Reverse image filtering using total derivative approximation and accelerated gradient descent

Abstract

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.

Fast image reverse filters through fixed point and gradient descent acceleration

“…To make the discussion of this paper self contained, we present a brief review of each method by pointing out its main idea and advantages/limitations. The review is similar to that presented in our previous work [7] and is included in Appendix A for easy reference.…”

“…In this table, we present these methods from both the fixed point iteration (detailed in Section 2.2) and gradient descent (detailed in our previous work [7]) point of views. In Section 2.3, we briefly discuss the implementations and relationships between the two point of views based on this table.…”

“…Since the filter is unknown and is potentially highly nonlinear, traditional methods such as image restoration techniques [1], which model the filter as a linear shift invariant system and try to estimate both the signal and the system, cannot be applied. This problem has been studied by researchers in recent years [2], [3], [4], [5], [6], [7]. Methods that have been developed to solve this problem are based on formulation of either a fixed point iteration or a minimization of a cost function using gradient descent.…”

“…In our recent work [7], we proposed a more efficient method called TDA-method which is a gradient free method justified trough total derivative approximation and gradient descent. Also we have explored the acceleration of the T-TDA-, and P-methods using some well known accelerated gradient descent (AGD) methods [10] such as momentum (MGD) Nesterov (NAG), RMSProp, Adadelta and Adam.…”

“…In order to use fixed-point acceleration techniques, we also need to re-develop the TDA and P-method as fixed point iterations. They are originally derived from gradient descent point of view [7], [5].…”