Three-dimensional phase retrieval in propagation-based phase-contrast imaging

We present a solution to the phase problem in near-field x-ray (propagation) imaging. The three-dimensional complex-valued index of refraction is reconstructed from a set of projections recorded in the near-field (Fresnel) setting at a single detector distance. The solution is found by an iterative algorithm based only on the measured data and the three-dimensional tomographic (Helgason-Ludwig) consistency constraint without the need for further a priori knowledge or other restrictive assumptions.

Figure 1

(a) Schematic of holographic phase-contrast imaging. A plane wave illuminates the object $\Omega$, leading to an exit wave in the xy plane as given by the projected optical indices, followed by free-space propagation, resulting in holographic phase-contrast formation as recorded in the detection plane at distance $z_d$. (b) The finite cross section $L$ of the reconstruction volume leads to an effective width $d\propto 1/L$ of the Fourier slices, characterizing correlated areas in reciprocal space. This width results in a coupling between different projections in particular at low spatial frequencies, providing a powerful constraint for phase retrieval. (c) Sketch of the iterative reconstruction algorithm devised to retrieve both the phase information and the three-dimensional distribution of the optical indices of the object. The Gerchberg-Saxton-like step explained in the text is represented by the dashed line.

Figure 2

Reconstruction quality for the standard approach of separated phase and tomographic reconstruction compared to the coupled reconstruction according to IRP, each for the case of a pure phase object and a mixed object. (a) Phase shift $\tilde{\delta }$ of the reference phantom. (b) Reconstructed phase shift of a pure phase object $\beta =0$ using the MGS algorithm (20 000 GS loops, followed by 128 ART iterations). The pure phase constraint has been enforced in the reconstruction. (c) Same as in (b) but with IRP reconstruction (${\kappa }_{max}=128$ ART iterations), and without using the fact that the phantom was a phase object. (d) Reconstructed phase shift of an object with fixed $\beta /\delta =0.02$ coupling, using the modified GS algorithm (20 000 GS loops, followed by 128 ART iterations). As in (b) the coupling value was enforced. (e) As in (d) but with IRP reconstruction (${\kappa }_{max}=128$ ART iterations), and without using any preknown coupling values. Slices of reconstructed volumes in the $yz$ plane (top row) are shown to illustrate the difference in data quality, along with the histograms of the $\tilde{\delta }$ values for the entire volume (bottom row). The scale bar applies to the entire row, the color bars to all images and the histograms. The frequency of occurrence in the histograms is shown in linear scale from 0 to 5000 counts.

Figure 3

Influence of Gaussian noise and reconstruction of experimental data. (a) $xy$ slice of the undisturbed pure phase object. (b) $xy$ slice of the MGS reconstruction from noisy pure phase object images ($\sigma =0.01$, 20 000 loops, followed by 128 ART iterations). (c) $xy$ slice of the IRP reconstruction from the same detector images as in (a) (${\kappa }_{max}=128$ ART iterations). (d) Typical holographic diffraction pattern (detector plane) of a bacterial cell [21]. (e) Slice though the modified HIO reconstruction as described in [21]. (f) Slice through the IRP reconstruction from the experimental data, based on 83 projections.