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Georges Matheron

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Georges Matheron (1930 - 2000) is regarded by some as the father of spatial statistics and/or geostatistics. He is also a co-founder (together with Jean Serra) of mathematical morphology. In 1968 he created the Centre de Géostatistique et de Morphologie Mathématique at the Paris School of Mines in Fontainebleau. He is known for his contributions on Kriging and mathematical morphology. His seminal work is posted for study and review to the Online Library of the Centre de Géostatistique, Fontainebleau, France.

Geostatistics

Matheron’s Formule des Minerais Connexes became his Note Statistique No 1. In this paper of November 25, 1954, Matheron derived the degree of associative dependence between lead and silver grades of core samples. In his Rectificatif of January 13, 1955, he revised the arithmetic mean lead and silver grades because his core samples varied in length. He did derive the length-weighted average lead and silver grades but failed to derive the variances of those central values. Neither did he derive the degree of associative dependence between metal grades of ordered core samples as a measure for spatial dependence in his sample space. He did not disclose his primary data set and worked mostly with symbols rather than measured values. Matheron's Interprétations des corrélations entre variables aléatoires lognormales of November 29, 1954, was marked Note statistisque No 2. In this paper, Matheron explored lognormal variables and set the stage for statistics by symbols. Primary data would have allowed him to assess whether or not lead and silver grades departed from the lognormal distribution, or displayed spatial dependence along his borehole.

Matheron coined the eponym krigeage (Kriging) for the first time in his 1960 Krigeage d’un Panneau Rectangulaire par sa Périphérie. In this Note géostatistique No 28, Matheron derived k*, his estimateur and a precursor to the kriged estimate or kriged estimator. In mathematical statistics, Matheron’s k* is the length-weighted average grade of a single panneau in his set. What Matheron failed to derive was var(k*), the variance of his estimateur. Matheron presented his Stationary Random Function at the first colloquium on geostatistics in the USA. He called on Brownian motion to conjecture the continuity of his Riemann integral. He did not explain what Brownian motion and ore deposits have in common. Matheron, unlike John von Neumann in 1941 and Anders Hald in 1952, never worked with Riemann sums. It was not Professor Dr Georges Matheron but Dr Frederik P Agterberg who derived the distance-weighted average of a set of measured values determined in samples selected at positions with different coordinates in a sample space. What Agterberg did not do was derive the variance of this function.

Matheron did indeed derive length-weighted average grades of core samples and ore blocks but did not derive the variance of these functions. In time, the length-weighted average grade for Matheron's three-dimensional block was replaced with the distance-weighted average grade for Agterberg's zero-dimensional point. Both central values turned into honorific kriged estimates or estimators. An infinite set of Agterberg's zero dimensional points fits within any ore block, along any borehole, or inside any sampling unit or sample space. Matheron's block grades and Agterberg's point grades are unique because both are variance-deprived functions.

Mathematical morphology

In 1963, Matheron took a break from geostatistics, and investigated new areas beyond the mining world[1]. Matheron posted another 52 Notes Géostatistiques before drifting into mathematical morphology in 1969 when he looked at the morphology of Brownian motion and observed likeness with ore deposits. At that time, he was supervising the PhD thesis of Jean Serra, dedicated to quantifying the ore properties of the iron deposit of Lorraine. Serra came up with the idea of using structuring elements for the analysis, which lead to the concept of hit-or-miss transform. The theoretical analysis of this transform lead Matheron to derive and investigate the concepts of erosion, dilation, opening and closing, which became known later as the basic morphological operators. He also developed a tool for granulometry, i.e., the computation of a "size distribution", where he mathematically characterizes the concept of size. In December 1964, Matheron and Serra, together with [Philippe Formery], named this approach mathematical morphology. It has since evolved into a theory and method that is applied in a variety of image processing problems and tasks, and is researched worldwide[2] (main article: Mathematical morphology). Matheron continued to contribute to mathematical morphology during the years, his best-known contribution being the morphological filtering theory, which he developed with Serra in the 1980s.

The Centre de Géostatistique et de Morphologie Mathématique

In 1968, the Paris School of Mines created the Centre de Morphologie Mathématique, located in Fontainebleau, France, and named Matheron its first director. In 1979, the center was renamed Centre de Géostatistique et de Morphologie Mathématique, and, in 1986, the latter was split into two separate centers: Centre de Géostatistique, directed by Matheron, and Centre de Morphologie Mathématique, directed by Serra.

Notes

  1. ^ See the below link to History of Mathematical Morphology by Matheron and Serra
  2. ^ See, e.g., (Serra and Soille (Eds.) 1994)

References

  • Mathematical Morphology and Its Applications to Image Processing, J. Serra and P. Soille (Eds.), proceedings of the 2nd international symposium on mathematical morphology (ISMM'93), ISBN 0-7923-3093-5 (1994)
  • Image Analysis and Mathematical Morphology by Jean Serra, ISBN 0126372403 (1982)
  • Image Analysis and Mathematical Morphology, Volume 2: Theoretical Advances by Jean Serra, ISBN 0-12-637241-1 (1988)
  • An Introduction to Morphological Image Processing by Edward R. Dougherty, ISBN 0-8194-0845-X (1992)
  • Morphological Image Analysis; Principles and Applications by Pierre Soille, ISBN 3540-65671-5 (1999)

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