APPENDIX B SPS-OS Algorithm

The one-subset version of the SPS-OS algorithm applied for the PWLS cost function with a continuously differentiable penalty function has the following preconditioned gradient descent from [29]:

ݔ^௞ାଵ ൌ ݔ^௞െ ܲ൫׏߶ሺݔ^௞ሻ ൅ ߚ׏ܴሺݔ^௞ሻ൯, ܲ ൌ †‹ƒ‰ሼܩ^்ܹܩ૚ ൅ ߚ׏^ଶܴሺݔ^௞ሻሽ^ିଵ (B.1) where ׏ܴሺݔ^௞ሻ ൌ σ ݀_{௜௝ ௝௜}߱_௜߮Ԣሺሾܦݔሿ_௜ሻ, ׏^ଶܴሺݔ^௞ሻ ൌ σ ห݀_{௜௝ ௝௜}ห߱_௜σ ȁ݀_{௞ ௞௜}ȁ, where ݀_௝௜s are the elements of matrix D and ߱௜ are weighting factors to control the strength of regularization at voxel i.

85

References

[1] J. Czernin, M. Allen-Auerbach, and H. R. Schelbert, “Improvements in cancer staging with PET/CT:

Literature-based evidence as of september 2006.,” J Nucl Med, vol. 48, no. 1_suppl, pp. 78S-88, January 1, 2007, 2007.

[2] M. F. Di Carli, S. Dorbala, J. Meserve, G. El Fakhri, A. Sitek, and S. C. Moore, “Clinical myocardial perfusion PET/CT,” J. Nucl. Med. , vol. 48, no. 5, pp. 783-793, 2007.

[3] J. C. Price, “Molecular brain imaging in the multimodality era.,” J Cereb Blood Flow Metab, vol. 32, pp.1377–139, 2012.

[4] H. Zaidi, and A. Alavi, “Current trends in PET and combined (PET/CT and PET/MR) systems design.,”

PET Clinics, vol. 2, no. 2, pp. 109-123, 2007/4, 2007.

[5] A. J. Reader, and H. Zaidi, “Advances in PET image reconstruction.,” PET Clinics, vol. 2, no. 2, pp. 173-190, 2007/4, 2007.

[6] R. M. Lewitt, and S. Matej, “Overview of methods for image reconstruction from projections in emission computed tomography.,” Proceedings of the IEEE, vol. 91, no. 10, pp. 1588-1611, 2003.

[7] J. Qi, and R. M. Leahy, “Iterative reconstruction techniques in emission computed tomography.,” Phys Med Biol, vol. 51, no. 15, pp. R541-R578, Aug 7, 2006.

[8] D. S. Lalush, and M. Wernick, "Iterative image reconstruction.," Emission Tomography: The Fundamentals of PET and SPECT., M. Wernick and J. Aarsvold, eds., pp. 443-472, San Diego: Academic Press, 2004.

[9] J. A. Fessler, “Penalized weighted least-squares image reconstruction for positron emission tomography.,” IEEE Trans Med Imaging, vol. 13, no. 2, pp. 290-300, 1994.

[10] W. C. Karl, "Regularization in Image Restoration and Reconstruction," Handbook of Image and Video Processing, A. Bovik, ed., San Diego: Academic Press, 2000.

[11] P. J. Green, “Bayesian reconstructions from emission tomography data using a modified EM algorithm.,”

IEEE Trans Med Imaging, vol. 9, no. 1, pp. 84-93, 1990.

[12] V. E. Johnson, W. H. Wong, X. Hu, and C. T. Chen, “Image restoration using Gibbs priors: Boundary modeling, treatment of blurring, and selection of hyperparameter,” IEEE Trans Pattern Anal Machine Intell, vol. 13, pp. 413-425, 1991.

[13] C. Bouman, and K. Sauer, “A generalized Gaussian image model for edge-preserving MAP estimation,”

IEEE Trans Image Process, vol. 2, pp. 296-310, 1993.

[14] S. J. Lee, A. Rangarajan, and G. Gindi, “Bayesian image reconstruction in SPECT using higher order mechanical models as priors,” IEEE Trans Med Imaging, vol. 14, pp. 669-680, 1995.

[15] P. Perona, and J. Malik, “Scale-space and edge detection using anisotropic diffusion.,” IEEE Trans Patt Analysis Machine Intelligence, vol. 12, no. 7, pp. 629-639, 1990.

[16] F. Catte, P.-L. Lions, J.-M. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J Numer Anal, vol. 29, pp. 182-193, 1992.

[17] S. T. Acton, “Locally monotonic diffusion,” IEEE Trans Signal Process, vol. 48, pp. 1379-1389, 2000.

[18] P. J. Huber, Robust Statistics, New York: Wiley, 1981.

[19] L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D, vol. 60, pp. 259-268, 1992.

[20] D. F. Yu, and J. A. Fessler, “Edge-preserving tomographic reconstruction with nonlocal regularization,”

IEEE Trans Med Imaging, vol. 21, no. 2, pp. 159-173, 2002.

[21] H. H. Feng, W. C. Karl, and D. A. Castanon, “A curve evolution approach to object-based tomographic reconstruction,” IEEE Trans Image Process, vol. 12, pp. 44-57, 2003.

[22] A. Sawatzky, “(Nonlocal) Total Variation in Medical Imaging,” Ph.D Thesis, Fachbereich Mathematik und Informatik der Mathematisch-Naturwissenschaftlichen Fakultät, Westfälischen Wilhelms-Universität Münster, Münster, Germany, 2011.

[23] J. E. Bowsher, V. E. Johnson, T. G. Turkington, R. J. Jaszczak, C. E. Floyd, and R. E. Coleman,

“Bayesian reconstruction and use of anatomical a priori information for emission tomography,” IEEE Trans Med Imaging, vol. 15, pp. 673-686, 1996.

86 [24] G. Gindi, M. Lee, A. Rangarajan, and I. G. Zubal, “Bayesian reconstruction of functional images using

anatomical information as priors.,” IEEE Trans Med Imaging, vol. 12, no. 4, pp. 670-680, 1993.

[25] J. A. Fessler, N. H. Clinthorne, and W. L. Rogers, "Regularized emission image reconstruction using imperfect side information." pp. 1991-1995 vol.3.

[26] K. Baete, J. Nuyts, W. Van Paesschen, P. Suetens, and P. Dupont, “Anatomical-based FDG-PET reconstruction for the detection of hypo-metabolic regions in epilepsy,” Medical Imaging, IEEE Transactions on, vol. 23, no. 4, pp. 510-519, 2004.

[27] J. Tang, and A. Rahmim, “Bayesian PET image reconstruction incorporating anato-functional joint entropy,” Phys. Med. Biol., vol. 54, pp. 7063-75, 2009.

[28] K. Vunckx, A. Atre, K. Baete, A. Reilhac, C. M. Deroose, K. Van Laere, and J. Nuyts, “Evaluation of Three MRI-Based Anatomical Priors for Quantitative PET Brain Imaging,” Medical Imaging, IEEE Transactions on, vol. 31, no. 3, pp. 599-612, 2012.

[29] Y. K. Dewaraja, K. F. Koral, and J. A. Fessler, “Regularized reconstruction in quantitative SPECT using CT side information from hybrid imaging,” Phys. Med. Biol., vol. 55, pp. 2523-2539, 2010.

[30] F. Bataille, C. Comtat, S. Jan, F. C. Sureau, and R. Trebossen, “Brain PET Partial-Volume Compensation Using Blurred Anatomical Labels,” Nuclear Science, IEEE Transactions on, vol. 54, no. 5, pp. 1606-1615, 2007.

[31] S. Somayajula, E. Asma, and R. M. Leahy, "PET image reconstruction using anatomical information through mutual information based priors." pp. 2722-2726.

[32] C. Comtat, P. E. Kinahan, J. A. Fessler, T. Beyer, D. W. Townsend, M. Defrise, and C. Michel,

“Clinically feasible reconstruction of 3D whole-body PET/CT data using blurred anatomical labels,”

Phys. Med. Biol., vol. 47, pp. 1-20, 2002.

[33] H. Erdogan, and J. A. Fessler, “Ordered subsets algorithms for transmission tomography,” Phys Med Biol, vol. 44, no. 11, pp. 2835-2851, 1999.

[34] T. M. Benson, B. K. B. De Man, F. Lin, and J. B. Thibault, "Block-based iterative coordinate descent."

pp. 2856-2859.

[35] J. A. Fessler, and S. D. Booth, “Conjugate-gradient preconditioning methods for shift-variant PET image reconstruction.,” IEEE Trans Image Process, vol. 8, no. 5, pp. 688-699, 1999.

[36] N. H. Clinthorne, and et al, “Preconditioning methods for improved convergence rates in iterative reconstructions,” IEEE Trans Med Imaging, vol. 12, pp. 78-83, 1993.

[37] A. H. Delaney, and Y. Bresler, “A fast and accurate Fourier algorithm for iterative parallel-beam tomography,” IEEE Trans Image Process, vol. 5, no. 5, pp. 740-753, 1996.

[38] J. Savage, and K. Chen, "On multigrids for Solving a Class of Improved Total Variation Based Staircasing Reduction Models," Mathematics and Visualization. pp. 60-94.

[39] A. Chambolle, “An algorithm for total variation minimization and applications,” J Math Imaging Vis, vol.

20, no. 1-2, pp. 89-97, 2004.

[40] S. Ramani, and J. A. Fessler, “A Splitting-Based Iterative Algorithm for Accelerated Statistical X-Ray CT Reconstruction,” IEEE Trans. Med. Imaging, vol. 31, no. 3, pp. 677-688, 2012.

[41] J. Eckstein, and D. Bertsekas, “On the DouglasRachford splitting method and the proximal point algorithm for maximal monotone operators,” Math Program, vol. 5, pp. 293-318, 1992.

[42] S. Setzer, “Operator splittings, bregman methods and frame shrinkage in image processing,” Int J Comput Vision, vol. 92, no. 3, pp. 265-280, 2011.

[43] K. Lange, D. R. Hunter, and I. Yang, “Optimization transfer using surrogate objective functions,” J.

Comput Graph Stat, vol. 9, pp. 1-20, 2000

[44] D. Hunter, and K. Lange, “A tutorial on MM algorithms,” The American Statistician, vol. 58, pp. 30-37, 2004.

[45] A. Beck, and M. Teboulle, "Gradient-based algorithms with applications in signal recovery problems,"

Convex Optimization in Signal Processing and Communications, D. Palomar and Y. Eldar, eds., pp. 33-88: Cambribge University Press, 2010.

[46] W. Zuo, and Z. Lin, “A generalized accelerated proximal gradient approach for total-variation-based image restoration,” IEEE Trans Image Process, vol. 20, no. 10, pp. 2748-2759, 2011.

[47] C. A. Johnson, J. Seidel, and A. Sofer, “Interior-point methodology for 3-D PET reconstruction,” IEEE Trans Med Imaging, vol. 19, pp. 271-285, 2000.

87 [48] L. Kaufman, “Maximum likelihood, least squares, and penalized least squares for PET.,” IEEE Trans

Med Imaging, vol. 12, no. 2, pp. 200-214, 1993.

[49] D. Kim, and J. A. Fessler, "Accelerated ordered-subsets algorithm based on separable quadratic surrogates for regularized image reconstruction in x-ray CT." pp. 1134-1137.

[50] E. P. Chong, and S. H. Zak, An Introduction to Optimization, 2 ed., New York: John Wiley & Sons, 2001.

[51] A. Chambolle, and T. Pock, “A first-order primal-dual algorithm for convex problems with applications to imaging,” J Math Imaging Vis, vol. 40, no. 1, pp. 120-145, 2010.

[52] S. Becker, J. Bobin, and E. J. Candes, “NESTA: A fast and accurate first-order method for sparse recovery,” SIAM J Imaging Sci, vol. 4, pp. 1-39, 2011.

[53] J. Kiusalaas, Numerical Methods in Engineering with MATLAB, 2 ed.: Cambridge University Press, 2010.

[54] J. Cheng-Liao, and J. Qi, “PET image reconstruction with anatomical edge guided level set prior,”

Physics in Medicine and Biology, vol. 56, no. 21, pp. 6899, 2011.

[55] J. F. Canny, “A computational approach to edge detection,” IEEE Trans Pattern Anal Machine Intell, vol.

8, no. 6, pp. 679-698, 1986.

[56] J. Fessler. "Image reconstruction toolbox, http://www.eecs.umich.edu/~fessler/code/index.html, University of Michigan."

[57] F. Jacobs, E. Sundermann, B. D. Sutter, M. Christiaens, and I. Lemahieu, “A fast algorithm to calculate the exact radiological path through a pixel or voxel space,” J Comput Inform Technol, vol. 6, no. 1, pp.

89-94, 1998.

[58] D. S. Lalush, and B. M. W. Tsui, “A fast and stable maximum a posteriori conjugate gradient reconstruction algorithm,” Med. Phys., vol. 22, no. 8, pp. 1273-1284, 1995.

[59] T. Jensen, J. Jørgensen, P. Hansen, and S. Jensen, “Implementation of an optimal first-order method for strongly convex total variation regularization,” BIT Numerical Mathematics,, vol. 52, no. 2, pp. 329-356, 2012.

[60] W. Guo, and F. Huang, “A local mutual information guided denoising technique and its application to self-calibrated partially parallel imaging,” in Med. Image. Comput. Comput. Assist. Interv., Part II, LNCS 5242, 2008, pp. 939-947.

88

Chapter 6

Smoothly clipped absolute deviation (SCAD)