Bivariate (Two-dimensional) Wavelets

Definitions

Throughout this article, ${ \mathbb{R}, \mathbb{C}, \mathbb{Z} }$ denote the real line, the complex plane, and the set of all integers, respectively. For ${ 1\leqslant p \leqslant \infty }$, ${ L_p(\mathbb{R}^2) }$ denotes the set of all Lebesguemeasurable bivariate functions f such that $\|f\|^p_{L_p(\mathbb{R}^2)}:=\int_{\mathbb{R}^2} |f(x)|^p \text{d} x<\infty$. In particular, the space ${ L_2(\mathbb{R}^2) }$ of square integrable functions isa Hilbert space under the inner product

$$ \langle f, g\rangle:=\int_{\mathbb{R}^2} f(x) \overline{g(x)} \text{d} x\:,\quad f,g\in L_2(\mathbb{R}^2)\:, $$

where ${ \overline{g(x)} }$ denotes the complex conjugate of the complex number g(x).

In applications such as image processing and computer graphics, the following are commonly used isotropic dilation matrices:

$$ \begin{aligned} M_{\sqrt{2}}&=\left[\begin{matrix}1 &1\\ 1&-1\end{matrix}\right]\:,\quad& Q_{\sqrt{2}}&=\left[\begin{matrix}1 &-1\\ 1...