Multichannel image denoising using color monogenic curvelet transform

Abstract

Color monogenic curvelet transform (CMCT) is a new multi-scale analysis tool for geometric image features. CMCT has useful properties that it behaves at the fine scales like curvelet transform and at the coarse scales like the color monogenic wavelet transform. CMCT has one magnitude and three phases which encode geometric information of color images. In order to demonstrate the properties of CMCT, new color image denoising algorithm is proposed based on CMCT and total variation. The experimental results demonstrate that the proposed algorithm is at par with or exceeds current state-of-the-art algorithms in both visual and quantitative performance.

Appendix

1.1 Proof of Theorem 1

Let $f\in L^{2}\left( {\mathfrak {R}^{3},\mathfrak {R}} \right)$, $M\lambda _{\mu \nu \theta } \left( x \right) =M\lambda _{\mu \nu \theta } \left( {x-\nu } \right)$, and $\hat{{M}}$ denote the reflection of M, then we have to show

$$\begin{aligned} Mf\left( x \right)= & {} {M\lambda _{\mu 0\theta } {*}M\lambda _{\mu 0\theta } {*}f\left( x \right) \mathrm{d}\theta \mathrm{d}\mu }\Big /{\mu ^{3}} \\= & {} {M\lambda _{\mu 0\theta } {*}\overline{\hat{{M}}\lambda _{\mu 0\theta }} {*}f\left( x \right) \mathrm{d}\theta \mathrm{d}\mu }\Big /{\mu ^{3}} \\= & {} \frac{\int {M\lambda _{\mu 0\theta } \left( {x-\nu } \right) \left( \overline{\hat{{M}}\lambda _{\mu 0\theta }}{*}f \right) } \left( x \right) \mathrm{d}\nu \mathrm{d}\theta \mathrm{d}\mu }{\mu ^{3}} \\= & {} \frac{\int {M\lambda _{\mu 0\theta } \left( {x-\nu } \right) \left( \overline{M\lambda _{\mu 0\theta } (x-\nu )}{*}f \right) } \left( x \right) \mathrm{d}\nu \mathrm{d}\theta d\mu }{\mu ^{3}} \\= & {} \int {{\left\langle {M\lambda _{\mu \nu \theta } ,f} \right\rangle M\lambda _{\mu \nu \theta } (x)\mathrm{d}\nu \mathrm{d}\theta \mathrm{d}\mu }\Big /{\mu ^{3}}} \end{aligned}$$