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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]

Estimation of the noncausal image model described by a symmetric spatial filter is important in the study of statistical image processing. However, the estimating methods proposed so far have some problems in that they require too much computation and are not necessarily good enough to estimate a white-noise-driven image model. To cope with such difficulties, this paper proposes a new method for noncausal model estimation where a test function is introduced called ‘whiteness’ which represents the degree of spectral whiteness of an image field. ‘Whiteness’ is a positive-valued functional of the sample correlation function and takes the minimum value 0 when the image is whitened. Thus, by minimizing the ‘whiteness’ by some optimization algorithm, the parameters of the whitening filter are estimated along with the support on which the model parameter is located. This method is applied successfully to various images to demonstrate its effectiveness; and three examples, i.e., two simulated images with single- and double-peaked spectra and a texture pattern, are treated by the proposed method. For the simulated images, original correlation functions and spectra are reproduced; and, for the texture image, an almost perfect whitening filter is obtained. The present method is shown to be more effective than the conventional most likelihood method based on AIC or BIC, not only in parameter estimation but also in determining the size of support of a noncausal image model. [ABSTRACT FROM AUTHOR]