Learning integral operators from diagonal-circulant neural networks

There are imaging applications (e.g. optical calibration, exoplanet detection) where we observe an image "convolved" with a kernel whose shape depends on its localization. These operators are non shift-invariant and therefore cannot be modeled by a simple convolution. Moreover, these operators can appear in the resolution of an inverse problem. To solve such an optimization problem, it is necessary to evaluate at each iteration the operator and its derivatives with respect to its inputs. If the operator or its derivatives are too costly to evaluate, this may make the optimization problem intractable. Therefore, if we have an approximation model for which we can efficiently compute the derivatives and the result for a given input, we can replace the real operator by the approximation in the solving of the inverse problem. The problem addressed in this paper concerns the approximation of these general integral operators based on a dataset containing the values of the operator for several inputs. The mathematical model used in this work to learn the integral operator is based on a feedforward neural network composed of an alternating product of diagonal and circulant matrices for one-dimensional applications, whereas it will be an alternating product of diagonal and doubly-block-circulant matrices for two-dimensional applications. In addition to the computational interest of these networks, this work has shown that they are useful for particular problems for which the physical system presents such factorization, for example in the context of exoplanet detection by direct imaging. In these problems, the amount of training data can be significantly reduced.