2-D Continuous Wavelet Transform

The 2-D continuous wavelet transform is a representation of 2-D data (image data) in 4 variables: dilation, rotation, and position. Dilation and rotation are real-valued scalars and position is a 2-D vector with real-valued elements. Let x denote a two-element vector of real-numbers. If

$$f(x)\in L^2({ℝ}^2)$$

is square-integrable on the plane, the 2-D CWT is defined as

$${\text{WT}}_f(a,b,\theta )={\displaystyle {\int }_{{ℝ}^2}f}(x)\frac{1}{a}\overline{\psi }\left(r_{-\theta }\left(\frac{x-b}{a}\right)\right)dx,a\in {ℝ}^{+},x,b\in {ℝ}^2,$$

where the bar denotes the complex conjugate and r_θ is the 2-D rotation matrix

$$r_{\theta }=\left(\begin{array}{cc}\cos (\theta )& -\sin (\theta )\\ \sin (\theta )& \cos (\theta )\end{array}\right)\theta \in [0,2\pi )$$

The 2-D CWT is a space-scale representation of an image. You can view the inverse of the scale and the rotation angle taken together as a spatial-frequency variable, which gives the 2-D CWT an interpretation as a space-frequency representation. For all admissible 2-D wavelets, the 2-D CWT acts as a local filter for an image in scale and position. If the wavelet is isotropic, there is no dependence on angle in the analysis. The Mexican hat wavelet, also known as the Ricker wavelet, is an example of an isotropic wavelet. Isotropic wavelets are suitable for pointwise analysis of images. If the wavelet is anisotropic, there is a dependence on angle in the analysis, and the 2-D CWT acts a local filter for an image in scale, position, and angle. The Cauchy wavelet is an example of an anisotropic wavelet. In the Fourier domain, this means that the spatial frequency support of the wavelet is a convex cone with the apex at the origin. Anisotropic wavelets are suitable for detecting directional features in an image. See Two-Dimensional CWT of Noisy Pattern for an illustration of the difference between isotropic and anisotropic wavelets.