My first question is about the quadratic functional that is used in generalized Tikhonov regularization: $$\Psi(f)=\frac{1}{2}\|f\|^2_\Gamma=f^T\Gamma f.$$ In the above equation what does $\Gamma$ represent, some special matrix?
Then, I don't understand why $\Psi(f)=\frac{1}{2}\|f'\|^2$ is preferable over $\Psi(f)=\frac{1}{2}\|f\|^2$ because it imposes a penalty on the oscillations in the solution directly, rather than just a penalty on the magnitude of the solution. Does that mean that the process is faster in the case we use derivative?
And one more question, I don't understand how to read the following figure:
Is the part on the right a result of denoising? Under the figure is written that we the covariance $C=\Gamma^{-1}=I$$ is used.