# Multi-Channel Surface EMG Spatio-Temporal Image Enhancement Using Multi-Scale Hessian-Based Filters
## 2 Materials and Methods
### Filter for Enhancing the MUAP Propagation Region  
In single differential sEMG spatial-temporal images, the bipolar (positive and negative) MUAP structure looks like a bright and dark region on a gray background, respectively (see Figure 1b) [3; 4]. The propagation of the positive amplitude appears as a bright line-like structure, while the negative EMG amplitude appears as a dark line-like structure in sEMG images, as shown in Figure 1b. These bright and dark regions in spatio-temporal sEMG images can be approximated by a Gaussian waveform with EMG amplitude maximum at the middle and decreasing gradually towards the end. Second-order derivative of the image with respect to the x- and y-axis, i.e., Hessian, is then used to design a filter whose response is maximum for the MUAP region, while minimum for the background, resulting in enhancing the MUAP structure. The filter is based on the eigenvalues of the Hessian matrix of each pixel of sEMG spatio-temporal sEMG images. The same method is applied in References [20; 21; 22; 23] to design a filter that enhances line-like Gaussian structures in digital images, like retinal images. Some of the methods use fixed scale and the (nonlinear) combinations of finite difference operators applied in a set of orientations for enhancement of digital images [24]. All these methods have shown great performance in enhancing linear Gaussian structures with different scales but none of these is applied to sEMG spatio-temporal image enhancement. In sEMG images, the width of the MUAP propagation region is not fixed due to the interference of other MUAPs and noises; thus, a fixed scale approach with a fixed sigma will not be able to enhance the EMG signals due to variations in the MUAP [3]. Therefore, the response of the enhancement filter may not be the highest for every pixel at a fixed scale. On the other hand, with the use of a multi-scale approach where a range of sigma  
Figure 3: (**a**) A synthetic multi-channel signal with a propagating Gaussian profile of \(\sigma=15\) pixels, (**b**) the corresponding spatio-temporal image, where the propagation of Gaussian appears as a line-like inclined structure, (**c**) the second eigenvalue, \(\lambda_{2}\), of the Hessian matrix of each pixel of the image in (**b**). values are used, whenever the sigma is equal to the width of the MUAP structure, the response of the enhancing filter is highest. In the proposed method, first, to achieve the multi-scale Hessian matrix at each pixel, the input sEMG spatio-temporal image is convolved with the first- and second-order derivative of a Gaussian at various scales (that is with multiple \(\sigma\) values) to estimate the first- and second-order derivative of each pixel with respect to the x- and y-axis. The second-order derivative approximates the Hessian. Eigenvalues \(\lambda\)1 and \(\lambda\)2 of the Hessian matrix are then computed for each pixel to find whether it is a background or foreground pixel. For a pixel in the MUAP region in a sEMG image, the first eigenvalue of the Hessian matrix is almost zero, while the second eigenvalue is negative and has a big absolute value for the bright portion of the MUAP. The second eigenvalue is positive and bigger for the dark portion of the MUAP. Norm of the eigenvalues is then computed, and the filter response depends on the value of the norm. In the MUAP region, the value of the norm is big because one of the eigenvalues is high, thus the filter response is high. For the spatio-temporal sEMG image I(x,y), the Taylor series up to the second order can be written as (Equation (5)):  
\[I(p)=I(p_{0})+\delta p^{T}\
abla_{I}(p_{0})+0.5\delta p^{T}H_{I}(p_{0})\delta p +error \tag{5}\]  
where, \(p\) is the current pixel, \(\delta p=p-p_{0}\) is the vector, while \(\
abla_{I}(p_{0})\) and \(H_{I}(p_{0})\) are the first- and second-order derivatives of the sEMG image computed at \(p_{0}\). The eigenvalues of the Hessian can represent the local variation of the intensity in the direction of the associated eigenvectors [20].