Reverse image filtering with clean and noisy filters
Lizhong Wang, Pierre‐Alain Fayolle, Alexander Belyaev
Abstract
Abstract Given an image filter $${{\varvec{y}}}={{\varvec{f}}}\,({{\varvec{x}}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mi>y</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> <mml:mspace/> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , where $${{\varvec{x}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> </mml:math> and $${{\varvec{y}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>y</mml:mi> </mml:mrow> </mml:math> are input and output images, respectively, reverse image filtering consists of rendering an approximation to $${{\varvec{x}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> </mml:math> from $${{\varvec{y}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>y</mml:mi> </mml:mrow> </mml:math> using the filter $${{\varvec{f}}}\,(\cdot )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> <mml:mspace/> <mml:mo>(</mml:mo> <mml:mo>·</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> itself as a black box, without knowing the internal structure of the filter. In this paper, we propose to use modified Landweber iterations for reverse image filtering, evaluate the performance of our approach, and present applications to image deblurring and super-resolution. An important advantage of our approach over the existing reverse image filtering methods is high robustness to noise.