Bücher von Gabor T. Herman

Goals of the Book Overthelast thirty yearsthere has been arevolutionindiagnostic radiology as a result oftheemergenceofcomputerized tomography (CT), which is the process of obtaining the density distribution within the human body from multiple x-ray projections. Since an enormous variety of possible density values may occur in the body, a large number of projections are necessary to ensure the accurate reconstruction oftheir distribution. There are other situations in which we desire to reconstruct an object from its projections, but in which we know that the object to be recon- structed has only a small number of possible values. For example, a large fraction of objects scanned in industrial CT (for the purpose of nonde- structive testing or reverse engineering) are made of a single material and so the ideal reconstruction should contain only two values: zero for air and the value associated with the material composing the object. Similar as- sumptions may even be made for some specific medical applications; for example, in angiography ofthe heart chambers the value is either zero (in- dicating the absence of dye) or the value associated with the dye in the chamber. Another example arises in the electron microscopy of biological macromolecules, where we may assume that the object to be reconstructed is composed of ice, protein, and RNA. One can also apply electron mi- croscopy to determine the presenceor absence ofatoms in crystallinestruc- tures, which is again a two-valued situation.

The conference was devoted to the discussion of present andfuture techniques in medical imaging, including 3D x-ray CT,ultrasound and diffraction tomography, and biomagnetic ima-ging. The mathematical models, their theoretical aspects andthe development of algorithms were treated. The proceedingscontains surveys on reconstruction in inverse obstacle scat-tering, inversion in 3D, and constrained least squares pro-blems.Research papers include besides the mentioned imagingtechniques presentations on image reconstruction in Hilbertspaces, singular value decompositions, 3D cone beam recon-struction, diffuse tomography, regularization of ill-posedproblems, evaluation reconstruction algorithms and applica-tions in non-medical fields. Contents: Theoretical Aspects:J.Boman: Helgason' s support theorem for Radon transforms-anewproof and a generalization -P.Maass: Singular value de-compositions for Radon transforms- W.R.Madych: Image recon-struction in Hilbert space -R.G.Mukhometov: A problem of in-tegral geometry for a family of rays with multiple reflec-tions -V.P.Palamodov: Inversion formulas for the three-di-mensional ray transform - Medical Imaging Techniques:V.Friedrich: Backscattered Photons - are they useful for asurface - near tomography - P.Grangeat: Mathematical frame-work of cone beam 3D reconstruction via the first derivativeof the Radon transform -P.Grassin,B.Duchene,W.Tabbara: Dif-fraction tomography: some applications and extension to 3Dultrasound imaging -F.A.Gr}nbaum: Diffuse tomography: a re-fined model -R.Kress,A.Zinn: Three dimensional reconstruc-tions in inverse obstacle scattering -A.K.Louis: Mathemati-cal questions of a biomagnetic imaging problem - InverseProblems and Optimization: Y.Censor: On variable blockalgebraic reconstruction techniques -P.P.Eggermont: OnVolterra-Lotka differential equations and multiplicativealgorithms for monotone complementary problems

This revised and updated edition presents the computational and mathematical procedures underlying data collection, image reconstruction, and image display in computerized tomography. New topics include fast calculation of a ray sum in a digitized image and the task-oriented comparison of reconstruction algorithm performance.