# Three-Dimensional Anatomical Analysis of Muscle-Skeletal Districts
## 3 Geometrical and Texture-Based Features of Anatomical Districts
### Texture-Based Features of Single and Follow-Up Exams  
Following [22], we generated a texture for the 3D bone surface extracted from an input image of a _single exam_. The segmented 3D surface is a triangle mesh, and the volume image is composed of a series of 2D slices and is represented as a voxel grid. The image is loaded into a 3D grid, whose elements have the same dimension as the image voxels. In this way, every grid cell has its 3D coordinates and each grid cell is associated with a voxel and its grey level. This data structure allows the navigation of the volume through the surface to find the correct correspondences between the volume image and the segmented 3D surface. For the grey value mapping, three different criteria are defined to analyse the grey values in the proximity of the surface. Such criteria depend on the method chosen to identify the correspondences between the surface vertices and the volume voxels. In the _Euclidean mapping_, the surface vertex \(\mathbf{p}\) obtains the grey level of the voxel closest to \(\mathbf{p}\). In the _internal mapping_, the closest voxels are searched only inside the surface, i.e., inside the object's volume. In the _external mapping_, the closest voxels are searched only outside the surface, i.e., outside the object's volume. Once every vertex of the surface has been associated with  
Figure 2: (**a**) Registration result between the 3D surface model of a single bone at baseline time and follow-up time. (**b,c**) Geometry-based feature extraction: distribution of local distances between the bones at baseline time and follow-up time. Red rectangles highlight pathological deformation localised by experts. The original value in millimetres was normalised between 0 and 1.  
a voxel, the same grey level is associated with the corresponding voxel. The result of the grey-level mapping is a textured surface, where each vertex coordinate is associated with its specific colour, representative of the information contained in the image (Figure 3).  
This result is satisfactory if we are evaluating a _single exam_, since we can directly consider the grey level characterising the tissue in the neighbourhood of the bone surface from the texture. The mean value of the texture in a particular region of the 3D surface model will be considered as the texture features in the following integration section. To consider all the volume surrounding the bone surface, the mean value of the grey levels can be computed for all the different mapping methods: internal, external, and Euclidean. In this way, the analysis will be more exhaustive.  
For _follow-up exams_, we compare the tissue evolution over time. For this reason, we now describe how to evaluate the tissue changes from the textured 3D surface. After co-registering the two surfaces, the follow-up 3D model is textured with the grey levels of both the image at baseline time (\(t_{1}\)) and at follow-up time (\(t_{2}\)), exploiting the grey-level mapping. Mapping the volume grey levels at time \(t_{1}\) on the surface at time \(t_{2}\), the follow-up 3D model shows either the volume situation at time \(t_{1}\) and at time \(t_{2}\). We focused on external mapping, which explores the volume toward the external normal directions from the surface vertices, since erosion typically results from a degeneration of articular regions. The variations in texture are highlighted by computing the difference between the grey levels mapped from the baseline image onto the follow-up surface and the grey levels mapped from the follow-up image onto the follow-up surface. A higher value of the difference indicates a region where the tissue has been substantially damaged, probably due to an erosion process. The colour map represents the tissue modification directly on the 3D surface.