  In this paper, we address a new problem of reversing the effect of an image
filter, which can be linear or nonlinear. The assumption is that the algorithm
of the filter is unknown and the filter is available as a black box. We
formulate this inverse problem as minimizing a local patch-based cost function
and use total derivative to approximate the gradient which is used in gradient
descent to solve the problem. We analyze factors affecting the convergence and
quality of the output in the Fourier domain. We also study the application of
accelerated gradient descent algorithms in three gradient-free reverse filters,
including the one proposed in this paper. We present results from extensive
experiments to evaluate the complexity and effectiveness of the proposed
algorithm. Results demonstrate that the proposed algorithm outperforms the
state-of-the-art in that (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, and
(2) it can reverse the same list of filters as that of the very complex reverse
filter, but its complexity is much smaller.
