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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • 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)

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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

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

A. L. Yuille and T. Poggio, "Fingerprints theorems for zero crossings," J. Opt. Soc. Am. A 2, 683-692 (1985)

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  1. D. Marr, Vision, A Computational Investigation into the Human Representation and Processing of Visual Information (Freeman, San Francisco, Calif., 1982).
  2. B. F. Logan, “Information in the zero crossings of bandpass signals,” Bell Syst. Tech. J. 56, 487–510 (1977).
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  4. D. Marr, T. Poggio, S. Ullman, “Bandpass channels, zero-crossings and early visual information processing,”J. Opt. Soc. Am. 70, 868–870 (1979). [CrossRef]
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  13. J. Babaud, A. Witkin, R. Duda, “Uniqueness of the Gaussian kernel for scale-space filtering,” (Fairchild Artificial Intelligence Laboratory, Palo Alto, Calif., 1983).
  14. A. L. Yuille, T. Poggio, “Scaling theorems for zero-crossings,” Artificial Intelligence Memo 722 (Massachusetts Institute of Technology, Cambridge, Mass., June1983).
  15. A. L. Yuille, T. Poggio, “Fingerprints theorems for zero-crossings,” Artificial Intelligence Memo 730 (Massachusetts Institute of Technology, Cambridge, Mass., October1983).
  16. J. Koenderink, University of Utrecht, Utrecht, The Netherlands (personal communication, 1984).
  17. H. K. Nishihara, “Intensity, visible-surface, and volumetric representations,” Art. Intell. 17, 265–284 (1981). [CrossRef]
  18. Clearly, the scale-map fingerprint cannot always be a more compact description of the signal than the signal itself, unless the signal is redundant in precisely the way that the fingerprint representation can exploit. We expect this to be the case for images, if an appropriate differential operator is used, because images are not a purely random array of numbers. Usually images consist of rather homogeneous regions that do not change much over significant scale intervals.
  19. H. Asada, M. Brady, “The curvature primal sketch,” Artificial Intelligence Memo 758 (Massachusetts Institute of Technology, Cambridge, Mass., 1984).
  20. A. L. Yuille, T. Poggio, “Fingerprints and the psychophysics of gratings,” Artificial Intelligence Memo 751 (Massachusetts Institute of Technology, Cambridge, Mass., 1984).
  21. A. L. Yuille, T. Poggio, “Fingerprints theorems,” presented at the Conference on Artificial Intelligence, Austin, Texas, 1984.
  22. S. W. Zucker, R. A. Hummel, “Receptive fields and the reconstruction of visual information,” (Courant Institute, New York, N.Y., 1983).
  23. This argument cannot be applied when all zero-crossing contours are vertical straight lines: It is impossible to reconstruct the signal.20 In this case the matrices in Eqs. (3.3.1) and (3.3.2) take simple forms.
  24. A. Albert, Regression and the Moore–Penrose Pseudoinverse (Academic, New York, 1982).
  25. W. E. L. Grimson, “Surface consistency constants in vision,” Comput. Vision Graph. Inf. Process, 24, 28–51 (1983). [CrossRef]
  26. A. L. Yuille, “Zero-crossings and lines of curvature,” Artificial Intelligence Memo 718 (Massachusetts Institute of Technology, Cambridge, Mass., 1983).
  27. R. A. Hummel, Courant Institute, New York, N.Y. (personal communication).
  28. A. L. Yuille, T. Poggio, “Fingerprints and their slope,” Artificial Intelligence Memo 752 (Massachusetts Institute of Technology, Cambridge, Mass., 1984).
  29. D. Marr, “Early processing of visual information,” Phil. Trans. R. Soc. London Ser. B 275, 483–524 (1976). [CrossRef]

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