## 2 Methods  
### 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 \(S(i;t)\), where \(i\) denotes the sensor number and \(t\) denotes time, indicating that at time \(t\), the \(i\)-th sensor receives a signal of \(S(i;t)\). For a specific source location \((x,y)\), each sensor receives the signal with a corresponding delay time \(\tau(x,y,i)\). The output signal \(Y_{DAS}(x,y)\) of the DAS algorithm at location \((x,y)\) can be defined as:  
\[Y_{DAS}(x,y)=\sum_{i=1}^{N}S(i,\tau(x,y;i)), \tag{1}\]  
the output \(Y_{DAS}(x,y)\) is the sum of the signals received by each transducer at the corresponding delayed time \(\tau(x,y,i)\); 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:  
\[\tau(x,y,i)=\frac{l(x,y,i)}{c}, \tag{2}\]  
here, \(l(x,y,i)\) in the formula represents the distance from the place \((x,y)\) 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.