## 2 Research Status and Development Trends for Computer Aided Detection and Diagnosis of MRI
### Tensor Based MRI Image Analysis and Classification
The set of all state vectors within one mode of the basis images over a pre-determined time course is defined as \(\mathrm{U}_{\mathrm{i}}=\left\{u^{t}_{i}\right\}\), with the number of pixels \(i\) in the basis image at a different mode denoted by \(\iota\). We used the symbol \(\varepsilon\) to label the mean of pixels in the basis images, the first-order covariance matrix of \(\mathrm{U}^{t}\), \(\mathrm{\Delta}^{t}\), was calculated according to: \(\mathrm{\Delta}^{t}=\frac{1}{\varepsilon}\sum\limits_{\mathrm{u}^{t}_{i}\in \mathrm{U}^{t}}\left(u^{t}_{i}-\overline{u}^{t}\right)\left(u^{t}_{i}-\overline {u}^{t}\right)^{T}\) and \(\overline{u}^{t}=\frac{1}{\varepsilon}\sum\limits_{u^{t}_{i}\in\mathrm{U}^{t} }u^{t}_{i}\)  
A linear principal component analysis (PCA) transformation can then be applied to obtain the corresponding eigenvectors \(\mathrm{E}^{t}_{\varepsilon}=\left\{\mathrm{e}^{t}_{\varepsilon}\right\}\) and eigenvalues \(\lambda=\left\{\lambda_{1},\,\lambda_{2},\,\ldots,\,\lambda_{6}\right\}\) by solving the generic equation \(\lambda\mathrm{E}=\lambda\mathrm{E}\). A PCA routine of dynamic basis image datasets at each of the image modes yields 6 eigenvectors. As a result, a new mode vector is reconstructed \(\mathrm{A}^{t}_{\varepsilon}=\mathrm{\Delta}^{t}\mathrm{E}^{t}_{\varepsilon}\) for each of the different channels (state points) (\(\xi=1,2,\,\ldots,6\)). Then A can be matricised to \(\mathrm{A}\in\mathrm{N}^{t}\times\mathrm{M}^{t}\) to generate \(\iota\) modes of basis images. In order to reconstruct a tensor for a 3D MRI approximation, the tensor product between the averaged core tensor and three modes of filtered basis images is calculated. This product is expressed as \(\Gamma_{\xi}=\mathrm{C}_{\tau a}\times_{1}\mathrm{A}^{1}_{\xi}\times_{2} \mathrm{A}^{2}_{\xi}\times_{3}\mathrm{A}^{3}_{\xi^{\prime}}\), where \(\mathrm{C}_{\tau a}=\frac{1}{3}\sum_{\tau=1}^{3}\mathrm{C}_{\tau}\). Finally, the spatio-temporal features are reconstructed in a 3D space. An example of the resulting volume image generated following reconstruction from multiple channels is illustrated in Figure 4c. One can clearly observe improved intensity consistency for the colour-coded blue tumourous regions, which are clearly separated from the red background region. Figure 4d illustrates the resultant segment of the tumour following reconstruction. This should be contrasted with Figure 4b, where the whole region of the tumour shows inhomogeneous intensity with missed and spurious edges. Moreover, segmentation through the multi-channel reconstruction approach shows homogeneous boundaries for the tumours, and the entire tumour shape and tumour position are clearly retrieved. This can be also further validated by observing the overlapping images, shown in Figure 4e, where the yellow region consists of voxels mainly from the reconstructed tumour segment, which is used to fill-in the missing voxels from the conventional fuzzy c means (FCM) classified image. Multi-channel reconstruction leads to better defined tumour boundaries with reduced ambiguity and eliminates artefacts generated through the standard FCM reconstruction process.