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Deep neural networks for inverse problems with pseudodifferential operators an application to limited-angle tomography

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Document pages: 30 pages

Abstract: We propose a novel convolutional neural network (CNN), called $ Psi$DONet,designed for learning pseudodifferential operators ($ Psi$DOs) in the contextof linear inverse problems. Our starting point is the Iterative SoftThresholding Algorithm (ISTA), a well-known algorithm to solvesparsity-promoting minimization problems. We show that, under rather generalassumptions on the forward operator, the unfolded iterations of ISTA can beinterpreted as the successive layers of a CNN, which in turn provides fairlygeneral network architectures that, for a specific choice of the parametersinvolved, allow to reproduce ISTA, or a perturbation of ISTA for which we canbound the coefficients of the filters. Our case study is the limited-angleX-ray transform and its application to limited-angle computed tomography(LA-CT). In particular, we prove that, in the case of LA-CT, the operations ofupscaling, downscaling and convolution, which characterize our $ Psi$DONet andmost deep learning schemes, can be exactly determined by combining theconvolutional nature of the limited angle X-ray transform and basic propertiesdefining an orthogonal wavelet system. We test two different implementations of$ Psi$DONet on simulated data from limited-angle geometry, generated from theellipse data set. Both implementations provide equally good and noteworthypreliminary results, showing the potential of the approach we propose andpaving the way to applying the same idea to other convolutional operators whichare $ Psi$DOs or Fourier integral operators.

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