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.
References
[1]
Cloetens, P., Barrett, R., Baruchel, J., Guigay, J.P. and Schlenker, M. (1996) Phase Objects in Synchrotron Radiation Hard X-Ray Imaging. Journal of Physics D: Applied Physics, 29, 133. http://dx.doi.org/10.1088/0022-3727/29/1/023
[2]
Wilkins, S.W., Gureyev, T.E., Gao, D., Pogany, A. and Stevenson, A.W. (1996) Phase Contrast Imaging Using Polychromatic Hard X-Rays. Nature, 384, 335-338. http://dx.doi.org/10.1038/384335a0
[3]
Momose, A. and Fukuda, J. (1995) Phase-Contrast Radiographs of Nonstained Rat Cerebellar Specimen. Medical Physics, 22, 375. http://dx.doi.org/10.1118/1.597472
[4]
Paganin, D., Mayo, S.C., Gureyev, T.E., Miller, P.R. and Wilkins, S.W. (2002) Simultaneous Phase and Amplitude Extraction from a Single Defocused Image of a Homogeneous Object. Journal of Microscopy, 206, 33-40. http://dx.doi.org/10.1046/j.1365-2818.2002.01010.x
[5]
Cloetens, P., Ludwig, W., Boller, E., Helfen, L., Salvo, L., Mache, R. and Schlenker, M. (2002) Quantitative Phase Contrast Tomography Using Coherent Synchrotron Radiation. Proceedings of SPIE, 4503, 82. http://dx.doi.org/10.1117/12.452867
[6]
Beleggia, M., Schofield, M.A., Volkov, V.V. and Zhu, Y. (2004) On the Transport of Intensity Technique for Phase Retrieval. Ultramicroscopy, 102, 37-49. http://dx.doi.org/10.1016/j.ultramic.2004.08.004
[7]
Wu, X., Liu, H. and Yan, A. (2005) X-Ray Phase-Attenuation Duality and Phase Retrieval. Optics Letters, 30, 379-381. http://dx.doi.org/10.1364/OL.30.000379
[8]
Guigay, J.P., Langer, M., Boistel, R. and Cloetens, P. (2007) A Mixed Contrast Transfer and Transport of Intensity Approach for Phase Retrieval in the Fresnel Region. Optics Letters, 32, 1617-1619. http://dx.doi.org/10.1364/OL.32.001617
[9]
Langer, M., Cloetens, P. and Peyrin, F. (2010) Regularization of Phase Retrieval with Phase-Attenuation Duality Prior for 3-D Holotomography. IEEE Trans. Image Processing, 19, 2428-2436. http://dx.doi.org/10.1109/TIP.2010.2048608
[10]
Beltran, M.A., Paganin, D.M., Uesugi, K. and Kitchen, M.J. (2010) 2D and 3D X-Ray Phase Retrieval of Multi-Material Objects Using a Single Defocus Distance. Optics Express, 18, 6423-6436. http://dx.doi.org/10.1364/OE.18.006423
[11]
Langer, M., Cloetens, P., Pacureanu, A. and Peyrin, F. (2012) X-Ray In-Line Phase Tomography of Multimaterial Objects. Optics Letters, 37, 2151-2153. http://dx.doi.org/10.1364/OL.37.002151
[12]
Ohlsson H., Yang A., Dong, R. and Sastry S. S. (2012) CPRL—An Extension of Compressive Sensing to the Phase Retrieval Problem. In: Bartlett, P., Pereira, F., Burges, C., Bottou, L. and Weinberger, K., Eds., Advances in Neural Information Processing Systems 25, 1376-1384.
[13]
Candes, E.J., Eldar, Y.C., Strohmer, T. and Voroninski, V. (2011) Phase Retrieval via Matrix Completion. http://arXiv:1109.0573v2
[14]
Gureyev, T.E. (2003) Composite Techniques for Phase Retrieval in the Fresnel Region. Optics Communications, 220, 49-58. http://dx.doi.org/10.1016/S0030-4018(03)01353-1
[15]
Moosmann, J., Hofmann, R., Bronnikov, A.V. and Baumbach, T. (2010) Nonlinear Phase Retrieval from Single-Distance Radiograph. Optics Express, 18, 25771-25785. http://dx.doi.org/10.1364/OE.18.025771
[16]
Davidoiu, V., Sixou, B., Langer, M. and Peyrin, F. (2011) Nonlinear Phase Retrieval Based on Frechet Derivative. Optics Express, 19, 22809-22819. http://dx.doi.org/10.1364/OE.19.022809
[17]
Davidoiu, V., Sixou, B., Langer, M. and Peyrin, F. (2012) Nonlinear Phase Retrieval Using Projection Operator and Iterative Wavelet Thresholding. IEEE Signal Processing Letters, 19, 579-582. http://dx.doi.org/10.1109/LSP.2012.2207113
[18]
Born, M. and Wolf, E. (1997) Principles of Optics. Cambridge University Press, Cambridge.
[19]
Davidoiu, V., Sixou, B., Langer, M. and Peyrin, F. (2013) Nonlinear Approaches for the Single-Distance Phase Retrieval Problem Involving Regularizations with Sparsity Constraints. Applied Optics, 52, 3977-3986. http://dx.doi.org/10.1364/AO.52.003977
[20]
Scherzer, O., Grasmair, M., Grossauer, H., Haltmeier, M. and Lenzen, F. (2008) Variational Methods in Imaging. Springer Verlag, New York.
[21]
Hanke, M., Neubauer, A. and Scherzer, O. (1995) A Convergence Analysis of the Landweber Iteration for Nonlinear Ill-Posed Problems. Numerische Mathematik, 72, 21-37. http://dx.doi.org/10.1007/s002110050158
[22]
Bakushinsky, A.B. (1992) The Problem of the Convergence of the Iteratively Regularized Gauss-Newton Method. Computational Mathematics and Mathematical Physics, 32, 1353-1359.
[23]
Bakushinsky, A.B. and Smirnova, A. (2005) On Application of Generalized Discrepancy Principle to Iterative Methods for Nonlinear Ill-Posed Problems. Numerical Functional Analysis and Optimization, 26, 35-48. http://dx.doi.org/10.1081/NFA-200051631
[24]
Kak, A.C. and Slaney, M. (1989) Principles of Computerized Tomographic Imaging. IEEE Press, New York.
[25]
Langer, M., Cloetens, P., Guigay, J.P. and Peyrin, F. (2008) Quantitative Comparison of Direct Phase Retrieval Algorithms in In-Line Phase Tomography. Medical Physics, 35, 4556. http://dx.doi.org/10.1118/1.2975224
[26]
Sanchez del Rio, M. and Dejus, R. (2004) Status of XOP: An X-Ray Optics Software Toolkit. SPIE Proceedings, 5536, 171-174. http://dx.doi.org/10.1117/12.560903
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. 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.
