On Numerical Analysis of View-Dependent Derivatives in Computed Tomography

We explore the numerical implementation of a view dependent derivative that occurs in $$\pi $$π-line reconstruction formulas for two- and three-dimensional computed tomography. Focusing on two-dimensional fan-beam tomography, we provide an error analysis and a common framework for the comparison of several schemes used to discretize this derivative. The leading error terms for each scheme are determined. The results demonstrate some advantages and drawbacks of the methods that are confirmed by numerical experiments.

[1]  K. Taguchi,et al.  Practical hybrid convolution algorithm for helical CT reconstruction , 2004, IEEE Transactions on Nuclear Science.

[2]  F. Natterer,et al.  Sampling in Fan Beam Tomography , 1993, SIAM J. Appl. Math..

[3]  Eric Todd Quinto,et al.  The Radon Transform, Inverse Problems, and Tomography (Proceedings of Symposia in Applied Mathematics) , 2006 .

[4]  Xiaochuan Pan,et al.  Exact image reconstruction on PI-lines from minimum data in helical cone-beam CT. , 2004, Physics in medicine and biology.

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

[6]  Steven H. Izen.  Sampling in Flat Detector Fan Beam Tomography , 2012, SIAM J. Appl. Math..

[7]  M. Glas,et al.  Principles of Computerized Tomographic Imaging , 2000 .

[8]  Alexander Katsevich,et al.  An improved exact filtered backprojection algorithm for spiral computed tomography , 2004, Adv. Appl. Math..

[9]  G. Herman,et al.  Fast Image Reconstruction Based on a Radon Inversion Formula Appropriate for Rapidly Collected Data , 1977 .

[10]  F. Noo,et al.  A two-step Hilbert transform method for 2D image reconstruction. , 2004, Physics in medicine and biology.

[11]  Frédéric Noo,et al.  Exact helical reconstruction using native cone-beam geometries. , 2003, Physics in medicine and biology.

[12]  Willi A. Kalender,et al.  Computed tomography : fundamentals, system technology, image quality, applications , 2000 .

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

[14]  Hiroyuki Kudo,et al.  Image reconstruction from fan-beam projections on less than a short scan , 2002, Physics in medicine and biology.

[15]  A note on computing the derivative at a constant direction. , 2011, Physics in medicine and biology.

[16]  Adel Faridani,et al.  Numerical and theoretical explorations in helical and fan-beam tomography , 2008 .

[17]  Adel Faridani,et al.  Regions of Backprojection and Comet Tail Artifacts for Pi-Line Reconstruction Formulas in Tomography , 2012, SIAM J. Imaging Sci..

[18]  Günter Lauritsch,et al.  A new scheme for view-dependent data differentiation in fan-beam and cone-beam computed tomography. , 2007, Physics in medicine and biology.

[19]  Hiroyuki Kudo,et al.  Truncated Hilbert transform and image reconstruction from limited tomographic data , 2006 .

[20]  Alexander Katsevich,et al.  Theoretically Exact Filtered Backprojection-Type Inversion Algorithm for Spiral CT , 2002, SIAM J. Appl. Math..

[21]  Xiaochuan Pan,et al.  An extended data function and its generalized backprojection for image reconstruction in helical cone-beam CT. , 2004, Physics in medicine and biology.

[22]  Xiaochuan Pan,et al.  Image reconstruction in regions-of-interest from truncated projections in a reduced fan-beam scan , 2005, Physics in medicine and biology.

[23]  Andreas Rieder,et al.  The Semidiscrete Filtered Backprojection Algorithm Is Optimal for Tomographic Inversion , 2003, SIAM J. Numer. Anal..