Fourier-wavelet regularization of phase retrieval in x-ray in-line phase tomography

Phase sensitive x-ray imaging extends standard x-ray microscopy techniques by offering up to a thousand times higher sensitivity than absorption-based techniques. If an object is illuminated with a sufficiently coherent beam, phase contrast is achieved by moving the detector downstream from the object. There is a quantitative relationship between the phase shift induced by the object and the recorded intensity. This relationship can be used to retrieve the phase shift induced by the object through the solution of an inverse problem. Since the phase shift can be considered as a projection through the 3D refractive index, the latter can be reconstructed using standard tomographic inversion techniques. However, the determination of the phase shift from the recorded intensity is an ill-posed inverse problem. We investigate the application of Fourier-wavelet regularized deconvolution (ForWaRD) to this problem. The method is evaluated using simulated and experimental data and is shown to increase the quality of reconstructions, in terms of normalized RMS error and compared with standard Tikhonov regularization, at a three times increase in computational cost.