The goal of this paper is not to make slick surface textures for computer games. Nontheless, as a tutorial device, texture synthesis using the PEF is valuble, since it concretely and intuitively illustrates in two dimensions some of the fundamental concepts of autoregression which are proved only in the one dimensional case Claerbout (1976). In fact, some of the results shown here and in Claerbout and Brown 1999 have recently been incorporated into Jon Claerbout's textbook, Geophysical Estimation by Example 1998a.
Both the Fourier transform and PEF-based texture synthesis operate under the assumption that the training image is sufficiently well characterized by amplitude spectrum alone. For some images (Figures 2, 5, and 7) the assumption holds, but for others (Figures 3, 6) it is obviously violated. Real digital images and earth phenomena alike often exhibit complex spatial correlation which are modelable only with multiple point templates Caers and Journel (1998); Malzbender and Spach (1993). Additionally, I have ignored the interesting subjects of nonstationarity and spatial scale variance. By scale-variant, I mean that the characteristic scale of an image's features is not constant with respect to spatial frequency. Many methods for characterizing scale-variant images appeal to the world of wavelets for a methodology known as multiresolution analysis Heeger and Bergen (1995); Simoncelli and Portilla (1998); Strang and Nguyen (1997). The notion of texture synthesis for nonstationary images is ill-defined, since it amounts to a random reordering of filters estimated on locally-stationary patches, followed by deconvolution on the correspondng patches.
When the training image has missing values, as in Figure 8, the PEF-based texture synthesis method performs favorably. As shown in the missing data interpolation example (Figure 10), the ability of the PEF to reliably estimate the data spectrum, even with missing data, makes it an ideal regularization operator. Figure 9 illustrates the fact that the PEF primarily predicts plane waves. I proposed using a PEF residual measure to determine the viability of a given migration velocity. In general, PEF estimation/convolution might have value as a preprocessing step for a variety of applications. For instance, a very small PEF (2 columns) has a relatively large residual in the presence of conflicting dips, and thus may help in determining local filter size or patch size.