View - Statistics - University of Washington

403.4.1 Mixture versus Componentwise ClassificationThe appropriate method for making our final classification depends on how wewill evaluate the classification when it is done. For instance, we might count thenumber of pixels in a feature of interest which are correctly classifed; clearly, a classificationwhich places all pixels into that feature regardless of all the estimatedparameters would be optimal, though rather unuseful. A more reasonable evaluationcriterion would be to simply count the number of pixels correctly classifiedin the whole image; the optimal classification method for this case would lead toa very different result than the previous example. The point here is that we canthink of many different evaluation methods; these may be driven by concerns of aparticular application, or they may just be different common sense approaches. Inthis section I consider two sensible evaluation criteria, mixture and componentwise,and give the classification methods which are appropriate for each.Mixture ClassificationFirst let us consider the case mentioned above in which we want to maximize thenumber of pixels which are correctly classified. This is the mixture classificationcase.Theorem 3.1: Optimality of Mixture ClassificationTo maximize the number of correctly classified pixels, the optimal classification ruleis to assign each pixel to the segment which has the largest posterior probability,as shown in equation 3.18.C i = argmax m P m Φ(Y i |θ m ) (3.18)In equation 3.18, Y is the observed image and C is the estimated classification.