Parametrization of bivariate nonseparable Haar wavelets

A parametrization of all orthogonal wavelet bases for Haar multiresolution analysis is derived. The bases generated by three piecewise constant wavelets $\{\eta_i(x,y)\}$, $i=1,2,3$, supported on $[0,1]\times[0,1]$, with values $a_{ij}\in\mathbb R$, $i=1,2,3$, $j=1,2,3,4$, are considered.