Inverse Problems最新文献

This paper presents a novel methodological framework to obtain superior reconstructions in limited data photoacoustic tomography. The proposed framework exploits the presence of Cauchy data on an accessible part of the observation domain and uses a Nash game-theoretic framework to complete the missing data on the inaccessible region. To solve the game-theoretic problem, a gradient-free sequential quadratic Hamiltonian scheme, which is based on Pontryagin's maximum principle characterization, is combined with physics-informed neural networks to obtain the initial guess, leading to a robust and accurate reconstruction scheme. Numerical simulations with various phantoms, choice of accessible observation domains, and noise, demonstrate the effectiveness of our proposed framework to obtain high contrast and resolution reconstructions.

This paper presents an accelerated preconditioned proximal gradient algorithm (APPGA) for effectively solving a class of positron emission tomography (PET) image reconstruction models with differentiable regularizers. We establish the convergence of APPGA with the generalized Nesterov (GN) momentum scheme, demonstrating its ability to converge to a minimizer of the objective function with rates of $o\left(1/k^{2\omega }\right)$ and $o\left(1/k^{\omega }\right)$ in terms of the function value and the distance between consecutive iterates, respectively, where $\omega \in (0,1]$ is the power parameter of the GN momentum. To achieve an efficient algorithm with high-order convergence rate for the higher-order isotropic total variation (ITV) regularized PET image reconstruction model, we replace the ITV term by its smoothed version and subsequently apply APPGA to solve the smoothed model. Numerical results presented in this work indicate that as $\omega \in (0,1]$ increase, APPGA converges at a progressively faster rate. Furthermore, APPGA exhibits superior performance compared to the preconditioned proximal gradient algorithm and the preconditioned Krasnoselskii-Mann algorithm. The extension of the GN momentum technique for solving a more complex optimization model with multiple nondifferentiable terms is also discussed.

Quantitative photoacoustic computed tomography (qPACT) is an emerging medical imaging modality that carries the promise of high-contrast, fine-resolution imaging of clinically relevant quantities like hemoglobin concentration and blood-oxygen saturation. However, qPACT image reconstruction is governed by a multiphysics, partial differential equation (PDE) based inverse problem that is highly non-linear and severely ill-posed. Compounding the difficulty of the problem is the lack of established design standards for qPACT imaging systems, as there is currently a proliferation of qPACT system designs for various applications and it is unknown which ones are optimal or how to best modify the systems under various design constraints. This work introduces a novel computational approach for the optimal experimental design of qPACT imaging systems based on the Bayesian Cramér-Rao bound (CRB). Our approach incorporates several techniques to address challenges associated with forming the bound in the infinite-dimensional function space setting of qPACT, including priors with trace-class covariance operators and the use of the variational adjoint method to compute derivatives of the log-likelihood function needed in the bound computation. The resulting Bayesian CRB based design metric is computationally efficient and independent of the choice of estimator used to solve the inverse problem. The efficacy of the bound in guiding experimental design was demonstrated in a numerical study of qPACT design schemes under a stylized two-dimensional imaging geometry. To the best of our knowledge, this is the first work to propose Bayesian CRB based design for systems governed by PDEs.

We develop a linearized boundary control method for the inverse boundary value problem of determining a density in the acoustic wave equation. The objective is to reconstruct an unknown perturbation in a known background density from the linearized Neumann-to-Dirichlet map. A key ingredient in the derivation is a linearized Blagoves̆c̆enskiĭ's identity with a free parameter. When the linearization is at a constant background density, we derive two reconstructive algorithms with stability estimates based on the boundary control method. When the linearization is at a non-constant background density, we establish an increasing stability estimate for the recovery of the density perturbation. The proposed reconstruction algorithms are implemented and validated with several numerical experiments to demonstrate the feasibility.