A new concept σ-inverse" of a matrix is proposed. The σ-inverse of a matrix A is defined as the matrix X satisfying the following two equations:AXAσ2X*A,(XA)*XA,were σ is a given positive real number. The σ-inverse of A is denoted by Aσ. This matrix gives minimum-variance estimation in linear systems, while in contrast the Moore-Penrose inverse gives best linear unbiased estimation. The spectral properties of the σ-inverse can be represented by a complex-valued function /(zσ2). This function is continuous and bounded, which reflects the properties of the σ-inverse.

For similarity methods to work well, the image must be blurred before being input. However, the relationship between the blurring operation and the similarity is not fully understood. To solve the problem of this relationship, in this paper, the effect of blurring is investigated by expressing figure f(x) in the form of the sum of higher derivatives of f (x,σ), and then a simple similarity between figures was mathematically formulated in terms of the relation between visual patterns. By modifying this formulation, we propose pluralized simple similarity to increase the allowance in different view of multiple similarity. The similarity maintains higher allowance without any discernible loss of distinguishing power. We verify the effectiveness of the pluralized simple similarity throughout some experiments.

One of the most basic characteristics of the image is accompanied by its blur. It was 1962 that I had discovered for the first time in the world that the blur was a Gaussian type. In this paper the outline is described about historical details concerning this circumstances.

The pressure waveforms indicated on a catheter manometer system are subject to serious distortion due to the resonance of the catheter itself, or the compliance of a particular transducer. Although several methods have been proposed for improving those characteristics, they ahave never been put into practice. We have focused on the transfer function of the catheter manometer, and made a pilot system, using the natural observation method. This method has been suggested as a means of studying the structure of the instantaneous waveform. In this manner, we were able to increace the bandwidth in the ferquency domain and reduce the ringing in the time domain. Correction was performed automatically, using a step wave. Reproduction of the waveform with a flushing device, was a task of equal simplicity, that allowed us to estimate the system parameters so that the response waveform became step-like. In the experiment, our system provided distortion-free left-ventricular pressure waveform measurements and exact evaluation of the cardiac pumping system. The values obtained came much closer to the original figures arrived at by the catheter-tip manometer system.

In this paper we discuss the limiting behavior of the search direction of the steepest descent method in minimizing the Rayleigh quotient. This minimization problem is equivalent to finding the smallest eigenvalue of a matrix. It is shown that the search direction asymptotically alternates between two directions represented by linear combinations of two eigenvectors of the matrix. This is similar to the phenomenon in minimizing the quadratic form. We also show that these eigenvectors correspond to the largest and second-smallest eigenvalues, unlike in the case of the quadratic form.