# Biomedical Photoacoustic Imaging Optimization with Deconvolution and EMD Reconstruction
## 2 Methods  
Given that the detected PA signals have low SNR that deteriorate the quality of PA images, de-noising operation is quite common as a pre-processing step. Theoretically, averaging signals of multiple frames can eliminate random background noise and raise the SNR. However, non-negligible time shifts among signals of different frames always exist for a variety of reasons, including the slight fluctuation of the water around the imaging target, the small movement from a living biological imaging target, and system defect (possibly random delay generated by inner circuit). To solve this issue, we adopted an effective averaging method to adjust time shifts based on cross-correlation method [14; 15]. First, we calculated the time shifts \((\Delta m_{1},\Delta m_{2},\cdots)\) between the reference frame and other frames by cross-correlation. For discrete PA signals \(p_{1}(n)\) and \(p_{2}(n)\), an estimate of the cross-correlation function is defined as [16]:  
\[\hat{R}_{p_{1}p_{2}}(m)=\left\{ \begin{array}{c}\frac{1}{L-|m|}\sum_{n=0}^{L- m-1}p_{1}(n+m)p_{2}^{*}(n),\;m\geq 0\\ \hat{R}_{p_{1}p_{2}}(-m),\;\;\;\;m<0\end{array} \right., \tag{1}\]  
where \(p_{2}^{*}(n)\) is the complex conjugate of \(p_{2}(n)\), \(L\) is the maximum length of \(p_{1}(n)\) and \(p_{2}(n)\), the normalization coefficient \(1/(L-|m|)\) can avoid the zero bias inherent in the standard cross-correlation function. According to this equation, the value of time shift \(\Delta m\) should equal to the value of \(m\) when the cross-correlation function \(\hat{R}_{p_{1}p_{2}}(m)\) is of maximum value. Assuming \(p_{1}(n)\) is the uniform reference, \(p_{2}(n)\), the signals of all the rest frames need to be shifted to the length of their corresponding \(|\Delta m|\) forwards or backwards based on the sign of \(\Delta m\). After that, shifted signals of multiple frames that all align is acquired, so we can obtain the final de-noised signal \(\hat{p}(n)\) by simply averaging them all. Mathematically, the convolution result of a single wave and positive pulses with different delay is the superposition of many same-shaped waves with different delay, and the specific size of each wave depends on the size of the corresponding positive pulse. Hence, we assumed that PA signal is the convolution result of a series of specific monopolar pulses and a single N-shape wave. With that assumption, the monopolar signal, which contains the amplitude information and time information, can theoretically be acquired through deconvolution. As we mentioned in the former section, the N-shape wave basis has defects that will reduce the quality of PA images, while the de-convolved monopolar signals possess equally important information for imaging as raw PA signals and perform better than raw N-shape PA signals when reconstructed into images. We intended to acquire de-convolved signals to replace raw PA signals. Generally, the discrete detected PA signal \(\mathrm{s}(n)\) can be represented in convolution form as:  
\[\mathrm{s}(n)=o(n)*h(n)+n(n), \tag{2}\]  
where \(*\) denotes convolution, \(h(n)\) is the kernel of this signal convolution system, \(\mathrm{s}(n)\) is the detected raw signal, \(o(n)\) refers to the signal which we aim to acquire and \(n(n)\) represents the additive noise. In our method, we took \(h(n)\) as a normalized single N-shape pulse which is acquired by detecting an approximately ideal optical absorbing particle, specifically, the smallest source which the PA data acquisition system can measure. For example, assuming that velocity is 1540 m/s, an L7-4 ultrasound probe whose center frequency is 5.208 MHz can detect the minimum source is approximately a diameter of 0.296 mm, so in this case, the kernel \(h(n)\) ought to be acquired by detecting the source whose size is closest to and bigger than 0.296 mm. The deconvolution algorithm we used was the Wiener deconvolution [17]. A typical simulated signal was used for demonstration. As shown in Figure 1b, the signal was converted from a bipolar signal to monopolar signal after deconvolution.