Hello Every one I have been trying to understand the intuitive meaning of using a regularizer in images. Specifically what does the Total Variation regularizer do in images and how is it able to prevent the edge smoothing?
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$\begingroup$ Regularization means to reduce the solution space to something one can handle $\endgroup$– mathreadlerCommented Jun 16, 2018 at 11:03
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One general form of Inverse Problem in Imaging which assumes Linear Operator is given by:
$$ \arg \min_{x} \frac{1}{2} {\left\| A x - b \right\|}_{2}^{2} + \lambda R \left( x \right) $$
Where $ R \left( x \right) $ is the regularization function.
One way, which is pretty intuitive in my opinion, is to treat the above as a solution to Maximum a Posteriori Estimation (MAP) problem.
The first term, the fidelity term is the result of the assumption additive white Gaussian noise.
The regularization is the term of the Prior Knowledge on the image.
In the case of Total Variation it is usually given by:
$$ R \left( x \right) = \sum_{i, j} \left| {x}_{i + 1, j} - {x}_{i, j} \right| + \left| {x}_{i, j + 1} - {x}_{i, j} \right| $$
If you think on it as a Derivative (Vertical and Horizontal) then the MAP tells you have the assumption of Laplace Distribution for the Derivative.
It is well known that Laplace Distribution is something which promotes sparsity.
Namely the assumption is having few large values for the Derivative but mostly having values of zeros.
This fits perfectly with the Piece Wise Smooth model of Imaging.
Hence, in case of Total Variation Denoising the use of this regularization tries to smooth image while assuming that small variation of the derivative is caused by noise hence should be smoothed out and larger variations should be sparse.