High spatial resolution is a central aspect of tomographic imaging. In photoacoustic tomography, this requires accurate models for acoustic wave propagation and corresponding efficient image reconstruction methods. In this article we consider such models 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 ill-posed operator equation in a Hilbert space framework that is tackled by iterative regularization methods. For that purpose we derive explicit expressions for the adjoint problem that can efficiently be implemented. Our approach comes with a clear convergence analysis that is not shared by existing approaches for the considered model class. 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. Indeed, if the attenuation is not too strong then the reconstruction results are even better than in the absence of attenuation. 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, where alternative methods cannot be directly applied.
The case of non-vanishing attenuation is much less investigated and existing methods are very different from our approach. One class of reconstruction methods uses the following two-stage procedure: In a first step, by solving an ill-posed integral equation the (idealized) un-attenuated pressure data @math are estimated from the attenuated data @math . In the second step, the standard PAT problem is solved. Such a two step method has been proposed and implemented for the power law in @cite_63 @cite_13 , and later been used in @cite_55 @cite_64 for various attenuation laws. Compared to two stage approaches, in the single step approach it is easier to include prior information available in the image domain, such as positivity of the PAT source (compare Section ). Furthermore, in the limited data case, where the measurement surface does not fully enclose the PA source, the second step in the two-stage approach is again a non-standard problem for PAT, for which iterative methods can be applied. In such a situation it seems reasonable to directly apply iterative methods to the attenuated data, as considered in the present paper.
@cite_55 @cite_64 @cite_13 @cite_63
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Two compressive sensing inspired approaches for the solution of non-linear inverse scattering problems are introduced and discussed. Differently from the sparsity promoting approaches proposed in most of the papers published in the literature, the two methods here tackle the problem in its full non-linearity, by adopting a contrast source inversion scheme. In the first approach, the @math -norm of the unknown is added as a weighted penalty term to the contrast source cost functional. The second, and (to the best of our knowledge) completely original, approach enforces sparsity by constraining the solution of the non-linear problem into a convex set defined by the @math -norm of the unknown. A numerical assessment against a widely used benchmark example (the “Austria” profile) is given to assess the capabilities of the proposed approaches. Notably, the two approaches can be applied to any kind of basis functions and they can successfully tackle both reduced number of data (with respect to Nyquist sampling) and or overcomplete dictionaries. We propose and analyze a discontinuous Galerkin approximation for the Stokes problem. The finite element triangulation employed is not required to be conforming and we use discontinuous pressures and velocities. No additional unknown fields need to be introduced, but only suitable bilinear forms defined on the interfaces between the elements, involving the jumps of the velocity and the average of the pressure. We consider hp approximations using ℚk′–ℚk velocity-pressure pairs with k′ = k + 2, k + 1, k. Our methods show better stability properties than the corresponding conforming ones. We prove that our first two choices of velocity spaces ensure uniform divergence stability with respect to the mesh size h. Numerical results show that they are uniformly stable with respect to the local polynomial degree k, a property that has no analog in the conforming case. An explicit bound in k which is not sharp is also proven. Numerical results show that if equal order approximation is chosen for the velocity and pressure, no spurious pressure modes are present but the method is not uniformly stable either with respect to h or k. We derive a priori error estimates generalizing the abstract theory of mixed methods. Optimal error estimates in h are proven. As for discontinuous Galerkin methods for scalar diffusive problems, half of the power of k is lost for p and hp pproximations independently of the divergence stability. Inspired by the recent advances on minimizing nonsmooth or bound-constrained convex functions on models using varying degrees of fidelity, we propose a line search multi-grid (MG) method for full-wave iterative image reconstruction in photoacoustic tomography (PAT) in heterogeneous media. To compute the search direction at each iteration, we decide between the gradient at the target level, or alternatively an approximate error correction at a coarser level, relying on some predefined criteria. To incorporate absorption and dispersion, we derive the analytical adjoint directly from the first-order acoustic wave system. The effectiveness of the proposed method is tested on a total-variation penalized Iterative Shrinkage Thresholding algorithm (ISTA) and its accelerated variant (FISTA), which have been used in many studies of image reconstruction in PAT. The results show the great potential of the proposed method in improving speed of iterative image reconstruction. We give an overview of recent developments in the problem of reconstructing a band-limited signal from nonuniform sampling from a numerical analysis view point. It is shown that the appropriate design of the finite-dimensional model plays a key role in the numerical solution of the nonuniform sampling problem. In the one approach (often proposed in the literature) the finite-dimensional model leads to an ill-posed problem even in very simple situations. The other approach that we consider leads to a well-posed problem that preserves important structural properties of the original infinite-dimensional problem and gives rise to efficient numerical algorithms. Furthermore, a fast multilevel algorithm is presented that can reconstruct signals of unknown bandwidth from noisy nonuniformly spaced samples. We also discuss the design of efficient regularization methods for ill-conditioned reconstruction problems. Numerical examples from spectroscopy and exploration geophysics demonstrate the performance of the proposed methods.