It is important to mention that in this work CSS is not used to describe the shape, since a PDM is the featured shape descriptor. Here, only a part of the CSS representation is used, in this case the evolutions of the curve and the extrema or zero-crossings. The following is an explanation of how these are employed, in the particular case of the proposed local shape model, to create contour partitions. Given an input contour (Figure 4.4-(a)), there is the need to select the appropriate level of smoothing σ. Applying different values of sigma to the contour produces the aforementioned evolutions of the curve (Figure 4.4-(b)), and for each evolved contour it is possible to find the zero-crossings (displayed as red dots). This are the points where there is a change in the sign of the curvature of the contour. Then, having selected a value of σ, the set of generated inflection points are used to create the contour partitions (Figure 4.4-(c)), where each point constitutes the start and end for each partition. So these points provide a basic but efficient way to create meaningful contour partitions.
4.4.1
Supervised Shape Partitioning and Clustering
Having stated how CSS is used to create partitions, the aim here is the localisation and clustering of the featured model presented on the right side of Figure 1.7 in Chapter 1. The first task is done using CSS and the second using an interface, both in a supervised fashion. This section then, describes the the way these two tasks are incorporated into the local shape modelling framework, for which an overview is shown in figure 4.5-(b). On the mentioned figure, part (a) corresponds to the data step in figure 1.7 and part (c)corresponds to the steps of registration, PCA and reconstruction, as well in figure 1.7.
To facilitate the research, a tool with a graphical interface was created, which was divided into 3 main sections corresponding to the three main steps of our methodology (Figure 4.5). Refer to Appendix A for a full description of this tool.
Chapter 4. Curvature Scale Space Representation for Contour Localisation 93
Figure 4.4: (a)Original contour, (b)CSS evolution of a white-matter brain contour. At some appropriate level of smoothing, a set of meaningful partitions can be identified. (c) Pairs of zero-crossings (red points) are used to create
contour partitions.
Successive steps of the the method will now be described. From the input data set of images, we obtain the contours for each of them as a set of (x, y) coordinates with the method explained in section 2.4.4.1 (Figure 2.10).
The first step is to select a shape that will be the reference one to create the partitions for the rest of the contours in the set. In Figure 4.6-(a) the reference contour is shown together with the set of zero-crossing points, and a pair of these is manually selected to create a reference partition. Afterwards, to create the set of partitions from all the contours in the set, each one is smoothed with the same scale factor σ as the one the reference contour. Here, zero-crossings are taken in combinations of point-pairs and occupying the points between each of these com- binations is how partitions are created. After obtaining the set of shapes, the next
Chapter 4. Curvature Scale Space Representation for Contour Localisation 94
Figure 4.5: Overview of the method for proposed LSM with the incorporation of CSS for contour partitioning and the supervised clustering: (a)Contours are obtained from MRI brain data;(b)CSS is used first to obtain a reference shape from the smoothing of a target contour at some large scale,σ. A pair of zero- crossings selects a prototype section and all others are ranked accordingly by search and rigid alignment; (c)Statistical Shape Modelling and reconstruction
based on selected modes.
major concern is deciding which of these partitions could be useful for the shape analysis. An MSE ranking plot (Figure 4.6-(c)) is constructed by doing a pose alignment of the partitions (Figure 4.6-(b)) and then calculating the mean square error of Euclidean distances for each partition of the set against the reference shape using:
D(Cm,Cn) =
q
Chapter 4. Curvature Scale Space Representation for Contour Localisation 95
Figure 4.6: CSS for contour partitioning and the supervised clustering process: (a) A shape is selected that will be the reference one to create the partitions for the rest of the contours in the set. The reference contour is shown together with the set of zero-crossing points, and a pair of these is manually selected to create a reference partition. Then, to create the set of partitions from all the contours in the set, each one is smoothed with the same scale factor σ as the one the reference contour. Here, zero-crossings are taken in combinations of point-pairs and occupying the points between each of these combinations is how partitions are created. After obtaining the set of shapes, a pose alignment of the partitions is performed using the reference partition from previous step (b)and then MSE ranking plot is constructed(c). This plot is used to indicate the number of partitions (by setting an error threshold) that are going to be
used in the statistical analysis (d).
This plot is used to indicate the number of partitions (by setting an error threshold) that are going to be used in the statistical analysis (Figure 4.6-(d)). Now, having a smaller set of shapes is possible to perform shape analysis in the way described in section 2.4.4.3. Finally, a reconstruction of the selected partitions can be visualised along with the modes of variation of the shape set (Details in section 2.4.4.4). In the next section this will illustrated in a more effective way with examples from two different data sets.