Another approach to deal with the extent problem is to use ideas from the field of active contours [BI98] and snakes [KWT88, JBU04]. Informally, a snake is a closed path on which two competing geometrical forces are applied (Figure 4.5a). One force attempts to minimize the internal energy, which can be visualized by interpreting the snake (red) as a set of springs with each point in the curve constantly pulling its neighbors (red arrows). The resulting effect is to reduce the curvature by making the shape flatter and smoother. The other force aims to minimize the external energy by pushing the snake away (grey arrows) from the image (light grey). The end result is that the snake path closely follows the shape boundary. This section aims to transfer these ideas to the field of EOT, which is not a straightforward task, as snakes are meant to work with dense measurement sets, such as pixel grid images, where a large amount of measurements are available simultaneously. In contrast, for this thesis we are also concerned with relatively few point measurements, which may arrive at different time steps. This makes finding a balance between shrinking and pushing difficult, as the snake may easily fall through sparsely
4.3 Active Models
observed regions, and thus, in the absence of external energy, the path will inevitably collapse into a single point.
(a) Snake applied on an image. (b) Active forces on a rectangle. Figure 4.5: An illustration of active models. We use an active force (red) as part of a shape dynamic
model, which pulls the shape towards a smaller, smoother form. The update step acts as a counterforce (dark gray) which pushes the estimate back towards the true shape (light gray).
A more appropriate approach is, instead, to take the concepts from active contours and apply them to the state parameter space. For recursive Bayesian estimators, this idea can be implemented as a ‘shape dynamic model’, which we denote as active models [2, 12]. The concept of energies is reinterpreted as follows. The reduction of internal energy is modeled as a prediction step which applies some sort of operation on the state, such as making the corresponding shape smaller or applying a force in a given direction. This effect is counteracted by the measurement update step, which pushes the shape towards its correct form and acts as the external energy. We denote the application of an active model to the state as the regularization step. Note that active models do not exclude other dynamic models, such as motion models, which can also be used as additional prediction steps. Furthermore, we want to emphasize that regularization is an operation on the state xk. This means that, for recursive estimators, the entire state pdf needs to be transformed, not just the mean or a representative value.
4 Active Random Hypersurface Models
For illustration, we will now implement an active model for the example rectangle in Figure 4.1a and we will show how it solves the extent problem for GAMs. The active model can be described with
xk+1p = (1 −ck) · xek,
where 0 ≤ck 1 is a coefficient that ensures that the predicted rectangle width xkp+1shrinks a bit from the estimated width xe
k at each time step (Figure 4.5b). Thus, the regularization step takes the form of a linear operation, allowing for the state pdf to be easily propagated. In order to better demonstrate the results, we implemented an evaluation using a Progressive Gaussian Filter [SH14b]. The state is Gaussian distributed and initialized with variance σx,02 = 0.01, with three different means ˆx0 = 2, 5, and 8. Furthermore, the measurement noise covariance matrix is Cvk = 0.1 · I, and the regularization coefficient is
ck = 0.015. There is a single measurement for each timestep. Figure 4.6a shows the results averaged over 100 runs. We can see that, effectively, the ground truth was approximately found after 80 measurements for all initial states.
The advantage of this approach is that it does not require knowledge of any probability distribution, and thus, does not suffer from numerical instability or low robustness in cases of high uncertainty or occlusions. Furthermore, a regularization approach tends to be helpful in case of bad initialization, as it pushes the estimate away from a potential local minimum. However, this mechanism raises several challenges. First, the state will not converge to the ground truth as long as regularization is applied, as it is constantly being pulled and pushed in different directions. This can be seen in Figure 4.6a, where the final value is always slightly below 2, even for the green line which was initialized with the correct value. Second, it is difficult to obtain an appropriate value for the regularization coefficient ck, as its relationship to the shape characteristics, measurement uncertainty, and process noise is not intuitive. This issue is made more difficult by the fact that all of these factors may change over time. Furthermore, in practice the specific mechanism for regularization and the selection of the coefficients are generally ad-hoc, and thus, cannot be generalized and may need to be reconfigured for each scenario.
In literature, the idea of ‘pulling’ and ‘shrinking’ a shape is, as mentioned before, a staple in active contours [BI98]. In the context of shape representation using Gaussian processes, it serves a similar function as the ‘forgetting factor’
4.4 Active Random Hypersurface Models