14.4.4 Data space

According to the definition of the image to be reconstructed, the Fourier data corresponding to  $\Phi_{\! s}$ are defined by the relationship:

$$\Psi_{\! s}(\mathbf{u}) = \widehat{\Theta}_s(\mathbf{u}) \Psi_{\! e}(\mathbf{u}) \quad\forall \mathbf{u} \in \mathcal{L}_e.$$ (14.9)

Clearly, $\Psi_{\! s}$ lies in the experimental data space K_e.

Let us now introduce the data vector:

$$\Psi_{\! d}(\mathbf{u}) = \begin{cases} \Psi_{\! s}(\mathbf{u... ...mathcal{L}_e$ ;}\\ 0 & \text{on $\mathcal{L}_r$ .} \end{cases}$$ (14.10)

This vector lies in the data space K_d, the real Euclidian space underlying the space of complex-valued functions $\psi$ on  $\mathcal{L}$, such that  $\psi(-\mathbf{u}) = \bar{\psi}(\mathbf{u})$. This space is equipped with the scalar product:

$$(\psi_1 \mid \psi_2)_d = \sum_{\mathbf{u}\in\mathcal{L}_e} \b... ..._r} \bar{\psi_1}(\mathbf{u}) \psi_2(\mathbf{u}) (\delta u )^2;$$ (14.11)

W(u) is a given weighting function that takes into account the reliability of the data via the standard deviation  $\sigma_e(\mathbf{u})$ of  $\Psi_{\! e}(\mathbf{u})$, as well as the local redundancy  $\rho(\mathbf{u})$ of  u up to the sampling interval $\delta u$.

The Fourier sampling operator A is the operator from the object space H_o into the data space K_d:

$$A: H_o \longrightarrow K_d, \qquad (A\phi)(\mathbf{u}) = \beg... ...hat{\phi}(\mathbf{u}) & \text{on $\mathcal{L}_r$ .} \end{cases}$$ (14.12)

As the experimental data  $\Psi_{\! e}(\mathbf{u})$ are blurred values of  $\widehat{\Phi}_o(\mathbf{u})$ on  $\mathcal{L}_e$, this operator will play a key role in the image reconstruction process. The definition of this Fourier sampling operator suggests that the action of Ashould be decomposed into two components: A_e on the experimental frequency list  $\mathcal{L}_e$, and A_r on the regularization frequency list  $\mathcal{L}_r$.