Search

We study linear inverse problems under the premise that the forward operator is not at hand but given indirectly through some input-output training pairs. We demonstrate that regularization by projection and variational regularization can be formulated by using the training data only and without making use of the forward operator. We study convergence and stability of the regularized solutions in view of Seidman (1980 J. Optim. Theory Appl. 30 535), who showed that regularization by projection is not convergent in general, by giving some insight on the generality of Seidman’s nonconvergence example. Moreover, we show, analytically and numerically, that regularization by projection is indeed capable of learning linear operators, such as the Radon transform.

We propose aNETT (augmented NETwork Tikhonov) regularization as a novel data-driven reconstruction framework for solving inverse problems. An encoder-decoder type network defines a regularizer consisting of a penalty term that enforces regularity in the encoder domain, augmented by a penalty that penalizes the distance to the signal manifold. We present a rigorous convergence analysis including stability estimates and convergence rates. For that purpose, we prove the coercivity of the regularizer used without requiring explicit coercivity assumptions for the networks involved. We propose a possible realization together with a network architecture and a modular training strategy. Applications to sparse-view and low-dose CT show that aNETT achieves results comparable to state-of-the-art deep-learning-based reconstruction methods. Unlike learned iterative methods, aNETT does not require repeated application of the forward and adjoint models during training, which enables the use of aNETT for inverse problems with numerically expensive forward models. Furthermore, we show that aNETT trained on coarsely sampled data can leverage an increased sampling rate without the need for retraining.

Photoacoustic tomography (PAT) is an imaging technique that enables us to visualize vasculature networks at high resolution at depths much greater than optical microscopy.1,2 PAT can also resolve tissue chromophores such as oxyhemoglobin and deoxyhemoglobin, which allows estimation of tissue oxygentation levels.1,3 Accurate estimation of tissue oxygentation is very important in the fields of neuroimaging, cancer diagnostics, and critical care medicine.4,5 PAT involves collection of broadband acoustic waves generated by thermoelastic expansion of tissue upon absorption of light (illuminated at near-infrared wavelengths).6,7 PAT image reconstruction involves recovery of the acoustic source distribution from the detected broadband acoustic waves, which can be performed using analytical or model-based methods.8 Analytical methods are computationally less expensive compared to model-based image reconstruction.8,9 In contrary, model-based schemes are known to generate accurate images compared to analytical schemes, specifically in scenarios where irregular acquisition geometry is involved or limited data is acquired.9,10 Even with the model-based schemes, PAT inversion becomes difficult when the number of ultrasound detector positions is limited, and the situation gets aggravated due to the presence of noise in the data. More recently deep learning-based image postprocessing schemes have been developed for artifact removal and for limited data situations in PAT/microscopy;11–13 however, these schemes were trained on simulated data and generating large experimental data is very difficult. Segmentation approaches were proposed to delineate skin lining in mesoscopic photoacoustic images.14 Active contours-based image segmentation was used to obtain the PAT region of interest (ROI), the obtained ROI was then used to perform fluence correction to make PAT more quantitative.15,16 Different computational techniques have been reported in the literature to obtain vascular network in photoacoustic imaging. Vesselness filter has been proposed to enable accurate recovery of vasculature in PAT;17–19 however, questions have been raised on the accuracy of using the vesselness filter (due to generation of artificial vasculature).20 Probabilistic approach involving different steps such as smoothing and filtering the data, clustering, vessel-segmentation using clusters, and morphological filling was developed for segmenting vessels in photoacoustic images.21 More recently convolution neural network-based approaches were also developed for jointly performing the image reconstruction and segmentation.22 Further, deep learning-based approach was proposed for segmenting animal boundary using multi-modal photoacoustic and ultrasound data.23 Further, deep learning-based method has also been developed for animal brain imaging.24 Notably segmenting vasculature is very important in the context of diagnosis and surgical guidance, specifically for cardiovascular diseases.25,26 In this work, we developed a binary tomography approach wherein we assume the reconstruction distribution contains only two unknowns namely the background and the absorbers. The major contributions of this work include: (a) application of binary tomographic methods for photoacoustic tomographic imaging involving spherical wave propagation as opposed to straight line approximation applied in x-ray computed tomography; (b) showing the utility of binary tomographic methods for improving photoacoustic tomographic imaging of vasculature; and (c) establishing the binary tomographic method superiority in photoacoustic tomographic imaging for providing approximate priors to improve model-based image reconstruction. Binary tomography problem is solved based on a dual optimization approach with an auxiliary variable posed as a constrained optimization problem. We compared the binary tomography algorithm against standard backprojection approach and model-based reconstruction schemes such as Tikhonov regularization and sparsity-based regularization.27 The performance of these different algorithms was validated with in-silico data, physical phantom, and in-vivo brain vasculature imaging data.

We introduce a framework for the reconstruction and representation of functions in a setting where these objects cannot be directly observed, but only indirect and noisy measurements are available, namely an inverse problem setting. The proposed methodology can be applied either to the analysis of indirectly observed functional images or to the associated covariance operators, representing second-order information, and thus lying on a non-Euclidean space. To deal with the ill-posedness of the inverse problem, we exploit the spatial structure of the sample data by introducing a flexible regularizing term embedded in the model. Thanks to its efficiency, the proposed model is applied to MEG data, leading to a novel approach to the investigation of functional connectivity., 40 pages