A discrete finite image I is a function assigning colors to a finite, rectangular array of discrete pixels. A dipole is a triple, ((d_R, d_C), α, β), where d_R and d_C are vertical and horizontal, integer-valued displacements and α and β are colors. For any such dipole, D_I((d_R, d_C), α, β) gives the number of pixel pairs ((r₁, c₁), (r₂, c₂)) of I such that I[r₁, c₁] = α, I[r₂, c₂] = β and (r₂, c₂) − (r₁, c₁) = (d_R, d_C). 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.

© 2002 Optical Society of America

[Optical Society of America ]

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