# Foundations of Pattern Recognition

Computational anatomy (CA) is a system of automatically generating medical descriptions of patients from their medical images and the medical description can be generated by classifying each pixel in the images into one of some pre-deflned classes. This classification problem can be solved by solving a pattern recognition problem in which, when representing data as points in a high-dimensional space, the space must be divided into regions, each of which represents each of the classes. A classifier then judges in which region a given data point is included and outputs the class that corresponds to the region.

## Bayes Decision Theory

The best classifier minimizes the classification error under the assumption that the probability distributions of the data points in the space are known. Assuming only two categories, *w** _{1}* and

*w*

_{2}, let

*x*denote a given datum. Then, the probability of misclassification of the data, p(error|x), is given as [34]

The probability of the classification error can be minimized by selecting the category that maximizes P(wj|x). This selection maximizes the average probability of misclassification for all data such that

Fig. 2.5 **Block diagram of a classifier in [35]**

It is called a *Bayes decision theory* that the average probability of the can be minimized by following this rule:

Rule: Decide *Wj* if *P.wjjx)* is larger than *P.w _{k}x)* for any

*k ф j.*