Quantitative photoacoustic tomography (QPAT) is an upsurging imaging modality which can provide high-resolution tissue images based on optical absorption. Classical reconstruction methods rely on sufficient prior information to overcome noisy and imperfect data. As these methods utilise computationally expensive forward models, the computation becomes slow, delimiting the possibilities of QPAT in time-critical applications. As an alternative approach, deep learning-based reconstruction methods have been proposed to allow fast computation of accurate reconstructions. In our work, we adopt the model-based learned iterative approach to solve the nonlinear optical problem of QPAT. In the learned iterative model-based approach, the forward operator and its derivative are iteratively evaluated to compute an update step direction, which is then fed to the network. The learning task is formulated as greedy, requiring iterate-wise optimality, as well as in an end-to-end manner, where all updating networks are trained jointly. We formulated these training schemes and evaluated the performances when the step direction was computed with gradient descent and with the Gauss-Newton method.

Cryo-electron microscopy is a non-linear inverse problem that aims to reconstruct 3-D molecular structures from randomly oriented tomographic projection images, taken at extremely low signal-to-noise-ratio.

This talk presents new results for using the method of moments to reconstruct sparse molecular structures. We prove that molecular structures modeled as sparse sums of Gaussians can be uniquely recovered from the autocorrelations of the images, which significantly lowers the sample complexity of the problem compared to previous results. Moreover, we provide practical reconstruction algorithms inspired by crystallographic phase retrieval.

The full reconstruction pipeline includes estimating autocorrelations from projection images, using rotation-invariant principal component analysis made possible by recent improvements to approximation algorithms into the Fourier-Bessel basis.

Solving inverse problems in a variational formulation requires repeated evaluation of the forward operator and its derivative. This can lead to a severe computational burden, especially so for nonlinear inverse problems, where the derivative has to be recomputed at every iteration. This motivates the use of faster approximate models to make computations feasible, but due to an arising approximation error the need to introduce a designated correction arises.

In this talk we first discuss the concept of learned model corrections applied to linear inverse problems, when computationally fast but approximate forward models are used. We then proceed to examine the possibility to approximate nonlinear models with a linear one and then solve the linear problem instead, avoiding differentiation of the nonlinear model. To correct for the arising approximation errors, we sequentially estimate the error between linear and nonlinear model and update a correction term in the variational formulation. In both cases we discuss convergence properties to solutions of the variational problem given the accurate models.