It is to be mentioned here that most of the imaging techniques in RI literature work under the assumption of absence of any calibration errors or availability of pre-calibrated data. In the more practical scenario of working in the presence of DDE calibration errors, these terms need to be accounted for within the measurement model, such as done in Bhatnagar et al. (2013) in the case when these terms are known. Particularly for joint full polarization imaging, we can generalize the model proposed in Birdi et al. (2018b) and employ it to solve inverse problem (7). The corresponding imaging strategy leverages the CS theory. The main idea is to obtain an estimation of the sought images by solving a minimization problem of the form (Birdi et al. 2018a, b)
(9)$$\begin{eqnarray}
\underset{^{{{\boldsymbol{\sf {S}}}}}}{\mathrm{minimize}}\,\,\,\,r ({\boldsymbol{\sf {S}}}) \, \, \text{subject to} \, \, \Vert \Phi ({\boldsymbol{\sf {S}}}) - \boldsymbol {y} \Vert _2 \leqslant \epsilon ,
\end{eqnarray}$$where ϵ is related to the norm of the additive noise, ‖ · ‖2 denotes the ℓ2 norm of its argument vector. The term $r({\boldsymbol{\sf {S}}})$ is the regularization term, injecting prior information in the reconstruction process, whereas the data fidelity is ensured by the constraint $\Vert \Phi ({\boldsymbol{\sf {S}}}) - \boldsymbol {y} \Vert _2 \leqslant \epsilon$, implying that the residual is situated within an ℓ2 ball whose radius is determined by ϵ. In the current work, due to technical assumptions related to the proposed joint imaging and calibration algorithm (Chouzenoux et al. 2016), we propose to solve an unconstrained version of problem (9), given by
(10)$$\begin{eqnarray}
\underset{^{{{\boldsymbol{\sf {S}}}}}}{\mathrm{minimize}}\,\,\,\,\bar{h} \big ({\boldsymbol{\sf {S}}} \big) + r({\boldsymbol{\sf {S}}}),
\end{eqnarray}$$where $\bar{h}({\boldsymbol{\sf {S}}})$ is the data fidelity term. Using this formulation, and under the assumption that the additive noise is i.i.d. Gaussian, $\bar{h}({\boldsymbol{\sf {S}}})$ is given by a least-squares criterion
(11)$$\begin{eqnarray}
(\forall {\boldsymbol{\sf {S}}} \in {\mathbb {R}} ^{2 \times 2 N}) \, \, \bar{h}({\boldsymbol{\sf {S}}}) = \frac{1}{2}\Vert \Phi ({\boldsymbol{\sf {S}}}) - \boldsymbol {y} \Vert _2^2.
\end{eqnarray}$$This formulation has been introduced in Repetti et al. (2017) in the particular case of joint Stokes I imaging and calibration. In order to incorporate various polarimetric imaging specific prior informations in the regularization function, Birdi et al. (2018b) proposed to use a hybrid regularization term of the form
(12)$$\begin{eqnarray}
r({\boldsymbol{\sf {S}}}) = g({\boldsymbol{\sf {S}}}) + r^\prime ({\boldsymbol{\sf {S}}}),
\end{eqnarray}$$where the function $g:{\mathbb {R}} ^{2 \times 2N} \rightarrow ]-\infty , +\infty ]$ imposes sparsity of the underlying Stokes images and the function $r^\prime :{\mathbb {R}} ^{2 \times 2N} \rightarrow ]-\infty , +\infty ]$ constrains the domains of the argument images as per some physical constraints. In particular, the first term is inspired by the CS framework which proposes to exploit the sparsity of the target images in some dictionary ${\boldsymbol{\Psi }} \in {\mathbb {C}} ^{N \times J}$ (Candès et al. 2006; Donoho 2006). The choice of this dictionary is dependent on the images under scrutiny. To give an intuitive idea, images which are already sparse, for instance point sources images, are well represented by dirac basis taking ${\boldsymbol{\Psi }}$ to be identity. Otherwise, sparsity can be imposed in some other domain, such as TV regularization for piece-wise constant images (Rudin et al. 1992; Wiaux, Puy & Vandergheynst 2010; Akiyama et al. 2017), wavelet basis for smooth images (Mallat 2009), etc. In particular for astronomical images, a collection of wavelet bases is shown to be a good candidate for sparsifying dictionary ${\boldsymbol{\Psi }}$ (Carrillo et al. 2012, 2014), both for Stokes I imaging (Onose et al. 2016, 2017) and specifically for polarization imaging in Birdi et al. (2018a, b). Formally, for any matrix $\widetilde{{\boldsymbol{{{\sf S}}}}} \in {\mathbb {R}} ^{N \times 4}$, the sparse representation of the sought images in a chosen dictionary ${\boldsymbol{\Psi }}$ is given by ${\boldsymbol{\Psi }} ^\dagger \widetilde{{\boldsymbol{{{\sf S}}}}}$. The regularization function is then defined to impose sparsity of this term. This can be achieved by using the ℓ0 pseudo-norm, counting the number of non-zero components of its argument (Donoho et al. 1995). A more common approach is to use ℓ1 norm which overcomes the problem of non-convexity of the ℓ0 norm (Chen, Donoho & Saunders 2001). An even better approximation of the ℓ0 norm is provided by the (re)weighted ℓ1 norm, which tends to diminish the magnitude dependence of the ℓ1 norm (Candès et al. 2008; Carrillo et al. 2012). The (re)weighted ℓ1 norm is given by
(13)$$\begin{eqnarray}
g({\boldsymbol{\sf {S}}}) = \Vert {\boldsymbol{\Psi }} ^\dagger \mathcal {R} \big ({\boldsymbol{\sf {S}}} \big) \Vert _{{\boldsymbol{\sf {W}}},1} = \displaystyle \sum _{i=1}^4 \sum _{j=1}^J {\mathsf {W}} _{j,i} \big |[{\boldsymbol{\Psi }} ^\dagger \mathcal {R} \big ({\boldsymbol{\sf {S}}} \big) ]_{j,i} \big | \, ,
\end{eqnarray}$$where the subscripts j and i in the notation [ · ]j,i stand respectively for the row and column indices of the argument matrix, and $\mathcal {R}:{\mathbb {R}} ^{2 \times 2 N} \rightarrow {\mathbb {R}} ^{N \times 4}$ is the operator consisting in placing the four Stokes images contained in the matrix ${\boldsymbol{\sf {S}}}$ in four columns, whereas its adjoint $\mathcal {R}^\dagger :{\mathbb {R}} ^{N \times 4} \rightarrow {\mathbb {R}} ^{2 \times 2 N}$ does the contrary, i.e. stores the four images in the rows of a 2 × 2 block matrix. In equation (13), ${\boldsymbol{\sf {W}}} \in {\mathbb {R}} _+^{J \times 4}$ is the weighting matrix. In the case when this matrix is chosen to be identity, the ℓ1 regularization term is obtained. Another useful piece of information, which can be used to regularize the problem is the polarization constraint on the recovered Stokes images (Birdi et al. 2018b). This constraint comes from a physical point of view, that the polarized intensity should not exceed the total intensity and mathematically, can be described by the following set:
(14)$$\begin{eqnarray}
\mathbb {P} = \big \lbrace \widetilde{{\boldsymbol{{{\sf S}}}}} = \mathcal {R}\big ({\boldsymbol{\sf {S}}} \big) \in {\mathbb {R}} ^{N \times 4} \, \big | \, (\forall n \in \lbrace 1,\!\!\!&amp;&amp;\ldots ,N\rbrace) \nonumber \\
&amp;&amp; - \, \widetilde{{\mathsf {S}}}_{n,1} + \Vert \widetilde{{\boldsymbol{{{\sf S}}}}}_{n, 2:4} \Vert _2 \leqslant 0 \big \rbrace ,
\end{eqnarray}$$where, for every n ∈ {1, …, N}, $\widetilde{{\mathsf {S}}}_{n,1}$ and $\Vert \widetilde{{\boldsymbol{{{\sf S}}}}}_{n, 2:4} \Vert _2$ respectively represent the total intensity and the polarized intensity. As can be noticed, this constraint also imposes implicitly the positivity of the total intensity image (Stokes I). The enforcement of this constraint amounts to incorporation of the indicator function of the set $\mathbb {P}$ in the regularization function, i.e.
(15)$$\begin{eqnarray}
r^\prime ({\boldsymbol{\sf {S}}}) = \iota _{\mathbb {P}} ({\boldsymbol{\sf {S}}}).
\end{eqnarray}$$The indicator function of any such set $\mathbb {P}$ is defined as
(16)$$\begin{eqnarray}
\iota _{\mathbb {P}}({\boldsymbol{\sf {S}}}) = \left\lbrace \begin{array}{@{}l@{\quad }l@{}}0, \quad \quad \text{if} \, {\boldsymbol{\sf {S}}} \in \mathbb {P} \, , \\
+\infty , \quad \text{otherwise}. \end{array}\right.
\end{eqnarray}$$
Citation: Wiaux, Puy & Vandergheynst 2010
Citation function: Uses