arXiv subjects

Alexander Katsevich

Publications and source records attributed to Alexander Katsevich.

At least 19 recordsLinked to original sources

arXiv2026

Microlocal analysis of non-linear artifacts in cone beam CT

We present a novel microlocal analysis of beam hardening artifacts arising in cone-beam X-ray CT, where the set of X-ray sources is restricted to a 1D smooth curve $ \gamma\subset \mathbb{R}^3$. We show that, when the CT data is modeled in the standard way using the Beer Lambert law, the exponential term in the model creates singularities in the data that are not present in the linear X-ray transform. We assume that the attenuation coefficient $\mu$ (the reconstruction target) has a jump discontinuity across a surface $\mathcal{S}\subset\mathbb{R}^3$, and is smooth otherwise. We prove that these additional singularities in the data occur when the X-ray beam is tangent to $\mathcal{S}$ at two points simultaneously. To investigate how the singularities in the data propagate to the reconstruction space, we apply Filtered Back Projection (FBP) type reconstruction. We prove that the artifacts due to beam hardening are locally of conormal type, and lie on a 2-D surface which is the union of all double tangent rays which intersect $\gamma$. The artifacts are notably weaker than the reconstructed jumps of $\mu$ (i.e., the desired singularities), and we quantify this using the order of the corresponding conormal distributions. While our primary theory applies to the regions of $\mathcal{S}$ that are smooth, we also extend our theory to non-smooth $\mathcal{S}$ with "ridges." These arise where $\mathcal{S}$ is locally the intersection of two smooth surface patches meeting transversely along a curve (e.g., the edge of a cuboid). In addition, we present simulated reconstructions of metal objects in circular cone-beam CT to validate our theory.

math.GM↗

arXiv2026

Discrete to continuum limits in Bayesian inverse problems

We develop a posterior-level discrete-to-continuum theory for Bayesian inverse problems whose finite-dimensional priors arise from local finite-difference regularization and whose likelihoods are of a general convex GLM-type form. In contrast to continuum-first approaches, the continuum prior and posterior are not assumed at the outset but are constructed as limits of the finite-dimensional probability measures used in numerical computation. First, with the total information parameter $\tau$ and the regularization strength $\kappa$ fixed, we prove weak convergence on $L^2$ of the reconstructed discrete Gaussian priors and posterior measures to well-defined continuum laws. We then consider a coupled grid-refinement and small-noise limit in which $N\to\infty$ and $\tau_N,\kappa_N\to\infty$, while $\kappa_N/\tau_N$ remains fixed. Under explicit growth conditions relating $N$ and $\tau_N$, we prove that the reconstructed discrete MAP estimates converge to the unique minimizer $u_*$ of the limiting continuum cost functional, that the reconstructed posterior measures concentrate at $u_*$, and that their centered and rescaled fluctuations converge to $\mathcal N(\vec 0,Q^{-1})$, where $Q$ is the Hessian of the continuum cost functional at $u_*$. Finally, we show that the same deterministic limit and Gaussian fluctuation law are obtained by first constructing the continuum posterior and then taking its small-noise, high-regularization limit.

math.ST↗

arXiv2026

High-dimensional Laplace asymptotics up to the concentration threshold

We study high-dimensional Laplace-type integrals $I(\lambda):=(\lambda/2\pi)^{d/2}\int_{\mathbb R^d} g(x)e^{-\lambda f(x)}dx$ in the regime where both $d$ and $\lambda$ are large. Existing rigorous Laplace-expansion results in growing dimension are largely confined to the "Gaussian-approximation" regime $d^2/\lambda\to0$, which excludes many practically relevant settings that lie beyond this threshold but still satisfy the concentration condition $d/\lambda\to0$. We close this gap by deriving an explicit asymptotic expansion for $\log I(\lambda)$ with quantitative remainder bounds that remain valid throughout this intermediate region, arbitrarily close to the concentration threshold. Fix $L\ge1$ and assume that, in a neighborhood of the global minimizer of $f$, the operator norms of derivatives of $f$ and $g$ are bounded independently of $d,\lambda$ up to orders $2L+2$ and $2L$, respectively. Assuming also some mild global growth conditions, we prove $$\log I(\lambda)=\sum_{k=1}^{L-1} b_k(f,g)\lambda^{-k}+O(d^{L+1}/\lambda^L), \qquad d^{L+1}/\lambda^L\to0,$$ with coefficients satisfying $b_k(f,g)=O(d^{k+1})$. Moreover, the $b_k(f,g)$ coincide with the coefficients from the formal cumulant expansion of $\log I(\lambda)$. We also study computation for concentrating densities $\pi(x)\propto e^{-\lambda f(x)}$. For smooth observables $g$, our expansion yields closed-form, analytic approximations of $\mathbb E_{X\sim\pi}[g(X)]$. For sampling, we construct explicit polynomial transports $x_L$ such that $\pi_L:=(x_L)_\# N(0,\lambda^{-1}I_d)$ satisfies $\mathrm{TV}(\pi,\pi_L)\lesssim d^{L+1}/\lambda^L$ for $L=1,2,3,\dots$, yielding an accurate procedure arbitrarily close to the concentration threshold $d=o(\lambda)$.

math.CA↗

arXiv2025

Asymptotic analysis of rare events in high dimensions

Understanding rare events is critical across domains ranging from signal processing to reliability and structural safety, extreme-weather forecasting, and insurance. The analysis of rare events is a computationally challenging problem, particularly in high dimensions $d$. In this work, we develop the first asymptotic high-dimensional theory of rare events. First, we exploit asymptotic integral methods recently developed by the first author to provide an asymptotic expansion of rare event probabilities. The expansion employs the geometry of the rare event boundary and the local behavior of the log probability density. Generically, the expansion is valid if $d^2\ll\lambda$, where $\lambda$ characterizes the extremity of the event. We prove this condition is necessary by constructing an example in which the first-order remainder is bounded above and below by $d^2/\lambda$. We also provide a nonasymptotic remainder bound which specifies the precise dependence of the remainder on $d$, $\lambda$, the density, and the boundary, and which shows that in certain cases, the condition $d^2\ll \lambda$ can be relaxed. As an application of the theory, we derive asymptotic approximations to rare probabilities under the standard Gaussian density in high dimensions. In the second part of our work, we provide an asymptotic approximation to densities conditional on rare events. This gives rise to simple procedure for approximately sampling conditionally on the rare event using independent Gaussian and exponential random variables.

math.PR↗

arXiv2025

Saddle Point Approximation and Central Limit Theorem for Densities in high dimensions

We study the saddlepoint approximation (SPA) for sums of $n$ i.i.d. random vectors $X_i\in\mathbb R^d$ in growing dimensions. SPA provides highly accurate approximations to probability densities and distribution functions via the moment generating function. Recent work by Tang and Reid extended SPA to cases where the dimension $d$ increases with $n$, obtaining an error rate of order $O(d^3/n)$. We refine this analysis and improve the SPA error rate to $O(d^2/n)$. We obtain a non-asymptotic bound for the multiplicative SPA error. As a corollary, we establish the first local central limit theorem for densities in growing dimensions, under the condition $d^2/n \to 0$, and provide explicit multiplicative error bounds. An example involving Gaussian mixtures illustrates our results.

math.PR↗

arXiv2025

Local Characterization of Noise in Iterative Reconstruction of the Generalized Radon Transform

We study noise in iterative reconstruction from discrete noisy data of a generalized Radon transform in the plane. Our approach builds on Local Reconstruction Analysis (LRA), a framework for analyzing reconstructions at the native scale. We establish that the rescaled reconstruction error converges in distribution to a zero-mean Gaussian random field with explicitly computable covariance, providing a complete local characterization of noise in iterative reconstruction. Numerical experiments show strong agreement with the theoretical predictions. Combined with earlier deterministic results, our findings complete the analysis of iterative reconstruction at the native scale with respect to the two most fundamental limitations: the discreteness of the data and the presence of noise.

math.NA↗

arXiv2025

Statistical microlocal analysis in two-dimensional X-ray CT

In many imaging applications it is important to assess how well the edges of the original object, $f$, are resolved in an image, $f^\text{rec}$, reconstructed from the measured data, $g$. In this paper we consider the case of image reconstruction in 2D X-ray Computed Tomography (CT). Let $f$ be a function describing the object being scanned, and $g=Rf + \eta$ be the Radon transform data in $\mathbb{R}^2$ corrupted by noise, $\eta$, and sampled with step size $\sim\epsilon$. Conventional microlocal analysis provides conditions for edge detectability based on the scanner geometry in the case of continuous, noiseless data (when $\eta = 0$), but does not account for noise and finite sampling step size. We develop a novel technique called Statistical Microlocal Analysis (SMA), which uses a statistical hypothesis testing framework to determine if an image edge (singularity) of $f$ is detectable from $f^\text{rec}$, and we quantify edge detectability using the statistical power of the test. Our approach is based on the theory we developed in previous work, which provides a characterization of $f^\text{rec}$ in local $O(\epsilon)$-size neighborhoods when $\eta \neq 0$. We derive a statistical test for the presence and direction of an edge microlocally given the magnitude of $\eta$ and data sampling step size. Using the properties of the null distribution of the test, we quantify the uncertainty of the edge magnitude and direction. We validate our theory using simulations, which show strong agreement between our predictions and experimental observations. Our work is not only of practical value, but of theoretical value as well. SMA is a natural extension of classical microlocal analysis theory which accounts for practical measurement imperfections, such as noise and finite step size, at the highest possible resolution compatible with the data.

math.ST↗

arXiv2025

Analysis of beam hardening streaks in tomography

The mathematical foundation of X-ray CT is based on the assumption that by measuring the attenuation of X-rays passing through an object, one can recover the integrals of the attenuation coefficient $\mu(x)$ along a sufficiently rich family of lines $L$, $\int_L \mu(x) \text{d} x$. This assumption is inaccurate because the energy spectrum of an X-ray beam in a typical CT scanner is wide. At the same time, the X-ray attenuation coefficient of most materials is energy-dependent, and this dependence varies among materials. Thus, reconstruction from X-ray CT data is a nonlinear problem. If the nonlinear nature of CT data is ignored and a conventional linear reconstruction formula is used, which is frequently the case, the resulting image contains beam-hardening artifacts such as streaks. In this work, we describe the nonlinearity of CT data using the conventional model accepted by all CT practitioners. Our main result is the characterization of streak artifacts caused by nonlinearity. We also obtain an explicit expression for the leading singular behavior of the artifacts. Finally, a numerical experiment is conducted to validate the theoretical results.

math.NA↗

arXiv2024

Local analysis of iterative reconstruction from discrete generalized Radon transform data in the plane

Local reconstruction analysis (LRA) is a powerful and flexible technique to study images reconstructed from discrete generalized Radon transform (GRT) data, $g=\mathcal R f$. The main idea of LRA is to obtain a simple formula to accurately approximate an image, $f_\epsilon(x)$, reconstructed from discrete data $g(y_j)$ in an $\epsilon$-neighborhood of a point, $x_0$. The points $y_j$ lie on a grid with step size of order $\epsilon$ in each direction. In this paper we study an iterative reconstruction algorithm, which consists of minimizing a quadratic cost functional. The cost functional is the sum of a data fidelity term and a Tikhonov regularization term. The function $f$ to be reconstructed has a jump discontinuity across a smooth surface $\mathcal S$. Fix a point $x_0\in\mathcal S$ and any $A>0$. The main result of the paper is the computation of the limit $\Delta F_0(\check x;x_0):=\lim_{\epsilon\to0}(f_\epsilon(x_0+\epsilon\check x)-f_\epsilon(x_0))$, where $f_\epsilon$ is the solution to the minimization problem and $|\check x|\le A$. A numerical experiment with a circular GRT demonstrates that $\Delta F_0(\check x;x_0)$ accurately approximates the actual reconstruction obtained by the cost functional minimization.

math.NA↗

arXiv2024

Analysis of reconstruction from noisy discrete generalized Radon data

We consider a wide class of generalized Radon transforms $\mathcal R$, which act in $\mathbb{R}^n$ for any $n\ge 2$ and integrate over submanifolds of any codimension $N$, $1\le N\le n-1$. Also, we allow for a fairly general reconstruction operator $\mathcal A$. The main requirement is that $\mathcal A$ be a Fourier integral operator with a phase function, which is linear in the phase variable. We consider the task of image reconstruction from discrete data $g_{j,k} = (\mathcal R f)_{j,k} + \eta_{j,k}$. We show that the reconstruction error $N_\epsilon^{\text{rec}}=\mathcal A \eta_{j,k}$ satisfies $N^{\text{rec}}(\check x;x_0)=\lim_{\epsilon\to0}N_\epsilon^{\text{rec}}(x_0+\epsilon\check x)$, $\check x\in D$. Here $x_0$ is a fixed point, $D\subset\mathbb{R}^n$ is a bounded domain, and $\eta_{j,k}$ are independent, but not necessarily identically distributed, random variables. $N^{\text{rec}}$ and $N_\epsilon^{\text{rec}}$ are viewed as continuous random functions of the argument $\check x$ (random fields), and the limit is understood in the sense of probability distributions. Under some conditions on the first three moments of $\eta_{j,k}$ (and some other not very restrictive conditions on $x_0$ and $\mathcal A$), we prove that $N^{\text{rec}}$ is a zero mean Gaussian random field and explicitly compute its covariance. We also present a numerical experiment with a cone beam transform in $\mathbb{R}^3$, which shows an excellent match between theoretical predictions and simulated reconstructions.

math.NA↗

arXiv2024

Local reconstruction analysis of inverting the Radon transform in the plane from noisy discrete data

In this paper, we investigate the reconstruction error, $N_\e^{\text{rec}}(x)$, when a linear, filtered back-projection (FBP) algorithm is applied to noisy, discrete Radon transform data with sampling step size $\epsilon$ in two-dimensions. Specifically, we analyze $N_\e^{\text{rec}}(x)$ for $x$ in small, $O(\e)$-sized neighborhoods around a generic fixed point, $x_0$, in the plane, where the measurement noise values, $\eta_{k,j}$ (i.e., the errors in the sinogram space), are random variables. The latter are independent, but not necessarily identically distributed. We show, under suitable assumptions on the first three moments of the $\eta_{k,j}$, that the following limit exists: $N^{\text{rec}}(\chx;x_0) = \lim_{\e\to0}N_\e^{\text{rec}}(x_0+\e\chx)$, for $\check x$ in a bounded domain. Here, $N_\e^{\text{rec}}$ and $ N^{\text{rec}}$ are viewed as continuous random variables, and the limit is understood in the sense of distributions. Once the limit is established, we prove that $N^{\text{rec}}$ is a zero mean Gaussian random field and compute explicitly its covariance. In addition, we validate our theory using numerical simulations and pseudo random noise.

math.NA↗

arXiv2023

Analysis of reconstruction of functions with rough edges from discrete Radon data in $\mathbb R^2$

We study the accuracy of reconstruction of a family of functions $f_\epsilon(x)$, $x\in\mathbb R^2$, $\epsilon\to0$, from their discrete Radon transform data sampled with step size $O(\epsilon)$. For each $\epsilon>0$ sufficiently small, the function $f_\epsilon$ has a jump across a rough boundary $\mathcal S_\epsilon$, which is modeled by an $O(\epsilon)$-size perturbation of a smooth boundary $\mathcal S$. The function $H_0$, which describes the perturbation, is assumed to be of bounded variation. Let $f_\epsilon^{\text{rec}}$ denote the reconstruction, which is computed by interpolating discrete data and substituting it into a continuous inversion formula. We prove that $(f_\epsilon^{\text{rec}}-K_\epsilon*f_\epsilon)(x_0+\epsilon\check x)=O(\epsilon^{1/2}\ln(1/\epsilon))$, where $x_0\in\mathcal S$ and $K_\epsilon$ is an easily computable kernel.

math.NA↗

arXiv2023

Analysis of view aliasing for the generalized Radon transform in $\mathbb R^2$

In this paper we consider the generalized Radon transform $\mathcal R$ in the plane. Let $f$ be a piecewise smooth function, which has a jump across a smooth curve $\mathcal S$. We obtain a formula, which accurately describes view aliasing artifacts away from $\mathcal S$ when $f$ is reconstructed from the data $\mathcal R f$ discretized in the view direction. The formula is asymptotic, it is established in the limit as the sampling rate $\epsilon\to0$. The proposed approach does not require that $f$ be band-limited. Numerical experiments with the classical Radon transform and generalized Radon transform (which integrates over circles) demonstrate the accuracy of the formula.

math.NA↗

arXiv2022

Novel resolution analysis for the Radon transform in $\mathbb R^2$ for functions with rough edges

Let $f$ be a function in $\mathbb R^2$, which has a jump across a smooth curve $\mathcal S$ with nonzero curvature. We consider a family of functions $f_ε$ with jumps across a family of curves $\mathcal S_ε$. Each $\mathcal S_ε$ is an $O(ε)$-size perturbation of $\mathcal S$, which scales like $O(ε^{-1/2})$ along $\mathcal S$. Let $f_ε^{\text{rec}}$ be the reconstruction of $f_ε$ from its discrete Radon transform data, where $ε$ is the data sampling rate. A simple asymptotic (as $ε\to0$) formula to approximate $f_ε^{\text{rec}}$ in any $O(ε)$-size neighborhood of $\mathcal S$ was derived heuristically in an earlier paper of the author. Numerical experiments revealed that the formula is highly accurate even for nonsmooth (i.e., only H{ö}lder continuous) $\mathcal S_ε$. In this paper we provide a full proof of this result, which says that the magnitude of the error between $f_ε^{\text{rec}}$ and its approximation is $O(ε^{1/2}\ln(1/ε))$. The main assumption is that the level sets of the function $H_0(\cdot,ε)$, which parametrizes the perturbation $\mathcal S\to\mathcal S_ε$, are not too dense.

math.NA↗

arXiv2021

Resolution of 2D reconstruction of functions with nonsmooth edges from discrete Radon transform data

Let $f$ be an unknown function in $\mathbb R^2$, and $f_ε$ be its reconstruction from discrete Radon transform data, where $ε$ is the data sampling rate. We study the resolution of reconstruction when $f$ has a jump discontinuity along a nonsmooth curve $\mathcal S_ε$. The assumptions are that (a) $\mathcal S_ε$ is an $O(ε)$-size perturbation of a smooth curve $\mathcal S$, and (b) $\mathcal S_ε$ is Holder continuous with some exponent $γ\in(0,1]$. We compute the Discrete Transition Behavior (or, DTB) defined as the limit $\text{DTB}(\check x):=\lim_{ε\to0}f_ε(x_0+ε\check x)$, where $x_0$ is generic. We illustrate the DTB by two sets of numerical experiments. In the first set, the perturbation is a smooth, rapidly oscillating sinusoid, and in the second - a fractal curve. The experiments reveal that the match between the DTB and reconstruction is worse as $\mathcal S_ε$ gets more rough. This is in agreement with the proof of the DTB, which suggests that the rate of convergence to the limit is $O(ε^{γ/2})$. We then propose a new DTB, which exhibits an excellent agreement with reconstructions. Investigation of this phenomenon requires computing the rate of convergence for the new DTB. This, in turn, requires completely new approaches. We obtain a partial result along these lines and formulate a conjecture that the rate of convergence of the new DTB is $O(ε^{1/2}\ln(1/ε))$.

math.NA↗

arXiv2021

Resolution analysis of inverting the generalized $N$-dimensional Radon transform in $\mathbb R^n$ from discrete data

Let $\mathcal R$ denote the generalized Radon transform (GRT), which integrates over a family of $N$-dimensional smooth submanifolds $\mathcal S_{\tilde y}\subset\mathcal U$, $1\le N\le n-1$, where an open set $\mathcal U\subset\mathbb R^n$ is the image domain. The submanifolds are parametrized by points $\tilde y\subset\tilde{\mathcal V}$, where an open set $\tilde{\mathcal V}\subset\mathbb R^n$ is the data domain. The continuous data are $g={\mathcal R} f$, and the reconstruction is $\check f=\mathcal R^*\mathcal B g$. Here $\mathcal R^*$ is a weighted adjoint of $\mathcal R$, and $\mathcal B$ is a pseudo-differential operator. We assume that $f$ is a conormal distribution, $\text{supp}(f)\subset\mathcal U$, and its singular support is a smooth hypersurface $\mathcal S\subset\mathcal U$. Discrete data consists of the values of $g$ on a lattice $\tilde y^j$ with the step size $O(ε)$. Let $\check f_ε=\mathcal R^*\mathcal B g_ε$ denote the reconstruction obtained by applying the inversion formula to an interpolated discrete data $g_ε(\tilde y)$. Pick a generic pair $(x_0,\tilde y_0)$, where $x_0\in\mathcal S$, and $\mathcal S_{\tilde y_0}$ is tangent to $\mathcal S$ at $x_0$. The main result of the paper is the computation of the limit $$ f_0(\check x):=\lim_{ε\to0}ε^κ\check f_ε(x_0+ε\check x). $$ Here $κ\ge 0$ is selected based on the strength of the reconstructed singularity, and $\check x$ is confined to a bounded set. The limiting function $f_0(\check x)$, which we call the discrete transition behavior, allows computing the resolution of reconstruction.

math.NA↗

arXiv2020

On the spectral properties of the Hilbert transform operator on multi-intervals

Let $J,E\subset\mathbb R$ be two multi-intervals with non-intersecting interiors. Consider the following operator $$A:\, L^2( J )\to L^2(E),\ (Af)(x) = \frac 1π\int_{ J } \frac {f(y)\text{d} y}{x-y},$$ and let $A^\dagger$ be its adjoint. We introduce a self-adjoint operator $\mathscr K$ acting on $L^2(E)\oplus L^2(J)$, whose off-diagonal blocks consist of $A$ and $A^\dagger$. In this paper we study the spectral properties of $\mathscr K$ and the operators $A^\dagger A$ and $A A^\dagger$. Our main tool is to obtain the resolvent of $\mathscr K$, which is denoted by $\mathscr R$, using an appropriate Riemann-Hilbert problem, and then compute the jump and poles of $\mathscr R$ in the spectral parameter $λ$. We show that the spectrum of $\mathscr K$ has an absolutely continuous component $[0,1]$ if and only if $J$ and $E$ have common endpoints, and its multiplicity equals to their number. If there are no common endpoints, the spectrum of $\mathscr K$ consists only of eigenvalues and $0$. If there are common endpoints, then $\mathscr K$ may have eigenvalues imbedded in the continuous spectrum, each of them has a finite multiplicity, and the eigenvalues may accumulate only at $0$. In all cases, $\mathscr K$ does not have a singular continuous spectrum. The spectral properties of $A^\dagger A$ and $A A^\dagger$, which are very similar to those of $\mathscr K$, are obtained as well.

math.FA↗

arXiv2020

Analysis of resolution of tomographic-type reconstruction from discrete data for a class of distributions

Let $f(x)$, $x\in\mathbb R^2$, be a piecewise smooth function with a jump discontinuity across a smooth surface $\mathcal S$. Let $f_{Λε}$ denote the Lambda tomography (LT) reconstruction of $f$ from its discrete Radon data $\hat f(α_k,p_j)$. The sampling rate along each variable is $\simε$. First, we compute the limit $f_0(\check x)=\lim_{ε\to0}εf_{Λε}(x_0+ε\check x)$ for a generic $x_0\in\mathcal S$. Once the limiting function $f_0(\check x)$ is known (which we call the discrete transition behavior, or DTB for short), the resolution of reconstruction can be easily found. Next, we show that straight segments of $\mathcal S$ lead to non-local artifacts in $f_{Λε}$, and that these artifacts are of the same strength as the useful singularities of $f_{Λε}$. We also show that $f_{Λε}(x)$ does not converge to its continuous analogue $f_Λ=(-Δ)^{1/2}f$ as $ε\to0$ even if $x\not\in\mathcal S$. Results of numerical experiments presented in the paper confirm these conclusions. We also consider a class of Fourier integral operators $\mathcal{B}$ with the same canonical relation as the classical Radon transform adjoint, and a class of distributions $g\in\mathcal{E}'(Z_n)$, $Z_n:=S^{n-1}\times\mathbb R$, and obtain easy to use formulas for the DTB when $\mathcal{B} g$ is computed from discrete data $g(α_{\vec k},p_j)$. Exact and LT reconstructions are particlular cases of this more general theory.

math.NA↗