Adaptive wavelet-Galerkin methods for limited angle tomography

This paper studied incomplete data problems of computed tomography that frequently occur in medical or industrial imaging, for example, when the high-density region of objects can only be penetrated by X-rays at a limited angular range. When projection data are available only in an angular range, the incomplete data problem can be attributed to the limited angle problem, which is a severely ill-posed inverse problem. In this paper, a numerical method for the treatment of inverse problems based on an adaptive wavelet-Galerkin method is introduced and investigated. The paper focuses especially on how to avoid inverse crimes in numerical simulations. The method used here combines numerical simplicity and characteristics of adapting to the unknown smoothness of a reconstructed image, which leads to significant reduction in the computational cost. The reconstruction strategy has a comparable performance with a significant reduction in computational time.

[1]  F. Grünbaum.  A study of fourier space methods for “limited angle” image reconstruction * , 1980 .

[2]  Emmanuel J. Candès,et al.  Signal recovery from random projections , 2005, IS&T/SPIE Electronic Imaging.

[3]  M. Persson,et al.  Total variation norm for three-dimensional iterative reconstruction in limited view angle tomography , 2001, Physics in medicine and biology.

[4]  R T Smith,et al.  Reconstruction of tomographic images from sparse data sets by a new finite element maximum entropy approach. , 1991, Applied optics.

[5]  Albert Macovski,et al.  Iterative Reconstruction-Reprojection: An Algorithm for Limited Data Cardiac-Computed Tomography , 1982, IEEE Transactions on Biomedical Engineering.

[6]  A K Louis,et al.  Incomplete data problems in x-ray computerized tomography , 1986 .

[7]  Rafal Zdunek,et al.  Kaczmarz extended algorithm for tomographic image reconstruction from limited-data , 2004, Math. Comput. Simul..

[8]  C. Byrne.  Block-iterative interior point optimization methods for image reconstruction from limited data , 2000 .

[9]  Emmanuel J. Candès,et al.  Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information , 2004, IEEE Transactions on Information Theory.

[10]  Yoram Bresler,et al.  Globally convergent edge-preserving regularized reconstruction: an application to limited-angle tomography , 1998, IEEE Trans. Image Process..

[11]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[12]  Henry Stark,et al.  Image recovery: Theory and application , 1987 .

[13]  Ken D. Sauer,et al.  Bayesian estimation of 3-D objects from few radiographs , 1994 .

[14]  M. E. Davison,et al.  The Ill-Conditioned Nature of the Limited Angle Tomography Problem , 1983 .

[15]  E. T. Quinto.  Singular value decompositions and inversion methods for the exterior Radon transform and a spherical transform , 1983 .

[16]  Hiroyuki Kudo,et al.  Sinogram recovery with the method of convex projections for limited-data reconstruction in computed tomography , 1992 .

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

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

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

[20]  J. Mixter.  Fast , 2012 .

[21]  Wojciech Zbijewski,et al.  Comparison of methods for suppressing edge and aliasing artefacts in iterative x-ray CT reconstruction , 2006, Physics in medicine and biology.

[22]  Victor Perez-Mendez,et al.  Versatility of the CFR algorithm for limited angle reconstruction , 1990 .

[23]  E Somersalo,et al.  Statistical inversion for medical x-ray tomography with few radiographs: I. General theory. , 2003, Physics in medicine and biology.

[24]  B. Sewell,et al.  The application of the maximum entropy method to electron microscopic tomography. , 1989, Ultramicroscopy.

[25]  G Muehllehner,et al.  Constrained Fourier space method for compensation of missing data in emission computed tomography. , 1988, IEEE transactions on medical imaging.

[26]  A. H. Andersen.  Algebraic reconstruction in CT from limited views. , 1989, IEEE transactions on medical imaging.

[27]  Beilei Wang,et al.  Fast, accurate and memory-saving ART based tomosynthesis , 2003, 2003 IEEE 29th Annual Proceedings of Bioengineering Conference.

[28]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[29]  M I Reis,et al.  Maximum entropy algorithms for image reconstruction from projections , 1992 .

[30]  R M Lewitt,et al.  Multidimensional digital image representations using generalized Kaiser-Bessel window functions. , 1990, Journal of the Optical Society of America. A, Optics and image science.

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

[32]  M W Vannier,et al.  Local computed tomography via iterative deblurring. , 1996, Scanning.

[33]  E Somersalo,et al.  Statistical inversion for medical x-ray tomography with few radiographs: II. Application to dental radiology. , 2003, Physics in medicine and biology.

[34]  Matti Lassas,et al.  Wavelet-based reconstruction for limited-angle X-ray tomography , 2006, IEEE Transactions on Medical Imaging.

[35]  W. R. Brody,et al.  Image Reconstruction From Limited Data , 1982, Other Conferences.

[36]  Jerry L. Prince,et al.  Constrained sinogram restoration for limited-angle tomography , 1990 .

[37]  Xiaochuan Pan,et al.  Effect of the data constraint on few-view, fan-beam CT image reconstruction by TV minimization , 2006, 2006 IEEE Nuclear Science Symposium Conference Record.

[38]  G. W. Wecksung,et al.  Bayesian approach to limited-angle reconstruction in computed tomography , 1983 .

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

[40]  Victor Perez-Mendez,et al.  Limited-Angle Three-Dimensional Reconstructions Using Fourier Transform Iterations And Radon Transform Iterations , 1981 .

[41]  R. Kress,et al.  Inverse Acoustic and Electromagnetic Scattering Theory , 1992 .