An optimal three-class linear observer derived from decision theory

Many attempts have been made to develop an optimal linear observer for classifying multiclass data. Most approaches either do not have a definite description of optimality or have regions of ambiguity in decision making. In this paper, we derive a three-class Hotelling observer (3-HO), inspired by the ideal observer that results from a decision theoretic solution to the three-class classification problem. Assuming the data vectors follow multivariate Gaussian distributions with equal covariance matrices for the three classes, it is shown that two two-class Hotelling templates construct a 3-HO which has the same performance as the three-class ideal observer (3-IO). We show that, without the Gaussian and equal covariance assumptions, the 3-HO is still applicable when the two-class Hotelling templates of each pair of the classes satisfy a certain linear relationship. In this case, the 3-HO simultaneously maximizes the signal-to-noise (SNR) of the test statistics between each pair of the classes. In conclusion, we developed a three-class linear mathematical observer that uses first- and second-order ensemble data statistics. This mathematical observer, which has clearly defined optimality for several data statistics conditions and has decision rules that have no ambiguous decision regions, is potentially useful in the optimization and evaluation of imaging techniques for performing three-class diagnostic tasks.