In this talk we overview recent results on the analysis of resolution of reconstruction from discrete Radon transform data. We call our approach Local Resolution Analysis, or LRA. LRA yields simple formulas describing the reconstruction from discrete data in a neighborhood of the singularities of $f$ in a variety of settings. We call these formulas the Discrete Transition Behavior (DTB). The DTB function provides the most direct, fully quantitative link between the data sampling rate and resolution. This link is now established for a wide range of integral transforms, conormal distributions $f$, and reconstruction operators. Recently the LRA was generalized to the reconstruction of objects with rough edges. Numerical experiments demonstrate that the DTB functions are highly accurate even for objects with fractal boundaries.

Tomographic X-ray imaging on the nano-scale is an important tool to visualise the structure of materials such as alloys or biological tissue. Due to the small scale on which the data acquisition takes place, small perturbances caused by the environment become significant and cause a motion of the object relative to the scanner during the scan.

An iterative reconstruction method called RESESOP-Kaczmarz was introduced in [1] which requires the motion to be estimated. However, since the motion is hard to estimate and its incorporation into the reconstruction process strongly increases the numerical effort, we investigate a learned version of RESESOP-Kaczmarz. Imaging data was programmatically simulated to train a deep network which unrolls the iterative image reconstruction of the original algorithm. The network therefore learns the back-projected image after a fixed number of iterations.

[1] S. E. Blanke et al. Inverse problems with inexact forward operator: iterative regularization and application in dynamic imaging, Inverse Problems 36 124001, 2020.

X-ray computed tomography on the nano-meter scale is a challenging imaging task. Tiny perturbations, such as environmental vibrations, technology imprecision or a thermal drift over time, lead to considerable deviations in the measured projections for nano-CT. Reconstruction algorithms must take into account the presence of these deviations in order to avoid strong artifacts. We study different learned post-processing approaches for nano-CT reconstruction on simulated datasets featuring relative object shifts and rotations. The initial reconstruction is provided by a classical method (FBP, Kaczmarz) or a deviation-aware method (Dremel method, RESESOP-Kaczmarz). Neural networks are then trained supervisedly to post-process such initial reconstructions. We consider (i) a directly trained U-Net post-processing, and (ii) conditional normalizing flows, which learn an invertible mapping between a simple random distribution and the image reconstruction space, conditioned on the initial reconstruction. Normalizing flows do not only yield a reconstruction, but also an estimate of the posterior density. As a simple indicator of reconstruction uncertainty one may evaluate the pixel-wise standard deviation over samples from the estimated posterior.

Conventional x-ray imaging, which measures the intensity of the transmitted x-ray wavefield, is extremely useful when imaging strongly-attenuating samples like bone, but of limited use when imaging weakly-attenuating samples like the lungs or brain. In recent years, it has been seen that the phase of the transmitted x-ray wavefield contains useful information about these weakly-attenuating samples, however it is not possible to directly measure x-ray phase. This necessitates the use of mathematical models that relate the observed x-ray intensity, which is measurable, to the x-ray phase. These models can then be solved to retrieve how the sample has changed the x-ray phase; the inverse problem of phase retrieval. A widely adopted example is the use of the Transport of Intensity Equation (TIE) to retrieve x-ray phase from an intensity image collected some distance downstream of the sample, a distance at which sample-induced phase variations have resulted in self-interference of the wave and changed the local observed intensity [1]. The use of a single-exposure ‘propagation-based’ set-up like this, where no optics (gratings, crystals etc.) are required, makes for a robust and simple x-ray imaging set-up, which is also compatible with time-sequence imaging.

In this talk, we present an extension to the TIE, the X-ray Fokker-Planck Equation [2, 3], and associated novel retrieval algorithms for extracting x-ray phase and dark-field [5-7] from propagation-based images.

The TIE describes how x-ray intensity evolves with propagation from the sample to a downstream camera, for a wavefield with given phase and intensity. The X-ray Fokker-Planck Equation adds an additional term that incorporates how dark-field effects from the sample will be seen in the observed intensity [2,3]. X-ray dark-field effects are present when the sample contains microstructures that are not directly resolved, but which scatter the wavefield in such a way as to locally reduce image contrast. Examples of dark-field-inducing microstructure include powders, carbon fibres and the air sacs in the lungs. Until very recently [4], it was considered that crystals or gratings were required optical elements in the experimental set-up in order to capture a dark-field image.

Using the X-ray Fokker-Planck Equation, we have derived several novel algorithms that allow dark-field retrieval from propagation-based images. Because phase and dark-field effects evolve differently with propagation, images captured at two different sample-to-detector distances allow the separation and retrieval of dark-field images and phase images [5]. Alternatively, dark-field and phase images can be retrieved by looking at sample-induced changes in a patterned illumination via a Fokker-Planck approach, either using a single short exposure [6], or by scanning the pattern across the sample to access the full spatial resolution of the detector [7]. Incorporating dark-field effects in the TIE not only allows a dark-field image to be extracted from propagation-based images, but also increases the potential spatial resolution of the retrieved phase image. These propagation-based Fokker-Planck approaches are best suited to small samples (e.g. under 10 cm), so may provide avenues for fast and simple phase and dark-field micro/nano-tomography.

[1] D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, S. W. Wilkins. Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object, Journal of Microscopy 206(1): 33-40, 2002.

[2] K. S. Morgan, D. M. Paganin. Applying the Fokker–Planck equation to grating-based x-ray phase and dark-field imaging, Scientific Reports 9(1): 17465, 2019.

[3] D.M. Paganin, K. S. Morgan. X-ray Fokker–Planck equation for paraxial imaging, Scientific Reports, 9(1): 17537, 2019.

[4] T.E. Gureyev, D.M. Paganin, B. Arhatari, S. T. Taba, S. Lewis, P. C. Brennan, H. M. Quiney. Dark-field signal extraction in propagation-based phase-contrast imaging, Physics in Medicine & Biology, 65(21): 215029, 2020.

[5] T. A. Leatham, D. M. Paganin, K. S. Morgan. X-ray dark-field and phase retrieval without optics, via the Fokker–Planck equation, IEEE Transactions on Medical Imaging (in press), 2023.

[6] M. A. Beltran, D. M. Paganin, M. K. Croughan, K. S. Morgan. Fast implicit diffusive dark-field retrieval for single-exposure, single-mask x-ray imaging, Optica, 10(4): 422-429, 2023.

[7] S. J. Alloo, K. S. Morgan, D. M. Paganin, K. M. Pavlov. Multimodal intrinsic speckle-tracking (MIST) to extract images of rapidly-varying diffuse X-ray dark-field, Scientific Reports, 13(1): 5424, 2023.