  A bounded, Riemann integrable and measurable set $K\subset \mathbb{R}^d$,
which fulfills \[\sum\limits_{\gamma\in\Gamma}\mathbb{1}_K(x-\gamma)=k\text{
almost everywhere, $x\in\mathbb{R}^d$}\] for a lattice
$\Gamma\subset\mathbb{R}^d$ is called $k$-tiling. If $K\subset\mathbb{R}^d$ is
$k$-tiling $L^2(K)$ will admit a Riesz basis of exponentials. We use this
result to construct generalized Riesz wavelet bases of $L^2(\mathbb{R}^2)$,
arising from the action of suitable subsets of the affine group. One example of
our construction is the first known shearlet Riesz basis.
