History

As the above summary of the theory indicates, problems and results that can now be considered as part of geometric tomography have been around for a century or more. However, Richard J. Gardner coined the term at the 1990 Oberwolfach meeting on tomography and made the first systematic attempt to bring its various ingredients together in the first edition of his book with that title in 1995. The scope of the subject in this book owes a great deal to input from Erwin Lutwak in the early 1990's.

It seems that the term geometric tomography was used independently at least twice shortly after 1990. The usage in [32, Chapter 7] has essentially no overlap, but that of Thirion [38] is quite close in spirit; he defines geometric tomography to be the process of reconstructing the external or internal boundaries of objects from their X-rays.

Even earlier, in the 1980's, a program of related research was underway, directed by Alan Willsky, an electrical engineer at MIT. The PhD thesis [33] of Rossi was the first of several supervised by Willsky that develop and implement algorithms specially suited for reconstructing planar convex (and more general) sets. None of these algorithms solve the theoretical problem of obtaining arbitrarily accurate images of convex sets from X-rays in a fixed finite set of directions. However, they do produce images from a limited set of X-rays that are often sufficient for detecting some particular feature of the object, such as the approximate position or orientation of an ellipse or a rectangle. Moreover, they can work even when the data is inaccurate and when noise is present. The general idea is to take advantage of a priori information that the unknown object belongs to a restricted class of homogeneous objects, an idea completely compatible with geometric tomography. As motivation, Rossi gives references for applications to geophysics, oceanography, meteorology, nondestructive testing (locating cracks in nuclear reactor cooling pipes, etc.), and medicine (nearly homogeneous regions such as kidneys and airspaces between organs).

A second edition of the book Geometric Tomography appeared in 2006, with 60 extra pages of text and about 300 additional references. This and other research of Richard Gardner on Geometric Tomography has been supported by the National Science Foundation under grants number DMS-9201508, DMS-9501289, DMS-9802388, DMS-0203527, DMS-0603307, DMS-1103612, and DMS-1402929. Any opinions, findings, and conclusions or recommendations expressed in this material or on this web site are those of the author and do not necessarily reflect the views of the National Science Foundation.

In October, 2005, the first international workshop solely devoted to Geometric Tomography took place in Alicante, Spain, and since then there have been several conferences focusing on Geometric Tomography held in Canada, Denmark, and Italy.