Abstract
The problem for image restoration is usually reduced to a constraint optimization problem. Different choice of optimization operator leads to various restoration models, e.g. least squares model and original total variation (TV) model. The TV model and its modified version can efficiently preserve the edge of the restored image well, but there exist obvious staircases in smooth area of the restored image. To reduce those staircases, we propose a new modified TV model, by adding a constraint term for smooth area protection as a penalty function. The numerical experiment shows our model can not only preserve the edge as well as TV model, but also efficiently reduce the staircase appearing in the smooth areas. Furthermore, It is shown that the restored image by our model has higher signal-to-noise ratio, less mean square error and better visual effect than those by the least squares model and by the TV models.
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Notes
Given a point (x, y), let \( m\;\left( {x,y} \right) \) be the mean value of its neighborhood excluding point (x, y) itself.
$$ {m_f}\left( {x,y} \right) = {{\left[ {\iint\limits_{\Omega p} {f\left( {x,y} \right)dxdy - f\left( {x,y} \right)}} \right]} \mathord{\left/{\vphantom {{\left[ {\iint\limits_{\Omega p} {f\left( {x,y} \right)dxdy - f\left( {x,y} \right)}} \right]} {\iint\limits_{{\Omega_p}} {I\left( {x,y} \right)dxdy}}}} \right.} {\iint\limits_{{\Omega_p}} {I\left( {x,y} \right)dxdy}}} $$(13-1)$$ \sigma_f^2 = {\raise0.7ex\hbox{${{{\iint\limits_{{\Omega_p}} {\left( {f\left( {x,y} \right)dxdy - {m_f}\left( {x,y} \right)} \right)}}^2}}$} \!\mathord{\left/{\vphantom {{{{\iint\limits_{{\Omega_p}} {\left( {f\left( {x,y} \right)dxdy - {m_f}\left( {x,y} \right)} \right)}}^2}} {\iint\limits_{{\Omega_p}} {I\left( {x,y} \right)dxdy}}}}\right.}\!\lower0.7ex\hbox{${\iint\limits_{{\Omega_p}} {I\left( {x,y} \right)dxdy}}$}} $$(13-2)where Ω p is the neighborhood of point (x, y). Let \( \sigma_{f.\max }^2 \) be the maximum mean-square error of all neighborhoods. If \( \sigma_f^2\left( {x,y} \right) < \beta .\sigma_{f.\max }^2 \), then this region is considered as a smooth area, assuming β is small enough.
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Acknowledgements
Supported by: National Natural Science Foundations of China, No. 10771022; by Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, No. 890 [2008]; and by major foundation of educational committee of Hunan Province.
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Wang, C., Liu, Z. Total Variation for Image Restoration with Smooth Area Protection. J Sign Process Syst 61, 271–277 (2010). https://doi.org/10.1007/s11265-010-0451-3
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DOI: https://doi.org/10.1007/s11265-010-0451-3