We introduce a method to infer a variational approximation to the posterior distribution of solutions in computational imaging inverse problems. Machine learning methods applied to computational imaging have proven very successful, but have so far largely focused on retrieving a single optimal solution for a given task. Such retrieval is arguably an incomplete description of the solution space, as in ill-posed inverse problems there may be many similarly likely reconstructions. We minimise an upper bound on the divergence between our approximate distribution and the true intractable posterior, thereby obtaining a probabilistic description of the solution space in imaging inverse problems with empirical prior. We demonstrate the advantage of our technique in quantitative simulations with the CelebA dataset and common image reconstruction tasks. We then apply our method to two of the currently most challenging problems in experimental optics: imaging through highly scattering media and imaging through multi-modal optical fibres. In both settings we report state of the art reconstructions, while providing new capabilities, such as estimation of error-bars and visualisation of multiple likely reconstructions.
A second emerging class of learning based methods in CI is that of maximum a posteriori (MAP) inference with generative models. In these frameworks, similarly to MAP inference with analytical priors, the solution is found by minimising the Euclidean distance to the observations @math . However, instead of regularising the solution with some penalty function, the signal of interest is assumed to be generated by a pre-trained generative model as @math , where @math is the generative model's latent variable @cite_18 . As the generative model is trained with an example set that the target signal is assumed to belong to, generated images are expected to retain empirically learned properties @cite_37 . The solution is then found by performing the following minimisation, In such a way, the solution is constrained to be within the domain of the generative model, as the recovered @math is by definition generated from @math , but at the same time agreement to the measurements is directly induced. Though the optimisation problem is not convex, certain theoretical bounds were derived for the cases of linear observation processes by and more recently phase-less linear observation processes @cite_4 .
@cite_18 @cite_4 @cite_37
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The goal of compressed sensing is to estimate a vector from an underdetermined system of noisy linear measurements, by making use of prior knowledge on the structure of vectors in the relevant domain. For almost all results in this literature, the structure is represented by sparsity in a well-chosen basis. We show how to achieve guarantees similar to standard compressed sensing but without employing sparsity at all. Instead, we suppose that vectors lie near the range of a generative model @math . Our main theorem is that, if @math is @math -Lipschitz, then roughly @math random Gaussian measurements suffice for an @math recovery guarantee. We demonstrate our results using generative models from published variational autoencoder and generative adversarial networks. Our method can use @math - @math x fewer measurements than Lasso for the same accuracy. While deep learning methods have achieved state-of-theart performance in many challenging inverse problems like image inpainting and super-resolution, they invariably involve problem-specific training of the networks. Under this approach, each inverse problem requires its own dedicated network. In scenarios where we need to solve a wide variety of problems, e.g., on a mobile camera, it is inefficient and expensive to use these problem-specific networks. On the other hand, traditional methods using analytic signal priors can be used to solve any linear inverse problem; this often comes with a performance that is worse than learning-based methods. In this work, we provide a middle ground between the two kinds of methods — we propose a general framework to train a single deep neural network that solves arbitrary linear inverse problems. We achieve this by training a network that acts as a quasi-projection operator for the set of natural images and show that any linear inverse problem involving natural images can be solved using iterative methods. We empirically show that the proposed framework demonstrates superior performance over traditional methods using wavelet sparsity prior while achieving performance comparable to specially-trained networks on tasks including compressive sensing and pixel-wise inpainting.