1.1 Proposal Distribution

We use shape similarity to define the initial proposal distribution. In order to achieve scale invariance, each model segments s_k and edge fragments e_j in the image is sampled with the same number of points, e.g., 20 points. We use a novel shape descriptor described in Section 3 to define shape similarity By normalizing we obtain the initial proposal distribution An example is shown in Fig. 3(right). By sampling (with repetition) from this distribution, we obtain the initial set of particles for i = 1, . . .,N.

Figure 3. Matrix of similarities between 8 model contour segments left and 16 edge fragments middle. It represents the initial proposal distribution .

We now describe the proposal distribution for the consecutive integrations of our PF, i.e., = Many object detection methods reported in the literature, e.g., [25], utilize the object centroid as the localization constraint of various object parts. They explore the fact that object parts hinge around the centroid, which significantly reduces object search space when the set of centroid hypotheses is small. Our proposal distribution is based on this idea. We use the shape similarity to define the center point function CP : S x E → I. It transforms the center point of the model shape to the image I for a given correspondence = (s_k, e_j), for some k and j and some particle (i). We observe that our model shape has a unique center point. Every possible correspondence transfers it to a object center hypothesis in the image. The centroid transfer is possible, since we estimate the scaling factor from the length ratio of fragments s_k, e_j. Thus, each pair (s_k, e_j) defines a potential center point of the model shape M on the image I. Since each particle = ((s₁, e₁), . . . (s_t-1, e_t-1)) is a sequence of such pairs, we can extend the definition of the center point to include all particles. Thus, denotes the average center point of the model shape M on the image I. Then, the proposal distribution is defined as a discrete distribution over the set S x E, where the probability of each x_t ∈ S x E is proportional to a Gaussian of the distance between and CP(x_t).

We define in this section the likelihood , which is needed for particle evaluation. We recall that = ((s₁, e₁), . . . (s_t, e_t)), i.e., it is a sequence of pairs of corresponding model segments and edge fragments. It is defined by similarity between the shape formed by segments of model contour M and the shape formed by the edge fragments

where is defined in Section 1.

For small t (t = 1, 2) this posterior is not particularly discriminative. However, already starting with t = 3 or 4, the shape of correctly selected edge fragments starts to resemble the model contour, and consequently, the descriptive power of increases significantly.