Guest Editorial: Discrete Tomography

What is discrete tomography? One may respond by giving an provided by the National Committee for Technological Develop-ment (OMFB), the Higher Education Development Programexample: It is the method used by a mouse to find the holes in

an Ementhaler cheese from its smell taken in different positions. (FEFA), and the Mayor of Szeged.Being professors, we formulate a more pedantic answer: Dis- Gabor T. Herman

crete tomography is an area of mathematics in which there is a University of Pennsylvania,domain—which may itself be discrete (such as the set of ordered Philadelphia, PA, USApairs of integers) or continuous (such as a Euclidean space) —

Attila Kuba,and an unknown function f whose range is known to be a givenJozsef Attila University,discrete set (usually of real numbers) . The problems of discreteSzeged, Hungarytomography have to do with determining f (perhaps only partially

or approximately) from weighted sums over subsets of its domainParticipants and Their Lectures at the Discretein the discrete case and from weighted integrals over subspacesTomography Workshop, August 1997, Szeged,of its domain in the continuous case. The essential aspect ofHungarydiscrete tomography is that knowing the discrete range of f mayFrom the USA:allow us to determine its values at points where without thisJolyon A. Browneknowledge, it could not be determined. Discrete tomography isAdvanced Research and Applications Corporationfull of mathematically fascinating questions (e.g., what shapes areSunnyvale, Californiauniquely determined by their X-rays?) and it has many interest-‘‘On some industrial applications of discrete tomography’’ing applications (e.g., electron microscopy and nondestructive

testing).Charles A. BoumanThe current interest in discrete tomography is indicated by thePurdue Universityfact that during a 2-year period, there have been three meetingsWest Lafayette, Indianadevoted to this field:‘‘Multiscale Bayesian reconstruction algorithms for discrete to-mography’’

1. DIMACS Mini-Symposium on Discrete Tomography,Rutgers University, USA, September 1995; Michael Chan

2. Seminar on Discrete Tomography: Algorithms and Com- Rockwell Internationalplexity, Dagstuhl, Germany, January 1997; and Thousand Oaks, California

3. Discrete Tomography Workshop, Szeged, Hungary, Au- ‘‘Image-modeling Gibbs priors for Bayesian image reconstruc-gust 1997. tion’’

Shi-Kuo ChangA list of the attendees of the workshop in Szeged, togetherUniversity of Pittsburghwith the titles of their lectures, follows this editorial. All attendeesPittsburgh, Pennsylvaniaof the workshop were invited to submit a contribution to this‘‘Active index for content-based medical image retrieval’’Special Issue. Some of the submitted papers were based on talks

presented at the workshop, while others were not. In fact, severalZang-Hee Chosubmitted papers were on new results obtained in consequenceUniversity of Californiaof interactions at the workshop. In any case, all submitted papersIrvine, Californiawere subjected to the refereeing process customary for archival‘‘Discrete version of computed tomography based on YING-journals. The resulting Special Issue contains up-to-date researchYANG theory’’contributions to various aspects of discrete tomography: theory,

algorithms, and applications.Gabor T. HermanIt was an honor to edit this Special Issue on Discrete Tomogra-University of Pennsylvaniaphy of the International Journal of Imaging Systems and Technol-Philadelphia, Pennsylvaniaogy. We thank the editors (Z. H. Cho and L. A. Shepp) for the‘‘Bayesian discrete tomography based on various tessellations ofinvitation, the contributors for their papers, and the referees forspace’’a job well done. We would also like to take this opportunity to

thank the sponsors of the Discrete Tomography Workshop inSzeged. Major funding was provided by the National Science Johannes H. B. Kemperman

Rutgers UniversityFoundation (Grant INT960213) and the Hungarian Academy ofSciences (Grant MTA-NSF/95/1997). Additional funding was New Brunswick, New Jersey

q 1998 John Wiley & Sons, Inc. CCC 0899–9457/98/020067-02

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‘‘Minimal systems of linear maps such that each subset of size ‘‘Template methods on projection pictures’’q at most is uniquely determined by its associated images’’

Peter HajnalJATE, Bolyai IntezetYung KongSzegedQueens College‘‘On a generalized graph theoretical notion of kernel’’Flushing, New York

‘‘On which grids can tomographic equivalence of binary picturesJanos Kincsesbe characterized in terms of elementary switching operations?’’JATE, Bolyai IntezetSzegedSamuel Matej‘‘Pattern recognition from shadow pictures’’University of Pennsylvania

Philadelphia, PennsylvaniaAttila Kuba‘‘Binary tomography on a hexagonal grid using Gibbs priors’’JATE, Informatikai IntezetSzegedSarah Patch‘‘Reconstruction of two-value matrices from their two projec-General Electric Corporate Research and Developmenttions’’Niskayuna, New York

‘‘Recursive recovery of Markov transition probabilities fromFrom Other Countries:boundary value data’’Richard AnsteeUniversity of British ColumbiaPablo M. SalzbergVancouver, CanadaUniversity of Puerto Rico‘‘Matrices with given row and column sums’’Rio Piedras, Puerto Rico

‘‘Two spatial limited angle models for binary tomography onJose-Maria Carazolattices, with applications to crystallography’’Autonoma UniversityMadrid, SpainKen Sauer‘‘Studying the three-dimensional structure of biological macro-University of Notre Damemolecules from transmission electron microscopy images’’Notre Dame, Indiana

‘‘Statistical methods in discrete tomography for radiographicYair Censor

nondestructive evaluation’’University of HaifaHaifa, Israel

Larry Shepp ‘‘Algorithms for binary matrix reconstruction motivated by ma-Rutgers University trix-balancing algorithms and their roots in algorithms which useNew Brunswick, New Jersey Bregman projections’’‘‘Linear programming in tomography, probability, and finance’’

Alberto Del LungoYehuda Vardi University of FirenzeRutgers University Firenze, ItalyNew Brunswick, New Jersey ‘‘Medians of polyominoes: A property for the reconstruction’’‘‘The discrete Radon transform and its approximate inversion viathe EM algorithm’’ Francoise Peyrin

INSAAndrew E. Yagle Lyon, FranceUniversity of Michigan ‘‘Two applications of truly 3D binary image reconstruction fromAnn Arbor, Michigan 2D projections’’‘‘An explicit closed-form solution to the limited-angle discretetomography problem for finite-support objects’’ Peter Schwander

Institute for Semiconductor PhysicsFrom Hungary: Frankfurt, GermanyKaroly Boroczky ‘‘Application of discrete tomography to electron microscopy ofMTA Matematikai Kutato Intezet crystals’’Budapest‘‘Random projections of regular polytopes’’ Aljosa Volcic

University of TriesteTrieste, ItalyAttila Fazekas

KLTE, Matematikai es Informatikai Intezet ‘‘Determination of three dimensional convex bodies from areasof plane sections’’Debrecen

68 Vol. 9, 67–68 (1998)

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