and for Total Variation optimization how to frame the objective function having L1 norm function
then what is the alternative way of imposing Total variational regularizer .
Dear Asim,
There are several parts to this question:
1. The L1 norm is non-differentiable by nature of the function. Consider the simple case of a one-dimensional input, in which case the L1 norm becomes the absolute value (which is non-differentiable).
2. For some output y = x (input) + n (noise), NxM matrix A, and scalar k, the objective function J* = min(x) ||y-x||22 + k||Ax||1
3. This is a constrained optimization problem that can be best understood through convex optimization, which I can't cover in a single answer... In general, one can take the dual formulation of the objective function via the Lagrangian and thus create a feasible set.
Skipping many math steps, this optimization becomes: J* = max(x
thank you Nathan Zavanelli for the response,
I am dealing with the problem of image denoising where a TV regularizer is used, can you recommend some literature for getting some insights into the usage of TV regularization in the least square estimation problem.
Article SEAGLE: Sparsity-Driven Image Reconstruction under Multiple Scattering
Here are some articles that may prove helpful:
Article Regularization of Linear Least Squares Problems by Total Bou...
https://arxiv.org/ftp/arxiv/papers/1603/1603.09599.pdf
Article Image denoising by generalized total variation regularizatio...
https://iopscience.iop.org/article/10.1088/0266-5611/27/6/065002/meta?casa_token=kHDkvAROgxQAAAAA:aKRcLxtusdl4IeYsbAxJtLglLwF8GxmmnftBhqrac7L1X9peSbHUvCWJ0Coj1gfbrFazwgAaUMze6BrJZXWf
Article Image denoising by generalized total variation regularizatio...
https://sciresol.s3.us-east-2.amazonaws.com/IJST/Articles/2016/Issue-30/Article17.pdf
Article Total Variation Regularization for Image Denoising, I. Geome...
We cannot find the derivative of an absolute value (as L1-norm is sum of absolute values) as its derivative does not exist at 0. So derivative L1-norm cannot be calculated by regular calculus....
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