Non-Linear Phase Tomography Based on Fréchet Derivative

Valentina Davidoiu^1*, Bruno Sixou¹, Max Langer^1,2, Franoise Peyrin^1,2

¹Centre de Recherche en Acquisition et Traitement de l’Image pour la Santé (CREATIS), Centre National de la Recherche Scientifique Unité Mixte de Recherche 5220—Institut National de la Santé et de la Recherche Médicale Unité 1044—Université Lyon 1—Institut National des Sciences Appliquées de Lyon, Lyon, France.
²Centre de Recherche en Acquisition et Traitement de l’Image pour la Santé (CREATIS), Centre National de la Recherche Scientifique Unité Mixte de Recherche 5220—Institut National de la Santé et de la Recherche Médicale Unité 1044—Université Lyon 1—Institut National des Sciences Appliquées de Lyon, Lyon, France.
²European Synchrotron Radiation Facility, Grenoble, France.

DOI: 10.4236/act.2014.34007                

Abstract

Phase imaging coupled to micro-tomography acquisition has emerged as a powerful tool to investigate specimens in a non-destructive manner. While the intensity data can be acquired and recorded, the phase information of the signal has to be “retrieved” from the data modulus only. Phase retrieval is an ill-posed non-linear problem and regularization techniques including a priori knowledge are necessary to obtain stable solutions. Several linear phase recovery methods have been proposed and it is expected that some limitations resulting from the linearization of the direct problem will be overcome by taking into account the non-linearity of the phase problem. To achieve this goal, we propose and evaluate a non-linear algorithm for in-line phase micro-tomography based on an iterative Landweber method with an analytic calculation of the Fréchet derivative of the phase-intensity relationship and of its adjoint. The algorithm was applied in the projection space using as initialization the linear mixed solution. The efficacy of the regularization scheme was evaluated on simulated objects with a slowly and a strongly varying phase. Experimental data were also acquired at ESRF using a propagation-based X-ray imaging technique for the given pixel size 0.68 μm. Two regularization scheme were considered: first the initialization was obtained without any prior on the ratio of the real and imaginary parts of the complex refractive index and secondly a constant a priori value was assumed on  . The tomographic central slices of the refractive index decrement were compared and numerical evaluation was performed. The non-linear method globally decreases the reconstruction errors compared to the linear algorithm and is achieving better reconstruction results if no prior is introduced in the initialization solution. For in-line phase micro-tomography, this non-linear approach is a new and interesting method in biomedical studies where the exact value of the a priori ratio is not known.

Keywords

Phase Retrieval, In-Line Phase Tomography, Inverse Problems, Non-Linear Problem, Non-Linear Optimization, Fréchet Derivative, Coherent Imaging, Fresnel Diffraction, Phase Contrast, X-Ray Imaging

Conflicts of Interest

The authors declare no conflicts of interest.

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