# Transform $\beta^* = \arg \min_\beta \|\beta \ \ln H -\ln J\|^2_2 +\lambda\|\nabla \beta\|^2_2$ to standard linear equation

I am struggling with transforming this eq. $$\beta^* = \arg \min_\beta \|\beta \ \ln H -\ln J\|^2_2 +\lambda\|\nabla \beta\|^2_2$$

to standard linear equation. I have looked into several papers that arg min part can be converted to least square problem, but I do not know, what to do with the rest of the formula.

On the other hand some of the solutions derivate a formula and set the equation to zero, but I am not really sure, if it is applicable to this example.

I found this formula in this paper http://www.cse.cuhk.edu.hk/leojia/projects/pencilsketch/npar12_pencil.pdf on the page 6.

• What do you mean by transforming it into a standard linear equation? What are you trying to get out at the end? – David M. Apr 15 at 1:57
• In the paper they transform the eq. to linear equation to get $\beta$. They use it then to apply conjugate gradient method to solve it. – Muffy Apr 15 at 5:58