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I was reading the recent book Variational Methods in Image Processing by Vese and Guyader which is quite interesting. In the book the authors discuss different types of image processing problems, from denoising and deblurring problems to image segmentation and beyond. The common theme is that they use Variational Methods, read as Calculus of Variations methods to take the noisy image and remove the noise. My question is, why do these scholars use Variational Methods (Euler-Lagrange Equation) to solve a problem that is not a Boundary Value Problem. Let me lay some foundation to the question.

Now these image problems are naturally optimization problems where I have some original image $f$ and I want to find a new image $u$ that best replicates $f$ but is subject to some constraints. The Rudin-Osher-Fatemi method sets up the problem this way:

$$ F(u)=\int_{\Omega}|D u|+\lambda \int_{\Omega}(K u-f)^{2} d x $$

Where the second term is the standard difference between the original image $f$ and the adjusted image $Ku$, and the first term is the Total Variation norm in the image, meaning the sum of the absolute value of the first difference between adjacent pixels.

The solution to this problem sets up the Euler-Lagrange equation for the above energy function, and then minimizes that energy using gradient descent or such algorithms.

BUT, in my mind Calculus of Variations was always taught as a way to solve a boundary value problem. I am given two points, and I want to find the least energy path between those two points. So I have a BVP and I can either solve the optimization problem directly by discretizing the space between the two points and using shooting methods + opimizer to find the minimal energy path. Or I would use the Euler-Lagrange equation to find the least energy path as the numerical solution ODE or PDE--and thus avoiding minimization.

In looking at image processing problems, I did not have a sense of a boundary value problem? The initial image is fine but the final image is the result of optimization and not known before hand. Hence, can anyone explain why is the Variational approach used, or Euler-Lagrange approach?

Are they just using the E-L to find a minimum without thinking of a boundary value, or is there some other intuition going on that I am just not catching?

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This turned out to be easier than I thought. Image processing is a boundary value problem, and the boundary is the set of pixels along the edge of the image. The common notation for this boundary in the Vese Guyader book is $\partial(\Omega)$, so that is not the most obvious notation for an image. Of course $\partial$ is often the boundary operator in differential topology too. I think for the Rudin-Osher-Fatemi method, there is a Neumann boundary condition, meaning that the first derivative at the boundary pixels is not changing.

So the answer was very easy after all.

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