Locality estimates for Fresnel-wave-propagation and stability of x-ray phase contrast imaging with finite detectors

Abstract

Coherent wave-propagation in the near-field Fresnel-regime is the underlying contrast-mechanism to (propagation-based) X-ray phase contrast imaging (XPCI), an emerging lensless technique that enables 2D-and 3D-imaging of biological soft tissues and other light-element samples down to nanometer-resolutions. Mathematically, propagation is described by the Fresnel-propagator, a convolution with an arbitrarily non-local kernel. As real-world detectors may only capture a finite field-of-view, this non-locality implies that the recorded diffraction-patterns are necessarily incomplete. This raises the question of stability of image-reconstruction from the truncated data -even if the complex-valued wave-field, and not just its modulus, could be measured. Contrary to the latter restriction of the acquisition, known as the phase-problem, the finite-detector-problem has not received much attention in literature. The present work therefore analyzes locality of Fresnelpropagation in order to establish stability of XPCI with finite detectors. Image-reconstruction is shown to be severely ill-posed in this setting -even without a phase-problem. However, quantitative estimates of the leaked wave-field reveal that Lipschitz-stability holds down to a sharp resolution limit that depends on the detector-size and varies within the field-of-view. The smallest resolvable lengthscale is found to be ≈ 1/f times the detector's aspect length, wheref is the Fresnel number associated with the latter scale. The stability results are extended to phaseless imaging in the linear contrast-transfer-function regime.(1) Although we will refer to "real-valued" signals throughout the work, note that all the results obtained for such trivially extend to signals given by real-functions multiplied by a global complex phase.

Fast algorithms for nonlinear and constrained phase retrieval in near-field X-ray holography based on Tikhonov regularization

“…As the naming suggests it, the modification enforces a higher penalization for spatial frequencies that scatter under angles exceeding the numerical aperture (NA) of the imaging system, as defined by the field-of-view covered by the detector. Note that the corresponding high-frequency Fourier-modes are thus not sufficiently represented in the acquired hologram data anyway so that these could not be stably recovered [37]. Thus, nothing is really lost by damping out these frequencies, while stability is gained on the algorithmic side.…”

Hard X-ray Microscopy Enhanced by Coherent Image Reconstruction

“…Finite-difference propagation for simulation of x-ray multilayer optics results in a particularly well posed problem for phase retrieval [Mar18]. These spatial frequencies are therefore easily phased based on the enlarged holographic signal in the +1 st and -1 st diffraction orders.…”

Inverse wave propagation problems without phase information

“…In [10] a stability analysis of phaseless linearized near-field scattering under the Fresnel approximation is derived for the case of finite detector sizes. The analysis is extended for infinite detectors in [11], which shows Lipschitz stability estimates in terms of the Fresnel number.…”