Dipole statistics of discrete finite images: two visually motivated representation theorems

A discrete finite image I is a function assigning colors to a finite, rectangular array of discrete pixels. A dipole is a triple, ((dR, dC), alpha, beta), where dR and dC are vertical and horizontal, integer-valued displacements and alpha and beta are colors. For any such dipole, D(I)((dR, dC), alpha, beta) gives the number of pixel pairs ((r1, c1), (r2, c2)) of I such that I[r1, c1] = alpha, I[r2, c2] = beta and (r2, c2) - (r1, c1) = (dR, dC). The function D(I) is called the dipole histogram of I. The information directly encoded by the image I is purely locational, in the sense that I assigns colors to locations in space. By contrast, the information directly encoded by D(I) is purely relational, in the sense that D(I) registers only the frequencies with which pairs of intensities stand in various spatial relations. Previously we showed that any discrete, finite image I is uniquely determined by D(I) [Vision Res. 40, 485 (2000)]. The visual relevance of dipole histogram representations is questionable, however, for at least two reasons: (1) Even when an image viewed by the eye nominally contains only a small number of discrete color values, photon noise and the random nature of photon absorption in photoreceptors imply that the effective neural image will contain a far greater (and unknown) range of values, and (2) D(I) is generally of much greater cardinality than I. First we introduce "soft" dipole representations, which forgo the perfect registration of intensity implicit in the definition of D(I), and show that such soft representations uniquely determine the images to which they correspond; then we demonstrate that there exists a relatively small dipole representation of any image. Specifically, we prove that for any discrete finite image I with N > 1 pixels, there always exists a restriction Q of D(I) (with the domain of Q dependent on I) of cardinality at most N - 1 sufficient to uniquely determine I, provided that one also knows N; thus there always exists a purely relational representation of I whose order of complexity is no greater than that of I itself.