*9.2.1 Structured matrices*

One solution is to use the particular structure of these matrices when possible. This is the case for deconvolution or image restoration, where these matrices have Toeplitz or Bloc-Toeplitz structures which can be well approximated by Circulant or Bloc-Circulant matrices and diagonalized using Fourier Transform (FT) and Fast FT

(FFT). The main idea here is using the properties of the circulant matrices: If *H* is a circulant matrix, then

$$H = F \Lambda F'\tag{86}$$

where *F* is the DFT or FFT matrix and *F*<sup>0</sup> the IDFT or IFFT and Λ is a diagonal matrix whose elements are the FT of the first line of the circulent matrix. As the first line of that circulent matrix contains the samples of the impulse response, the vector of the diagonal elements represents the spectrum of the impulse response (transfer function). Using this property, we have:

$$\left[\mathbf{H}\mathbf{'}\mathbf{H}+\lambda\mathbf{I}\right]^{-1}=\left[\mathbf{F}\mathbf{'}\boldsymbol{\Lambda}\mathbf{F}\mathbf{'}\boldsymbol{\Lambda}+\lambda\right]^{-1}=\left[\mathbf{F}\boldsymbol{\Lambda}^2\mathbf{F}+\lambda\mathbf{I}\right]^{-1}=\mathbf{F}\left[\boldsymbol{\Lambda}^2+\lambda\mathbf{I}\right]^{-1}\mathbf{F}\mathbf{'}\tag{87}$$
