Journal of Mathematical Imaging and Vision

, Volume 61, Issue 4, pp 504–514 | Cite as

Nonconvex Mixed TV/Cahn–Hilliard Functional for Super-Resolution/Segmentation of 3D Trabecular Bone Images

  • Y. LiEmail author
  • B. Sixou
  • F. Peyrin


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.


Super-resolution/segmentation Nonconvex Nonsmooth Cahn–Hilliard Total variation 3D CT image Bone micro-architecture 



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.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Univ Lyon, INSA-Lyon, Université Claude Bernard Lyon 1, UJM-Saint Etienne, CNRS, Inserm, CREATIS UMR 5220, U1206LyonFrance
  2. 2.ESRFGrenobleFrance

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