Linearized Riesz transform and quasi-monogenic shearlets (Q2875664)

The Riesz transform \(\mathcal{R}={e}_1 \mathcal{R}_1+{e}_2\mathcal{R}_2: L_2(\mathbb{R}^2,\mathbb{R})\rightarrow L_2(\mathbb{R}^2,\mathbb{C})\) is given by NEWLINE\[NEWLINE \widehat{\mathcal{R}f}(\omega):= \frac{-i\omega_1+\omega_2}{|\omega|} \hat f(\omega) = -i e^{i\varphi(\omega)}\hat f(\omega),\quad \omega = \omega_1+i\omega_2, NEWLINE\]NEWLINE where \(\varphi\in(-\pi,\pi]\) is defined by \(\varphi(\omega) = \arccos(\omega_1/|\omega|)\) if \(\omega_2\geq0\) and \(\varphi(\omega) = -\arccos(\omega_1/|\omega|)\) if \(\omega_2<0\). The Riesz transform is a quadrature operator of order 2. In addition, the Riesz transform corresponds with the rotation operation in the sense NEWLINE\[NEWLINE \mathcal{R}(f(R_\theta^{-1}\cdot)) = e^{i\theta}\mathcal{R}f(R_\theta^{-1}\cdot), NEWLINE\]NEWLINE where \(R_\theta\) is the usual rotation matrix with angle \(\theta\). However, the Riesz transform does not correspond with a shear operation. In this paper, the authors modify the Riesz transform so that it is a quadrature operator of order \(2\) corresponding with the shear operation. The modified Riesz tranform is called linearized Riesz transform given by NEWLINE\[NEWLINE \widehat{\mathcal{R}_Lf}(\omega):= -i e^{i\varphi_L(\omega)}\hat f(\omega),\quad \omega = \omega_1+i\omega_2, NEWLINE\]NEWLINE where \(\varphi_L(\omega) = (1-\mathrm{sgn}(\omega_1))\mathrm{sgn}(\omega_2)\frac{\pi}{2}+\frac{\omega_2}{\omega_1}\frac{\pi}{4}\) if \(|\omega_2/\omega_1|\leq 1\) and \(\varphi_L(\omega) =\mathrm{sgn}(\omega_2)\frac{\pi}{2}-\frac{\omega_1}{\omega_2}\frac{\pi}{4}\) if \(|\omega_2/\omega_1|\geq 1\). The idea comes from the approximation of \(x\) by \(\frac{\pi}{4}\tan x\). \(\varphi\) and \(\varphi_L\) is related by \(\varphi_L(\omega) = \pi/4 \tan\varphi(\omega)\) for \(|\omega_2/\omega_1|\leq 1\) and \(\omega_1>0\). The authors show that the linearized Riesz transform is a quadrature operator that corresponds with a shear operation: NEWLINE\[NEWLINE \mathcal{R}_L(f(S_s^{-1}\cdot)) = e^{is\frac{\pi}{4}}\mathcal{R}_Lf(S_s^{-1}\cdot), NEWLINE\]NEWLINE where \(S_s\) is the shear operator. The authors compare the application of the Riesz transform and the linearized Riesze transform for cartoon image and image edge enhancement. Finally, the authors consider the construction of discrete quasi-monogenic shearlets, which is obtained by applying the linearized Riesz transform to a discrete shearlet. Applications to texture decomposition of images are shown in the last part.