4 The Likelihood

  Because of the variable lighting effects, it is difficult to directly model the image intensity given prior information about a fish outline. Therefore, we adopt an ad hoc approach, describing a partial set of features in the image. We investigated the approach of matching grey-level profiles, as used for example by [ 8 ], but found that it was not successful in this application. The following method provides useful directional information even when the model is at moderate distances from the true object.

The log-likelihood contains a contribution from each landmark, , in the template. At landmark l , a profile or line segment of moderate length ( pixels) is constructed in a normal direction to the outline. At a set of equally spaced points along this profile, labelled , there are directional edge strengths , which sum along the profile to give , say. Define a probability distribution , , by

and consider a hypothetical experiment for each fixed landmark l . First, choose a profile point with probability distribution . Secondly, let denote a random variable giving the position along this profile of the edge point relative to the landmark l . Assume that follows a normal distribution, . In this work, we have taken pixels. Let denote the squared distance between point j and landmark l along this profile. Then the log-likelihood , say, with respect to the random variables is given by

Since is not observed, it is convenient to maximise the log-likelihood over j , yielding the ``profile'' log-likelihood:

Then the overall profile log-likelihood for the template is given by:

The maximization in Eq. 4 amounts to using the best component in a mixture distribution. A better approach might be to use the EM algorithm or MCMC methods. The value pixels was found to provide a compromise between sensitivity when the model is at moderate distances from the true fish (allowing the model to find the neighbourhood of a fish) and precision when the model is close to the true fish (allowing the model to home in on the exact edges). Note that only that subset of the data which lies on the lines normal to the landmarks enters the likelihood, and this subset changes as changes. Further, the likelihood gives only limited influence to missing data and artifacts by limiting the search to a finite interval along each profile, so that it provides some robustness for the analysis.