## 2 Proposed Fusion Scheme
The low-frequency sub-bands at level k + 1 are denoted by the notation Low\(\_\)Low\(\_\)Low\(\_\)K+1, while Low\(\_\)Low\(\_\)K denotes those at level k.

Figure 1: Hypothesized \(SWT\) and \(D\)-\(TCWT\) with \(TEM\)-based image fusion. Low_HighK+1 represents the signal's horizontal-detail coefficient, High_LowK+1 represents the signal's vertical-detail coefficient, and High_HighK+1 represents the signal's diagonal-detail coefficient. _SWT_ improves shift invariance and low computational time because the algorithm does not need to down-sample. In _SWT_, filters are applied first to rows and secondly to columns. Due to these advantages, _SWT_ is selected for the proposed method. ### DTCWT Based Decomposition

In this subsection, concerning the base of _SWT_ approximate sub-bands, _DTCWT_[6; 7; 8; 9] is implemented for further decomposition, decomposing the information into two bands, _LL_ and _HH_, respectively. _TEM_ laws are applied for the bands of _LL_, which is discussed in Section 2.3. When implementing the _DTCWT_, extract the sub = bands using Equation (5). \[\{(L_{t}^{X}H_{t,d}^{X})|X=A,B;levels=1,2,3;t-orientation=1,\;2;d-orientation=1,2,3,4,5,6\} \tag{5}\]

where \(L_{t}^{X}\) stands for the _L-level_ sub-bands at low frequencies (_t orientation_) and \(H_{t,d}^{x}\) for the _H-level_ sub-bands at high frequencies (_d orientation_); Low and High are constructed. ### Implementation of TEM Laws for Final Fusion Process

In general, approximate sub-bands consist of much data from a source image. It is essential to transfer this information into a fused image, extracting features such as edges, ripples, waves, and sparks [15]. The following three primary vectors are used. \(L=[1,\;2,1]\), \(E=[-1,\;0,\;1]\), \(S=[-1,\;2,-1]\). With these three vectors, nine convolution matrixes are listed as follows:

\[matrix_{1}= \begin{bmatrix}1&2&1\\ 2&4&2\\ 1&2&1\end{bmatrix}  \tag{6}\]

\[matrix_{2}= \begin{bmatrix}-1&0&1\\ -2&0&2\\ -1&0&1\end{bmatrix}  \tag{7}\]

\[matrix_{3}= \begin{bmatrix}-1&2&-1\\ -2&4&-2\\ -1&2&-1\end{bmatrix}  \tag{8}\]

\[matrix_{4}= \begin{bmatrix}1&0&-1\\ 0&0&0\\ -1&0&-1\end{bmatrix}  \tag{9}\]

Figure 2: _SWT_ decomposition structure. \[matrix_{5}= \begin{bmatrix}1&-2&1\\ 0&0&0\\ -1&2&-1\end{bmatrix}  \tag{10}\]

\[matrix_{6}= \begin{bmatrix}-1&-2&-1\\ 0&0&0\\ 1&2&1\end{bmatrix}  \tag{11}\]

\[matrix_{7}= \begin{bmatrix}1&-2&1\\ -2&4&-2\\ 1&-2&1\end{bmatrix}  \tag{12}\]

\[matrix_{8}= \begin{bmatrix}-1&-2&-1\\ 2&4&2\\ -1&-2&-1\end{bmatrix}  \tag{13}\]

\[matrix_{9}= \begin{bmatrix}1&0&-1\\ -2&0&2\\ 1&0&-1\end{bmatrix}  \tag{14}\]

The mask of \(matrix_{1}\) is non-zero-sum, and the remaining are zero-sum. The mask \(matrix_{1}\) produces average energy. \(matrix_{2}\) and \(matrix_{3}\) produce first and second-order vertical edges. \(matrix_{6}\) gives horizontal edges. \(matrix_{4}\) refers to waves. \(matrix_{5}\), \(matrix_{7}\), and \(matrix_{9}\) represents ripples. The proposed image fusion process uses 3 \(\times\) 3 masks, providing the fused image with detailed information. Using these 3 \(\times\) 3 texture masks, the coefficient saliency measurement for the approximation sub-band is created. Assume that \(ASX\) represents the source image \(X\)'s approximate sub-band. Further, \(TEM\), represented as \(TEM_{1}\) to \(TEM_{9}\), relates to convolution matrices: \(matrix_{1}\) to \(matrix_{9}\) are used in the following Equations (15)-(23). \[TEM_{1}(r,c)=\sum_{m=-1}^{1}\sum_{m=-1}^{1}ASX(r+m,c+n)^{2}matrix_{1}(m+2,n+2) \tag{15}\]

The convolution matrix\({}_{1}\) should be applied on the approximate sub-band corresponding block; the result is stored in the \(TEM_{1}(r,c)\). Similarly, the remaining convolution matrices will be applied to the corresponding blocks of approximate sub-bands.