The Mathematics of Computerized Tomography (Classics in Applied Mathematics, Vol. 32)

Abstract

Sixty-two years passed between the publication of Radon's inversion formula inthe Berichte Sächsische Akademie der Wissenschaften in Leipzig in1917 and the 1979 Nobel prize in medicine awarded to Allen M Cormack andGodfrey N Hounsfield for their pioneering contributions to the development ofcomputerized tomography (CT). The field of computerized tomography has sincethen witnessed progress and development which can be encapsulated in no otherwords than scientific explosion. Transmission, emission, ultrasound, optical,electrical impedance and magnetic resonance are all CT imaging modalitiesbased on different physical models. But not only diagnostic radiology has beenrevolutionized by CT, many other scientific and technological areas fromnon-destructive material testing to seismic imaging in geophysics and fromelectron microscopy for biological studies to radiation therapy treatmentplanning have all been transformed and seen new paths being broken by theintroduction of the principles of CT.The mathematical formulation of CT commonly leads to an inverseproblem putting the underlying physical phenomenon and its model at the mercyof mathematics and mathematical techniques. Inadequate modelling due toinsufficient understanding of the physics or due to practical limitationswhich result in incomplete data collection make the mathematical inversiondifficult, and sometimes impossible. Two fundamentally different approaches areavailable. One way is to use `continuous' modelling in which quantities arerepresented by functions and their relations by operators between functionspaces. In this approach the inversion problem at hand is solved and then thesolution formula(e) are discretized for computational implementation. Anotherroute is to first fully discretize the problem at the modelling stage andrepresent quantities by finite-dimensional vectors and the relations betweenthem by functions over the vector space. Then a solution of the fullydiscretized inverse problem is reached which does not need furtherdiscretization of formulae for the computer implementation.Natterer's book handles the mathematics of CT in the `continuous' approach.In the preface to the original 1986 book the author wrote: `In this book Ihave made an attempt to collect some mathematics which is of possible interestboth to the research mathematician who wants to understand the theory andalgorithms of CT and to the practitioner who wants to apply CT in his specialfield of interest'. This attempt, one must say, was indeed very successful.In spite of the further tremendous progress that occurred since the originalbook appeared, the book is still a treasure for anyone joining or alreadyworking in the field. This proves that the choice of topics and theorganization of material were very well done and are still useful andrelevant. After a brief introduction (Chapter I), the book treats the followingtopics: the Radon transform and related transforms (Chapter II), sampling andresolution (Chapter III), ill-posedness and accuracy (Chapter IV),reconstruction algorithms (Chapter V), incomplete data (Chapter VI) and,finally, an appendix of mathematical tolls (Chapter VII). Except for theaddition of a table of errata, the book is an unabridged republication of theoriginal book. Therefore, it is the reader's responsibility to bridge theknowledge and literature gaps from 1986 until today with the aid of othersources. Nonetheless this book is an excellent starting point for such ajourney into the mathematics of computerized tomography, together with someother books from that period that withstood the `teeth of time' such asthose of Herman (New York: Academic Press 1980) and of Kak and Slaney (Piscataway, NJ: IEEEPress 1988) (see also: Classics in Applied Mathematics, Vol. 33 (Philadelphia, PA: SIAM)).Yet another interesting facet of the development of the field of CT was, andstill is, the continuous stream of mathematical problems it generates. Somemathematical problems aim at reaching practical solutions for either a`continuous' model or a fully discretized model of the ever newly emergingreal-world CT problems. Others are more theoretical extensions to integralgeometry, such as reconstruction from integrals over arbitrary manifolds, andto a variety of other fields in pure mathematics (see, e.g., Grinberg E andQuinto E T (ed) 1990 Integral Geometry and Tomography (Providence, RI: AmericanMathematical Society)). Natterer's book, althoughadmittedly not handling such extensions, is an indispensable tool for anyoneplanning to direct his efforts in those directions.