Two iterative methods for reverse image filtering

Abstract

We consider the problem of recovering an original image $${\varvec{x}}$$ x from its filtered version $${\varvec{y}}={\varvec{f}}({\varvec{x}})$$ y = f ( x ) , assuming that the internal structure of the filter $${\varvec{f}}(\cdot )$$ f ( · ) is unknown to us (i.e., we can only query the filter as a black-box and, for example, cannot invert it). We present two new iterative methods to attack the problem, analyze, and evaluate them on various smoothing and edge-preserving image filters.

Reverse image filtering using total derivative approximation and accelerated gradient descent

“…Para. Complexity T [17] x k+1 = x k + q k N/A max (O(n), C) R [18] x k+1 = αx k + λq k α, λ max (O(n), C) P [19] x…”

“…We follow the terminology used in [19] which called a particular reverse filter a method and briefly comment on each of them in this section. We present a summary of related previous work in Table 1 where we define the following variables which will be used in rest of this paper.…”

“…The P-method and the S-method [19] are inspired by gradient descent methods studied by Polak [32] and Steffensen [33]. They deliver successful results and converge for a larger number of linear and non-linear filters than other methods.…”

“…The 2-norm is the largest singular value of the matrix and has a computational complexity of O(n 2 ). Because the P-method is more stable than the S-method [19], we only consider the P-method in this work. The F-method, which is formulated in the frequency domain [20], uses the Newton-Raphson technique to invert unknown linear filters.…”

“…A brief review of current work on this new problem is presented in section 2 which includes three spatial domain methods: the T-method [17], R-method [18] and P-method [19] and a frequency domain method called F-method [20]. Except the T-method, which can be motivated as a fixed-point algorithm, the other three share a similar idea of formulating a cost function and use gradient approximation to derive the solution to the optimization problem.…”