# Segmentation of Infant Brain Using Nonnegative Matrix Factorization
## 2 Materials and Methods
### The Joint MGRF Segmentation Model
#### 2.4.1 The NMF-Based Visual Appearance Model  
After the DTI estimation, we extract various scalar features from the estimated diffusion tensor. These features contain a lot of information about the microstructure of the brain, which is beneficial in significantly distinguishing between different tissues of the brain. Throughout this work, we used the 3D Slicer software [35] to calculate 5 different anisotropy features: fractional anisotropy (FA), mean diffusivity (MD), axial diffusivity (\(\lambda_{\parallel}\)), relative anisotropy (RA), and radial diffusivity (\(\lambda_{\perp}\)). The two features FA and RA will help to separate WM from other brain tissues, while the other three features (MD, \(\lambda_{\parallel}\), and \(\lambda_{\perp}\)) can differentiate between GM and CSF tissues. However, the most meaningful information needs to be extracted from that large dimensional space of the DTI. It has been shown that NMF has strong capabilities to cluster complex data based on extracting features that are capable of learning data classes characteristics [39]. As a result, in the last decade, the use of NMF has been extensively applied to the area of medical image segmentation. A weight matrix \(\mathbf{W}\) is determined in NMF-based image segmentation to transform a vector from an input space into a new feature space (H-space) by the factorization of the input matrix \(\mathbf{A}\). NMF has been used in a few DTI-based segmentation systems. For instance, Xie et al. [40] utilized NMF for the segmentation of the corpus callosum, spinal cord, and hippocampus from DTI images by k-means clustering [41] in H-space. NMF has been used in some segmentation problems with promising success; however, further investigations are still needed to verify NMF segmentation potential. In this study, NMF-based feature fusion is used to extract new significant features from the wide-dimensional DTI feature space (\(\mathbf{A}\)), which consists of one appearance feature (b0) and five anisotropic features (FA, MD, \(\lambda_{\parallel}\), \(\lambda_{\perp}\), RA), to more precisely segment various structures of the brain from DTI data. Particularly, our segmentation approach uses NMF to determine the weights for each of the features of the input DTI (\(\mathbf{W}\)), for the purpose of constructing a new feature space (\(\mathbf{H}\)) with a decreased number of dimensions, while encoding the information in the training dataset and increasing the separability among different classes. With NMF, two matrices could be factorized out of the input data matrix \(\mathbf{A}\in\mathbb{R}^{+I\times XYZ}\):  
\[\mathbf{A}\approx\mathbf{W}\mathbf{H} \tag{2}\]where the columns of \(\mathbf{W}\in\mathbb{R}^{+I\times J}\) define the basis vectors, and the encoding of the features in H-space is \(\mathbf{H}\in\mathbb{R}^{+I\times XYZ}\)[39]. The 3D DTI features are processed using NMF by transforming each brain voxel \((x,y,z)\) into vectors of the input data matrix \(\mathbf{A}\). To calculate \(\mathbf{W}\) and \(\mathbf{H}\), the Euclidean distance between \(\mathbf{A}\) and \(\mathbf{WH}\) is minimized under the constraint that \(\mathbf{W}\) and \(\mathbf{H}\) have only non-negative elements; formally,  
\[ \begin{split}\underset{\mathbf{W},\mathbf{H}}{\text{minimize}}& \frac{1}{2}\left\|\mathbf{A}-\mathbf{WH}\right\|^{2}\\ \text{subject to}& w_{i,j},\;h_{j,n}\geq 0\text{ for all }i,j,n\end{split}  \tag{3}\]  
Since the introduction of NMF, numerous approaches have been applied for optimizing Equation (3). The most outstanding algorithms applied are multiplicative update, projected gradient descent (PGD), and the alternating least square (ALS) algorithm [42]. In this study, we selected the ALS algorithm, since it is more flexible and has higher speed than other competing algorithms. The ALS algorithm takes advantage from the fact that, although Equation (3) is non-convex in both \(\mathbf{H}\) and \(\mathbf{W}\), it is convex in either \(\mathbf{H}\) or \(\mathbf{W}\) when the other is considered as constant.