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Relaxation Labelling

Relaxation labelling techniques can be applied to many areas of computer vision.

Relaxation techniques have been applied to many other areas of computation in general, particularly to the solution of simultaneous nonlinear equations.

We shall first describe the basic principles behind relaxation labelling methods and then discuss various applications.

The basic elements of the relaxation labelling method are a set of features belonging to an object and a set of labels.

In the context of vision, these features are usually points, edges and surfaces.

Normally, the labelling schemes used are probabilistic in that for each feature, weights or probabilities are assigned to each label in the set giving an estimate of the likelihood that the particular label is the correct one for that feature.

Probabilistic approaches are then used to maximise (or minimise) the probabilities by iterative adjustment, taking into account the probabilities associated with neighbouring features.

Relaxation strategies do not necessarily guarantee convergence, and thus, we may not arrive at a final labelling solution with a unique label having probability one for each feature.

Let us now consider the labelling approach in more detail. Let us assume:

Let tex2html_wrap_inline8554 be the probability that the label tex2html_wrap_inline8556 is the correct label for object feature tex2html_wrap_inline8558.

The usual probability axioms can be applied that:

The labelling process starts with an initial, and perhaps arbitrary, assignment of probabilities for each label for each feature.

The basic algorithm then transforms these probabilities into to a new set according to some relaxation schedule.

This process is repeated until the labelling method converges or stabilises.

This occurs when little or no change occurs between successive sets of probability values.

Popular methods often take stochastic approaches to update the probability functions.

Here an operator considers the compatibility of label probabilities as constraints in the labelling algorithm.

The compatibility tex2html_wrap_inline8572 is a correlation between labels defined as the conditional probability that feature tex2html_wrap_inline8558 has a label tex2html_wrap_inline8556 given that feature tex2html_wrap_inline8578 has a label tex2html_wrap_inline8580, i.e. tex2html_wrap_inline8582.

Thus, updating the probabilities of labels is done by considering the probabilities of labels for neighbouring features.

Let us assume that we have changed all probabilities up to some step, S, and we now seek an updated probability for the next step S+1.

We can estimate the change in confidence of tex2html_wrap_inline8554 by:
where N is the set of features neighbouring tex2html_wrap_inline8558, and tex2html_wrap_inline8594 is a factor that weights the labellings of these neighbours, defined in such a way that

The new probability for label tex2html_wrap_inline8554 in generation S+1 can be computed from the values from generation S using

next up previous
Next: Statistical Relaxation Techniques Up: Vision II Previous: Labelling an Image