We investigate the measurement of a thin sample's optical thickness using the transport of intensity equation (TIE) and demonstrate a version of the TIE, valid for partially coherent illumination, that allows the measurement of a sample's optical path length by the removal of illumination effects.

We demonstrate a nonlinear diffusion (NLD) regularization method to solve the transport of intensity equation (TIE). A novel NLD regularization function is proposed to enforce piecewise-constant priors and to remove low-frequency artifacts in the TIE solution.

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We develop and implement a compressive reconstruction method for tomographic recovery of refractive index distribution for weakly attenuating objects in a microfocus X-ray system. This is achieved through the development of a discretized operator modeling both the transport of intensity equation and X-ray transform that is suitable for iterative reconstruction techniques.Traditional tomography with hard X-rays recovers the attenuation of an object. Attenuation does not always provide good contrast when imaging objects made of materials with low electron density, e.g. soft tissues. In these cases, richer information is often contained in the phase, i.e. the optical thickness of the sample [1][2][3]. Propagation based techniques are particularly suitable for X-ray phase imaging because they allow phase to be recovered from intensity images taken at multiple propagation distances without the need for optical elements [4,5]. Here we adopt the transport of intensity equation (TIE) which relates the measured intensity to the Laplacian of the phase under a weakly-attenuating sample approximation. Implementing TIE at many angles while rotating the object allows tomographic reconstruction of the refractive index distribution.TIE tomographic reconstruction first requires a suitable forward model, consisting of 1) a projection of refractive index through the sample and 2) modeling diffraction after the sample with the TIE. Recovering the phase then amounts to inverting these operations on the measured data. A straightforward method of reconstruction is to invert the forward model in two steps [7]. For the first step, the TIE can be solved by a Poisson equation solver. The TIE is ill-posed because the transfer function relating the intensity measurement to the phase tends to zero as the spatial frequency decreases. As a result, reconstructions are often corrupted by significant low-frequency noise, requiring regularization. Tikhonov regularization is most commonly employed to reduce these artifacts [7,8]. For the second step, a standard tomographic reconstruction is carried out, e.g. using the filtered back-projection (FBP) method. In order for FBP to yield a result free from high-frequency "streaking" artifacts, projections must be taken at many angles. It is often desirable to use fewer projections in order to reduce dose or acquisition time, in which case the tomographic inversion problem is underdetermined. Rather than solve these two inverse problems independently, the forward and inverse models may be adapted into a single-step op- Fig. 1: Imaging geometry for TIE tomography eration combining TIE and tomography [6,[9][10][11]. However, a single step inversion still requires many projection angles in order to avoid artifacts. Iterative solvers have recently been proposed to reduce these artifacts when attempting a reconstruction from a small number of projections. Myers et. al. propose inversion of TIE tomography measurements to obtain a sample distribution using prior knowledge that the sample consists of a single mat...

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