## 2 Research Status and Development Trends for Computer Aided Detection and Diagnosis of MRI
### Tensor Based MRI Image Analysis and Classification  
As stated earlier, DCE-MRI provides a time series of three-dimensional magnetic resonance imaging. According to geometric algebra [58], one can match complex geometric objects of different dimensions through a range of simple transformations. Thus, it is possible to generate a series of three-dimensional spatial image data (Volumes), after clustering information, from multiple block-related regions (a trivector). These three-dimensional blocks are a series of two-dimensional spatial image stacks (Planes) arranged in three different directions. The two-dimensional images in each direction combine the time frame sequences to form a three-dimensional space-time image (trivector); each row or column on the image, combined with the time series, forms a two-dimensional space-time image (bivector). Figure 2 shows the basic components of 3D magnetic resonance image structure data based on a third-order tensorial representation. The formal expression of the geometric object of the three-dimensional MRI based on multiple vectors is represented as Figure 3.  
Figure 3: Illustration of the formal expression of the geometric object of the three-dimensional MRI based on multiple vectors is represented. Figure 2: Basic components of 3D magnetic resonance image structure data based on third-order tensor [52]. From the equation presented above, a three dimensional MRI object (MRIobj) consists of object pixels (obj.Points), object rows or columns (obj.Lines), object planes (obj.Planes) and object volumes (obj.Volumes), along with three orientations in a spatial space. Considering that an image pixel corresponds to a point that is a scalar to be presented by 1; rows or columns of MRI are vectors to be labelled by \(\sigma_{i}\), \(i\in 1,2,3\); planes of MRI are bivectors, represented by \(\sigma_{i}\wedge\sigma_{j}=\sigma_{i}\sigma_{j}\), \(i\
eq j\); volumes of MRI object are trivectors, indicated by \(\sigma_{i}\wedge\sigma_{j}\wedge\sigma_{k}=\sigma_{i}\sigma_{j}\sigma_{k}\), \(i\
eq j\
eq k\). Tensors enable multilinear mappings over a set of vector spaces. Under a tensorial framework, the four-dimensional objects registered in DCE-MRI are represented by a fourth order tensor (with time associated with an additional dimension), and the 3D shape geometry is treated as a third order tensor. A third order tensor has a directional definition in space, so any associated spatial matrix consists of three directional slices: horizontal, vertical and frontal. Tensor factorisation of a 3D spatial matrix is a universal methodology that aids the analysis of tomographic images. The tensor decomposition method adopted by [PERSON] et al. (2018) is based on the standard Tucker decomposition ([PERSON] et al., 2018; [PERSON] et al., 2018; [PERSON] et al., 2018), and is also referred to as multilinear singular value decomposition (SVD) or higher-order SVD (HOSVD). The advantage of the HOSVD approach is that it allows estimation of the dimension of the core tensor by analysing the distribution of singular values ([PERSON] et al., 2018). Through the use of HOSVD, the dynamic MRI dataset (represented by a dynamic tensor \(\chi_{\tau}\)) is decomposed into three-mode basis image matrices \(\mathrm{A}^{t}_{\tau}\) and a core tensor \(C_{\tau}\), where \(\iota=1,2,3\). The core tensor is associated with each mode of basis images; \(\tau=1,2,\ldots,6\) corresponds to a single time frame. PCA is applied on a temporal set of basis images. The temporal signal intensity variations \(\mathrm{v}^{t}_{ri}\) for each pixel \(i\) within the decomposed basis image at each mode are associated with a state vector: \(\mathrm{u}^{t}_{i}=\mathrm{v}^{t}_{1i^{\prime}}\), \(\mathrm{v}^{t}_{2i^{\prime}}\), \(\mathrm{v}^{t}_{3i^{\prime}}\), \(\mathrm{v}^{t}_{4i^{\prime}}\), \(\mathrm{v}^{t}_{5i^{\prime}}\), \(\mathrm{v}^{t}_{ni}\) (\(n=6\) to account for different time frames).