Reverse Image Filtering Using Total Derivative Approximation and Accelerated Gradient Descent

In this paper, a new problem of reverse image filtering is addressed. The problem is to reverse the effect of an image filter given the observation
$\boldsymbol {b}=g(\boldsymbol {x})$
. The filter
$g$
is modelled as an available black box. An inverse method is proposed to recover the input image
$\boldsymbol {x}$
. The key idea is to formulate this inverse problem as minimizing a local patch-based cost function. To use gradient descent for the minimization, total derivative is used to approximate the unknown gradient. This paper presents a study of factors affecting the convergence and quality of the output in the Fourier domain when the filter is linear. It discusses the convergence property for nonlinear filters by using contraction mapping as a tool. It also presents applications of the accelerated gradient descent algorithms to three gradient-free reverse filters, including the one proposed in this paper. Results from extensive experiments are used to evaluate the complexity and effectiveness of the proposed algorithm. The proposed algorithm outperforms the state-of-the-art in two aspects. (1) It is at the same level of complexity as that of the fastest reverse filter, but it can reverse a larger number of filters. (2) It can reverse the same list of filters as that of a very complex reverse filter, but its complexity is much lower.