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Linh V. Nguyen

Publications and source records attributed to Linh V. Nguyen.

At least 19 recordsLinked to original sources
arXiv2024

Full Field Inversion of the Attenuated Wave Equation: Theory and Numerical Inversion

Standard photoacoustic tomography (PAT) provides data that consist of time-dependent signals governed by the wave equation, which are measured on an observation surface. In contrast, the measured data from the recently invented full-field PAT is the Radon transform of the solution of the wave equation on a spatial domain at a single instant in time. While reconstruction using classical PAT data has been extensively studied, not much is known about the full-field PAT problem. In this paper, we study full-field photoacoustic tomography with spatially variable sound speed and spatially variable damping. In particular, we prove the uniqueness and stability of the associated single-time full-field wave inversion problem and develop algorithms for its numerical inversion using iterative and variational regularization methods. Numerical simulations are presented for both full-angle and limited-angle data cases

math.NA↗
arXiv2023

The time dimensional reduction method to determine the initial conditions without the knowledge of damping coefficients

This paper aims to reconstruct the initial condition of a hyperbolic equation with an unknown damping coefficient. Our approach involves approximating the hyperbolic equation's solution by its truncated Fourier expansion in the time domain and using a polynomial-exponential basis. This truncation process facilitates the elimination of the time variable, consequently, yielding a system of quasi-linear elliptic equations. To globally solve the system without needing an accurate initial guess, we employ the Carleman contraction principle. We provide several numerical examples to illustrate the efficacy of our method. The method not only delivers precise solutions but also showcases remarkable computational efficiency.

math.NA↗
arXiv2023

Mitigating stimulated Brillouin scattering in multimode fibers with focused output via wavefront shaping

The key challenge for high-power delivery through optical fibers is overcoming nonlinear optical effects. To keep a smooth output beam, most techniques for mitigating optical nonlinearities are restricted to single-mode fibers. Moving out of the single-mode paradigm, we show experimentally that wavefront-shaping of coherent input light that is incident on a highly multimode fiber can increase the power threshold for stimulated Brillouin scattering (SBS) by an order of magnitude, whilst simultaneously controlling the output beam profile. The theory reveals that the suppression of SBS is due to the relative weakness of intermodal scattering compared to intramodal scattering, and to an effective broadening of the Brillouin spectrum under multimode excitation. Our method is efficient, robust, and applicable to continuous waves and pulses. This work points toward a promising route for suppressing detrimental nonlinear effects in optical fibers, which will enable further power scaling of high-power fiber systems for applications to directed energy, remote sensing, and gravitational-wave detection.

physics.optics↗
arXiv2021

Analysis for Full Field Photoacoustic Tomography with Variable Sound Speed

Photoacoustic tomography (PAT) is a non-invasive imaging modality that requires recovering the initial data of the wave equation from certain measurements of the solution outside the object. In the standard PAT measurement setup, the used data consist of time-dependent signals measured on an observation surface. In contrast, the measured data from the recently invented full-field detection technique provide the solution of the wave equation on a spatial domain at a single instant in time. While reconstruction using classical PAT data has been extensively studied, not much is known for the full field PAT problem. In this paper, we build mathematical foundations of the latter problem for variable sound speed and settle its uniqueness and stability. Moreover, we introduce an exact inversion method using time-reversal and study its convergence. Our results demonstrate the suitability of both the full field approach and the proposed time-reversal technique for high resolution photoacoustic imaging.

math.AP↗
arXiv2020

Regularization of Inverse Problems by Neural Networks

Inverse problems arise in a variety of imaging applications including computed tomography, non-destructive testing, and remote sensing. The characteristic features of inverse problems are the non-uniqueness and instability of their solutions. Therefore, any reasonable solution method requires the use of regularization tools that select specific solutions and at the same time stabilize the inversion process. Recently, data-driven methods using deep learning techniques and neural networks demonstrated to significantly outperform classical solution methods for inverse problems. In this chapter, we give an overview of inverse problems and demonstrate the necessity of regularization concepts for their solution. We show that neural networks can be used for the data-driven solution of inverse problems and review existing deep learning methods for inverse problems. In particular, we view these deep learning methods from the perspective of regularization theory, the mathematical foundation of stable solution methods for inverse problems. This chapter is more than just a review as many of the presented theoretical results extend existing ones.

math.NA↗
arXiv2019

An Efficient Model for Sentiment Analysis of Electronic Product Reviews in Vietnamese

In the past few years, the growth of e-commerce and digital marketing in Vietnam has generated a huge volume of opinionated data. Analyzing those data would provide enterprises with insight for better business decisions. In this work, as part of the Advosights project, we study sentiment analysis of product reviews in Vietnamese. The final solution is based on Self-attention neural networks, a flexible architecture for text classification task with about 90.16% of accuracy in 0.0124 second, a very fast inference time.

cs.CL↗
arXiv2019

An Inversion Formula for Horizontal Conical Radon Transform

In this paper, we consider the conical Radon transform on all cones with horizontal central axis whose vertices are on a straight line. We derive an explicit inversion formula for such transform. The inversion makes use of the vertical slice transform on a sphere and V-line transform on a plane.

math.CA↗
arXiv2019

Photoacoustic image reconstruction from full field data in heterogeneous media

We consider image reconstruction in full-field photoacoustic tomography, where 2D projections of the full 3D acoustic pressure distribution at a given time T>0 are collected. We discuss existing results on the stability and uniqueness of the resulting image reconstruction problem and review existing reconstruction algorithms. Open challenges are also mentioned. Additionally, we introduce novel one-step reconstruction methods allowing for a variable speed of sound. We apply preconditioned iterative and variational regularization methods to the one-step formulation. Numerical results using the one-step formulation are presented, together with a comparison with the previous two-step approach for full-field photoacoustic tomography

math.NA↗
arXiv2018

Reconstruction algorithms for photoacoustic tomography in heterogenous damping media

In this article, we study several reconstruction methods for the inverse source problem of photoacoustic tomography (PAT) with spatially variable sound speed and damping. The backbone of these methods is the adjoint operators, which we thoroughly analyze in both the $L^2$- and $H^1$-settings. They are casted in the form of a nonstandard wave equation. We derive the well-pawedness of the aforementioned wave equation in a natural functional space, and also prove the finite speed of propagation. Under the uniqueness and visibility condition, our formulations of the standard iterative reconstruction methods, such as Landweber's and conjugate gradients (CG), achieve a linear rate of convergence in either $L^2$- or $H^1$-norm. When the visibility condition is not satisfied, the problem is severely ill-posed and one must apply a regularization technique to stabilize the solutions. To that end, we study two classes of regularization methods: (i) iterative, and (ii) variational regularization. In the case of full data, our simulations show that the CG method works best; it is very fast and robust. In the ill-posed case, the CG method behaves unstably. Total variation regularization method (TV), in this case, significantly improves the reconstruction quality.

math.AP↗
arXiv2018

Full field inversion in photoacoustic tomography with variable sound speed

Recently, a novel measurement setup has been introduced to photoacoustic tomography, that collects data in the form of projections of the full 3D acoustic pressure distribution at a certain time instant. Existing imaging algorithms for this kind of data assume a constant speed of sound. This assumption is not always met in practice and thus leads to erroneous reconstructions. In this paper, we present a two-step reconstruction method for full field detection photoacoustic tomography that takes variable speed of sound into account. In the first step, by applying the inverse Radon transform, the pressure distribution at the measurement time is reconstructed point-wise from the projection data. In the second step, one solves a final time wave inversion problem where the initial pressure distribution is recovered from the known pressure distribution at the measurement time. For the latter problem, we derive an iterative solution approach, compute the required adjoint operator, and show its uniqueness and stability.

math.NA↗
arXiv2017

Analysis of Iterative Methods in Photoacoustic Tomography with Variable Sound Speed

In this article, we revisit iterative methods for solving the inverse problem of photoacoustic tomography in free space. Recently, there have been interesting developments on explicit formulations of the adjoint operator, demonstrating that iterative methods are an attractive choice for photoacoustic image reconstruction. In this work, we propose several modifications of current formulations of the adjoint operator which help speed up the convergence and yield improved error estimates. We establish a stability analysis and show that, with our choices of the adjoint operator, the iterative methods can achieve a linear rate of convergence, in the $L^2$-norm as well as in the $H^1$-norm. In addition, we analyze the normal operator from the microlocal analysis point of view. This gives insight into the convergence speed of the iterative methods and choosing proper weights for the mapping spaces. Finally, we present numerical results using various iterative reconstruction methods for full as well as limited view data. Our results demonstrate that Nesterov's fast gradient and the CG methods converge faster than Landweber's and iterative time reversal methods in the visible as well as the invisible case.

math.AP↗
arXiv2017

Iterative Methods for Photoacoustic Tomography in Attenuating Acoustic Media

The development of efficient and accurate reconstruction methods is an important aspect of tomographic imaging. In this article, we address this issue for photoacoustic tomography. To this aim, we use models for acoustic wave propagation accounting for frequency dependent attenuation according to a wide class of attenuation laws that may include memory. We formulate the inverse problem of photoacoustic tomography in attenuating medium as an ill-posed operator equation in a Hilbert space framework that is tackled by iterative regularization methods. Our approach comes with a clear convergence analysis. For that purpose we derive explicit expressions for the adjoint problem that can efficiently be implemented. In contrast to time reversal, the employed adjoint wave equation is again damping and, thus has a stable solution. This stability property can be clearly seen in our numerical results. Moreover, the presented numerical results clearly demonstrate the Efficiency and accuracy of the derived iterative reconstruction algorithms in various situations including the limited view case.

math.NA↗
arXiv2017

Analysis of the Linearized Problem of Quantitative Photoacoustic Tomography

Quantitative image reconstruction in photoacoustic tomography requires the solution of a coupled physics inverse problem involvier light transport and acoustic wave propagation. In this paper we address this issue employing the radiative transfer equation as accurate model for light transport. As main theoretical results, we derive several stability and uniqueness results for the linearized inverse problem. We consider the case of single illumination as well as the case of multiple illuminations assuming full or partial data. The numerical solution of the linearized problem is much less costly than the solution of the non-linear problem. We present numerical simulations supporting the stability results for the linearized problem and demonstrate that the linearized problem already gives accurate quantitative results.

math.AP↗
arXiv2015

On the strength of streak artifacts in limited angle weighted X-ray transform

In this article, we study the limited angle problem for the weighted X-ray transform. We consider the approximate reconstructions by applying two filtered back projection formulas to the limited data. We prove that each resulted operator can be decomposed into the sum of three Fourier integral operators whose symbols are of types $(\varrho,δ) \neq (1,0)$. The first operator, being a pseudo-differential operator, is responsible for the reconstruction of visible singularities. The other two are responsible for the generation of the artifacts. The theory of Fourier integral operators then implies, in particular, the continuity of the reconstruction operator and geometry of the artifacts. We then extend the technique developed by the author in [Inverse Problems 31 (2015) 055003] to obtain more refined microlocal estimates for the strength of the artifacts.

math-ph↗
arXiv2015

A dissipative time reversal technique for photo-acoustic tomography in a cavity

We consider the inverse source problem arising in thermo- and photo-acoustic tomography. It consists in reconstructing the initial pressure from the boundary measurements of the acoustic wave. Our goal is to extend versatile time reversal techniques to the case of perfectly reflecting boundary of the domain. Standard time reversal works only if the solution of the direct problem decays in time, which does not happen in the setup we consider. We thus propose a novel time reversal technique with a non-standard boundary condition. The error induced by this time reversal technique satisfies the wave equation with a dissipative boundary condition and, therefore, decays in time. For larger measurement times, this method yields a close approximation; for smaller times, the first approximation can be iteratively refined, resulting in a convergent Neumann series for the approximation.

math.AP↗
arXiv2015

On Artifacts in Limited Data Spherical Radon Transform: Curved Observation Surface

In this article, we consider the limited data problem for spherical mean transform. We characterize the generation and strength of the artifacts in a reconstruction formula. In contrast to the third's author work [Ngu15b], the observation surface considered in this article is not flat. Our results are comparable to those obtained in [Ngu15b] for flat observation surface. For the two dimensional problem, we show that the artifacts are $k$ orders smoother than the original singularities, where $k$ is vanishing order of the smoothing function. Moreover, if the original singularity is conormal, then the artifacts are $k+\frac{1}{2}$ order smoother than the original singularity. We provide some numerical examples and discuss how the smoothing effects the artifacts visually. For three dimensional case, although the result is similar to that [Ngu15b], the proof is significantly different. We introduce a new idea of lifting the space.

math.AP↗
arXiv2015

On Artifacts in Limited Data Spherical Radon Transform: Flat Observation Surfaces

In this article, we characterize the strength of the reconstructed singularities and artifacts in a reconstruction formula for limited data spherical Radon transform. Namely, we assume that the data is only available on a closed subset $Γ$ of a hyperplane in $\mathbb{R}^n$ ($n=2,3$). We consider a reconstruction formula studied in some previous works, under the assumption that the data is only smoothened out to a finite order $k$ near the boundary. For the problem in the two dimensional space and $Γ$ is a line segment, the artifacts are generated by rotating a boundary singularity along a circle centered at an end point of $Γ$. We show that the artifacts are $k$ orders smoother than the original singularity. For the problem in the three dimensional space and $Γ$ is a rectangle, we describe that the artifacts are generated by rotating a boundary singularity around either a vertex or an edge of $Γ$. The artifacts obtained by a rotation around a vertex are $2k$ orders smoother than the original singularity. Meanwhile, the artifacts obtained by a rotation around an edge are $k$ orders smoother than the original singularity. For both two and three dimensional problems, the visible singularities are reconstructed with the correct order. We, therefore, successfully quantify the geometric results obtained recently by J. Frikel and T. Quinto.

math.CA↗