arXiv subjects

Clifford J. Nolan

Publications and source records attributed to Clifford J. Nolan.

5 recordsLinked to original sources

arXiv2026

Microlocal analysis of Radon transforms over quadric surfaces

We study the microlocal properties of generalized Radon transforms over a family of quadric hypersurfaces whose centers lie on an orientable hypersurface $S$. The quadric surfaces we consider are level sets of the quadratic form associated to a symmetric, invertible matrix $A$, with real entries. We study the singularities of the right and left projections of the canonical relation associated with these operators and show that they are determined by the signature of the matrix $A$ and the hypersurface $S$. If the matrix is positive/negative definite (i.e., the surface of integration is an ellipsoid) and $S$ is strictly convex, we prove that the singularities are folds. If the matrix is indefinite (i.e., the surface of integration is a hyperboloid-type quadric) and $S$ is either strictly convex or a cylinder, then cusp, fold, or blowdown singularities are present. We also study the case when the surface of integration is a paraboloid and show that the Bolker condition is satisfied.

math.CA↗

arXiv2025

Mitigation of Artifacts in Multistatic & Passive Radar Imaging Using Microlocal Analysis

In the analysis of many synthetic aperture radar (SAR) experiments the possibility of passive background signals being recorded simultaneously and corrupting the image is often overlooked. Our work addresses this by considering the multistatic experiment where two stationary emitters are "always on" so there is "crosstalk" between their signals. The model for the radar data is given by a Fourier integral operator, and we assume that the data cannot be separated into contributions from individual emitters. Using techniques of microlocal analysis, we show that "crosstalk" between emitters leads to artifacts in the image and we determine their locations relative to the scatterers that produced the data. To combat the harmful effects of crosstalk, we develop methods that allow us to create an image of a region of interest (ROI) that is free from such artifacts. The first method makes use of a carefully designed data acquisition geometry to localise artifacts away from a ROI, and the second is an image processing technique that displaces artifacts away from a ROI. These methods are verified via numerical implementation in MATLAB. The analysis carried out here is valuable in bistatic and multistatic radar experiments, where an unwanted, passive source is also being detected, as well as in passive imaging, where one wishes to produce a high-quality image purely from uncontrolled sources of illumination.

math.AP↗

arXiv2021

Time-harmonic diffuse optical tomography: Hölder stability of the derivatives of the optical properties of a medium at the boundary

We study the inverse problem in Optical Tomography of determining the optical properties of a medium $Ω\subset\mathbb{R}^n$, with $n\geq 3$, under the so-called diffusion approximation. We consider the time-harmonic case where $Ω$ is probed with an input field that is modulated with a fixed harmonic frequency $ω=\frac{k}{c}$, where $c$ is the speed of light and $k$ is the wave number. Under suitable conditions that include a range of variability for $k$, we prove a result of Hölder stability of the derivatives of the absorption coefficient $μ_a$ of any order at the boundary $\partialΩ$ in terms of the measurements, in the case when the scattering coefficient $μ_s$ is assumed to be known. The stability estimates rely on the construction of singular solutions of the underlying forward elliptic system, which extend results obtained in J. Differential Equations 84 (2): 252-272 for the single elliptic equation.

math.AP↗

arXiv2020

Lipschitz stability at the boundary for time-harmonic diffuse optical tomography

We study the inverse problem in Optical Tomography of determining the optical properties of a medium $Ω\subset\mathbb{R}^n$, with $n\geq 3$, under the so-called diffusion approximation. We consider the time-harmonic case where $Ω$ is probed with an input field that is modulated with a fixed harmonic frequency $ω=\frac{k}{c}$, where $c$ is the speed of light and $k$ is the wave number. We prove a result of Lipschitz stability of the absorption coefficient $μ_a$ at the boundary $\partialΩ$ in terms of the measurements in the case when the scattering coefficient $μ_s$ is assumed to be known and $k$ belongs to certain intervals depending on some a-priori bounds on $μ_a$, $μ_s$.

math.AP↗

arXiv2017

Singular FIOs in SAR Imaging, II: Transmitter and Receiver at Different Speeds

In this article, we consider two bistatic cases arising in synthetic aperture radar imaging: when the transmitter and receiver are both moving with different speeds along a single line parallel to the ground in the same direction or in the opposite directions. In both cases, we classify the forward operator $\Fc$ as a Fourier integral operator with fold/blowdown singularities. Next we analyze the normal operator $\Fc^*\Fc$ in both cases (where $\Fc^{*}$ is the $L^{2}$-adjoint of $\Fc$). When the transmitter and receiver move in the same direction, we prove that $\Fc^*\Fc$ belongs to a class of operators associated to two cleanly intersecting Lagrangians, $I^{p,l} (Δ, C_1)$. When they move in opposite directions, $\Fc^*\Fc$ is a sum of such operators. In both cases artifacts appear and we show that they are, in general, as strong as the bona-fide part of the image. Moreover, we demonstrate that as soon as the source and receiver start to move in opposite directions, there is an interesting bifurcation in the type of artifact that appears in the image.

math.AP↗