Optimal reconstruction kernels in medical imaging

In this paper we present techniques for deriving inversion algorithms in medical imaging. To this end we present a few imaging technologies and their mathematical models. They essentially consist of integral operators. The reconstruction is then recognized as the solution of an inverse problem. General strategies, the socalled approximate inverse, for deriving a solution are adapted. Results from real data are presented.

[1]  A. Louis.  Inverse und schlecht gestellte Probleme , 1989 .

[2]  E. T. Quinto.  Tomographic reconstructions from incomplete data-numerical inversion of the exterior Radon transform , 1988 .

[3]  E. T. Quinto,et al.  An Introduction to X-ray tomography and Radon Transforms , 2006 .

[4]  Pierre Grangeat.  Analyse d'un systeme d'imagerie 3d par reconstruction a partir de radiographies x en geometrie conique , 1987 .

[5]  M. E. Davison,et al.  Tomographic reconstruction with arbitrary directions , 1981 .

[6]  H. Tuy.  AN INVERSION FORMULA FOR CONE-BEAM RECONSTRUCTION* , 1983 .

[7]  Alfred K. Louis,et al.  A unified approach to regularization methods for linear ill-posed problems , 1999 .

[8]  Hengyong Yu,et al.  A unified framework for exact cone-beam reconstruction formulas. , 2005, Medical physics.

[9]  Alfred K. Louis.  Approximate inverse for linear and some nonlinear problems , 1995 .

[10]  Peter Maass,et al.  Contour reconstruction in 3-D X-ray CT , 1993, IEEE Trans. Medical Imaging.

[11]  Rolf Clackdoyle,et al.  A cone-beam reconstruction algorithm using shift-variant filtering and cone-beam backprojection , 1994, IEEE Trans. Medical Imaging.

[12]  D. Slepian.  Prolate spheroidal wave functions, fourier analysis, and uncertainty — V: the discrete case , 1978, The Bell System Technical Journal.

[13]  Andreas Rieder.  On filter design principles in 2D computerized tomography , 2001 .

[14]  A. K. Louis,et al.  Approximate inversion of the 3 D radon transform , 1983 .

[15]  Thomas Schuster.  The 3D Doppler transform: elementary properties and computation of reconstruction kernels , 2000 .

[16]  A. Cormack.  Representation of a Function by Its Line Integrals, with Some Radiological Applications , 1963 .

[17]  Peter Maass,et al.  The x-ray transform: singular value decomposition and resolution , 1987 .

[18]  L. Feldkamp,et al.  Practical cone-beam algorithm , 1984 .

[19]  A K Louis,et al.  A novel filter design technique in 2D computerized tomography , 1996 .

[20]  Andreas Rieder,et al.  Incomplete data problems in X-ray computerized tomography , 1989 .

[21]  Andreas Rieder,et al.  Wavelets: Theory and Applications , 1997 .

[22]  P. Grangeat.  Mathematical framework of cone beam 3D reconstruction via the first derivative of the radon transform , 1991 .

[23]  R. Marr,et al.  On Two Approaches to 3D Reconstruction in NMR Zeugmatography , 1981 .

[24]  Eric Todd Quinto,et al.  Mathematical Methods in Tomography , 2006 .

[25]  A. Katsevich.  Analysis of an exact inversion algorithm for spiral cone-beam CT. , 2002, Physics in medicine and biology.

[26]  Peter Maaß,et al.  Contour reconstructions for 3D X-Ray CT-Images , 1993 .

[27]  F. Natterer.  The Mathematics of Computerized Tomography , 1986 .

[28]  J. Zweier,et al.  Mapping the spin-density and lineshape distribution of free radicals using 4D spectral-spatial EPR imaging. , 1995, Journal of magnetic resonance. Series B.

[29]  H. Kuo.  Gaussian Measures in Banach Spaces , 1975 .

[30]  Peter Maass,et al.  A mollifier method for linear operator equations of the first kind , 1990 .

[31]  Eric Todd Quinto,et al.  Singularities of the X-ray transform and limited data tomography , 1993 .

[32]  Alfred K. Louis.  Orthogonal Function Series Expansions and the Null Space of the Radon Transform , 1984 .

[33]  Alfred K. Louis,et al.  Nonuniqueness in inverse radon problems: The frequency distribution of the ghosts , 1984 .

[34]  A. Louis.  Filter design in three-dimensional cone beam tomography: circular scanning geometry , 2003 .

[35]  A. Louis,et al.  Picture reconstruction from projections in restricted range , 1980 .

[36]  Frank Natterer,et al.  Mathematical methods in image reconstruction , 2001, SIAM monographs on mathematical modeling and computation.