Tikhonov regularization

The Tikhonov regularization is a simple method to deal with ill-conditionned inverse problems. The measured data \(y\) is modeled by \[ y = Hx + n\] where \(n\) is a measurement noise, usually assumed white and gaussian. The estimate \(\hat x_\lambda\) is found by solving the optimization problem \[ \hat x_\lambda = \mathrm{argmin}_x \|y - Hx\|_2^2 + \lambda \|\Gamma x\|_{2}^2 \] where \(\Gamma\) is matrix modeling priori information about \(x\), and \(\lambda\) is the regularization parameter.

Here, \(\Gamma\) is chosen to be the identity operator, first derivative, or second derivative