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In this article we study the problem of recovering the initial data of the two-dimensional wave equation from Neumann measurements on a convex domain with smooth boundary in the plane. We derive an explicit inversion formula of a so-called back-projection type and deduce exact inversion formulas for circular and elliptical domains. In addition, for circular domains, we show that the initial data can also be recovered from any linear combination of its solution and its normal derivative on the boundary. Numerical results of our implementation of the derived inversion formulas are presented demonstrating their accuracy and stability.

Photothermal measurements with an infrared camera enable a fast and contactless part inspection. The main drawback of existing reconstruction methods is the degradation of the spatial resolution with increasing imaging depth, which results in blurred images for deeper lying structures. In this work, we propose an efficient image reconstruction strategy that allows prior information to be included to overcome the diffusion-based information loss. Following the virtual wave concept, in a first step we reconstruct from the measured photothermal signal an acoustic wave field that satisfies the standard wave equation. This wave is called a virtual one, because it is not the measured acoustic wave but mathematically calculated from the temperature signal measured on the sample surface. In the second step, stable and efficient reconstruction methods developed for photoacoustic tomography are used. We compensate for the loss of information in thermal measurements by incorporating the prior information positivity and sparsity. For that purpose we combine circular projections with an iterative regularization scheme. Using experimental data, this work demonstrates that the quality of the reconstruction based on photothermal measurements can be significantly enhanced. The main goal of this work was to illustrate that prior information significantly improves the regularized solution and, hence, the reconstructed field. Using an iterative non-linear regularization method, the prior information positivity and sparsity could be incorporated. The regularization and reconstruction results show that respecting information available about the data significantly increases the quality of the regularized solution.

PURPOSE Iterative convolutional neural networks (CNNs) which resemble unrolled learned iterative schemes have shown to consistently deliver state-of-the-art results for image reconstruction problems across different imaging modalities. However, because these methods include the forward model in the architecture, their applicability is often restricted to either relatively small reconstruction problems or to problems with operators which are computationally cheap to compute. As a consequence, they have not been applied to dynamic non-Cartesian multi-coil reconstruction problems so far. METHODS In this work, we propose a CNN architecture for image reconstruction of accelerated 2D radial cine MRI with multiple receiver coils. The network is based on a computationally light CNN component and a subsequent conjugate gradient (CG) method which can be jointly trained end-to-end using an efficient training strategy. We investigate the proposed training strategy and compare our method with other well-known reconstruction techniques with learned and non-learned regularization methods. RESULTS Our proposed method outperforms all other methods based on non-learned regularization. Further, it performs similar or better than a CNN-based method employing a 3D U-Net and a method using adaptive dictionary learning. In addition, we empirically demonstrate that even by training the network with only iteration, it is possible to increase the length of the network at test time and further improve the results. CONCLUSIONS End-to-end training allows to highly reduce the number of trainable parameters of and stabilize the reconstruction network. Further, because it is possible to change the length of the network at the test time, the need to find a compromise between the complexity of the CNN-block and the number of iterations in each CG-block becomes irrelevant.

Deep learning based reconstruction methods deliver outstanding results for solving inverse problems and are therefore becoming increasingly important. A recently invented class of learning-based reconstruction methods is the so-called NETT (for Network Tikhonov Regularization), which contains a trained neural network as regularizer in generalized Tikhonov regularization. The existing analysis of NETT considers fixed operators and fixed regularizers and analyzes the convergence as the noise level in the data approaches zero. In this paper, we extend the frameworks and analysis considerably to reflect various practical aspects and take into account discretization of the data space, the solution space, the forward operator and the neural network defining the regularizer. We show the asymptotic convergence of the discretized NETT approach for decreasing noise levels and discretization errors. Additionally, we derive convergence rates and present numerical results for a limited data problem in photoacoustic tomography.

We propose a sparse reconstruction framework (aNETT) for solving inverse problems. Opposed to existing sparse reconstruction techniques that are based on linear sparsifying transforms, we train an autoencoder network $D \circ E$ with $E$ acting as a nonlinear sparsifying transform and minimize a Tikhonov functional with learned regularizer formed by the $\ell^q$-norm of the encoder coefficients and a penalty for the distance to the data manifold. We propose a strategy for training an autoencoder based on a sample set of the underlying image class such that the autoencoder is independent of the forward operator and is subsequently adapted to the specific forward model. Numerical results are presented for sparse view CT, which clearly demonstrate the feasibility, robustness and the improved generalization capability and stability of aNETT over post-processing networks., Comment: The original proceeding is part of the ISBI 2020 and only contains 4 pages due to page restrictions

We analyze sparse frame based regularization of inverse problems by means of a diagonal frame decomposition (DFD) for the forward operator, which generalizes the SVD. The DFD allows to define a non-iterative (direct) operator-adapted frame thresholding approach which we show to provide a convergent regularization method with linear convergence rates. These results will be compared to the well-known analysis and synthesis variants of sparse l1-regularization which are usually implemented thorough iterative schemes. If the frame is a basis (non-redundant case), the three versions of sparse regularization, namely synthesis and analysis variants of l1-regularization as well as the DFD thresholding are equivalent. However, in the redundant case, those three approaches are pairwise different.

The development of fast and accurate image reconstruction algorithms is a central aspect of computed tomography. In this paper, we investigate this issue for the sparse data problem in photoacoustic tomography (PAT). We develop a direct and highly efficient reconstruction algorithm based on deep learning. In our approach, image reconstruction is performed with a deep convolutional neural network (CNN), whose weights are adjusted prior to the actual image reconstruction based on a set of training data. The proposed reconstruction approach can be interpreted as a network that uses the PAT filtered backprojection algorithm for the first layer, followed by the U-net architecture for the remaining layers. Actual image reconstruction with deep learning consists in one evaluation of the trained CNN, which does not require time-consuming solution of the forward and adjoint problems. At the same time, our numerical results demonstrate that the proposed deep learning approach reconstructs images with a quality comparable to state of the art iterative approaches for PAT from sparse data.

Photothermal measurements with an infrared camera enable a fast and contactless part inspection. The main drawback of existing reconstruction methods is the degradation of the spatial resolution with increasing imaging depth, which results in blurred images for deeper lying structures. In this paper, we propose an efficient image reconstruction strategy that allows prior information to be included to overcome the diffusion-based information loss. Following the virtual wave concept, in a first step we reconstruct an acoustic wave field that satisfies the standard wave equation. Therefore, in the second step, stable and efficient reconstruction methods developed for photoacoustic tomography can be used. We compensate for the loss of information in thermal measurements by incorporating the prior information positivity and sparsity. Therefore, we combine circular projections with an iterative regularization scheme. Using simulated and experimental data, this work demonstrates that the quality of the reconstruction from photothermal measurements can be significantly enhanced.

Filtered backprojection (FBP) is an efficient and popular class of tomographic image reconstruction methods. In photoacoustic tomography, these algorithms are based on theoretically exact analytic inversion formulas which results in accurate reconstructions. However, photoacoustic measurement data are often incomplete (limited detection view and sparse sampling), which results in artefacts in the images reconstructed with FBP. In addition to that, properties such as directivity of the acoustic detectors are not accounted for in standard FBP, which affects the reconstruction quality, too. To account for these issues, in this papers we propose to improve FBP algorithms based on machine learning techniques. In the proposed method, we include additional weight factors in the FBP, that are optimized on a set of incomplete data and the corresponding ground truth photoacoustic source. Numerical tests show that the learned FBP improves the reconstruction quality compared to the standard FBP., Comment: This paper is a proceedings to our presentation (Paper 1087837) at the Photons Plus Ultrasound: Imaging and Sensing, (within the SPIE Photonics West), Poster Sunday, February 03, 2019

We discuss several methods for image reconstruction in compressed sensing photoacoustic tomography (CS-PAT). In particular, we apply the deep learning method of [H. Li, J. Schwab, S. Antholzer, and M. Haltmeier. NETT: Solving Inverse Problems with Deep Neural Networks (2018), arXiv:1803.00092], which is based on a learned regularizer, for the first time to the CS-PAT problem. We propose a network architecture and training strategy for the NETT that we expect to be useful for other inverse problems as well. All algorithms are compared and evaluated on simulated data, and validated using experimental data for two different types of phantoms. The results on the one the hand indicate great potential of deep learning methods, and on the other hand show that significant future work is required to improve their performance on real-word data.

We propose a sparse reconstruction framework for solving inverse problems. Opposed to existing sparse regularization techniques that are based on frame representations, we train an encoder-decoder network by including an $\ell^1$-penalty. We demonstrate that the trained decoder network allows sparse signal reconstruction using thresholded encoded coefficients without losing much quality of the original image. Using the sparse synthesis prior, we propose minimizing the $\ell^1$-Tikhonov functional, which is the sum of a data fitting term and the $\ell^1$-norm of the synthesis coefficients, and show that it provides a regularization method., Comment: Presented at the SAMPTA 2019 conference

In this paper we present a generalized Deep Learning-based approach for solving ill-posed large-scale inverse problems occuring in medical image reconstruction. Recently, Deep Learning methods using iterative neural networks (NNs) and cascaded NNs have been reported to achieve state-of-the-art results with respect to various quantitative quality measures as PSNR, NRMSE and SSIM across different imaging modalities. However, the fact that these approaches employ the application of the forward and adjoint operators repeatedly in the network architecture requires the network to process the whole images or volumes at once, which for some applications is computationally infeasible. In this work, we follow a different reconstruction strategy by strictly separating the application of the NN, the regularization of the solution and the consistency with the measured data. The regularization is given in the form of an image prior obtained by the output of a previously trained NN which is used in a Tikhonov regularization framework. By doing so, more complex and sophisticated network architectures can be used for the removal of the artefacts or noise than it is usually the case in iterative NNs. Due to the large scale of the considered problems and the resulting computational complexity of the employed networks, the priors are obtained by processing the images or volumes as patches or slices. We evaluated the method for the cases of 3D cone-beam low dose CT and undersampled 2D radial cine MRI and compared it to a total variation-minimization-based reconstruction algorithm as well as to a method with regularization based on learned overcomplete dictionaries. The proposed method outperformed all the reported methods with respect to all chosen quantitative measures and further accelerates the regularization step in the reconstruction by several orders of magnitude.

We develop a data-driven regularization method for the severely ill-posed problem of photoacoustic image reconstruction from limited view data. Our approach is based on the regularizing networks that have been recently introduced and analyzed in [J. Schwab, S. Antholzer, and M. Haltmeier. Big in Japan: Regularizing networks for solving inverse problems (2018), arXiv:1812.00965] and consists of two steps. In the first step, an intermediate reconstruction is performed by applying truncated singular value decomposition (SVD). In order to prevent noise amplification, only coefficients corresponding to sufficiently large singular values are used, whereas the remaining coefficients are set zero. In a second step, a trained deep neural network is applied to recover the truncated SVD coefficients. Numerical results are presented demonstrating that the proposed data driven estimation of the truncated singular values significantly improves the pure truncated SVD reconstruction. We point out that proposed reconstruction framework can straightforwardly be applied to other inverse problems, where the SVD is either known analytically or can be computed numerically., Comment: This paper is a proceedings to our presentation (Paper 10878-111) at the Photons Plus Ultrasound: Imaging and Sensing, (within the SPIE Photonics West), Poster Sunday, February 03, 2019

Recently, a large number of efficient deep learning methods for solving inverse problems have been developed and show outstanding numerical performance. For these deep learning methods, however, a solid theoretical foundation in the form of reconstruction guarantees is missing. In contrast, for classical reconstruction methods, such as convex variational and frame-based regularization, theoretical convergence and convergence rate results are well established. In this paper, we introduce deep synthesis networks for regularizing inverse problems (DESYRE) using neural networks as nonlinear synthesis operator bridging the gap between these two worlds. The proposed method allows to exploit the deep learning benefits of being well adjustable to available training data and on the other hand comes with a solid mathematical foundation. We present a complete convergence analysis with convergence rates for the proposed deep synthesis regularization. We present a strategy for constructing a synthesis network as part of an analysis–synthesis sequence together with an appropriate training strategy. Numerical results show the plausibility of our approach.

In this tutorial, we combine the different scientific fields of information theory, thermodynamics, regularization theory and non-destructive imaging, especially for photoacoustic and photothermal imaging. The goal is to get a better understanding of how information gaining for subsurface imaging works and how the spatial resolution limit can be overcome by using additional information. Here, the resolution limit in photoacoustic and photothermal imaging is derived from the irreversibility of attenuation of the pressure wave and of heat diffusion during propagation of the signals from the imaged subsurface structures to the sample surface, respectively. The acoustic or temperature signals are converted into so-called virtual waves, which are their reversible counterparts and which can be used for image reconstruction by well-known ultrasound reconstruction methods. The conversion into virtual waves is an ill-posed inverse problem which needs regularization. The reason for that is the information loss during signal propagation to the sample surface, which turns out to be equal to the entropy production. As the entropy production from acoustic attenuation is usually small compared to the entropy production from heat diffusion, the spatial resolution in acoustic imaging is higher than in thermal imaging. Therefore, it is especially necessary to overcome this resolution limit for thermographic imaging by using additional information. Incorporating sparsity and non-negativity in iterative regularization methods gives a significant resolution enhancement, which was experimentally demonstrated by one-dimensional imaging of thin layers with varying depth or by three-dimensional imaging, either from a single detection plane or from three perpendicular detection planes on the surface of a sample cube., 20 pages, 14 figures, submission as a tutorial to JAP

We investigate compressed sensing (CS) techniques for reducing the number of measurements in photoacoustic tomography (PAT). High resolution imaging from CS data requires particular image reconstruction algorithms. The most established reconstruction techniques for that purpose use sparsity and $\ell^{1}$ -minimization. Recently, deep learning appeared as a new paradigm for CS and other inverse problems. In this paper, we compare a recently invented joint $\ell^{1}$ -minimization algorithm with two deep learning methods, namely a residual network and an approximate nullspace network. We present numerical results showing that all developed techniques perform well for deterministic sparse measurements as well as for random Bernoulli measurements. For the deterministic sampling, deep learning shows more accurate results, whereas for Bernoulli measurements the $\ell^{1}$ -minimization algorithm performs best. Comparing the implemented deep learning approaches, we show that the nullspace network uniformly outperforms the residual network in terms of the mean squared error (MSE).

Recovering a function or high-dimensional parameter vector from indirect measurements is a central task in various scientific areas. Several methods for solving such inverse problems are well developed and well understood. Recently, novel algorithms using deep learning and neural networks for inverse problems appeared. While still in their infancy, these techniques show astonishing performance for applications like low-dose CT or various sparse data problems. However, there are few theoretical results for deep learning in inverse problems. In this paper, we establish a complete convergence analysis for the proposed NETT (network Tikhonov) approach to inverse problems. NETT considers nearly data-consistent solutions having small value of a regularizer defined by a trained neural network. We derive well-posedness results and quantitative error estimates, and propose a possible strategy for training the regularizer. Our theoretical results and framework are different from any previous work using neural networks for solving inverse problems. A possible data driven regularizer is proposed. Numerical results are presented for a tomographic sparse data problem, which demonstrate good performance of NETT even for unknowns of different type from the training data. To derive the convergence and convergence rates results we introduce a new framework based on the absolute Bregman distance generalizing the standard Bregman distance from the convex to the non-convex case.

The development of accurate and efficient image reconstruction algorithms is a central aspect of quantitative photoacoustic tomography (QPAT). In this paper, we address this issues for multi-source QPAT using the radiative transfer equation (RTE) as accurate model for light transport. The tissue parameters are jointly reconstructed from the acoustical data measured for each of the applied sources. We develop stochastic proximal gradient methods for multi-source QPAT, which are more efficient than standard proximal gradient methods in which a single iterative update has complexity proportional to the number applies sources. Additionally, we introduce a completely new formulation of QPAT as multilinear (MULL) inverse problem which avoids explicitly solving the RTE. The MULL formulation of QPAT is again addressed with stochastic proximal gradient methods. Numerical results for both approaches are presented. Besides the introduction of stochastic proximal gradient algorithms to QPAT, we consider the new MULL formulation of QPAT as main contribution of this paper., Compared to the published journal version, in this manuscript we corrected some typos (e.g. in Theorem 2.8)

We introduce a new iterative regularization method for solving inverse problems that can be written as systems of linear or nonlinear equations in Hilbert spaces. The proposed averaged Kaczmarz (AVEK) method can be seen as a hybrid method between the Landweber and the Kaczmarz methods. As the Kaczmarz method, the proposed method only requires evaluation of one direct and one adjoint subproblem per iterative update. On the other hand, similar to the Landweber iteration, it uses an average over previous auxiliary iterates which increases stability. We present a convergence analysis of the AVEK iteration. Further, detailed numerical studies are presented for a tomographic image reconstruction problem, namely the limited data problem in photoacoustic tomography. Thereby, the AVEK is compared with other iterative regularization methods including standard Landweber and Kaczmarz iterations, as well as recently proposed accelerated versions based on error minimizing relaxation strategies.

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., 28 pages, 4 figures

Recently, deep learning based methods appeared as a new paradigm for solving inverse problems. These methods empirically show excellent performance but lack of theoretical justification; in particular, no results on the regularization properties are available. In particular, this is the case for two-step deep learning approaches, where a classical reconstruction method is applied to the data in a first step and a trained deep neural network is applied to improve results in a second step. In this paper, we close the gap between practice and theory for a new network structure in a two-step approach. For that purpose, we propose so called null space networks and introduce the concept of M-regularization. Combined with a standard regularization method as reconstruction layer, the proposed deep null space learning approach is shown to be a M-regularization method; convergence rates are also derived. The proposed null space network structure naturally preserves data consistency which is considered as key property of neural networks for solving inverse problems.

Purpose In real-time MRI serial images are generally reconstructed from highly undersampled datasets as the iterative solutions of an inverse problem. While practical realizations based on regularized nonlinear inversion (NLINV) have hitherto been surprisingly successful, strong assumptions about the continuity of image features may affect the temporal fidelity of the estimated reconstructions. Theory and Methods The proposed method for real-time image reconstruction integrates the deformations between nearby frames into the data consistency term of the inverse problem. The aggregated motion estimation (AME) is not required to be affine or rigid and does not need additional measurements. Moreover, it handles multi-channel MRI data by simultaneously determining the image and its coil sensitivity profiles in a nonlinear formulation which also adapts to non-Cartesian (e.g., radial) sampling schemes. The new method was evaluated for real-time MRI studies using highly undersampled radial gradient-echo sequences. Results AME reconstructions for a motion phantom with controlled speed as well as for measurements of human heart and tongue movements demonstrate improved temporal fidelity and reduced residual undersampling artifacts when compared with NLINV reconstructions without motion estimation. Conclusion Nonlinear inverse reconstructions with aggregated motion estimation offer improved image quality and temporal acuity for visualizing rapid dynamic processes by real-time MRI. Magn Reson Med 72:1039–1048, 2014. © 2013 Wiley Periodicals, Inc.

Line detectors integrate the measured acoustic pressure over a straight line and can be realized by a thin line of a piezoelectric film or by a laser beam as part of an interferometer. Photoacoustic imaging with integrating line detectors is performed by rotating a sample or the detectors around an axis perpendicular to the line detectors. The subsequent reconstruction is a two-step procedure: first, two-dimensional (2D) projections parallel to the line detector are reconstructed, then the three-dimensional (3D) initial pressure distribution is obtained by applying the 2D inverse Radon transform. The first step involves an inverse problem for the 2D wave equation. Wave propagation in two dimensions is significantly different from 3D wave propagation and reconstruction algorithms from 3D photoacoustic imaging cannot be used directly. By integrating recently established 3D formulae in the direction parallel to the line detector we obtain novel back-projection formulae in two dimensions. Numerical simulations demonstrate the capability of the derived reconstruction algorithms, also for noisy measurement data, limited angle problems and 3D reconstruction with integrating line detectors.

The development of efficient and accurate image reconstruction algorithms is one of the cornerstones of computed tomography. Existing algorithms for quantitative photoacoustic tomography currently operate in a two-stage procedure: First an inverse source problem for the acoustic wave propagation is solved, whereas in a second step the optical parameters are estimated from the result of the first step. Such an approach has several drawbacks. In this paper we therefore propose the use of single-stage reconstruction algorithms for quantitative photoacoustic tomography, where the optical parameters are directly reconstructed from the observed acoustical data. In that context we formulate the image reconstruction problem of quantitative photoacoustic tomography as a single nonlinear inverse problem by coupling the radiative transfer equation with the acoustic wave equation. The inverse problem is approached by Tikhonov regularization with a convex penalty in combination with the proximal gradient iteration for minimizing the Tikhonov functional. We present numerical results, where the proposed single-stage algorithm shows an improved reconstruction quality at a similar computational cost., Comment: 23 pages, 6 figures

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., Comment: 24 pages, 7 figures

Many modern imaging and remote sensing applications require reconstructing a function from spherical averages (mean values). Examples include photoacoustic tomography, ultrasound imaging or SONAR. Several formulas of the back-projection type for recovering a function in $n$ spatial dimensions from mean values over spheres centered on a sphere have been derived in [D. Finch, S. K. Patch, and Rakesh, SIAM J. Math. Anal. 35(5), pp. 1213--1240, 2004] for odd spatial dimension and in [D. Finch, M. Haltmeier, and Rakesh, SIAM J. Appl. Math. 68(2), pp. 392--412, 2007] for even spatial dimension. In this paper we generalize some of these formulas to the case where the centers of integration lie on the boundary of an arbitrary ellipsoid. For the special cases $n=2$ and $n=3$ our results have recently been established in [Y. Salman, J. Math. Anal. Appl., 2014, in press]. For the higher dimensional case $n > 3$ we establish proof techniques extending the ones in the above references. Back-projection type inversion formulas for recovering a function from spherical means with centers on an ellipsoid have first been derived in [F. Natterer, Inverse Probl. Imaging 6(2), pp. 315--320, 2012] for $n=3$ and in [V. Palamodov, Inverse Probl. 28(6), 065014, 2012] for arbitrary dimension. The results of Natterer have later been generalized to arbitrary dimension in [M. Haltmeier, SIAM J. Math. Anal. 46(1), pp. 214--232, 2014]. Note that these formulas are different from the ones derived in the present paper., 13 pages

Recovering an infinite dimensional parameter from incomplete and noisy observations is a fundamental task in many branches of mathematical and engineering science. Reasonable solution approaches require the use of regularization techniques, which incorporate a-priori knowledge about the desired unknown. For that purpose a frequently used property is the (block) sparsity of the coefficients with respect to some sparsifying transformation. In this paper we review regularization methods for sparse inverse problems and derive linear stability estimates for block-sparse analysis regularization implemented via l2,1-minimization.

Imaging the full acoustic field around an object by use of an optical phase contrast method is used to accelerate the data acquisition in photoacoustic tomography. Images obtained with a CCD-camera at a certain time show a projection of the instantaneous pressure field in a given direction. In this work a reconstruction method is presented to obtain the two-dimensional initial pressure distribution by back propagating the observed wave pattern in Radon space. Numerical simulations are used to prove the accuracy of the reconstruction algorithm and to demonstrate a method for correcting limited data artifacts. Finally, the overall performance is shown with experimentally obtained data.

A method for photoacoustic tomography is presented that uses circular integrals of the acoustic wave for the reconstruction of a three-dimensional image. Image reconstruction is a two-step process: In the first step data from a stack of circular integrating are used to reconstruct the circular projection of the source distribution. In the second step the inverse circular Radon transform is applied. In this article we establish inversion formulas for the first step, which involves an inverse problem for the axially symmetric wave equation. Numerical results are presented that show the validity and robustness of the resulting algorithm., Comment: 16 pages (6 figures). we now also present reconstructions with equation (3.5). we modified remark 3.2

In photoacoustic (also called optoacoustic or thermoacoustic) tomography acoustic pressure waves are generated by illumination of a semitransparent sample with pulsed electromagnetic radiation. Subsequently the waves propagate toward the detection surface enclosing the sample. The inverse problem consists of reconstructing the initial pressure sources from those measurements. By combining the high spatial resolution of ultrasonic imaging with the high contrast of optical imaging it offers new potentials in medical diagnostics. In certain applications of photoacoustic imaging one has to deal with media with spatially varying sound velocity, e.g. bones in soft tissue. These inhomogeneities have a strong influence on the propagation of photoacoustically generated sound waves. Image reconstruction without any compensation of this effect leads to a poor image quality. It is therefore essential to develop reconstruction algorithms that take spatially varying sound velocity into account and are able to reveal small structures in acoustically heterogeneous media. A model-based time reversal reconstruction method is presented that is capable of reconstructing the initial pressure distribution despite variations of sound speed. This reconstruction method calculates the time reversed field directly with a second order embedded boundary method by retransmitting the measured pressure on the detector positions in reversed temporal order. With numerical simulations the effect of heterogenous media on sound propagation and the consequences for image reconstruction without compensation are shown. It is demonstrated how time reversal can lead to a correct reconstruction if the distribution of sound speed is known. Corresponding experiments with phantoms consisting of areas with spatially varying sound velocity are carried out and the algorithm is applied to the measured signals.

Photoacoustic imaging is based on the generation of acoustic waves in a semitransparent sample after illumination with short pulses of light or radio waves. The goal is to recover the spatial distribution of absorbed energy density inside the sample from acoustic pressure signals measured outside the sample (photoacoustic inverse problem). We have proposed a numerical method to calculate directly the time reversed field by retransmitting the measured pressure on the detection surface in reversed temporal order. This model-based time reversal method can solve the photoacoustic inverse problem exactly for an arbitrary closed detection surface. Recently we presented a set up which requires a single rotation axis and line detectors perpendicular to the rotation axis. Using a two-dimensional reconstruction method, such as time reversal in two dimensions, and applying the inverse two-dimensional radon transform afterwards gives an exact reconstruction of a three-dimensional sample with this set up. The resolution in photoacoustic imaging is limited by the acoustic bandwidth and therefore by acoustic attenuation, which can be substantial for high frequencies. This effect is usually ignored in reconstruction algorithms but has a strong impact on the resolution of small structures. It is demonstrated that the model based time reversal method allows to partly compensate this effect.

In photoacoustic (also called optoacoustic or thermoacoustic) tomography acoustic pressure waves are generated by illumination of a semitransparent sample with pulsed electromagnetic radiation. Subsequently the waves propagate towards the detection surface enclosing the sample. The inverse problem consists of reconstructing the initial pressure sources from those measurements. In certain applications of photoacoustic imaging one has to deal with media with spatially varying sound velocity, e.g. bones in soft tissue. Image reconstruction without any compensation of this effect leads to a poor image quality. It is therefore essential to develop reconstruction algorithms that take spatially varying sound velocity into account and are able to reveal small structures in acoustically heterogeneous media. A model‐based time reversal reconstruction method is presented that is capable of reconstructing the initial pressure distribution despite variations of sound speed. This reconstruction method calculates the ti...

We consider the stable approximation of sparse solutions to non-linear operator equations by means of Tikhonov regularization with a subquadratic penalty term. Imposing certain assumptions, which for a linear operator are equivalent to the standard range condition, we derive the usual convergence rate $O(\sqrt{\delta})$ of the regularized solutions in dependence of the noise level $\delta$. Particular emphasis lies on the case, where the true solution is known to have a sparse representation in a given basis. In this case, if the differential of the operator satisfies a certain injectivity condition, we can show that the actual convergence rate improves up to $O(\delta)$., Comment: 15 pages

In part I we introduced modified Landweber-Kaczmarz methods and have established a convergence analysis. In the present work we investigate three applications: an inverse problem related to thermoacoustic tomography, a nonlinear inverse problem for semiconductor equations, and a nonlinear problem in Schlieren tomography. Each application is considered in the framework established in the previous part. The novel algorithms show robustness, stability, computational efficiency and high accuracy.

Thermoacoustic computerized tomography (TCT) is a hybrid imaging technique exploiting the thermoacoustic effect. TCT has many parallels with X-ray CT, including frequency dependent attenuation of the measured signal. However, the nature of these attenuation phenomena is nearly opposite: X-ray beams "harden", whereas ultrasound pulses "soften" An analytic formula for an attenuated TCT pulse generat.ed by a homogeneous spherical inclusion is presented. Resulting artifacts in reconstructed X-ray CT and TCT images are compared and contrasted.