# Adaptive Visual Saliency Feature Enhancement of CBCT for Image-Guided Radiotherapy
## 2 Materials and Methods
### Structural Similarity-Based Low-Rank Approximation  
Based on the singular value decomposition (SVD) algorithm [24; 25; 26; 27], we propose a structural similarity-based low-rank approximation method (SSLRA) to reconstruct the input CBCT images so as to denoise the source image. The decomposition and reconstruction of the SVD of an image \(\mathbf{X}\) with size \(m\times n\) can be expressed as:  
\[\mathbf{X}=U\sum V^{T} \tag{1}\]  
where \(\mathbf{X}\in\mathbb{R}^{m\times n}\), \(U\in\mathbb{R}^{m\times m}\), \(\sum\in\mathbb{R}^{m\times n}\), and \(V\in\mathbb{R}^{n\times n}\); the columns of the \(m\times n\) matrix \(U\) are the left singular vectors; and the rows of the \(n\times n\) matrix \(V^{T}\) contain the elements of the right singular vectors. \(\sum\) is the diagonal matrix, where \(\sum=(\sum_{n},\mathbf{0})\) and \(\sum_{n}=diag(\sigma_{1},\cdots,\sigma_{n})\). In addition, \(\sigma_{k}>0\) if \(1\leq k\leq r\) and \(\sigma_{k}=0\) if \(r+1\leq k\leq n\), where \(r\) is the rank of \(\mathbf{X}\). Using Equation (1), the source image can be computed by three matrices. Then, the image \(\mathbf{X}\) can be reconstructed by:  
\[\mathbf{X}=\sigma_{1}u_{1}v_{1}^{T}+\sigma_{2}u_{2}v_{2}^{T}+\cdots+\sigma_{ r}u_{r}v_{r}^{T} \tag{2}\]  
Figure 2: Flowchart of the proposed method for CBCT image enhancement method, in which SSLRA denotes the proposed structural similarity-based low-rank approximation algorithm and VSFE denotes the proposed visual saliency feature based enhancement algorithm. where \(u\) and \(v\) are the left singular vectors and right singular vectors in \(U\) and \(V\), respectively, and \(\sigma\) is the singular value in \(\sum\). The reconstruction result \(\mathbf{X}_{p}\) of different features can be obtained by:  
\[\mathbf{X}_{p}=\sum_{i=1}^{p}\sigma_{i}u_{i}v_{i}^{T} \tag{3}\]  
where \(1\leq p\leq r\). When \(p=r\), the full rank reconstruction result is \(\mathbf{X}_{p}=\mathbf{X}\). That is, the denoised \(\mathbf{X}_{p}\) can be estimated from \(\mathbf{X}\) by solving:  
\[\mathbf{X}_{p}=\underset{\mathbf{Z}}{\operatorname{argmin}}\|\mathbf{X}- \mathbf{Z}\|_{F}^{2}\text{ s.t. }\text{rank}(\mathbf{Z})\ =\text{k} \tag{4}\]  
where \(\text{rank}(\cdot)\) denotes the rank of the given data. Equation (4) can be solved by the theorem of Eckart-Young-Mirsky. That is, for the given matrix \(\mathbf{A}\) with \(k<r=rank(\mathbf{A})\), the truncated matrix can be expressed by:  
\[\mathbf{A}_{k}=\sum_{i=1}^{k}\sigma_{i}u_{i}v_{i}^{T} \tag{5}\]  
Next, for any given matrix \(\mathbf{B}\) with rank \(k\), the minimal error can be computed with \(\mathbf{A}_{k}\):  
\[\underset{rank(\mathbf{B})=k}{\operatorname{min}}=\left\|\mathbf{A}-\mathbf{B }\right\|_{2}=\left\|\mathbf{A}-\mathbf{A}_{k}\right\|_{2}=\sigma_{k+1} \tag{6}\]  
The above Equations shows that \(\mathbf{A}_{k}\) can be as the optimal solution of \(\mathbf{A}\) in the Frobenius norms sense. That is, Equation (6) can be computed by:  
\[\underset{rank(\mathbf{B})=k}{\operatorname{min}}=\left\|\mathbf{A}-\mathbf{B }\right\|_{F}=\left\|\mathbf{A}-\mathbf{A}_{k}\right\|_{F}=\sum_{i=k+1}^{r} \sqrt{\sigma_{i}^{2}} \tag{7}\]  
By an optimal rank, the reconstruction result can be denoised with high imaging quality and more detailed information. The proposed SSLRA are summarized in Algorithm 1, in which \(SS(\cdot)\) is defined as:  
\[SS(I,J)=((2\mu_{I}\mu_{J}+C_{1})(2\sigma_{IJ}+C_{2}))/((\mu_{I}^{2}+\mu_{J}^{ 2}+C_{1})(\sigma_{I}^{2}+\sigma_{J}^{2}+C_{2})) \tag{8}\]  
where \(\mu_{I},\ \mu_{J}\) are the local means of image \(I\) and \(J\). \(\sigma_{I},\ \sigma_{J}\) and \(\sigma_{IJ}\) are the standard deviations and cross-covariance of \(I\) and \(J\). \(W_{mc}(I_{p})\) can be computed by:  
\[W_{mc}(I_{p})=2\|\
abla U\|_{2}H(U) \tag{9}\]  
where \(\
abla\) is the gradient operator. Equation (9) is derived from total variation (TV) regularization.