Search

One of the important features that characterize the physical random field or the image texture is Taylor's correlation length. However, there has not beet presented a clear-cut theory as to the foundation of the correlation length from the viewpoint of probability theory or statistical theory. Another aspect is that the feature is one-dimensional, and, therefore, the application to the random field is limited to the isotropic random field. This paper defines the nth order statistical number of degrees of freedom as the equivalent number of uncorrelated points in the square finite region in the homogeneous random field. From the definition, the correlation area and the correlation length are derived. The correlation area is referred to as the average area of the region statistically affected by an arbitrary point on a two-dimensional field. The correlation length represents the effective distance between adjacent independent or uncorrelated points on the line parallel to the x- or y- axis. The first-order correlation length is the same as the conventional Taylor correlation length. The first-order correlation area is a natural extension of Taylor's correlation length to the two-dimensional random field. The correlation length along the principal axis, which is independent of the definition of the coordinate axis, is also derived. Those features are based on the theory of statistics and are expected to be useful for the analysis of the actual random physical field and the texture analysis of the image. [ABSTRACT FROM AUTHOR]

First, the number of nth order degrees of freedom (n-th order NDF) of stationary random process is defined in terms of the sample moment of order n of a finite length sample. Correlation times and equivalent band- widths are derived from the NDFs. It is shown that the second-order NDF asymptotically approaches the degrees of freedom of a gamma distribution by which the distribution of the ample variance is approximated it is shown also for the case of a Gaussian process that the first-order NDF is the same as the standardized Fisher information about the mean; and, furthermore, the second-order NDF of a Gaussian AR process is the same as the standardized Fisher information about the variance. A problem with the Fisher information is that it cannot always be calculated, which is not convenient from the practical viewpoint. Since the NDF defined in this paper is based on the sample moments, it can easily be calculated from the observed data. Further, from the NDF, useful features can be derived such as the correlation time and the equivalent bandwidth. Therefore, it is of practical use. Finally, for a Gaussian process, the significance of the approximation of the NDF by 2 WT, where W denotes the equivalent bandwidth and T is time duration, is discussed. It is shown, in particular, that the approximation error for the second-order NDF decreases only slowly, depending on the Logarithm of 2 WT. This result is interesting in examining the dimension of the signal space for each moment of a random process since the situation is the same as in the case where the dimension of the signal space for the deterministic signal is approximated by 2 WT. [ABSTRACT FROM AUTHOR]