14.4.6 Objective functional

The reconstructed image is defined as the function $\Phi _E$minimizing on E the objective functional:

$$q(\phi) = \Vert \Psi_{\! d} - A \phi\Vert^2_d.$$ (14.13)

According to the definition of the data vector  $\Psi_{\! d}$ and to that of the Fourier sampling operator A, this quantity can be written in the form:

$$q(\phi) = q_e(\phi) + q_r(\phi) \text{ with } \begin{cases} q... ...\widehat{\phi}(\mathbf{u}) \vert^2 (\delta u)^2. & \end{cases}$$ (14.14)

The experimental criterion q_e constraints the object model $\phi$ to be consistent with the damped Fourier data  $\Psi_{\! s}$, while the regularization criterion q_r penalizes the high-frequency components of $\phi$.

Let now F be the image of E by A (the space of the $A\phi$'s, $\phi$ spanning E), A_E be the operator from E into F induced by A, and $\Psi _F$ the projection of  $\Psi_{\! d}$ onto F (see Fig. 14.7). The vectors $\phi$ minimizing q on E, the solutions of the problem, are such that  $A_E\phi = \Psi_F$. They are identical up to a vector lying in the kernel of A_E (by definition, the kernel of A_E is the space of vectors $\phi$ such that  $A_E\phi = 0$).

As  $\Psi_{\! d} - \Psi_F$ is orthogonal to F, the solutions $\phi$ of the problem are characterized by the property: $\forall \varphi \in E, (A\varphi \mid \Psi_{\! d} - A\phi)_d = 0$. On denoting by A^* the adjoint of A, this property can also be written in the form:

$$\forall \varphi \in E, \quad (\varphi \mid r)_o = 0, \text{ with } r = A^*(\Psi_{\! d} - A\phi).$$ (14.15)

where r is regarded as a residue. This condition is of course equivalent to P_E r = 0, where P_E is the projector onto the object representation space E. The solutions of the problem are therefore the solutions of the normal equation on E:

$$A^*_E A^{\vphantom{*}}_E \phi = A^*_E \Psi_{\! d},$$ (14.16)

where  A^*_E = P_E A^*.

Many different techniques can be used for solving the normal equation (or minimizing q on E). Some of these are certainly more efficient than others, but this is not a crucial choice.

REMARK 2: beams and maps.

The action of A^*A involved in  $A^*_E A^{\vphantom{*}}_E$ is that of a convolutor. As the two lists  $\mathcal{L}_e$ and  $\mathcal{L}_r$ are disjoints, we have: A^*A = A_e^*A_e + A_r^*A_r. Thus, the corresponding point-spread function, called the dusty beam, has two components: the traditional dirty beam $\Theta_d$ and the regularization beam. The latter corresponds to the action of A_r^*A_r, the former to that of A_e^*A_e (see Fig. 14.5). Likewise, according to the definition of the data vector, $A^*\Psi_{\! d} = A_e^*\Psi_{\! s}$ is called the dusty map (as opposed to the traditional dirty map  $A_e^*\Psi_{\! d}$ because it is damped by the neat beam).

REMARK 3: construction of the object representation space.

With regard to the construction of the object representation space E, CLEAN and WIPE are very similar: it is defined through the choice of the (discrete) object support. It is important to note that this space may be constructed, in a global manner or step by step, interactively or automaticaly. In the last version of WIPE implemented at IRAM, the image reconstruction process is initialized with a few iterations of CLEAN. The support selected by CLEAN is refined throughout the iterations of WIPE by conducting a matching pursuit process at the level of the components of r in the interpolation basis of H_o: the current support is extended by adding the nodes of the object grid  $\text{\boldmath $G$\unboldmath}\,\delta x$ for which these coefficients are the largest above a given threshold (half of the maximum value, for example). The objective functional is then minimized on that new support, and the global residue r updated accordingly. The object representation space of the reconstructed image is thus obtained step by step in a natural manner.

The simulation presented on Fig.14.5-14.6 corresponds to the conditions of Fig. 14.4. The Fourier data  $\Psi_{\! e}$ were blurred by adding a Gaussian noise: for all  $\mathbf{u}\in\mathcal{L}_e$, the standard deviation of  $\Psi_{\! e}(\mathbf{u})$ was set equal to $5 \%$ of the total flux of the object ( $\widehat{\Phi}_o(\mathbf{0)}/20$). The image reconstruction process was initialized with a few iterations of CLEAN, and the construction of the final support of the reconstructed image was made as indicated in Remark 3. At the end of the reconstruction process, a final smoothing of the current object support was performed. In this classical operation of mathematical morphology, the effective support of $\Theta _s$, $\mathcal{D}_s$, is of course used as a structuring element. The boundaries of the effective support of the reconstructed neat map are thus defined at the appropriate resolution. In particular, the connected entities of size smaller than that of  $\mathcal{D}_s$ are eliminated.

   

Figure: Dirty beam (left hand) corresponding to the experimental frequency list  $\mathcal{L}_e$ of Fig. 14.4, and dusty map (right hand) of a simulated data set (the simulated Fourier data  $\Psi_{\! e}$ were blurred by adding a Gaussian noise with a standard deviation $\sigma _e$ equal to $5 \%$ of the total flux of the object  $\Phi_{\! o}$).
Figure: Image to be reconstructed  $\Phi_{\! s}$(left hand) at the resolution level defined in Fig. 14.4, and reconstructed neat map $\Phi _E$ (right hand) at the same resolution: the final condition number $\kappa _E$is equal to 2.46 (cf. Eq. 14.17 and 14.18).