Table 2.

Workflow for SC-based CT and MRI Image Estimation.

Input:	MRI image uMRI, CT image uCT, the number of similar patches K
Data transformation	According to Fig. 4, deform CT and MRI datasets based on given CT and MRI images.
Estimation:	1. Decompose uCT into patches ${\mathbf{p}}_{CT}^j={\mathbf{E}}_j{\mathbf{u}}_{CT}$ ;
	2. Decompose uMRI into patches ${\mathbf{p}}_{MRI}^j={\mathbf{E}}_j{\mathbf{u}}_{MRI}$ ;
	3. Optimize Eq. (4) by finding K-best suitable patches $\left\{{\mathbf{p}}_{CT,i}^j\right\}$ and $\left\{{\mathbf{p}}_{MRI,i}^j\right\}$, and their weights {wi} in TCT and TMRI to represent ${\mathbf{p}}_{CT}^j$ and ${\mathbf{p}}_{MRI}^j$ ;
	4. Estimate CT and MRI image patches with the corresponded CT and MRI image pairs in TCT–MRI and new weights {wi}, ${\mathbf{p}}_{new,CT}^j\approx {\sum }_i^K{w}_i{\mathbf{p}}_{CT,i}^j$, ${\mathbf{p}}_{new,MRI}^j\approx {\sum }_i^K{w}_i{\mathbf{p}}_{MRI,i}^j$ ;
	5. Estimate the corresponded CT image with ${\mathbf{p}}_{new,CT}^j$, ${\mathbf{u}}_{CT}^{est}={\left({\sum }_j{\mathbf{E}}_j^T{\mathbf{E}}_j\right)}^{-1}\left({\sum }_j{\mathbf{E}}_j^T{\mathbf{p}}_{new,CT}^j\right)$.
	6. Estimate the corresponded MRI image with ${\mathbf{p}}_{new,MRI}^j$, ${\mathbf{u}}_{MRI}^{est}={\left({\sum }_j{\mathbf{E}}_j^T{\mathbf{E}}_j\right)}^{-1}\left({\sum }_j{\mathbf{E}}_j^T{\mathbf{p}}_{new,MRI}^j\right)$.