## 3 High-Dimensional Medical Imaging Analysis Based on Deep Learning
### Deep Learning in Spatio-Temporal MRI  
Based on geometric algebra, [PERSON] et al. (2017) proposed a framework for deep learning of high-dimensional medical images. By decomposing high-dimensional images into simple geometric elements, such as points, lines, areas and volumes, and combining geometric algebra to analyse the multidimensional characteristics spatiotemporal data, a multi-scale deep classification system can be designed to perform feature extraction in medical images [57]. As discussed in [76], features originally extracted as scalar products can be conveniently combined to generate Clifford (geometric) products with additional discriminatory power from the original scalar products alone. An example of this is illustrated in Figure 5 where time-domain THz spectroscopic imaging data combined with MRI data are used for the design of a time-space unified deep learning scheme. A general representation of the associated geometric multilayered perceptron is shown in Figure 6.  
Figure 4: Tumour segment reconstruction from multiple channels. (**a**) Illustration of intensity variation for breast tumour tissue images; the yellow arrows indicate a high intensity whereas the red arrows low intensity. (**b**) Illustration of fuzzy c means (FCM) based segmentation on (**a**) assuming inhomogeneous boundaries; the yellow arrows indicate an irregular ring-shaped region and a green arrow indicates missing pixels in the segmented object. (**c**) Reconstructed volume image from multiple channels. (**d**) The tumour segment after reconstruction. (**e**) Overlapped images (brown) of the original tumour segment (blue) and reconstructed tumour segment (yellow). The green arrows indicate the fuzzy edges associated with the original image. Images depict an in-situ ductal carcinoma based on [52]. In order to create inputs to a DCNN, it is necessary to cluster dynamic image data associated with different dimensions. The purpose is to increase the depth of the deep learning network while removing redundant information so as to improve the convergence of the network to provide greater diagnostic accuracy. There are several hyper-dimensional image analysis and clustering strategies that may be considered for this purpose: one possibility is to adopt feature analysis and clustering methods for one-dimensional time series [77; 78; 79] or use clustering algorithms for two-dimensional spatial images, i.e., super voxel [80], Gray Level Co-ocurrence Matrix (GLCM) and [PERSON]'s statistical measures [81], 2D wavelet transforms [82; 83] or two dimensional functional principal component analysis [84]. Another possibility is to use an image segmentation algorithm for three-dimensional spatial images, i.e., LCCP super-voxel [85], 3D wavelet transform [86]; or image segmentation algorithms for 3D time-space images, i.e., SLIC super-voxel [87] approaches. Image sequences of different dimensions may use different image analysis strategies to form a collection of scale-dependent features to be further analysed using a multi-task deep learning network framework. Figure 7 shows a flowchart of a semi-supervised tumour segmentation (SSTS) technique for the analysis of 2D and 3D spatial MRI, which uses these features in a super-voxel context. Figure 5: Geometric neuron based on the McCulloch-Pitts neuron for magnetic resonance imaging (MRI) and terahertz (THz) pulse imaging datasets based on the generic framework discussed in Ref. [57]. Figure 6: A simplification of the learning rule discussed in Ref. [57], where it is suggested that in the training of geometric feedforward networks, the weights of the output layer could be real values (the output weight multivectors could be scalars of \(k\)-grade). In SSTS, Module 1 delineates the approximate area captured by the scan using intensity based Otsu thresholding. As imaged tumours in MRI normally show higher intensity levels than the background, clustered pixels with low intensity associated with normal tissue are easily removed.