An extended fisher criterion for feature extraction ‐ Malina's method and its problems
Toshihiko Okada, Shingo Tomita · Electronics and Communications in Japan (Part I Communications) · 1984
Abstract In the discussion of the two‐class feature extraction problem in pattern recognition, Malina [7] proposed recently a new criterion for overcoming the defects of Fisher's criterion. the discussion in this paper focuses on the effectiveness of Malina's criterion and at the same time points out the problem in his feature extraction method, indicating how to avoid the difficulties. First, no clear definition is made in Malina's method for the coordinate system in the feature space, and the coordinate system used is not optimal. Second, the eigenvalue problem must be solved including the solution of inverse matrix in order to determine the feature axis. For the first problem, this paper points out tha the coordinate axis di and dj in the coordinate system in Malina's method satisfies diTdj = 0, where is the sum of covariance matrices in the class. Then the optimal coordinate system from that viewpoint is given. For the second problem, the inverse matrix is eliminated by performing a certain transformation (A‐1 transformation) to the coordinate system of the original pattern space. the validity of A‐1 transformation is shown by the invariance of the distances among classes, and it is shown that the above optimal coordinate system under A‐1 transformation is the optimal orthogonal coordinate system satisfying diTdj = 0, i j. Finally, Malina's criterion is extended to the multiclass case and the problems are discussed.