AdaBoost Training and Classification

At each vertex in the medial model, I build a model of local appearance, which is subse- quently used to drive image segmentation. Recall that for most medial pseudo-landmarks there are two corresponding boundary nodes, one on each side. At these two nodes, the maximal inscribed ball which centers at the medial pseudo-landmark is tangent to the model boundary. The exceptions are the medial pseudo-landmarks on the branching curves at the two ends of the interventricular septum, which have three corresponding boundary nodes. The triple tangency branching points are treated as the limit case, where three bitangency points meet together. Each one of these three bitangency points belongs to one branch of the medial axis and is treated normally during the training and boundary detection. After boundary detection, the triple tangency medial pseudo-landmark and the radius will be updated as the average of the three independent bitangency detection results. Below I only deal with bitangency cases.

At each boundary node, I train an AdaBoost classifier to discriminate between a “well- placed” boundary node and a “misplaced” boundary node, as illustrated in Figure 5.3. A

(a) Training boundary classifiers

(b) Detecting a pair of boundary points

Figure 5.1: Illustration of appearance matching using the AdaBoost classifier. (a). The classifier is trained to differentiate between boundary nodes located at the correct anatom- ical boundary and displaced boundary nodes. In the figure, the yellow bars show samples drawn from correct anatomical boundary, while the red bars are samples that are displaced. During training, each boundary node is displaced along the chord direction (illustrated in Figure 5.2), and samples from the image neighborhood are used to generate appearance features. Combining features from different subjects, at each boundary node, I train an Ad- aBoost classifier with two classes (displaced node vs. not displaced). (b). The deformable model is shown in red color while the underlying object is shown in green color. During segmentation, the classifier is used to position boundary nodes close to anatomical bound- aries. Pairs of boundary nodes that share a medial pseudo-landmark are displaced along the chord direction, governed by the AdaBoost classifiers corresponding to the nodes. Fol- lowing these displacements, the deformable model is updated so as to satisfy the necessary geometric constraints and to abide by the shape priors.

Figure 5.2: Illustration of chord direction. For boundary nodes b± which correspond to the medial pseudo-landmark m, the chord direction which crosses b+ and b− shown as purple line in the figure.

well-placed boundary node lies within a certain distance to the corresponding anatomi- cal boundary in the training image, and a misplaced boundary node lies some distance away from the anatomical boundary, as illustrated in Figure 5.3. Well-placed boundary nodes are obtained by fitting models to manual segmentations of the myocardium in the training data. Misplaced boundary nodes are obtained by applying displacements to the well-placed boundary nodes along the direction between the two corresponding boundary nodes. This displacement direction, called chord direction since it is a chord of the MIB, is illustrated in Figure 5.2. Therefore training exemplars for each classifier include well- placed and misplaced versions of a given boundary node across all subjects included in the training subset. To further increase the number of training exemplars and make classifiers less sensitive to location, I include, as training exemplars for each classifier, misplaced and well-placed versions of the boundary nodes in the two-ring neighborhood of the boundary node associated with the classifier.

During local boundary detection, each sample corresponding to a pseudo-landmark go through two classifiers, one for each of the two corresponding boundary nodes. According to the classification scores, a pair of points satisfying the following conditions is selected to be the new candidates of boundary nodes: (1) they are classified as correct boundary nodes according to the two classifiers respectively; (2) their order is consistent with the right order of the boundary nodes (otherwise the boundaries would intersect); and (3) the

Figure 5.3: Illustration of training exemplars of a “well-placed” boundary node (class 0) and a “misplaced” boundary node (class 1) in AdaBoost training. The manual segmenta- tion of the anatomical structure is shown in gray. The medial model is fitted to the man- ual segmentation to obtain medial pseudo-landmarks and corresponding boundary points. The left figure shows a “well-placed” boundary node centered at the exact boundary of the manual segmentation. Note that since the manual segmentation can not be perfect, I actually place three “well-placed” boundary nodes for each boundary location: one is on the exact boundary of the manual segmentation as illustrated in the left figure, the other two are on two sides of the first one and are obtained by applying a small displacement to it along the chord direction. The right figure shows a “misplaced” boundary nodes, which is obtained by applying displacements to the well-placed boundary nodes along chord direction.

overall classification score is the highest for all pairs satisfying condition (1) and (2). If no such pair can be located for a particular medial pseudo-landmark, the local boundary detection algorithm will return a void result for this medial pseudo-landmark.