Linh V. Nguyen

Linh V. Nguyen, Markus Haltmeier

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.

Gerhard Zangerl, Markus Haltmeier, Linh V. Nguyen et al.

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.

Markus Haltmeier, Linh V. Nguyen

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.

Suong N. Hoang, Linh V. Nguyen, Tai Huynh et al.

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.

Markus Haltmeier, Richard Kowar, Linh V. Nguyen

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.