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.
Highlights
Hard X-ray imaging with a high spatial resolution is nowadays a powerful tool to investigate specimens in 2D or 3D in a non-destructive manner
Phase retrieval and tomography can be coupled by a two-step process: first, the phase information is retrieved for all the projections, and secondly the three-dimensional tomographic reconstruction is performed on the retrieved phase images obtained for each angle of view
We have considered a non-linear phase retrieval method for phase tomography
Summary
Hard X-ray imaging with a high spatial resolution is nowadays a powerful tool to investigate specimens in 2D or 3D in a non-destructive manner. A new inversion method where a prior phase estimate at each projection angle is obtained from a measured absorption index map evaluated with the intensity measured for a propagation distance D1 = 0 m is described in [11]. This prior is introduced in the low-frequency range only. New algorithms which take into account the nonlinearity of the inverse problem for the radiographic case have been proposed recently [14]-[17] These nonlinear approaches are very promising and lead to a large decrease of the reconstruction errors. L2 paper, we first summarize this new multi-image non-linear ( NL) scheme, and detail the results obtained on simulated images and for a tomographic reconstruction on a real multi-material 3D object
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