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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Vol. 2, Iss. 5 — May. 1, 1985
  • pp: 683–692

Fingerprints theorems for zero crossings

A. L. Yuille and T. Poggio  »View Author Affiliations


JOSA A, Vol. 2, Issue 5, pp. 683-692 (1985)
http://dx.doi.org/10.1364/JOSAA.2.000683


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Abstract

We prove that the scale map of the zero crossings of almost all signals filtered by a Gaussian filter of variable size determines the signal uniquely, up to a constant scaling. The proof assumes that the filtered signal can be represented as a polynomial of finite, albeit possibly high, order. The result applies to zero and level crossings of linear differential operators of Gaussian filters. In this case the signal is determined uniquely, modulus the null space of the linear operator. The theorem can be extended to two-dimensional functions. These results are reminiscent of Logan’s theorem [ Bell Syst. Tech. J. 56, 487 ( 1977)]. They imply that extrema of derivatives at different scales are a complete representation of a signal. They are especially relevant for computational vision in the case of the Laplacian operator acting on image intensities, and they suggest rigorous foundations for the primal sketch.

© 1985 Optical Society of America

History
Original Manuscript: April 5, 1984
Manuscript Accepted: December 18, 1984
Published: May 1, 1985

Citation
A. L. Yuille and T. Poggio, "Fingerprints theorems for zero crossings," J. Opt. Soc. Am. A 2, 683-692 (1985)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-2-5-683

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