Abstract
In this work, we investigate an inverse problem approach to 3D super-resolution/segmentation for an application to the analysis of trabecular bone micro-architecture from in vivo 3D X-ray CT images. The problem is expressed as the minimization of a functional including a data term and a prior. We consider here a regularization term combining total variation (TV) and a double-well potential to enforce the quasi-binarity of the resulting image. Three different schemes to minimize this nonconvex functional are presented and compared. The methods are applied to experimental new high-resolution peripheral quantitative CT images (voxel size \(82\,\upmu \hbox {m}\)) and evaluated with respect to a micro-CT image at higher spatial resolution (voxel size \(41\,\upmu \hbox {m}\)) considered as a ground truth. Our results show that a combination of double-well functional and TV term improves the contrast and the quality of the restoration even if the connectivity may be degraded.
Similar content being viewed by others
Notes
\(f_p\) is obtained as follows: (1) Given a restored image f, its binary image is \(f_b\). \(f_b\) has only 0 and 1 values. (2) If we regard \(f_b\) as a mask of f, we could calculate the average values \(f_0\) and \(f_1\) of f over zeros regions and ones regions of \(f_b\). (3) The voxels of value 0 in \(f_b\) are assigned to \(f_0\), the voxels of value 1 in \(f_b\) are set to \(f_1\).
References
Afonso, M.V., Bioucas-Dias, J.M., Figueiredo, M.A.: An augmented lagrangian approach to the constrained optimization formulation of imaging inverse problems. IEEE Trans. Image Process. 20(3), 681–695 (2011)
Artina, M., Fornasier, M., Solombrino, F.: Linearly constrained nonsmooth and nonconvex minimization. SIAM J. Optim. 23(3), 1904–1937 (2013)
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)
Beck, A., Teboulle, M.: Gradient-based algorithms with applications to signal recovery. In: Convex Optimization in Signal Processing and Communications, pp. 42–88 (2009)
Bertozzi, A., Esedoglu, S., Gillette, A.: Analysis of a two-scale Cahn–Hilliard model for binary image inpainting. Multiscale Model. Simul. 6(3), 913–936 (2007)
Bertozzi, A.L., Esedoglu, S., Gillette, A.: Inpainting of binary images using the Cahn–Hilliard equation. IEEE Trans. Image Process. 16(1), 285–291 (2007)
Boutroy, S., Bouxsein, M.L., Munoz, F., Delmas, P.D.: In vivo assessment of trabecular bone microarchitecture by high-resolution peripheral quantitative computed tomography. J. Clin. Endocrinol. Metab. 90(12), 6508–6515 (2005)
Burghardt, A.J., Link, T.M., Majumdar, S.: High-resolution computed tomography for clinical imaging of bone microarchitecture. Clin. Orthop. Relat. Res. 469(8), 2179–2193 (2011)
Combettes, P.L., Pesquet, J.C.: Proximal splitting methods in signal processing. In: Fixed-Point Algorithms for Inverse Problems in Science and Engineering, pp. 185–212. Springer, New York (2011)
Elliott, C.M.: The Cahn–Hilliard model for the kinetics of phase separation. In: Mathematical Models for Phase Change Problems, pp. 35–73. Springer, New York (1989)
Eyre, D.J.: An unconditionally stable one-step scheme for gradient systems. Unpublished article (1998)
Foucart, S., Lai, M.J.: Sparsest solutions of underdetermined linear systems via \(l_q\)-minimization for \(0<q\le 1\). Appl. Comput. Harmon. Anal. 26(3), 395–407 (2009)
Kijewski, M.F., Judy, P.F.: The noise power spectrum of ct images. Phys. Med. Biol. 32(5), 565 (1987)
Klein, S., Staring, M., Murphy, K., Viergever, M.A., Pluim, J.P.: Elastix: a toolbox for intensity-based medical image registration. IEEE Trans. Med. Imaging 29(1), 196–205 (2010)
Li, H., Lin, Z.: Accelerated proximal gradient methods for nonconvex programming. In: Advances in Neural Information Processing Systems, pp. 379–387 (2015)
Li, Y., Sixou, B., Burghardt, A., Peyrin, F.: Super-resolution/segmentation of 3d trabecular bone images with total variation and nonconvex Cahn–Hilliard functional. In: IEEE 14th International Symposium on Biomedical Imaging (ISBI 2017), pp. 1193–1196. IEEE (2017)
Li, Y., Sixou, B., Peyrin, F.: Estimation of the blurring kernel in experimental HR-pQCT images based on mutual information. In: Signal Processing Conference (EUSIPCO), 2017 25th European, pp. 2086–2090. IEEE (2017)
Ochs, P., Chen, Y., Brox, T., Pock, T.: iPiano: inertial proximal algorithm for nonconvex optimization. SIAM J. Imaging Sci. 7(2), 1388–1419 (2014)
Odgaard, A.: Three-dimensional methods for quantification of cancellous bone architecture. Bone 20(4), 315–328 (1997)
Ohser, J., Nagel, W., Schladitz, K.: Miles formulae for boolean models observed on lattices. Image Anal. Stereol. 28(2), 77–92 (2011)
Otsu, N.: A threshold selection method from gray-level histograms. Automatica 11(285–296), 23–27 (1975)
Parikh, N., Boyd, S.P., et al.: Proximal algorithms. Found. Trends Optim. 1(3), 127–239 (2014)
Peyrin, F., Engelke, K.: CT imaging: Basics and new trends. In: Handbook of Particle Detection and Imaging, pp. 883–915. Springer (2012)
Samson, C., Blanc-Féraud, L., Aubert, G., Zerubia, J.: A variational model for image classification and restoration. IEEE Trans. Pattern Anal. Mach. Intell. 22(5), 460–472 (2000)
Seeman, E., Delmas, P.D.: Bone quality the material and structural basis of bone strength and fragility. New Engl. J. Med. 354(21), 2250–2261 (2006)
Shamonin, D.P., Bron, E.E., Lelieveldt, B.P., Smits, M., Klein, S., Staring, M.: Fast parallel image registration on CPU and GPU for diagnostic classification of Alzheimer’s disease. Front. Neuroinform. 7, 50 (2014)
Shen, J., Chan, T.F.: Mathematical models for local nontexture inpaintings. SIAM J. Appl. Math. 62(3), 1019–1043 (2002)
Toma, A., Denis, L., Sixou, B., Pialat, J.B., Peyrin, F.: Total variation super-resolution for 3D trabecular bone micro-structure segmentation. In: 2014 22nd European Signal Processing Conference (EUSIPCO), pp. 2220–2224. IEEE (2014)
Toma, A., Sixou, B., Peyrin, F.: Iterative choice of the optimal regularization parameter in TV image restoration. Inverse Probl. Imaging 9(4), 1171–1191 (2015)
Wang, Y., Yang, J., Yin, W., Zhang, Y.: A new alternating minimization algorithm for total variation image reconstruction. SIAM J. Imaging Sci. 1(3), 248–272 (2008)
Acknowledgements
This work is financed by China Scholarship Council and was performed within the framework of the LABEX PRIMES (ANR-11-LABX-0063) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). Here we also want to acknowledgment Andrew Burghardt for offering us experimental images in this study, thanks to Alina TOMA for her contributions in image registration.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Li, Y., Sixou, B. & Peyrin, F. Nonconvex Mixed TV/Cahn–Hilliard Functional for Super-Resolution/Segmentation of 3D Trabecular Bone Images. J Math Imaging Vis 61, 504–514 (2019). https://doi.org/10.1007/s10851-018-0858-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10851-018-0858-1