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

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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.

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Notes

  1. \(f_p\) 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 \(f_1\).

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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.

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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

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