# Biomedical Photoacoustic Imaging Optimization with Deconvolution and EMD Reconstruction
## 2 Methods
However, directly using de-convolved signal to reconstruct images creates defects. As shown in Figure 1c, when reconstructing images by conventional delay-and-sum algorithm, assuming the velocity is uniform, the value corresponding with the position of the imaging target was also added to the other positions of the same distance to the transducer element as imaging target. Normally, artifacts can be restrained as the signal has both positive part and negative part to cancel each other out. Since the de-convolved signal is monopolar, unexpected artifacts will appear around the imaging object because amplitude cannot be cancelled out when summing up as they are all positive. To alleviate this phenomenon and further optimize the images, EMD was adopted subsequent to signal deconvolution. The principle of EMD [18], is to decompose a signal into several intrinsic mode functions (IMFs) without setting any basis function in advance, and each of them reserves the local features of the original signal with different time scales. Given that different IMFs possess different features, we were able to rebuild new signals to meet our requirements by depressing some of the IMFs and enhancing other IMFs. Specifically, we applied weight coefficients to each IMF and refactor new signal automatically under the constraints which is set in advance. In our proposed method, we apply the de-convolved signal to EMD and refactor it with two constraints, positive polarity and spectrum consistence. Given that EMD can only process one-dimensional signals at a time, a one-dimensional signal \(o_{1}(n)\) (detected by a single transducer element of the probe) of de-convolved signal \(\mathrm{o}(n)\) should be considered as an example, the procedure [19] is described in detail as follows:  
Figure 1: **(a)** Diagram of the normalized deconvolution kernel \(h(n)\) in simulation; (b) comparison of simulated signal before and after deconvolution; (c) illustration of the widespread artifacts forming process in delay-and-sum algorithm. Step 1: Use EMD method to decompose signal \(o_{1}(n)\) into \(k\) IMFs (\(k\) is a changeable number) and the residue:  
\[o_{1}(n)=\sum_{i=1}^{k}IMF_{i}+res(n), \tag{3}\]  
Step 2: Set weight coefficient value of each IMF to refactor a set of new signals:  
\[new_{o_{1}(n)}=\left[a_{1}\;a_{2}\;\cdots\;a_{k}\right]\times\left[ \begin{array} []{c}IMF_{1}\\ IMF_{2}\\ \vdots\\ IMF_{k}\end{array} \right], \tag{4}\]  
where \(a=\left[a_{1}\;a_{2}\;\cdots\;a_{k}\right]\) refers to the weight coefficients of IMFs. As the number of IMFs is \(k\), the number of weight coefficient is also \(k\) and each weight coefficient corresponds with an IMF. The value of each weight coefficient ranges from [0, 1] with step 0.1, so that the number of iteration is \(11^{k}\). The coefficient of residue remains 1. Step 3: Calculate two constraints in each iteration:  
The first constraint is positive polarity. As the original de-convolved signal is monopolar, positive polarity is set to avoid distortion between the refactor signal and the original signal. Positive polarity is defined as:  
\[p=\frac{\min(new_{-}o_{1}(n))}{\max(new_{-}o_{1}(n))}>\text{threshold}, \tag{5}\]  
where the threshold here should equal the minimum amplitude of \(o_{1}(n)\) divided by maximum amplitude of \(o_{1}(n)\). Within each iteration, the second constraint can be calculated if the refactored signal satisfies the positive polarity. Otherwise, this iteration should be skipped and to begin the next iteration. The second constraint is spectrum consistence. As the high frequency components usually retain detailed information, we applied the spectrum consistence to refactoring a new signal, with more detail and less aliasing. Spectrum consistence is defined as:  
\[c=\sum_{\omega=f_{1}}^{f_{2}}f(\omega)\Bigg{/}\sum_{\omega=f_{0}}^{f_{1}}f( \omega),\;\omega<\frac{f_{s}}{2}, \tag{6}\]  
where \(f(\omega)\) is the FFT result of refactored signal, \(f_{s}\) refers to the sampling frequency.