How to Use the DFT (FFT) to Solve a Least Squares Regularization Problem (Inverse Problem)?
The question really depends on $ f \left( \cdot \right) $.
Yet in order to show how to use FFT we can even use 1D signals.

Let's rewrite the problem:

$$ \hat{x} = \arg \min_{x} \frac{1}{2} \left\| K x - b \right\|_{2}^{2} + \frac{\lambda}{2} \left\| f \left( x \right) \right\|_{2}^{2} $$

The derivative is given by:

$$ g = {K}^{T} \left( K x - b \right) + \lambda f' \left( x \right)^{T} f \left( x \right) $$

Now, $ K $ as a matrix is given by a Circulant Convolution Matrix.
Hence the operation $ K v $ or $ {K}^{T} v $ can be done in the Fourier Domain.
Under some circumstances it might accelerate the operation significantly.