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On the importance of mutual coherence for image reconstruction with neural networks

Loohuis, P. (2020) On the importance of mutual coherence for image reconstruction with neural networks.

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Abstract:We consider the issue of solving inverse problems with a convolutional neural network (CNN). We introduce a left preconditioning method that lowers the mutual coherence of the preconditioned forward mapping as much as possible and examine whether this is beneficial when the inverse problem is solved with a CNN. We consider a CNN because this network contains less trainable parameters, which makes it a popular option for learning the inverse mapping. This type of network, however, can only be used when the forward mapping is a local mapping. If this is not the case, e.g. when the Radon transform is used, then preconditioners can be used in order to transform the forward mapping to one that is close to the identity matrix. These mappings can have a low mutual coherence, which can be beneficial when solving inverse problems with neural networks. Besides that, we propose a framework based on information theory that connects inverse problems, compressed sensing and preconditioning via the Bayesian perspective on inverse problems. This framework enables us to analyze the effects of our preconditioning method. We show that the mutual coherence depends on the amount of regularization applied to the inverse problem, where a low mutual coherence implies small amounts of regularization. This has a big effect on the noise present in the inverse problem, leading to less detailed reconstructions. We show that when only left preconditioners are used, lowering the mutual coherence as much as possible is therefore not beneficial. In the discussion, we explain that lowering the mutual coherence can be beneficial when right preconditioners are also taken into account.
Item Type:Essay (Master)
Faculty:EEMCS: Electrical Engineering, Mathematics and Computer Science
Subject:31 mathematics
Programme:Applied Mathematics MSc (60348)
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