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David Bate

Publications and source records attributed to David Bate.

At least 19 recordsLinked to original sources
arXiv2026

Universal Phase Contrast in Micro-CT Systems

Conventional high-resolution micro-CT systems are regarded as attenuation-based unless visible Fresnel fringes reveal the presence of propagation-based phase contrast. Here we show that this interpretation is incomplete. When propagation-induced phase transfer is non-negligible relative to the system blur, micro-CT operates in a phase-transfer regime in which propagation improves spatial resolution relative to that expected from source and detector blur alone, irrespective of whether Fresnel fringes remain visible. Because micro-CT systems possess finite source and detector blur, this phase transfer is accompanied by a degree of hardware-induced phase retrieval (HIPR), ranging from under-HIPR to matched-HIPR and over-HIPR. We further identify detector-induced phase retrieval (DIPR) as the optimal case in which the detector provides all of the filtering required for phase retrieval, thereby preserving the phase-induced resolution enhancement while introducing spatial correlations between detected photons that reduce high-spatial-frequency noise. We derive analytical conditions for HIPR and DIPR, introduce frequency-domain metrics quantifying preserved phase transfer and detector compatibility, and validate the framework numerically with simulations and experimentally using custom and commercial micro-CT systems. Our results demonstrate that phase-induced resolution enhancement can occur in conventional micro-CT systems even in the absence of visible Fresnel fringes, changing how image formation, spatial resolution, and optimization should be interpreted.

physics.optics↗
arXiv2025

Laboratory-based x-ray microtomography with directional dark-field sensitivity

We demonstrate dark-field x-ray microtomography in a compact, laboratory-based system capable of resolving attenuation, phase, and anisotropic scattering signals with micrometer-scale resolution across centimetre-scale samples. The method is based on two-directional beam tracking (2DBT), which requires only a single optical element and is compatible with standard x-ray sources and detectors. We validate the system's capabilities through imaging of a custom-built phantom, a fibre-reinforced composite and ex-vivo biological tissues, including a bovine intervertebral disc, a rat heart, and a porcine meniscus. The results show that dark-field tomography provides complementary information to attenuation as well as to phase tomography, by revealing sub-resolution features such as fibre orientation and microstructural heterogeneity at length scales that are well below the voxel size. A key element of our system is its sensitivity to scattering along two orthogonal directions in the image plane, enabling the measurement of scattering anisotropy with a single exposure. As well as simple and robust, our approach is sensitive and precise. These findings demonstrate the potential of 2DBT for non-destructive and three-dimensional structural characterisation of samples and materials in engineering, materials science and biomedical applications.

physics.optics↗
arXiv2025

Shortcut Laakso spaces, pure PI unrectifiability and differentiability of Lipschitz functions

We construct a family of purely PI unrectifiable Lipschitz differentiability spaces and investigate the possible of Banach spaces targets for which Lipschitz differentiability holds. We provide a general investigation into the geometry of \emph{shortcut} metric spaces and characterise when such spaces are PI rectifiable, and when they are $Y$-LDS, for a given $Y$. The family of spaces arises as an example of our characterisations. Indeed, we show that Laakso spaces satisfy the required hypotheses.

math.FA↗
arXiv2025

Structure of Metric $1$-currents: approximation by normal currents and representation results

We prove the $1$-dimensional flat chain conjecture in any complete and quasiconvex metric space, namely that metric $1$-currents can be approximated in mass by normal $1$-currents. The proof relies on a new Banach space isomorphism theorem, relating metric $1$-currents and their boundaries to the Arens-Eells space. As a by-product, any metric $1$-current in a complete and separable metric space can be represented as the integral superposition of oriented $1$-rectifiable sets, thus dropping a finite dimensionality condition from previous results of Schioppa [Schioppa Adv. Math. 2016, Schioppa J. Funct. Anal. 2016]. The connection between the flat chain conjecture and the representation result is provided by a structure theorem for metric $1$-currents in Banach spaces, showing that any such current can be realised as the restriction to a Borel set of a boundaryless normal $1$-current. This generalizes, to any Banach space, the $1$-dimensional case of a recent result of Alberti-Marchese in Euclidean spaces [Alberti-Marchese 2023]. The argument of Alberti-Marchese requires the strict polyhedral approximation theorem of Federer for normal $1$-currents, which we obtain in Banach spaces.

math.MG↗
arXiv2025

On 1-regular and 1-uniform metric measure spaces

A metric measure space $(X,\mu)$ is 1-regular if \[0< \lim_{r\to 0} \frac{\mu(B(x,r))}{r}<\infty\] for $\mu$-a.e $x\in X$. We give a complete geometric characterisation of the rectifiable and purely unrectifiable part of a 1-regular measure in terms of its tangent spaces. A special instance of a 1-regular metric measure space is a 1-uniform space $(Y,\nu)$, which satisfies $\nu(B(y,r))=r$ for all $y\in Y$ and $r>0$. We prove that there are exactly three 1-uniform metric measure spaces.

math.MG↗
arXiv2025

Alberti representations, rectifiability of metric spaces and higher integrability of measures satisfying a PDE

We give a sufficient condition for a Borel subset $E\subset X$ of a complete metric space with $\mathcal{H}^n(E)<\infty$ to be $n$-rectifiable. This condition involves a decomposition of $E$ into rectifiable curves known as an Alberti representation. Precisely, we show that if $\mathcal{H}^n|_E$ has $n$ independent Alberti representations, then $E$ is $n$-rectifiable. This is a sharp strengthening of prior results of Bate and Li. It has been known for some time that such a result answers many open questions concerning rectifiability in metric spaces, which we discuss. An important step of our proof is to establish the higher integrability of measures on Euclidean space satisfying a PDE constraint. These results provide a quantitative generalisation of recent work of De Philippis and Rindler and are of independent interest.

math.MG↗
arXiv2024

Fragment-wise differentiable structures

The $p$-modulus of curves, test plans, upper gradients, charts, differentials, approximations in energy and density of directions are all concepts associated to the theory of Sobolev functions in metric measure spaces. The purpose of this paper is to give an analogous geometric and ``fragment-wise'' theory for Lipschitz functions and Weaver derivations, where $\infty$-modulus of curve fragments, $\ast$-upper gradients and Alberti representations play a central role. We give a new definition of fragment-wise charts and prove that they exists for spaces with finite Hausdorff dimension. We give a replacement for $p$-duality in terms of Alberti representations and $\infty$-modulus and present the theory of $\ast$-upper gradients. Further, we give new and sharper results for approximations of Lipschitz functions, which yields the density of directions. Our results are applicable to all complete and separable metric measure spaces. In the process, we show that there are strong parallels between the Sobolev and Lipschitz worlds.

math.CA↗
arXiv2023

On the closability of differential operators

We discuss the closability of directional derivative operators with respect to a general Radon measure $\mu$ on $\mathbb{R}^d$; our main theorem completely characterizes the vectorfields for which the corresponding operator is closable from the space of Lipschitz functions $\mathrm{Lip}(\mathbb{R}^d)$ to $L^p(\mu)$, for $1\leq p\leq\infty$. We also discuss the closability of the same operators from $L^q(\mu)$ to $L^p(\mu)$, and give necessary and sufficient conditions for closability, but we do not have an exact characterization. As a corollary we obtain that classical differential operators such as gradient, divergence and Jacobian determinant are closable from $L^q(\mu)$ to $L^p(\mu)$ only if $\mu$ is absolutely continuous with respect to the Lebesgue measure. We finally consider the closability of a certain class of multilinear differential operators; these results are then rephrased in terms of metric currents.

math.CA↗
arXiv2023

Multi-contrast x-ray identification of inhomogeneous materials and their discrimination through deep learning approaches

Recent innovations in x-ray technology (namely phase-based and energy-resolved imaging) offer unprecedented opportunities for material discrimination, however they are often used in isolation or in limited combinations. Here we show that the optimized combination of contrast channels (attenuation at three x-ray energies, ultra-small angle scattering at two, standard deviation of refraction) significantly enhances material identification abilities compared to dual-energy x-ray imaging alone, and that a combination of off-the-shelf machine learning approaches can effectively discriminate e.g., threat materials in complex datasets. The methodology is validated on a range of materials and image dataset that are both an order of magnitude larger than those used in previous studies. Our results can provide an effective methodology to discriminate, and in some cases identify, different materials in complex imaging scenarios, with prospective applications across the life and physical sciences. While the detection of threat materials is used as a demonstrator here, the methodology could be equally applied to e.g., the distinction between diseased and healthy tissues or degraded vs. pristine materials.

physics.app-ph↗
arXiv2023

Direct x-ray scattering signal measurements in edge-illumination/beam-tracking imaging and their interplay with the variance of the refraction signals

X-ray dark-field or ultra-small angle scatter imaging has become increasingly important since the introduction of phase-based x-ray imaging and is having transformative impact in fields such as in vivo lung imaging and explosives detection. Here we show that dark-field images acquired with the edge-illumination method (either in its traditional double mask or simplified single mask implementation) provide a direct measurement of the scattering function, which is unaffected by system-specific parameters such as the autocorrelation length. We show that this is a consequence both of the specific measurement setup and of the mathematical approach followed to retrieve the dark-field images. We show agreement with theoretical models for datasets acquired both with synchrotron and laboratory x-ray sources. We also introduce a new contrast mechanism, the variance of refraction, which is extracted from the same dataset and provides a direct link with the size of the scattering centres. We show that this can also be described by the same theoretical models. We study the behaviour of both signals vs. key parameters such as x-ray energy and scatterer radius. We find this allows quantitative, direct, multi-scale scattering measurements during imaging, with implications in all fields where dark-field imaging is used.

physics.optics↗
arXiv2023

Uniformly rectifiable metric spaces: Lipschitz images, Bi-Lateral Weak Geometric Lemma and Corona Decompositions

In their 1991 and 1993 foundational monographs, David and Semmes characterized uniform rectifiability for subsets of Euclidean space in a multitude of geometric and analytic ways. The fundamental geometric conditions can be naturally stated in any metric space and it has long been a question of how these concepts are related in this general setting. In this paper we prove their equivalence. Namely, we show the equivalence of Big Pieces of Lipschitz Images, Bi-lateral Weak Geometric Lemma and Corona Decomposition in any Ahlfors regular metric space. Loosely speaking, this gives a quantitative equivalence between having Lipschitz charts and approximations by nicer spaces. En route, we also study Reifenberg parameterizations.

math.MG↗
arXiv2023

Typical Lipschitz images of rectifiable metric spaces

This article studies typical 1-Lipschitz images of $n$-rectifiable metric spaces $E$ into $\mathbb{R}^m$ for $m\geq n$. For example, if $E\subset \mathbb{R}^k$, we show that the Jacobian of such a typical 1-Lipschitz map equals 1 $\mathcal{H}^n$-almost everywhere and, if $m>n$, preserves the Hausdorff measure of $E$. In general, we provide sufficient conditions, in terms of the tangent norms of $E$, for when a typical 1-Lipschitz map preserves the Hausdorff measure of $E$, up to some constant multiple. Almost optimal results for strongly $n$-rectifiable metric spaces are obtained. On the other hand, for any norm $|\cdot|$ on $\mathbb{R}^m$, we show that, in the space of 1-Lipschitz functions from $([-1,1]^n,|\cdot|_\infty)$ to $(\mathbb{R}^m,|\cdot|)$, the $\mathcal{H}^n$-measure of a typical image is not bounded below by any $\Delta>0$.

math.MG↗
arXiv2022

X-ray phase-contrast micro tomography of soft tissues using a compact laboratory system with two-directional sensitivity

X-ray micro tomography is a non-destructive, three-dimensional inspection technique applied across a vast range of fields and disciplines, ranging from research to industrial, encompassing engineering, biology and medical research. Phase-contrast imaging extends the domain of application of X-ray micro tomography to classes of samples that exhibit weak attenuation, thus appear with poor contrast in standard X-ray imaging. Notable examples are low-atomic-number materials, like carbon-fibre composites, soft matter and biological soft tissues. We report on a compact and cost effective system for X-ray phase-contrast micro tomography. The system features high sensitivity to phase gradients and high resolution, requires a low-power sealed X-ray tube, a single optical element, and fits in a small footprint. It is compatible with standard X-ray detector technologies: single-photon-counting offers higher sensitivity whereas flat-panels are preferred for a larger field of view. The system is benchmarked against known-material phantoms and its potential for soft-tissue three-dimensional imaging is demonstrated on small-animal organs: a piglet oesophagus and a rat heart.

physics.ins-det↗
arXiv2022

Bi-Lipschitz embeddings of the space of unordered $m$-tuples with a partial transportation metric

Let $\Omega\subset \mathbb{R}^n$ be non-empty, open and proper. Consider $Wb(\Omega)$, the space of finite Borel measures on $\Omega$ equipped with the partial transportation metric introduced by Figalli and Gigli that allows the creation and destruction of mass on $\partial \Omega$. Equivalently, we show that $Wb(\Omega)$ is isometric to a subset of all Borel measures with the ordinary Wasserstein distance, on the one point completion of $\Omega$ equipped with the shortcut metric \[\delta(x,y)= \min\{\|x-y\|, \operatorname{dist}(x,\partial \Omega)+\operatorname{dist}(y,\partial\Omega)\}.\] In this article we construct bi-Lipschitz embeddings of the set of unordered $m$-tuples in $Wb(\Omega)$ into Hilbert space. This generalises Almgren's bi-Lipschitz embedding theorem to the setting of optimal partial transport.

math.MG↗
arXiv2021

Characterising rectifiable metric spaces using tangent spaces

We characterise rectifiable subsets of a complete metric space $X$ in terms of local approximation, with respect to the Gromov--Hausdorff distance, by an $n$-dimensional Banach space. In fact, if $E\subset X$ with $\mathcal{H}^n(E)<\infty$ and has positive lower density almost everywhere, we prove that it is sufficient that, at almost every point and each sufficiently small scale, $E$ is approximated by a bi-Lipschitz image of Euclidean space. We also introduce a generalisation of Preiss's tangent measures that is suitable for the setting of arbitrary metric spaces and formulate our characterisation in terms of tangent measures. This definition is equivalent to that of Preiss when the ambient space is Euclidean, and equivalent to the measured Gromov--Hausdorff tangent space when the measure is doubling.

math.MG↗
arXiv2019

Cheeger's differentiation theorem via the multilinear Kakeya inequality

Suppose that $(X,d,\mu)$ is a metric measure space of finite Hausdorff dimension and that, for every Lipschitz $f \colon X \to \mathbb R$, $\operatorname{Lip}(f,\cdot)$ is dominated by every upper gradient of $f$. We show that $X$ is a Lipschitz differentiability space, and the differentiable structure of $X$ has dimension at most $\dim_{\mathrm{H}} X$. Since our assumptions are satisfied whenever $X$ is doubling and satisfies a Poincar\'e inequality, we thus obtain a new proof of Cheeger's generalisation of Rademacher's theorem. Our approach uses Guth's multilinear Kakeya inequality for neighbourhoods of Lipschitz graphs to show that any non-trivial measure with $n$ independent Alberti representations has Hausdorff dimension at least $n$.

math.MG↗