Curvelet

Definition

This is the second-generation curvelet system on $\R^2$. Let $W \colon (0,\infty) \to [0,\infty)$ and $V \colon \R \to [0,\infty)$ be smooth windows with $W$ supported in $(1/2, 2)$ and $V$ supported in $[-1, 1]$, such that $$\sum_{j \in \Z} W(2^j r)^2 = 1 \text{ for } r > 0, \qquad \sum_{l \in \Z} V(t - l)^2 = 1 \text{ for } t \in \R.$$ For an integer $j \ge 0$, let $\varphi_j$ be the function whose Fourier transform is, in polar coordinates $(r, \theta)$ of the frequency plane, $$\hat\varphi_j(r, \theta) = 2^{-3j/4} \, W(2^{-j} r) \, V\left(\frac{2^{\floor{j/2}}\theta}{2\pi}\right).$$ Let $R_\theta$ be the rotation of $\R^2$ by the angle $\theta$. For $l$ with $0 \le l < 2^{\floor{j/2}}$, put $\theta_l = 2\pi \cdot 2^{-\floor{j/2}} \, l$. For $k = (k_1, k_2) \in \Z^2$, put $x_k^{(j,l)} = R_{\theta_l}^{-1}(k_1 2^{-j}, k_2 2^{-j/2})$. The curvelet at scale $2^{-j}$, orientation $\theta_l$ and position $x_k^{(j,l)}$ is $$\varphi_{j,l,k}(x) = \varphi_j\left(R_{\theta_l}\left(x - x_k^{(j,l)}\right)\right), \qquad x \in \R^2.$$ The curvelet coefficients of $f \in L^2(\R^2)$ are $c(j,l,k) = \ip{f}{\varphi_{j,l,k}}$.