# Multi-Channel Surface EMG Spatio-Temporal Image Enhancement Using Multi-Scale Hessian-Based Filters
## 3 Results and Discussions  
sEMG images have poor MUAP visualization due to different noises and destructive interference of nearby MUAPs. Sometimes, the width of the MUAP propagation region is nonuniform due to a mix of destructive and constructive interferences and because of the presence and absence of noises at different electrodes [3]. Thus, a multi-scale filter is needed to preserve and enhance the MUAP regions at different scales proportional to the MUAP width and to attenuate the background noise. The performance of the multi-scale Hessian-based filter is studied and analyzed in different scenarios, such as synthetic images with gaussian propagating profile, simulated EMG signals using the model proposed by the authors in Reference [26], and the experimental EMG signals recordedfrom different human muscles. As discussed in the previous section, the MUAP propagation across the electrodes with respect to time appears as a Gaussian line-like structure. The second eigenvalue of the Hessian matrix is negative for the MUAP propagation region (see Equation (4) and Figure 3c). To verify this, a synthetic multi-channel signal with a propagating Gaussian profile is generated and contaminated with additive white gaussian noise. In the presence of this high-level noise with SNR of 6.996 dB, the signal is passed through the proposed multi-scale filter. The results show that the multi-scale filter has preserved and enhanced the region with Gaussian profile and has suppressed the background noise. This is because the second eigenvalue of Hessian is negative and its magnitude is higher in the Gaussian region, and the output of the proposed filter is also high, while in the background region, both the R\({}_{\text{B}}\) and \(N\) parameters of the eigenvalues of the Hessian matrix are very small; thus, the filter output is also nearly zero. To further quantify the performance of the proposed filter for various noise levels, noises are added to the synthetic signal of Figure 4a and the input signal to interference ratio (\(SIR_{imp}\)) is computed. The \(SIR_{imp}\) is the ratio between the power of the clean input signal and the noise added to the signal. The signal is then enhanced using the proposed filter and the remaining interference, \(R_{i}\)_(\(t\))_, after enhancement for each channel is calculated, as follows, by subtracting the clean simulated sEMG \(V(t)\) from the enhanced signal \(V\left(x,y,\sigma_{0}\right)\), as follows:  
\[R(t)=V(t)-V\left(x,y,\sigma_{0}\right) \tag{15}\]  
Using this \(R(t)\), the performance of the proposed filter is evaluated by calculating the output signal to the interference ratio (\(SIR_{out}\)) of the enhanced signals. This is named as the output SIR. The output SIR is the ratio of the power of the simulated signal (clean original signal) to the power of  
Figure 4: (**a**) A synthetic multi-channel signal with a propagating Gaussian profile of \(\sigma\) = 15 pixels contaminated by random noise with a SNR of 6.996 dB, (**b**) the corresponding spatio-temporal image interpolated with an interpolation factor of 10, where the propagation of Gaussian appears as a line-like structure, (**c**) the second eigenvalue, \(\lambda_{2}\), of the Hessian matrix of each pixel of the image in (**b**), (**d**) the enhanced image using the proposed multi-scale filter. the remaining interference (noise left in the signal after enhancement, i.e., \(R(t)\)). The output SIR is computed as:  
\[SIR_{out}=10 log10\left(\frac{P_{v(t)}}{P_{R(t)}}\right) \tag{16}\]  
where \(P_{v(t)}\) is the power of the original simulated signal and \(P_{R(t)}\) is the power of the remaining interference left in the signal after filtering.