Sparse-View CT Reconstruction Based on Nonconvex L1 − L2 Regularizations

The reconstruction from sparse-view projections is one
of important problems in computed tomography (CT) limited by
the availability or feasibility of obtaining of a large number of
projections. Traditionally, convex regularizers have been exploited
to improve the reconstruction quality in sparse-view CT, and the
convex constraint in those problems leads to an easy optimization
process. However, convex regularizers often result in a biased
approximation and inaccurate reconstruction in CT problems. Here,
we present a nonconvex, Lipschitz continuous and non-smooth
regularization model. The CT reconstruction is formulated as a
nonconvex constrained L1 − L2 minimization problem and solved
through a difference of convex algorithm and alternating direction
of multiplier method which generates a better result than L0 or L1
regularizers in the CT reconstruction. We compare our method with
previously reported high performance methods which use convex
regularizers such as TV, wavelet, curvelet, and curvelet+TV (CTV)
on the test phantom images. The results show that there are benefits in
using the nonconvex regularizer in the sparse-view CT reconstruction.

Authors:

• Ali Pour Yazdanpanah
• Farideh Foroozandeh Shahraki
• Emma Regentova

Keywords:

• Computed tomography
• sparse-view reconstruction
• L1 −L2 minimization
• non-convex
• difference of convex functions.

References:

[1] Brenner DJ J., Hall EJ.: Computed tomography: an increasing source of
radiation exposure. N Engl J Med. 357(22), 2277–2284 (2007).
[2] Natarajan, B.K.: Sparse approximate solutions to linear systems. SIAM
J. Com- put., 227234 (1995).
[3] Goldstein T., Osher S.:The split Bregman method for L1 regular- ized
problems. SIAM J. Imag. Sci. 2(2), 323–343 (2009).
[4] Vandeghinste B., Goossens B., Holen R. V., Vanhove C., Pizurica
A., Vandenberghe S., Staelens S.: Iterative CT Reconstruction Using
Shearlet-Based Regularization. IEEE Trans Nuclear Science. 60(5),
3305–3317 (2013).
[5] Oliveira J.P., Bioucas-Dias J. M., Figueiredo M.A.T.: Adaptative total
variation image deblurring: A majorization-minimization approach.
Signal Process., 89, 1683–1693 (2009).
[6] Beck A., Teboulle M.: Fast gradient-based algorithms for constrained total
variation image denoising and deblurring problems. IEEE Trans. Image
Process. 18(11), 2419–2434 (2009).
[7] Sidky E. Y., Pan X.: Image reconstruction in circular cone-beam computed
tomography by constrained, total-variation minimization. Phys. Med.
Biol. 53(17), 4777–4807 (2008).
[8] Herman G. T., Davidi R.: On image reconstruction from a small number
of projections. Inv. Probl. 24(4), 45011–45028 (2008).
[9] Yang J., Yu H., Jiang M., Wang G.: High-order total variation
minimization for interior tomography. Inv. Probl. 26(3), 035013 (2010).
[10] Starck JL, Candes EJ, Donoho DL.:The curvelet transform for image
denoising. IEEE Trans Image Process. 11(6), 670–84 (2002).
[11] Wu, H., Maier, A., Hornegger, J.: Iterative CT reconstruction using
curvelet-based regularization. In: Meinzer, H.-P., Deserno, T.M., Handels,
H., Tolxdorff, T. (eds.) Bildverarbeitung fur die Medizin. Springer,
Heidelberg (2013).
[12] Pour Yazdanpanah A., Regentova E. E., Bebis G.: Algebraic
iterative reconstruction-reprojection (AIRR) method for high performance
sparse-view CT reconstruction. Applied mathematics & information
sciences. 10(6), 2007–2014 (2016).
[13] Pour Yazdanpanah A., Regentova E. E.: Compressed sensing MRI using
curvelet sparsity and nonlocal total variation: CS-NLTV. IS&T 29th
International Symposium on Electronic Imaging (2017).
[14] Pour Yazdanpanah A., Regentova E. E.: Sparse-View CT Reconstruction
using Curvelet and TV-based Regularization. 13th International
Conference Image Analysis and Recognition (ICIAR), Vol. 9730,
672–677 (2016).
[15] Yin, P., Lou, Y., He, Q., Xin, J.: Minimization of L1 L2 for compressed
sensing. SIAM J. Sci. Comput., 37(1), A536–A563 (2014).
[16] Lou, Y., Yin, P., He, Q., Xin, J.: Computing sparse representation in
a highly coherent dictionary based on difference of L1 and L2. J. Sci.
Comput. 64(1), 178–196 (2014).
[17] Yifei, L., Osher, S., and Xin, J.: Computational cspects of constrained
L1-L2 minimization for compressive sensing. Modelling, Computation
and Optimization in Information Systems and Management Sciences,
169–180 (2015).
[18] R. Chartrand: Exact reconstruction of sparse signals via nonconvex
minimization, IEEE Signal Process. Lett., 14, 707–710 (2007).
[19] R. Chartrand and V. Staneva Restricted isometry properties and
nonconvex compressive sensing, Inverse Problems, 24, 1–14 (2008).
[20] E. Candes, M. Wakin, and S. Boyd: Enhancing sparsity by reweighted
L1 minimization, J. Fourier Anal. Appl., 14, 877–905 (2008).
[21] E. Esser, Y. Lou, and J. Xin, A method for finding structured sparse
solutions to non-negative least squares problems with applications, SIAM
J. Imaging Sci., 6, 2010–2046 (2013).
[22] Pham-Dinh T., Le-Thi H.A.: Convex analysis approach to d.c.
programming: Theory, algorithms and applications. Acta Math. Vietnam.
22(1), 289–355 (1997).
[23] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, Distributed
optimization and statistical learning via the alternating direction method
of multipliers, Foundations and Trends in Machine Learning, 3(1), 1–122
(2011).
[24] Hestenes M. R., Stiefel E.: Methods of conjugate gradients for solving
linear systems. J. Res. Nat. Bureau Stand. 49(6), 409–436 (1952).
[25] Shepp L.A., Logan B. F.: Reconstructing interior head tissue from X-ray
transmissions. IEEE Trans. Nucl. Sci. 21(1), 228–236 (1974).
[26] FORBILD group, http://www.imp.uni-erlangen.de/phantoms/head/head.
html, Accessed on 01/10/2016.