Questions tagged [variational-methods]

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Justification for Squared $ {L}_{2} $ Data and Smoothness Term as an Error Bound

Often in variational methods (and not only) we have an energy that is of the form: $$E(u) = \frac{1}{2}\|f-u\|^2_2 + \frac{\alpha}{2}\|\psi(u)\|^2_2,$$ where the first term is referred to as the data ...
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Laplacian of Gaussian Approximation and Gaussian Blur as the Solution of Heat Equation

While I was reading SIFT paper(Lowe, 2004), I came across the method that he apply "heat diffusion equation" to Gaussian function to derive that $$ \frac{∂G}{∂σ} = σ∇^2G $$ I searched Wikipedia and ...
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Is the behaviour of the following filter plausible?

I've designed a filter based on this equation $$ E(u) = \frac{1}{2}\int_{\Omega} \left[(u-f)^2 + g(\lVert \nabla u \rVert)\right]dxdy $$ Euler-Lagrange equations in this case are (should be) given ...