Efficient solution of linear inverse problems using an iterative linear neural network with a generalization training approach

Abstract

Inverse problems are some of the most important mathematical problems in science and mathematics because their solution yields information about parameters that are not directly observable. Artificial neural networks have long been used as a mathematical modelling method and have been used successfully to solve inverse problems for application including denoising and medical image reconstruction. Many inverse problems result from integral processes that can be modelled using a linear formulation. These can be efficiently solved via simple networks which are easily trained with reasonable datasets. An innovative simple neural network architecture, the iterative linear neural network (ILNN), consisting of two non-hidden layer networks, one for the forward model and one for the inverse model, is proposed to solve linear inverse problems. Iteration between the two models refines network outcomes with greater accuracy than a network with only the inverse model. A training procedure accompanying the network is also introduced. The network needs to train only the inverse model with one-hot vectors as targets. The training inputs of the inverse model define the weights of the forward model. The number of targets is finite and equal to the length of the vector. With the defined targets, the training process ensures that the inverse model is at least a left inverse of the forward model. This leads to generalizable networks. The experimental results show that the ILNN produces good results even if its inverse model is not perfectly trained. The proposed network is applied to solve two linear inverse problems, deconvolution and the inverse Radon transform. The network successfully reconstructed original data following blurring and Radon transformation.