# Is the Bilateral Filter a Solution of Some Variational Method?

I've been watching these video lectures about variational methods in computer vision.
In one of such video lectures a it is explained for example that the Gaussian filtering is a solution of the diffuse equation (so Gaussian filtering is a specific class of diffuse filtering). I do wonder if the same applies to the bilateral filter, because if so it can means that all such filters can be derived by some functional to be minimized. This is because the diffusion equation is a solution of a variational problem like

$$E(u) = \int_{\Omega} \left[(u-f)^2 + \lambda\lVert \nabla u\rVert^2 \right]d\Omega$$

Again, is the bileteral filter a similar case?

$$\arg \min_{x} \frac{1}{2} {\left\| x - y \right\|}_{2}^{2} + \frac{\lambda}{2} \sum_{n = 1}^{N} { \left( x - {D}^{n} x \right) }^{T} W \left( y, n \right) \left( x - {D}^{n} x \right)$$
Where ${D}^{n}$ is Image Gradient Operator (Raising by $n$ simply shifts it) and $W$ is the weights matrix of the Regularized Least Squares.
• Can you explain what all the parameters mean? ($W$ is a window function I assume, what about $D^n$?) – user8469759 Jul 6 '18 at 22:06