Related papers: Singularly Weighted X-ray Tensor Tomography

Using the machinery of unitary spherical harmonics due to Koornwinder, Folland and other authors, we~obtain expansions for the Szeg\"o and the weighted Bergman kernels of $M$-harmonic functions, i.e.~functions annihilated by the invariant…

In this article, we study the properties of the geodesic X-ray transform for asymptotically Euclidean or conic Riemannian metrics and show injectivity under non-trapping and no conjugate point assumptions. We also define a notion of lens…

We consider integral geometry inverse problems for unitary connections and skew-Hermitian Higgs fields on manifolds with negative sectional curvature. The results apply to manifolds in any dimension, with or without boundary, and also in…

Differential geometric approaches to the analysis and processing of data in the form of symmetric positive definite (SPD) matrices have had notable successful applications to numerous fields including computer vision, medical imaging, and…

The mechanism of describing quantum states by standard probability (tomographic one) instead of wave function or density matrix is elucidated. Quantum tomography is formulated in an abstract Hilbert space framework, by means of the identity…

This article studies the problem of approximating functions belonging to a Hilbert space $\mathcal H_d$ with a reproducing kernel of the form $$\tilde K_d(\boldsymbol x,\boldsymbol t):=\prod_{\ell=1}^d…

For smooth compact connected manifolds with strictly convex boundary, no conjugate points and a hyperbolic trapped set, we prove an equivalence principle concerning the injectivity of the X-ray transform $I_m$ on symmetric solenoidal…

We use methods of algebraic geometry to find new, effective methods for detecting the identifiability of symmetric tensors. In particular, for ternary symmetric tensors T of degree 7, we use the analysis of the Hilbert function of a finite…

This article surveys recent results aiming at obtaining refined mapping estimates for the X-ray transform on a Riemannian manifold with boundary, which leverage the condition that the boundary be strictly geodesically convex. These…

Let $\mathscr T$ be a rooted directed tree with finite branching index $k_{\mathscr T}$ and let $S_{\lambda} \in B(l^2(V))$ be a left-invertible weighted shift on ${\mathscr T}$. We show that $S_{\lambda}$ can be modelled as a…

We realize the relative discrete series of a weighted $L^2$-space on a bounded symmetric doamin as kernels of invariant Cauchy-Riemann operator, and thus as the spaces of nearly holomorphic functions.

We prove that the geodesic X-ray transform is injective on scalar functions and (solenoidally) on one-forms on simple Riemannian manifolds $(M,g)$ with $g \in C^{1,1}$. In addition to a proof, we produce a redefinition of simplicity that is…

Consider the product of $M$ quadratic random matrices with complex elements and no further symmetry, where all matrix elements of each factor have a Gaussian distribution. This generalises the classical Wishart-Laguerre Gaussian Unitary…

We study the variation of weighted Szeg\H{o} and Garabedian kernels on planar domains as a function of the weight. A Ramadanov type theorem is shown to hold as the weights vary. As a consequence, we derive properties of the zeros of the…

We study the Pareto frontier for two competing norms $\|\cdot\|_X$ and $\|\cdot\|_Y$ on a vector space. For a given vector $c$, the pareto frontier describes the possible values of $(\|a\|_X,\|b\|_Y)$ for a decomposition $c=a+b$. The…

We study embeddings between reproducing kernel Hilbert spaces $H(K)$ of functions of $d \in \mathbb{N} \cup \{\infty\}$ variables. The kernels $K$ are superpositions of weighted finite tensor products of a fixed univariate kernel. The basic…

This article extends the author's past work [Inv. Probl. Imaging, 10:2 (2016), 433--459] to attenuated X-ray transforms, where the attenuation is complex-valued and only depends on position. We give a positive and constructive answer to the…

The development of energy selective, photon counting X-ray detectors allows for a wide range of new possibilities in the area of computed tomographic image formation. Under the assumption of perfect energy resolution, here we propose a…

We give a Riemannian structure to the set $\Sigma$ of positive invertible unitized Hilbert-Schmidt operators, by means of the trace inner product. This metric makes of $\Sigma$ a nonpositively curved, simply connected and metrically…

The eleven-dimensional gravitational action invariant under local Poincare transformations is given by the dimensional continuation of the Euler class of ten dimensions. Here we show that the supersymmetric extension of this action leads,…