I am trying to form an approximation of the wavelet transform from a partially sampled image.
Reconstruction in the 1D case is easy. We have $w = h x$, with $w$ as the wavelet coefficients, $h$ as the inverse of the Haar matrix and $x$ our signal. $w$ and $x$ are $n \times 1$ and $h$ is $n \times n$. Where we don't sample the signal, we delete those columns from $h$ and those rows/elements from the full signal, $x$. Since we have to have as many equations as unknowns, we sample at $m$ locations and approximate the first $m$ wavelet coefficients. All we do is delete the corresponding columns of where we do not sample.
I have $w = c x r$, where $c$ is the matrix that performs the wavelet transform on the columns, and $r$ the rows. I am having a hard time forming in approximation in this 2D case. I know this is a simple solution and am having a hard time getting the math behind it. I know it's just solving a linear system of equations, but am having a hard time coming up with the matrix to represent that linear system.