2D TV Denosing with ADMM -- Mathematics && Implementation
December 25, 2020
For 2D Total variation denosing, we have the following objective, where $x$ is the clean image we want to optimize, $y$ is the origin noisy image, and $D_r $and $D_c$ are doubly block circulant matrices for two 2D convolution.
In detail, both of $x$ and $y$ are vectorized into 1D vectors, and the total variation terms are expressed as two convolutions in Matrix-vector form.
$$
\operatorname{minimize} \enspace \frac{1}{2} \|x-y\|_{2}^{2} + \lambda\|D_r x\|_{1} + \lambda\|D_c x\|_{1}
$$
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