FOCNet: A Fractional Optimal Control Network for Image Denoising

Deep convolutional neural networks (DCNN) have been successfully used in many low-level vision problems such as image denoising. Recent studies on the mathematical foundation of DCNN has revealed that the forward propagation of DCNN corresponds to a dynamic system, which can be described by an ordinary differential equation (ODE) and solved by the optimal control method. However, most of these methods employ integer-order differential equation, which has local connectivity in time space and cannot describe the long-term memory of the system. Inspired by the fact that the fractional-order differential equation has long-term memory, in this paper we develop an advanced image denoising network, namely FOCNet, by solving a fractional optimal control (FOC) problem. Specifically, the network structure is designed based on the discretization of a fractional-order differential equation, which enjoys long-term memory in both forward and backward passes. Besides, multi-scale feature interactions are introduced into the FOCNet to strengthen the control of the dynamic system. Extensive experiments demonstrate the leading performance of the proposed FOCNet on image denoising. Code will be made available.