## 2 Methods
### Multiple Delay and Sum with Enveloping
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]:  
\[Env_{multi-DAS}(x,y;t)=|H(Y_{multi-DAS}(x,y;t))|, \tag{5}\]  
\[H(f(t))=f(t)*\frac{1}{\pi t}=F^{-1}\bigg{(}F(f(t))\times F\bigg{(}\frac{1}{ \pi t}\bigg{)}\bigg{)}, \tag{6}\]  
\[F\bigg{(}\frac{1}{\pi t}\bigg{)}=-j\;\text{sgn}(\omega), \tag{7}\]  
where \(H(f(t))\) is the Hilbert transform of \(f(t)\), \(F(f(t))\) is the Fourier transform of \(f(t)\), and \(\text{sgn}(\omega)\) 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:  
\[t_{peak}(x,y)=\text{argmax}(Env_{multi-DAS}(x,y;t)). \tag{8}\]  
The black lines in Figure 3b,c represent the envelopes obtained, respectively. It is easy to find that when \((x,y)\) is the pixel at the source, the \(t_{peak}\) of the envelope signal is the initial time \(t_{0}=0\). When \((x,y)\) is not the location of the source, the \(t_{peak}\) corresponding to the envelope signal is no longer the initial time. \(\big{|}t_{peak}\big{|}\) is introduced here to represent  
\[\tau(x,y,i)=\frac{l(x,y,i)}{c}, \tag{2}\]0 The N-shaped PA signal and signal envelope. (**a**) A circular source with a radius of 0.2 mm. (**b**) The recovered N-shaped signal and signal envelope of the pixel at the center of the circle. (**c**) The recovered N-shaped signal and signal envelope pixels at non-source locations. the difference between \(t_{peak}\) and \(t_{0}\). It was found through experimentation that when the distance between \((x,y)\) 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  
\[\tau(x,y,i)=\frac{l(x,y,i)}{c}, \tag{2}\]1  
\(Env_{multi-DAS}\left(x,y;t_{peak}\right)\) 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 \(inh(x,y;t)\), and the envelope signal will be suppressed by the following expression  
\[\tau(x,y,i)=\frac{l(x,y,i)}{c}, \tag{2}\]2  
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  
\[\tau(x,y,i)=\frac{l(x,y,i)}{c}, \tag{2}\]3  
\(Y_{multi-DAS}(x,y)\) represents the data that is ultimately used for reconstruction at the position \((x,y)\).