The Annals of Statistics

Maximal meaningful events and applications to image analysis

Agnès Desolneux, Lionel Moisan, and Jean-Michel Morel

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Abstract

We discuss the mathematical properties of a recently introduced method for computing geometric structures in a digital image without any a priori information. This method is based on a basic principle of perception which we call the Helmholtz principle. According to this principle, an observed geometric structure is perceptually "meaningful" if the expectation of its number of occurrences (in other words, its number of false alarms, NF) is very small in a random image. It is "maximal meaningful" if its NF is minimal among the meaningful structures of the same kind which it contains or is contained in. This definition meets the gestalt theory requirement that parts of a whole are not perceived. We explain by large-deviation estimates why this definition leads to an a priori knowledge-free method, compatible with phenomenology. We state a principle according to which maximal structures do not meet. We prove this principle in the large-deviations framework in the case of alignments in a digital image. We show why these results make maximal meaningful structures computable and display several applications.

Article information

Source
Ann. Statist., Volume 31, Number 6 (2003), 1822-1851.

Dates
First available in Project Euclid: 16 January 2004

Permanent link to this document
https://projecteuclid.org/euclid.aos/1074290328

Digital Object Identifier
doi:10.1214/aos/1074290328

Mathematical Reviews number (MathSciNet)
MR2036391

Zentralblatt MATH identifier
1046.62104

Subjects
Primary: 33B20: Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals) 62H15: Hypothesis testing 62H35: Image analysis 62M40: Random fields; image analysis 68U10: Image processing 68T45: Machine vision and scene understanding 91E30: Psychophysics and psychophysiology; perception

Keywords
Image analysis perception alignment tail of the binomial distribution rare events large deviations

Citation

Desolneux, Agnès; Moisan, Lionel; Morel, Jean-Michel. Maximal meaningful events and applications to image analysis. Ann. Statist. 31 (2003), no. 6, 1822--1851. doi:10.1214/aos/1074290328. https://projecteuclid.org/euclid.aos/1074290328


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