Quantifying Admissible Undersampling for Sparsity-Exploiting Iterative Image Reconstruction in X-Ray CT

Iterative image reconstruction with sparsity-exploiting methods, such as total variation (TV) minimization, investigated in compressive sensing claim potentially large reductions in sampling requirements. Quantifying this claim for computed tomography (CT) is nontrivial, because both full sampling in the discrete-to-discrete imaging model and the reduction in sampling admitted by sparsity-exploiting methods are ill-defined. The present article proposes definitions of full sampling by introducing four sufficient-sampling conditions (SSCs). The SSCs are based on the condition number of the system matrix of a linear imaging model and address invertibility and stability. In the example application of breast CT, the SSCs are used as reference points of full sampling for quantifying the undersampling admitted by reconstruction through TV-minimization. In numerical simulations, factors affecting admissible undersampling are studied. Differences between few-view and few-detector bin reconstruction as well as a relation between object sparsity and admitted undersampling are quantified.

[1]  Vijay K. Madisetti,et al.  The fast discrete Radon transform , 1992, [Proceedings] ICASSP-92: 1992 IEEE International Conference on Acoustics, Speech, and Signal Processing.

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

[3]  Robert M. Lewitt,et al.  Practical considerations for 3-D image reconstruction using spherically symmetric volume elements , 1996, IEEE Trans. Medical Imaging.

[4]  H H Barrett,et al.  Cone-beam tomography with discrete data sets. , 1994, Physics in medicine and biology.

[5]  B. De Man,et al.  Distance-driven projection and backprojection in three dimensions. , 2004, Physics in medicine and biology.

[6]  Hiroyuki Kudo,et al.  An accurate iterative reconstruction algorithm for sparse objects: application to 3D blood vessel reconstruction from a limited number of projections. , 2002, Physics in medicine and biology.

[7]  R H Huesman,et al.  The effects of a finite number of projection angles and finite lateral sampling of projections on the propagation of statistical errors in transverse section reconstruction. , 1977, Physics in medicine and biology.

[8]  Kyle J. Myers,et al.  Foundations of Image Science , 2003, J. Electronic Imaging.

[9]  Xiaochuan Pan,et al.  Sampling and aliasing consequences of quarter-detector offset use in helical CT , 2004, IEEE Transactions on Medical Imaging.

[10]  Jie Tang,et al.  Prior image constrained compressed sensing (PICCS): a method to accurately reconstruct dynamic CT images from highly undersampled projection data sets. , 2008, Medical physics.

[11]  D. Sorensen.  Numerical methods for large eigenvalue problems , 2002, Acta Numerica.

[12]  William H Press,et al.  Discrete Radon transform has an exact, fast inverse and generalizes to operations other than sums along lines , 2006, Proceedings of the National Academy of Sciences.

[13]  Gregory Beylkin,et al.  Discrete radon transform , 1987, IEEE Trans. Acoust. Speech Signal Process..

[14]  A. Ramm,et al.  The RADON TRANSFORM and LOCAL TOMOGRAPHY , 1996 .

[15]  Jeffrey A. Fessler,et al.  Statistical image reconstruction for polyenergetic X-ray computed tomography , 2002, IEEE Transactions on Medical Imaging.

[16]  Emil Y. Sidky,et al.  Toward optimal X-ray flux utilization in breast CT , 2011 .

[17]  R. F. Wagner,et al.  Objective assessment of image quality. II. Fisher information, Fourier crosstalk, and figures of merit for task performance. , 1995, Journal of the Optical Society of America. A, Optics, image science, and vision.

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

[19]  A. Macovski,et al.  A Natural Pixel Decomposition for Two-Dimensional Image Reconstruction , 1981, IEEE Transactions on Biomedical Engineering.

[20]  R M Nishikawa,et al.  Task-based assessment of breast tomosynthesis: effect of acquisition parameters and quantum noise. , 2010, Medical physics.

[21]  Jeffrey A. Fessler,et al.  A Splitting-Based Iterative Algorithm for Accelerated Statistical X-Ray CT Reconstruction , 2012, IEEE Transactions on Medical Imaging.

[22]  Marc Teboulle,et al.  Fast Gradient-Based Algorithms for Constrained Total Variation Image Denoising and Deblurring Problems , 2009, IEEE Transactions on Image Processing.

[23]  Hiroyuki Kudo,et al.  Statistical image reconstruction from limited projection data with intensity priors , 2012, Physics in medicine and biology.

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

[25]  G. Gullberg,et al.  A reconstruction algorithm using singular value decomposition of a discrete representation of the exponential Radon transform using natural pixels , 1993, 1993 IEEE Conference Record Nuclear Science Symposium and Medical Imaging Conference.

[26]  Lei Zhu,et al.  Compressed sensing based cone-beam computed tomography reconstruction with a first-order methoda). , 2010, Medical physics.

[27]  Emmanuel J. Candès,et al.  Templates for convex cone problems with applications to sparse signal recovery , 2010, Math. Program. Comput..

[28]  M. Vannier,et al.  Why do commercial CT scanners still employ traditional, filtered back-projection for image reconstruction? , 2009, Inverse problems.

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

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

[31]  Michael Elad,et al.  Sparse and Redundant Representations - From Theory to Applications in Signal and Image Processing , 2010 .

[32]  Avinash C. Kak,et al.  Principles of computerized tomographic imaging , 2001, Classics in applied mathematics.

[33]  P. L. Combettes.  The foundations of set theoretic estimation , 1993 .

[34]  J. Hadamard,et al.  Lectures on Cauchy's Problem in Linear Partial Differential Equations , 1924 .

[35]  Antonin Chambolle,et al.  A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging , 2011, Journal of Mathematical Imaging and Vision.

[36]  E. Candès,et al.  Stable signal recovery from incomplete and inaccurate measurements , 2005, math/0503066.

[37]  Stephen J Glick,et al.  Breast CT. , 2007, Annual review of biomedical engineering.

[38]  Xiaochuan Pan,et al.  Enhanced imaging of microcalcifications in digital breast tomosynthesis through improved image-reconstruction algorithms. , 2009, Medical physics.

[39]  Michael W. Berry,et al.  Large-Scale Sparse Singular Value Computations , 1992 .

[40]  Emil Y. Sidky,et al.  Algorithm-Enabled Low-Dose Micro-CT Imaging , 2011, IEEE Transactions on Medical Imaging.

[41]  Michael Unser,et al.  A review of wavelets in biomedical applications , 1996, Proc. IEEE.

[42]  E. Sidky,et al.  Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT , 2009, 0904.4495.

[43]  M. Defrise,et al.  An algorithm for total variation regularization in high-dimensional linear problems , 2011 .

[44]  David P. Rohler,et al.  Exploiting Symmetry in Fan Beam CT: Overcoming Third Generation Undersampling , 2005, SIAM J. Appl. Math..

[45]  David L. Donoho,et al.  Precise Undersampling Theorems , 2010, Proceedings of the IEEE.

[46]  E. Sidky,et al.  Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization , 2008, Physics in medicine and biology.

[47]  Xiaochuan Pan,et al.  Evaluation of sparse-view reconstruction from flat-panel-detector cone-beam CT , 2010, Physics in medicine and biology.

[48]  J. Boone,et al.  Dedicated breast CT: radiation dose and image quality evaluation. , 2001, Radiology.

[49]  E. Sidky,et al.  Convex optimization problem prototyping for image reconstruction in computed tomography with the Chambolle–Pock algorithm , 2011, Physics in medicine and biology.

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

[51]  Xiaochuan Pan,et al.  Image reconstruction exploiting object sparsity in boundary-enhanced X-ray phase-contrast tomography. , 2010, Optics express.

[52]  M. Kachelriess,et al.  Improved total variation-based CT image reconstruction applied to clinical data , 2011, Physics in medicine and biology.

[53]  E.J. Candes,et al.  An Introduction To Compressive Sampling , 2008, IEEE Signal Processing Magazine.

[54]  Jiayu Song,et al.  Sparseness prior based iterative image reconstruction for retrospectively gated cardiac micro-CT. , 2007, Medical physics.

[55]  S. Frick,et al.  Compressed Sensing , 2014, Computer Vision, A Reference Guide.

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

[57]  E. Somersalo,et al.  Statistical and computational inverse problems , 2004 .

[58]  Yoram Bresler,et al.  Multiresolution tomographic reconstruction using wavelets , 1994, Proceedings of 1st International Conference on Image Processing.

[59]  Søren Holdt Jensen,et al.  Implementation of an optimal first-order method for strongly convex total variation regularization , 2011, ArXiv.