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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.

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  • $\begingroup$ Why do you want to formulate the matrix? It's enough to note that it's a linear system and then solve the system. In this field, the matrix-vector notation is a neat tool to see what is going on, you never explicitly compute the matrix. If you want to use one of the (vast array) of techniques for solving such linear inverse problems, you're unlikely to need to explicitly represent the matrix. $\endgroup$ – Henry Gomersall Aug 20 '13 at 16:12
  • $\begingroup$ I'm formulating the matrix because we don't have every sample, meaning it's easier (I think!) to form the matrix. We're trying to solve a linear system of equations, and I'm having a hard time coming up with the equations. $\endgroup$ – Scott Aug 20 '13 at 19:09
  • $\begingroup$ It makes no difference that it's 1D or 2D, the maths is the same. You have the Haar matrix (which you don't really need to know explicitly) and the sampling matrix and that should be enough. $\endgroup$ – Henry Gomersall Aug 21 '13 at 9:26

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