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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 19, Iss. 5 — May. 1, 2002
  • pp: 825–832

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

Charles Chubb and John I. Yellott  »View Author Affiliations

JOSA A, Vol. 19, Issue 5, pp. 825-832 (2002)

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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 ( ( r 1 ,   c 1 ) ,   ( r 2 ,   c 2 ) ) of I such that I [ r 1 ,   c 1 ] = α , I [ r 2 ,   c 2 ] = β and ( r 2 ,   c 2 ) - ( r 1 ,   c 1 ) = ( 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

OCIS Codes
(100.2960) Image processing : Image analysis
(100.5010) Image processing : Pattern recognition
(330.5000) Vision, color, and visual optics : Vision - patterns and recognition
(330.7310) Vision, color, and visual optics : Vision

Original Manuscript: August 27, 2001
Revised Manuscript: October 12, 2001
Manuscript Accepted: October 12, 2001
Published: May 1, 2002

Charles Chubb and John I. Yellott, "Dipole statistics of discrete finite images: two visually motivated representation theorems," J. Opt. Soc. Am. A 19, 825-832 (2002)

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