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Numerical Implementation: Gradient Descent on the Euler-Lagrange Equation--Rudin Osher Fatemi Total Variation Denoising Model

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$begingroup$

I am watching some wonderful videos on Variational Methods in Image Processing, and the presenter talks about how variational methods are used to de-blur or denoise images, as well as other applications--including "active contours." Variational image processing has been an active research program for some time. One example model the presenter discusses is the Rudin, Osher, Fatemi model for image denoising. The energy function for this minimization problem is given below.

$$
E(u) = int_Omega(u - f)^2 + lambda |nabla u|dx
$$

The presenter indicated that the Variational problem is minimized using gradient descent methods for optimization.

My question was, does anyone know of a good tutorial on numerically implementing gradient descent on this type of optimization problem? I want to understand exactly how the computer will compute the solution to this problem. Gradient Descent is easy enough to compute, but I was not sure about optimizing over the discretized infinite dimensional space of functions. The functional analysis is not the problem, just understand how the computer algorithm is coded to make this work.

I found an article by Getreuer which talks about the optimization algorithm using the Split Bregman method, which also has some C code. In looking at the code though, it is a bit hard to understand the underlying algorithm versus all of the different function calls and typdefs, and all.

I was hoping someone might know a good online tutorial or video, etc., that discusses how to implement this type of optimization routine. Thanks.

optimization numerical-methods calculus-of-variations image-processing gradient-descent

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asked 2 mins ago

krishnabkrishnab

454415

$endgroup$

add a comment | 

0

$begingroup$

I am watching some wonderful videos on Variational Methods in Image Processing, and the presenter talks about how variational methods are used to de-blur or denoise images, as well as other applications--including "active contours." Variational image processing has been an active research program for some time. One example model the presenter discusses is the Rudin, Osher, Fatemi model for image denoising. The energy function for this minimization problem is given below.

$$
E(u) = int_Omega(u - f)^2 + lambda |nabla u|dx
$$

The presenter indicated that the Variational problem is minimized using gradient descent methods for optimization.

My question was, does anyone know of a good tutorial on numerically implementing gradient descent on this type of optimization problem? I want to understand exactly how the computer will compute the solution to this problem. Gradient Descent is easy enough to compute, but I was not sure about optimizing over the discretized infinite dimensional space of functions. The functional analysis is not the problem, just understand how the computer algorithm is coded to make this work.

I found an article by Getreuer which talks about the optimization algorithm using the Split Bregman method, which also has some C code. In looking at the code though, it is a bit hard to understand the underlying algorithm versus all of the different function calls and typdefs, and all.

I was hoping someone might know a good online tutorial or video, etc., that discusses how to implement this type of optimization routine. Thanks.

optimization numerical-methods calculus-of-variations image-processing gradient-descent

share|cite

asked 2 mins ago

krishnabkrishnab

454415

$endgroup$

add a comment | 

$begingroup$

I am watching some wonderful videos on Variational Methods in Image Processing, and the presenter talks about how variational methods are used to de-blur or denoise images, as well as other applications--including "active contours." Variational image processing has been an active research program for some time. One example model the presenter discusses is the Rudin, Osher, Fatemi model for image denoising. The energy function for this minimization problem is given below.

$$
E(u) = int_Omega(u - f)^2 + lambda |nabla u|dx
$$

The presenter indicated that the Variational problem is minimized using gradient descent methods for optimization.

My question was, does anyone know of a good tutorial on numerically implementing gradient descent on this type of optimization problem? I want to understand exactly how the computer will compute the solution to this problem. Gradient Descent is easy enough to compute, but I was not sure about optimizing over the discretized infinite dimensional space of functions. The functional analysis is not the problem, just understand how the computer algorithm is coded to make this work.

I found an article by Getreuer which talks about the optimization algorithm using the Split Bregman method, which also has some C code. In looking at the code though, it is a bit hard to understand the underlying algorithm versus all of the different function calls and typdefs, and all.

I was hoping someone might know a good online tutorial or video, etc., that discusses how to implement this type of optimization routine. Thanks.

optimization numerical-methods calculus-of-variations image-processing gradient-descent

share|cite

asked 2 mins ago

krishnabkrishnab

454415

$endgroup$

share|cite

asked 2 mins ago

krishnabkrishnab

454415

share|cite

asked 2 mins ago

krishnabkrishnab

454415

asked 2 mins ago

krishnabkrishnab

454415

asked 2 mins ago

krishnabkrishnab

454415