arXiv subjects

Fatma Terzioglu

Publications and source records attributed to Fatma Terzioglu.

12 recordsLinked to original sources

arXiv2026

An analytic inversion formula for the n-dimensional cylindrical Radon transform

We study the overdetermined problem of inverting the n-dimensional cylindrical Radon transform, which maps a function to its integrals over cylindrical surfaces. In practice, the cylindrical Radon transform arises in photoacoustic tomography as a measurement model for integrating line detectors. By exploiting its relationship with the Funk and Radon transforms, we derive an analytic inversion formula for the cylindrical Radon transform. We also present numerical implementations of the proposed inversion algorithm, demonstrating its stability.

math.FA↗

arXiv2026

Stability of the $k$-Plane Transform on Measures and H\"older-Type Comparisons of Wasserstein Metrics

We establish stability estimates for the $k$-plane transform on finite positive Radon measures, with emphasis on Fourier and Wasserstein metrics. We first introduce a metric on $k$-plane transform data and prove a bi-Lipschitz stability estimate showing that this metric is equivalent to a generalized Fourier metric obtained by augmenting the Fourier distance between centered normalized measures with separate barycenter and total mass difference terms. Building on a H\"older-type comparison between Fourier and Wasserstein metrics due to Carrillo and Toscani, we extend this comparison to positive Radon measures under uniform bounds on centered moments of order slightly larger than $2$. This yields H\"older-type stability for the $k$-plane transform in a generalized $2$-Wasserstein metric and, in particular, a $W_2$-stability estimate for centered probability measures. We also compare the $2$-Wasserstein distance with its max-sliced analogue. For centered probability measures with uniformly bounded moments of order slightly larger than $2$, we prove a two-sided H\"older-type comparison between these distances. We then extend the result to positive Radon measures by applying it to centered normalized measures and adding separate barycenter and mass terms. Finally, for absolutely continuous compactly supported probability measures with bounded densities, we prove a strong equivalence between the $2$-Wasserstein distance of the measures and the $(k/2-1)$-order Sobolev norm of the $k$-plane transform data of the difference of their densities.

math.FA↗

arXiv2026

Range Characterization of the Weighted Divergent Beam and Cone Integral Transforms

We establish range characterizations, or data consistency conditions, for an integral transform that maps a function to its weighted integrals over conical surfaces in $\mathbb{R}^n$. We consider two different geometries for the cone vertices, which lead to mathematically distinct range conditions. We use the term \emph{conical Radon transform} when the vertex set is a bounded convex subset of $\mathbb{R}^n$ including support of the input function. The second geometry is motivated by Compton camera imaging where the vertex set represents planar detector locations and is disjoint from the support of the input function representing the radiation density. We refer to the associated transform as the \emph{Compton transform}. Our approach is based on a factorization into the $k$-weighted divergent beam transform and the spherical section transform. In the bounded convex vertex geometry, the range of the divergent beam component is described by a higher-order transport boundary-value problem, as studied by Derevtsov, Volkov, and Schuster [7]. In the planar detector geometry, we derive range conditions for the $k$-weighted divergent beam transform that generalize the planar cone-beam consistency conditions of Clackdoyle and Desbat [5]. Combining these results with the range characterization of the spherical section transform yields complete range descriptions for both the $k$-weighted conical Radon transform and the $k$-weighted Compton transform.

math.FA↗

arXiv2026

THE ROLE OF FOURIER ANALYSIS IN TWO DIMENSIONAL TOMOGRAPHY

We highlight the important role of the Fourier transform in deriving inversion formulas for the integral transforms of tomographic imaging. We demonstrate this principle by deriving inversion formulas for the divergent beam transform and the V-line transform, the latter arising in contemporary models of single-scattering optical tomography.

physics.optics↗

arXiv2026

Mapping estimates for the $k$-plane transform in Sobolev, Besov, and Triebel--Lizorkin Spaces

We study mapping properties of the $k$-plane transform in Sobolev, Besov, and Triebel--Lizorkin spaces. For $1\le k\le d-1$, the $k$-plane transform integrates a function over $k$-dimensional affine planes in $\mathbb{R}^d$, yielding a function on the affine Grassmannian $\mathcal{G}_{k,d}$. First, we establish Sobolev stability estimates for compactly supported functions, extending classical results of Natterer for the X-ray ($k=1$) and Radon ($k=d-1$) transforms to the general $k$-plane transform. Second, we extend isometry identities for the Radon and X-ray transforms, due to Reshetnyak, Sharafutdinov, and Kindermann--Hubmer, to the $k$-plane transform. Finally, we prove boundedness of the $k$-plane transform in Besov and Triebel--Lizorkin spaces.

math.FA↗

arXiv2024

Non-unique water and contrast agent solutions in dual-energy CT

The goal of this work is to study occurrences of non-unique solutions in dual-energy CT (DECT) for objects containing water and a contrast agent. Previous studies of the Jacobian of nonlinear systems identified that a vanishing Jacobian determinant indicates the existence of multiple solutions to the system. Vanishing Jacobian determinants are identified for DECT setups by simulating intensity data for practical thickness ranges of water and contrast agent. Once existence is identified, non-unique solutions are found by simulating scan data and finding intensity contours with that intersect multiple times. With this process non-unique solutions are found for DECT setups scanning iodine and gadolinium, including setups using tube potentials in practical ranges. Non-unique solutions demonstrate a large range of differences and can result in significant discrepancies between recovered and true material mapping.

physics.med-ph↗

arXiv2023

Optimizing dual-energy CT technique for iodine-based contrast-to-noise ratio

Purpose: This study proposes a systematic method for determining the optimal x-ray tube settings/energy windows and fluence for minimal noise and maximum CNR in material density images obtained from DECT scans by fixing the subject size and the total radiation dose. Methods: The noise propagation in the process of sinogram and image reconstruction from DECT measurements is analyzed. Analytic estimates for the sinogram and monochromatic image pixel variances and the CNR as functions of tube potentials, fluence, and virtual monochromatic image (VMI) energy are derived, and then used in a phantom experiment as an objective function for optimizing the tube settings to minimize the image noise and maximize the CNR. Results: A non-trivial example that shows the existence of singular solutions to the inversion of sinograms-to-DECT measurements map was presented. Additionally, the optimal VMI energy for maximal CNR was determined. The optimal energy VMI was found to be the least noisy monochromatic image synthesized from the iodine and water density images, and it was shown that using more general weights in combining the two images linearly does not improve image quality. When the x-ray beam filter material was fixed at 2mm of Aluminum and the photon fluence for low and high kV scans were considered equal, the tube potential pair of 60/120 kV led to the maximal CNR in the VMI formed at energy 55 KeV. Conclusions: Optimizing DECT scan parameters to maximize the CNR can be done in a systematic way. Also choosing the parameters that maximize the Jacobian determinant over the sinogram domain would lead to more stable reconstructions due to the reduced amplification of the measurement noise. Since the values of the Jacobian determinant depend strongly on the imaging task, careful consideration of all of the relevant factors is needed when implementing the proposed framework.

physics.med-ph↗

arXiv2021

An inversion algorithm for P-functions with applications to Multi-energy CT

Multi-energy computed tomography (ME-CT) is an x-ray transmission imaging technique that uses the energy dependence of x-ray photon attenuation to determine the elemental composition of an object of interest. Mathematically, forward ME-CT measurements are modeled by a nonlinear integral transform. In this paper, local conditions for global invertibility of the ME-CT transform are studied, and explicit stability estimates quantifying the error propagation from measurements to reconstructions are provided. Motivated from the inverse problem of image reconstruction in ME-CT, an iterative inversion algorithm for the so-called P-functions is proposed. Numerical simulations for ME-CT, in two and three materials settings with an equal number of energy measurements, confirm the theoretical predictions.

physics.med-ph↗

arXiv2020

Uniqueness criteria in multi-energy CT

Multi-Energy Computed Tomography (ME-CT) is a medical imaging modality aiming to reconstruct the spatial density of materials from the attenuation properties of probing x-rays. For each line in two- or three-dimensional space, ME-CT measurements may be written as a nonlinear mapping from the integrals of the unknown densities of a finite number of materials along said line to an equal or larger number of energy-weighted integrals corresponding to different x-ray source energy spectra. ME-CT reconstructions may thus be decomposed as a two-step process: (i) reconstruct line integrals of the material densities from the available energy measurements; and (ii) reconstruct densities from their line integrals. Step (ii) is the standard linear x-ray CT problem whose invertibility is well-known, so this paper focuses on step (i). We show that ME-CT admits stable, global inversion provided that (a well-chosen linear transform of) the differential of the transform in step (i) satisfies appropriate orientation constraints that makes it a P-matrix. We introduce a notion of quantitative P-function that allows us to derive global stability results for ME-CT in the determined as well as over-determined (with more source energy spectra than the number of materials) cases. Numerical simulations based on standard material properties in imaging applications (of bone, water, contrast agents) and well accepted models of source energy spectra show that ME-CT is often (always in our simulations) either (i) non-globally injective because it is non-injective locally (differential not of full rank), or (ii) globally injective as soon as it is locally injective (differentials satisfy our proposed constraints).

math.NA↗

arXiv2016

Inversion of Weighted Divergent Beam and Cone Transforms

In this paper, we investigate the relations between the Radon and weighted divergent beam and cone transforms. Novel inversion formulas are derived for the latter two. The weighted cone transform arises, for instance, in image reconstruction from the data obtained by Compton cameras, which have promising applications in various fields, including biomedical and homeland security imaging and gamma ray astronomy. The inversion formulas are applicable for a wide variety of detector geometries in any dimension. The results of numerical implementation of some of the formulas in dimensions two and three are also provided.

math.NA↗

arXiv2016

3D Image Reconstruction from Compton camera data

In this paper, we address analytically and numerically the inversion of the integral transform (\emph{cone} or \emph{Compton} transform) that maps a function on $\mathbb{R}^3$ to its integrals over conical surfaces. It arises in a variety of imaging techniques, e.g. in astronomy, optical imaging, and homeland security imaging, especially when the so called Compton cameras are involved. Several inversion formulas are developed and implemented numerically in $3D$ (the much simpler $2D$ case was considered in a previous publication). An admissibility condition on detectors geometry is formulated, under which all these inversion techniques will work.

physics.data-an↗

arXiv2015

Some Inversion Formulas for the Cone Transform

Several novel imaging applications have lead recently to a variety of Radon type transforms, where integration is done over a family of conical surfaces. We call them \emph{cone transforms} (in 2D they are also called \emph{V-line} or \emph{broken ray} transforms). Most prominently, they are present in the so called Compton camera imaging that arises in medical diagnostics, astronomy, and lately in homeland security applications. Several specific incarnations of the cone transform have been considered separately. In this paper, we address the most general (and overdetermined) cone transform, obtain integral relations between cone and Radon transforms in $\mathbb{R}^n$, and a variety of inversion formulas. In many applications (e.g., in homeland security), the signal to noise ratio is very low. So, if overdetermined data is collected (as in the case of Compton imaging), attempts to reduce the dimensionality might lead to essential elimination of the signal. Thus, our main concentration is on obtaining formulas involving overdetermined data.

math.FA↗