment of window function parameters, we selected the Gaussian window for improvement attempts. The Gaussian window demonstrates excellent smoothness in both the time domain and frequency domain. It can effectively suppress abrupt changes and side-lobe effects in the signal, reduce the interference from high-frequency components, significantly decrease spectral leakage, and thereby improve resolution. It is defined as

$$g\left(x\right)=a\times \exp {\displaystyle \frac{-{\left(x-b\right)}^2}{2c^2}}$$

(12)

where $a$ is the amplitude, which determines the maximum height of the curve; $c$ is the standard deviation, which controls the width of the curve; and the Gaussian function is symmetric with respect to $x=b$. After the signal is smoothed by the Gaussian window, the envelope can be obtained to effectively avoid the overshoot; as shown in Figure 5, the initial signal remains the one shown in Figure 4, which contains the mutant signal. The images in Figure 5 present, respectively, the envelope signal directly obtained through Hilbert transform and the envelope signal obtained after windowing processing.

3. Results

We performed numerical simulations using the k-WAVE toolbox (version 1.4) to demonstrate the reconstruction results using the DAS algorithm and multi-DASE algorithm, respectively [29]. The simulation was based on a homogeneous propagation medium with the sound speed set to 1500 m/s. The computing area is a square area with borders (21.6 × 21.6 mm²) as shown in Figure 6a. In total, 108 transducers are uniformly placed at the top of the area, and the circular sound source is placed in the middle of the area with a radius of 1.5 mm. Firstly, the DAS algorithm is used for reconstruction, and there are two obvious cambered artifacts in the reconstruction results, as shown in Figure 6b. In order to better illustrate the superiority of the multi-DASE method, we also used the DMAS method to reconstruct the source, and the reconstruction result is shown in Figure 6c. Although the artifacts have been effectively suppressed, it is evident that the reconstructed images exhibit discontinuities, and the reconstruction results still demonstrate a certain degree of shape distortion. When applying Equation (11) for reconstruction, the value of k needs to be determined first. The suppression effect corresponding to the smaller k is not obvious, while the larger k may lead to excessive suppression, which results in unsatisfactory reconstruction results. To ensure the scientific validity of selecting the k value, a quantitative criterion was established: when $inv\left(x,y;t\right)=0.5$, introducing k should enable the suppression factor to reach at least ${inv}^k\left(x,y;t\right)=0.1$. Based on this criterion, and after conducting numerical calculations and experimental validations, the optimal parameter value of k = 3 was ultimately determined. The reconstruction result is shown in Figure 6d. From this result, it can be seen that, compared with the first two methods, artifacts are effectively suppressed, and the shape of the reconstructed result is well preserved. However, the internal structure of the source is also suppressed, leading to a darker appearance in the reconstructed region and causing a distinct “gap” to appear.

Since different k values correspond to different degrees of suppression, we attempt to use different values of k for reconstruction. This was carried out to investigate whether the selection of k values affected the internal reconstruction results. To make the results more clear, we chose a circular sound source with a radius of 1.5 mm for verification. Figure 7 shows reconstructed images using Equation (11) with different k values, respectively. When k = 0, the reconstruction result of multi-DASE, compared to the reconstruction result of DAS, eliminates the negative components, resulting in more complete boundaries. However, in terms of image quality, there is no significant improvement. Since it does not suppress artifacts, the reconstruction results of multi-DASE still exhibit strong artifacts, as shown in Figure 7a. As the value of k increases, artifacts are increasingly suppressed, with larger k values leading to the stronger suppression of artifacts. We observed that when the value of k is relatively small, the problem of missing the internal structure of the reconstructed source also exists, as shown in Figure 7b. As k increases, the problem of the missing internal structure of the reconstructed source becomes more and more obvious, as shown in Figure 7c,d. Through this experiment, it can be determined that the problem of the missing internal structure of the reconstructed source is not directly determined by the value of k.

We restored the complete signal at the pixel point at the center of the circular sound source shown in Figure 6, and calculated its envelope. The result is shown in Figure 8b. The envelope signal observed is not a single peak but instead exhibits a bimodal distribution, which is what we refer to as signal overshoot. The envelope value at t = 0 corresponds precisely to the “concavity” between the two peaks. For comparison, we also calculated the complete signal and envelope signal at the center of a circular sound source with a radius of 0.5 mm, as shown in Figure 8a. It is evident that when the volume of the signal source is larger, the “concavity” between the two peaks in the envelope signal becomes more pronounced. If we directly suppress the envelope signal using Equation (10), the envelope value at t = 0 will naturally be subject to suppression. The final result is that the source’s internal region is also suppressed. And, as mentioned earlier, the larger the signal source volume, the more pronounced the suppression of its internal region.

The recovered N-shaped PA signal for each pixel can be understood as returning the data received by transducer to the pixel with a time delay. In short, when the sensor receives the PA signal, it records data only at the moment the PA signal arrives at the sensor, and records 0 at all other times, so the signal indicates a sudden change at the moment the sensor starts receiving it and at the moment it stops. Conversely, when we want to obtain the complete signal of a certain pixel over the entire imaging time, the data returned by the sensor will also suddenly change at the moment the signal reaches that pixel point and at the moment it ends, and will be 0 at all other times. In this way, if we directly calculate the envelope of an N-shaped signal containing abrupt changes, it will cause overshoot in the envelope signal, manifesting as the two peaks we observe. This phenomenon is particularly evident when the source volume is large. Therefore, it needs to be smoothed before we calculate its envelope.

The amplitude of the Gaussian function will affect the final value. Considering that the purpose of adding the window function is to smooth the signal as much as possible without changing the amplitude of the envelope signal, in our experiment, the amplitude of the Gaussian function is set to 1 (a = 1). The standard deviation of the Gaussian function determines the width of the Gaussian curve. A wider Gaussian curve results in insufficient smoothing, while a narrower curve leads to over-smoothing. To achieve optimal smoothing, we aim to match the width of the Gaussian curve with that of the complete signal. In our experiment, as all calculations involve discrete data, the width of the Gaussian window can be adaptively adjusted based on the width of the obtained complete signal. For simplicity, we set the standard deviation to 1 (c = 1). Additionally, the improper selection of the position of the symmetry axis may cause the window function to fail to play a smoothing role. The symmetry axis is set to the middle position of the double peaks of the envelope signal with overshoot. Subsequently, we perform a computer simulation of a circular source with a radius of 1.5 mm.

Figure 9 shows the envelope signal obtained directly and the envelope signal obtained by multiplying the window function. It is obvious that, after the window processing, the envelope signal obtained by the Hilbert transform will not overshoot even if it is not a completely smooth signal. The ultimate purpose of obtaining the envelope signal is to determine a suitable inhibitory factor, so even if the amplitude of the envelope signal obtained after multiplied by the window function is greatly changed compared with the previous, it will not have much impact on the final imaging results. In order not to alter the amplitude, we still choose to use the data corresponding to the moment $t_0=0$ of the original envelope signal as the initial imaging data. Then, the final output result of the method we propose is as follows:

$$Y_{w-inv}=Env\left(x,y;t_0\right)\times {inv}^k\left(x,y;t_0\right),$$

(13)

the definition of the inhibitory factor is as follows:

$$inv\left(x,y;t\right)={\displaystyle \frac{{Env}_{win}\left(x,y;t\right)}{{Env}_{win}\left(x,y;t_{peak}\right)}},$$

(14)

where ${Env}_{win}\left(x,y;t\right)$ refers to the envelope signal obtained after combining the window function, the calculation method and formula of $t_{peak}$ are the same as those in Equation (8). The value of k is set at 3 based on the established criteria discussed earlier.

Therefore, when imaging again, the reconstruction result will not have the distortion problem caused by overshoot. To facilitate a visual comparison, we have integrated the reconstruction results obtained using four different methods into one figure, as shown in Figure 10.

The DAS algorithm, which merely calculates the initial values of signals during the imaging process, leads to prominent artifacts in the reconstructed images. Additionally, there exists a certain degree of deviation in the shape of the reconstructed sources as shown in Figure 10a. The results obtained by using the DMAS method for reconstruction have reduced artifacts, but the reconstructed images have obvious discontinuities and distortions in shape, which is shown in Figure 10b. Compared with DAS, the multi-DASE algorithm shows significant improvement and effectively reduces the presence of artifacts. However, it has the problem of internal structure omission caused by the overshoot of the envelope signal, as shown in Figure 10c. The method we proposed not only ensures that the artifacts are suppressed, but also retains the internal information of the source more comprehensively, as shown in Figure 10d.

We calculated the Structural Similarity Index (SSIM) of the reconstructed images obtained using the DAS algorithm, the multi-DASE algorithm, and our proposed method to quantify the superiority of our proposed method. The SSIM has a good correlation with human subjective visual perception and can intuitively reflect the effectiveness of the reconstructed images through the data. Before calculating the SSIM, we need to normalize the reconstructed images obtained by different methods. The data are presented in

Table 1. SSIM for circular source using various methods.

	DAS	DMAS	DASE	Proposed Method
SSIM	0.0187	0.5486	0.7518	0.8660

.

The table above presents the SSIM values calculated by comparing the reconstruction results from different methods with the initial sound pressure field. The DAS algorithm simply sums the data for imaging, and the reconstructed image has a lot of artifacts, and there are negative values in the reconstructed data, so its SSIM value is very low, it is just 0.0187. The DMAS method can suppress artifacts and the SSIM value has been improved to 0.5486. However, due to the problems of slice and shape distortion, there is still much room for improvement. Although the DASE algorithm suppresses the artifacts to a large extent, its source is also suppressed, so its SSIM value is improved compared with the former two methods, reaching 0.7518, but there is still a lot of room for improvement. The reconstructed image obtained by our method can suppress the artifacts and distinguish the source at the same time, so it is most similar to the original sound source field, and the SSIM index reaches 0.8660.

We extracted the lateral profile at y = 10.8 mm and the vertical profile at x = 10.8 mm, respectively, of the reconstructed image using the DASE algorithm and the method proposed in this paper. Figure 11a,c show the lateral profile and vertical profile at the center of the circle source of the reconstruction imaging using multi-DASE, and they show that there is clearly inhibition within the source. Figure 11b,d are the lateral profile and vertical profile at the center of the circle source of the reconstruction imaging using the method proposed in this paper, and show that there are no obvious faults in the reconstructed images processed by our method. The signal distribution within the sound source is more uniform, no abnormal suppression phenomenon occurs, the overall signal strength remains good, and there is no obvious signal loss. Although this section focuses solely on presenting the reconstruction results of a sound source with a radius of 1.5 mm using various methods, as previously discussed, our approach can adaptively adjust the Gaussian window size based on the sound source dimensions. So our method is applicable to sound sources of different sizes.

To verify the robustness of our method under different noise levels, we added random Gaussian white noise of different levels for simulation. Figure 12 shows the reconstructed images at different noise levels. Figure 12a shows the reconstruction results obtained using different methods when the SNR ratio is 34 dB. The noise level is relatively low, and the reconstruction result is not very different from that in the noise-free case. Figure 12b shows the reconstruction results obtained using different methods when the SNR ratio is 20 dB. It can be seen that the reconstructed images obtained by the four methods are all affected by noise, resulting in a decrease in reconstruction quality. Figure 12c shows the reconstructed images obtained using different methods when the signal-to-noise ratio is 17 dB. Due to the high noise level, the reconstruction results obtained by various methods have all been affected. To quantitatively compare the robustness of different methods in dealing with noise, we calculated their respective SSIM values for comparison, and the results are shown in

Table 2. SSIM of various methods at different SNRs.

SNR	DAS	DMAS	DASE	Proposed Method
34 dB	0.0185	0.5173	0.7360	0.8602
20 dB	0.0168	0.4908	0.6980	0.8328
17 dB	0.0167	0.3625	0.6105	0.8164

.

The data shown in Table 2 can demonstrate that, under different noise levels, the quality of the reconstructed images obtained by our proposed method remains the highest. Additionally, as the noise level increases, the SSIM of the reconstruction results of the DMAS method decreases by 0.1548, that of the DASE method decreases by 0.1255, and that of our method decreases by 0.0442, verifying the robustness of our method. After a simple analysis, we concluded that the key characteristic of our proposed method lies in its ability to smooth the initial signal, which can eliminate the influence of noise to a certain extent and thus has high robustness.

We also conducted simulations for multiple sources. The distribution of multiple sources is shown in Figure 13a. When there are multiple sound sources within the imaging area, the complete photoacoustic signal obtained through DASE is no longer a distinct N-shaped signal, as shown in Figure 13b. Throughout the whole imaging period, the signal shape appears as an irregular and fluctuating one. Considering that the signals occurring at times far from $t=0$ can be regarded as interference signals, we eliminate this part of the signal and only process the portion deemed as useful signals, as shown in Figure 13c.

Figure 14 shows the comparison results of the reconstructions of multiple sources using different methods. Figure 14a is the reconstruction result obtained by using the DAS method, which exhibits strong artifacts. Figure 14b shows the reconstruction result obtained by the DMAS method. It can be seen that the artifacts in the image have been suppressed to a certain extent, but there are some relatively obvious layers in the reconstruction result, which makes the reconstruction quality not very high. Figure 14c presents the reconstruction using the multi-DASE. Compared with the previous two methods, the artifacts have been well suppressed. However, at the same time, the internal structure of the source has also been suppressed, resulting in the loss of internal structural information. Figure 14d is the reconstruction result using our proposed method. It can be observed that our method not only preserves the artifact suppression capability of the multi-DASE method but also addresses the issue of missing internal structural information in the reconstructed source, thereby substantially enhancing the quality of the reconstructed image. We also calculated the SSIM for the reconstruction results of the irregular source to quantify the differences between images reconstructed using different methods, as presented in

Table 3. SSIM for multiple sources using various methods.

	DAS	DMAS	DASE	Proposed Method
SSIM	0.0106	0.3424	0.7881	0.9783

. The SSIM value for the DAS method is only 0.0106. In comparison, the DMAS method demonstrates a stronger ability to suppress artifacts, increasing the SSIM to 0.3424. Nevertheless, there remains significant room for improvement. The multi-DASE method exhibits superior performance in artifact suppression, with the SSIM significantly improving to 0.7881. Our proposed method not only preserves the advantages of the multi-DASE method but also addresses the issue of partial information loss observed in the multi-DASE method. Consequently, the SSIM for our proposed method reaches 0.9783, indicating that the reconstruction results achieved using our method represent the optimal outcomes currently available.

We also applied the method proposed in this paper to image point-like sources, and the results are shown in Figure 15. In the same imaging area, four point sources are vertically distributed at x = 10.8 mm in the center of the imaging area. The vertical distance between the first point source from the top is 6.8 mm, and the distance between each subsequent point source is 3 mm. The results show that our method can not only deal with the distortion problem that may occur when imaging large-volume sources, but also, like many existing methods, can still exhibit excellent imaging performance when imaging small-volume sources.

4. Conclusions

In this paper, we analyzed the principle of delay and sum beamforming in photoacoustic imaging. Meanwhile, on the basis of the multi-DASE method, we also analyzed the shortcomings of this method in imaging large-volume sources. In response to these shortcomings, we proposed a solution. Through verification, the method we proposed can effectively solve the problem of sudden changes in the signal, thereby avoiding the overshoot phenomenon that may occur in the envelope signal. The distortion problem of large-volume source imaging by multi-DASE method is solved. Our improved method retains the advantage of the multi-DASE method in suppressing artifacts while compensating for the problem of losing internal source information during imaging of larger volume sources. Additionally, our method also preserves the high parallelism advantage of DAS, making it applicable for real-time imaging. Simulation results verified the effectiveness of our approach. The proposed method is primarily designed for imaging sources with a certain volume, effectively addressing the limitations of the multi-DASE method. The photoacoustic imaging of large-volume tissues, such as the liver and breast, can help doctors observe the size, shape, boundaries, and internal structure of tumors more comprehensively. However, when applied to small-volume and complexly distributed biological tissues, such as blood vessels, the method exhibits certain limitations. Overcoming these limitations will be a key focus of our future research. Additionally, while the suppression method described in this paper is relatively straightforward, we plan to explore more sophisticated suppression techniques in subsequent studies to satisfy diverse imaging requirements. Furthermore, since we employed a relatively simple window function, its advantages were not particularly evident when dealing with some more complex problems. More suitable window functions and parameters can be explored to meet different application requirements and obtain higher-quality reconstructed images.