This thesis develops theoretical formalisms and experimental procedures for “Multimodal Intrinsic Speckle- Tracking (MIST)”, a complementary signal retrieval approach appropriate for the speckle-based phase-contrast X-ray imaging (SB-PCXI) technique. SB-PCXI exploits random speckle modulations embedded into an X-ray wavefield to retrieve sample attenuation, phase-shift, and diffusion information. The three retrieved signals provide complementary sample information to one another and, hence, provide a more complete description of the sample compared to just one mechanism alone (for example, clinical radiography, which only considers X-ray attenuation). Contrast generated when just considering X-ray attenuation alone works well for materials that have significantly different X-ray attenuation properties, that is, have different densities – take, for instance, bone compared to soft tissues in a dental radiography image. When the X-ray phase is utilized for image contrast, sample features that weakly attenuate the beam but refract the beam differently are well resolved – for example, two different types of soft tissue like fat compared to glandular tissue. Finally, diffusion information is useful for visualising structures that impose intensity variations in the X-ray wavefield that are smaller than the spatial resolution of the imaging system – for example, unresolved sample microstructure like calcifications in a malignant breasttissue sample. SB-PCXI has the appealing feature of being experimentally simple and adaptable. The mask in SB-PCXI simply needs to be spatially random and generate a well-resolved reference speckle pattern on the X-ray imaging system’s detector. Distance, i.e. free space propagation, is introduced between the sample and the detector such that the refracted speckles will drift across the detector, and the diffusive dark-field (DDF)-affected speckles will diffuse/blur across a local area of the detector. It follows that the capabilities or sensitivity of an SB-PCXI system can be modified simply by changing the speckle-generating mask, as well as the sample-to-detector distance and X-ray energy – low-energy X-rays will be refracted and diffused more easily than X-rays of higher energy. The theoretical approach of MIST makes SB-PCXI even more attractive for widespread use as it is computationally rapid; the MIST algorithm operates on a whole-image level rather than tracking each speckle with local windows, which has been the dominant approach until now. MIST employs the Fokker-Planck equation for paraxial X-ray imaging and combines it with a geometric-flow formalism for SB-PCXI. In essence, MIST analyses sample-imposed speckle changes by considering local energy conservation for each speckle in the SB-PCXI regime. MIST was first realised in 2020 but under stringent sample requirements. Throughout this doctoral study, significant improvements to the MIST algorithm, as well as supporting experimental and theoretical procedures, have been developed and implemented to relax these early requirements. The overarching motive for this thesis study is to assist with the translation of SB-PCXI from a synchrotronbased technique into a user-friendly multimodal X-ray imaging technique that can be utilized in multiple diverse applications.

Theoretical derivations of formalisms, experimental data collections, software implementations, and presentation of results are provided for each of the sub-studies constituting this thesis. The primary focus is on the technique of synchrotron-based SB-PCXI, which involves coherent X-ray wavefields and nearfield speckles. Chapter 1 and Chapter 2 of this thesis provide a theoretical and experimental introduction, respectively, to the relevant concepts of SB-PCXI and MIST. Chapters 3–6 contain the principal developments and findings of this thesis.

In Chapter 3, an algorithm capable of recovering true X-ray attenuation, β, and refraction, δ, coefficients of the complex refractive index of distinct materials within an unknown sample (no a priori information about constituting materials in the sample is required) is presented. Extracting a sample’s β and δ (i) allows full characterisation of the sample and (ii) allows retrieval algorithms that require these values as a priori information to be performed optimally, for example, MIST. In short, the algorithm operates on an incorrectly phase-retrieved axial computed tomography (CT) slice and curve-fits an error-function model to distinguishable interfaces in the sample. The curve-fit parameters can then be used to determine exact sample information. The general approach was successfully applied to two biological data sets collected using a propagation-based phase-contrast X-ray imaging (PB-PCXI) technique on both a synchrotron Xray source and a laboratory microfocus source.

Chapters 4 and 5 each derive, implement, and discuss generalised MIST formalisms. The first MIST variant described in Chapter 4 (abbreviated to MISTAtten.&Slow−Varying in this section) alleviates the assumption of no sample X-ray attenuation, which was assumed in the first-published MIST approach. The second developed MIST approach in Chapter 5 (abbreviated to MISTW.−Atten.&Rapid−Varying in this section) relieves the assumption that sample-induced X-ray diffusion is spatially slowly varying within the sample, an assumption that was present in all earlier MIST formalisms. Within both of these developed MIST variants, associated numerical stabilisation techniques were also presented such that the ill-posed multimodal inverse problem was sufficiently regularised and optimal images could be retrieved. The MISTAtten.&Slow−Varying formalism, which considers X-ray attenuation but assumes the DDF to be slowly varying, reconstructed sufficiently high-quality DDF images of a wood sample such that CT could be performed on the recovered projection set. Optimised (in terms of pre- and post-processing) 3D multimodal images are presented, and the complementarity of the phase-shift and DDF information of the wood sample is particularly noticeable in these images. The MISTW.−Atten.&Rapid−Varying formalism developed in the thesis can describe samples with unresolved microstructure which varies rapidly with transverse position – something that no prior MIST approach is capable of doing. This generalised approach is restricted to weakly attenuating samples, however, it was found to work effectively on a red currant sample that had moderate attenuation characteristics. The new variant of MIST, MISTW.−Atten.&Rapid−Varying, was successfully applied to two sets of synchrotron SB-PCXI data. The results reveal that the generalised rapidly-varying MIST approach reconstructs higher-quality images in terms of signal-to-noise ratio, perceived image quality, and spatial resolution compared to previous MIST approaches. The development of the MISTW.−Atten.&Rapid−Varying approach means that a wider class of sample types can be characterized by MIST, thus increasing its breadth of applicability.

Chapter 6 is comprised of two comparison studies: (i) the two evolution models of the SB-PCXI Fokker- Planck equation in the context of solving the inverse problem using the MIST approach and (ii) MIST and a selection of other emerging DDF-sensitive imaging techniques. The two evolution models are called “evolving” and “devolving”, and they consider the blurring of speckles due to the introduction of a sample and the sharpening of speckles due to the removal of a sample from the system, respectively. The two primary contributions to DDF contrast in MIST approaches are local-small-angle X-ray scattering (SAXS) and sharp-edge diffusion thought to result from the Young-Maggi-Rubinowicz boundary-wave mechanism. Two different MIST approaches were reformulated, beginning with the devolving SB-PCXI Fokker-Planck equation, to explore the differences in the associated inverse problem. The different MIST approaches for the evolving and devolving models were applied to three synchrotron SB-PCXI data sets. All of the reconstructed DDF signals revealed the same result: the devolving SB-PCXI Fokker-Planck model can separate the two mechanisms that generate DDF contrast, and the evolving cannot. The reconstructed DDF values when using the evolving model were primarily positive; in particular, local- SAXS and sharp-edge-induced diffusion were both attributed to positive DDF values. Conversely, the devolving-retrieved DDF signals had a significant and equal amount of both positive and negative DDF values; specifically, the local-SAXS-induced X-ray diffusion was reconstructed with positive DDF values and diffusion from sharp edges in a sample was attributed to the negative DDF signal. The devolving reconstructions, therefore, provide an added level of complementarity on top of what is already provided by the multimodal signals themselves. Mentions of negative diffusion in a Fokker-Planck-type model are sparse, even in the most recent research literature. Within the presented thesis, a hypothesised justification for this concept is provided, based on the fundamentals of X-ray optics as well as analogies made to the commonly employed forward and backward Kolmogorov equations (which find similarities to the presented evolving and devolving Fokker-Planck equations, respectively). These results provide an entirely new perspective on retrieving and interpreting DDF signals – an exciting avenue for future research. The second comparison analysis in Chapter 6 is presented to determine the MIST approach’s “position” within current emerging DDF techniques. DDF images were retrieved using a dual-energy PB-PCXI approach, a single two-dimensional (2D) periodic grid-based approach, an alternative single-exposure speckle-tracking algorithm, and a multi-exposure speckle-tracking algorithm derived within this thesis (namely, MISTW.−Atten.&Rapid−Varying). The reconstructed DDF signals of two different samples showed a mutual agreement between the approaches for low-spatial-frequency features in the samples. At high spatial frequencies, the retrieved DDF signals diverged for the approaches, revealing the idea of implicit “rulers” for each approach. These rulers describe a technique’s sensitivity to X-ray diffusion mechanisms, that is, whether or not particular sample features are captured in the recovered DDF image. Each approach has distinct underlying assumptions and performs different mathematical operations to retrieve multimodal images. It therefore follows that the approaches have different sensitivities, particularly at length scales close to the threshold of being either resolved or unresolved.

This thesis has shown that the MIST approach can retrieve high-quality complementary signals from a sample in 2D projection and 3D CT. In addition, more-generalised MIST variants (in terms of the sample restrictions) reconstruct superior images, however, achieving numerical stability of the associated inverse problem gets slightly more difficult as the number of underlying sample constraints is reduced. Some of the potential future research avenues stemming from the results of this doctoral study include: (a) Developing a fully generalised MIST approach for isotropic X-ray diffusion that is numerically stable as well as computationally simple and rapid. (b) Refining the associated DDF-CT procedure in terms of a suitable reconstruction algorithm as well as pre- and post-processing techniques. (c) Understanding the mechanism of sharp-edge induced X-ray diffusion and the associated devolving MIST-retrieved negative diffusion coefficient. (d) Applying the MIST algorithm to a wide class of sample types and SB-PCXI experimental configurations such that a sample–versus–set-up space can be deduced. This would allow SB-PCXI users to arrange the setup optimally for their sample type and ensure that the retrieved multimodal signals are suitable for their intended purpose. (e) Developing and implementing SB-PCXI experimental set-ups in synchrotron beamlines and laboratories all around the world, as well as user-friendly software that could perform MIST algorithms and other related approaches. In the current research literature, multimodal imaging has proven to be particularly beneficial in clinical and industrial applications. Thus, designing and building an experimental system capable of implementing SB-PCXI in these applications would be particularly exciting. In conclusion, SB-PCXI with MIST image retrieval is an experimentally simple multimodal technique that is capable of rapidly retrieving high-quality complementary X-ray images.