4 On-the-fly training

 

In the spirit of [1] we propose on-the-fly training for the most appropriate local potential of the form

 

While has no effect on the location of any extrema of P(x,y), it allows us to consider (7) as a single layer network with weights , and . We can then train the network using a pseudo-inverse [2] and training samples gathered from around the boundary. We define target values for the training samples as follows:

where d is the distance in pixels of from the boundary. Thus the target values are -1 on the boundary and approach zero asymptotically away from the boundary. When trained, P(x,y) should have local minima on the segmentation boundary, as required. Since we are training local boundary models, , and are calculated for each spline segment. We found that the performance of the system was not sensitive to the parameter , which we conveniently set to unity. We also found it advantageous to discard the moduli signs in equation (5): this allows the algorithm to exploit the polarity of the boundary. Note how on-the-fly training will automatically find the right sign for .

While such training is of no benefit for single-frame segmentation, it can greatly speed-up the process of segmenting many slices through a 3D data set. Since the boundary statistics generally change slowly from one slice to the next, optimal segmentation potentials (7) learned in one slice will also work well on the next slice. What emerges is a segmentation paradigm with the following structure:

1. The user specifies an initial B-spline by selecting control points in slice 1. Initially, in the absence of any texture models, the snake's potential function (7) is calculated with and .
2. The spline is sampled at n locations distributed evenly around its length. Corresponding target points are located using a 1D search normal to the spline for a local minimum of the potential (7).
3. The user corrects any mis-positioned target points. Segmentation (based on the target points) is now complete for this slice.
4. The B-spline control points are adjusted to minimise in (1).
5. Local texture potentials (one per spline segment) are calculated by sampling patches around the boundary (Figure 3) and using (2)-(4) and (6).
6. Local , and are calculated using training samples and a pseudo-inverse [2] such that P(x,y) in (7) is minimised at boundaries.
7. Move on to the next slice and repeat from 2, using the snake's current position as a starting point in the new slice.