Image compression by rectangular wavelet transform

We study image compression by a separable wavelet basis $\big\{\psi(2^{k_1}x-i)\psi(2^{k_2}y-j),$ $\phi(x-i)\psi(2^{k_2}y-j),$ $\psi(2^{k_1}(x-i)\phi(y-j),$ $\phi(x-i)\phi(y-i)\big\},$ where $k_1, k_2 \in \mathbb{Z}_+$; $i,j\in\mathbb{Z}$; and $\phi,\psi$ are elements of a standard biorthogonal wavelet basis in $L_2(\mathbb{R})$. Because $k_1\ne k_2$, the supports of the basis elements are rectangles, and the corresponding transform is known as the {\em rectangular wavelet transform}. We prove that if one-dimensional wavelet basis has $M$ dual vanishing moments then the rate of approximation by $N$ coefficients of rectangular wavelet transform is $\mathcal{O}(N^{-M}\log^C N)$ for functions with mixed derivative of order $M$ in each direction. The square wavelet transform yields the approximation rate is $\mathcal{O}(N^{-M/2})$ for functions with all derivatives of the total order $M$. Thus, the rectangular wavelet transform can outperform the square one if an image has a mixed derivative. We provide experimental comparison of image compression which shows that rectangular wavelet transform outperform the square one.