Paraxial diffusion-field retrieval

Unresolved spatially-random microstructure, in an illuminated sample, can lead to position-dependent blur when an image of that sample is taken using an incoherent imaging system. For a small propagation distance, between the exit surface of the sample and the entrance surface of a position-sensitive detector, the paraxial approximation implies that the blurring influence of the sample may be modeled using an anomalous-diffusion field. This diffusion field may have a scalar or tensor character, depending on whether the random microstructure has an autocorrelation function that is rotationally isotropic or anisotropic, respectively. Partial differential equations are written down and then solved, in a closed-form manner, for several variants of the inverse problem of diffusion-field retrieval given suitable intensity images. Both uniform-illumination and structured-illumination schemes are considered. Links are made, between the recovered diffusion field and certain statistical properties of the unresolved microstructure. The developed theory -- which may be viewed as a crudely parallel form of small-angle scattering under the Guinier approximation -- is applicable to a range of paraxial radiation and matter fields, such as visible light, x rays, neutrons, and electrons

Estimating uncertainty in multiple fibre reconstructions

Diffusion magnetic resonance imaging (MRI) is a technique that allows us to probe the microstructure of materials. The standard technique in diffusion MRI is diffusion tensor imaging (DTI). However, DTI can only model a single fibre orientation and fails in regions of complex microstructure. Multiple-fibre algorithms aim to overcome this limitation of DTI, but there remain many questions about which multiple-fibre algorithms are most promising and how best to exploit them in tractography. This work focuses on exploring the potential of multiple-fibre reconstructions and preparing them for transfer to the clinical arena. We provide a standardised framework for comparing multiple-fibre algorithms and use it for a robust comparison of standard algorithms, such as persistent angular structure (PAS) MRI, spherical deconvolution (SD), maximum entropy SD (MESD), constrained SD (CSD) and QBall. An output of this framework is the parameter settings of the algorithms that maximise the consistency of reconstructions. We show that non-linear algorithms, and CSD in particular, provide the most consistent reconstructions. Next, we investigate features of the reconstructions that can be exploited to improve tractography. We show that the peak shapes of multiple-fibre reconstructions can be used to predict anisotropy in the uncertainty of fibre-orientation estimates. We design an experiment that exploits this information in the probabilistic index of connectivity (PICo) tractography algorithm. We then compare PICo tractography results using information about peak shape and sharpness to estimate uncertainty with PICo results using only the peak sharpness to estimate uncertainty and show structured differences. The final contribution of this work is a robust algorithm for calibrating PICo that overcomes some of the limitations of the original algorithm. We finish with some early exploratory work that aims to estimate the distribution of fibre-orientations in a voxel using features of the reconstruction

Computationally Tractable Riemannian Manifolds for Graph Embeddings

Representing graphs as sets of node embeddings in certain curved Riemannian manifolds has recently gained momentum in machine learning due to their desirable geometric inductive biases, e.g., hierarchical structures benefit from hyperbolic geometry. However, going beyond embedding spaces of constant sectional curvature, while potentially more representationally powerful, proves to be challenging as one can easily lose the appeal of computationally tractable tools such as geodesic distances or Riemannian gradients. Here, we explore computationally efficient matrix manifolds, showcasing how to learn and optimize graph embeddings in these Riemannian spaces. Empirically, we demonstrate consistent improvements over Euclidean geometry while often outperforming hyperbolic and elliptical embeddings based on various metrics that capture different graph properties. Our results serve as new evidence for the benefits of non-Euclidean embeddings in machine learning pipelines.Comment: Submitted to the Thirty-fourth Conference on Neural Information Processing System