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.