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Image Solvers

Since the connection between estimating the sky brightness and deconvolution is becoming less and less straightforward as our approaches gain in sophistication, I will use the term Image Solver for what we previously called a deconvolution algorithm. An Image Solver estimates the sky brightness $\vec{\cal I}\,$ for a fixed set of calibration matrices. The principal algorithms that we need to accomodate are CLEAN and MEM. However, there should be flexibility sufficient to allow other algorithms. I will assume that all image solvers need:

Either

a residual image, $\vec{\cal I}^{R}_{k}$

Or

A measure of the misfit between a model for the sky brightness and the observed visibility data, $\chi^{2}_{}$, and the derivatives with respect to the sky brightness, $\partial$$\chi^{2}_{}$$\over$$\partial$$\vec{\cal I}_{k}^{}$ and $\partial^{2}_{}$$\chi^{2}_{}$$\over$$\partial$$\vec{\cal I}^{2}_{k}$.

The main point of this memo is to take this definition and demonstrate that it does indeed fit nearly all of the image solvers that we now use. This is convenient since it provides a simple interface between an image solver and the machinery for the measurement equation. A well-defined interface is useful for designing software but it is also helpful in understanding algorithms. Thus this memo should address both subjects.

As shown in the first memo:

$${{\partial \chi^2\over\partial \vec{\cal I}_k}} \Re \sum_{ij}^{}$$

$$\left[\vphantom{{{F}_i {\left(\underline{\rho}_k\right)} \otimes {F}^*_j{\left(\underline{\rho}_k\right)}}}\right. \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \otimes \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \left.\vphantom{{{F}_i {\left(\underline{\rho}_k\right)} \otimes {F}^*_j{\left(\underline{\rho}_k\right)}}}\right]^{*T}_{} \left[\vphantom{{{P}_i {\left(\underline{\rho}_k\right)} \otimes {P}^*_j{\left(\underline{\rho}_k\right)}}}\right. \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \otimes \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \left.\vphantom{{{P}_i {\left(\underline{\rho}_k\right)} \otimes {P}^*_j{\left(\underline{\rho}_k\right)}}}\right]^{*T}_{} \left[\vphantom{{{E}_i {\left(\underline{\rho}_k\right)} \otimes {E}^*_j{\left(\underline{\rho}_k\right)}}}\right. \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \otimes \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \left.\vphantom{{{E}_i {\left(\underline{\rho}_k\right)} \otimes {E}^*_j{\left(\underline{\rho}_k\right)}}}\right]^{*T}_{}$$

$$\left[\vphantom{{{C}_i\otimes{C}^*_j}}\right. \otimes \left.\vphantom{{{C}_i\otimes{C}^*_j}}\right]^{*T}_{} \left[\vphantom{{{D}_i\otimes{D}^*_j}}\right. \otimes \left.\vphantom{{{D}_i\otimes{D}^*_j}}\right]^{*T}_{} \left[\vphantom{{{G}_i\otimes{G}^*_j}}\right. \otimes \left.\vphantom{{{G}_i\otimes{G}^*_j}}\right]^{*T}_{}$$

$$\Delta \vec{V}_{\rm ij}^{} \pi \left(\vphantom{{\underline{r}_i-\underline{r}_j}}\right. \underline{r}_{i} \underline{r}_{j} \left.\vphantom{{\underline{r}_i-\underline{r}_j}}\right) \underline{\rho}_{k}$$ (2)

$${\partial^2 \chi^2\over\partial \vec{\cal I}_k\partial \vec{\cal I}_k^T} \Re \sum_{ij}^{}$$

$$\left[\vphantom{{{F}_i {\left(\underline{\rho}_k\right)} \otimes {F}^*_j{\left(\underline{\rho}_k\right)}}}\right. \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \otimes \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \left.\vphantom{{{F}_i {\left(\underline{\rho}_k\right)} \otimes {F}^*_j{\left(\underline{\rho}_k\right)}}}\right]^{*T}_{} \left[\vphantom{{{P}_i {\left(\underline{\rho}_k\right)} \otimes {P}^*_j{\left(\underline{\rho}_k\right)}}}\right. \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \otimes \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \left.\vphantom{{{P}_i {\left(\underline{\rho}_k\right)} \otimes {P}^*_j{\left(\underline{\rho}_k\right)}}}\right]^{*T}_{} \left[\vphantom{{{E}_i {\left(\underline{\rho}_k\right)} \otimes {E}^*_j{\left(\underline{\rho}_k\right)}}}\right. \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \otimes \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \left.\vphantom{{{E}_i {\left(\underline{\rho}_k\right)} \otimes {E}^*_j{\left(\underline{\rho}_k\right)}}}\right]^{*T}_{}$$

$$\left[\vphantom{{{C}_i\otimes{C}^*_j}}\right. \otimes \left.\vphantom{{{C}_i\otimes{C}^*_j}}\right]^{*T}_{} \left[\vphantom{{{D}_i\otimes{D}^*_j}}\right. \otimes \left.\vphantom{{{D}_i\otimes{D}^*_j}}\right]^{*T}_{} \left[\vphantom{{{G}_i\otimes{G}^*_j}}\right. \otimes \left.\vphantom{{{G}_i\otimes{G}^*_j}}\right]^{*T}_{}$$ W_ij/TD>

$$\left[\vphantom{{{G}_i\otimes{G}^*_j}}\right. \otimes \left.\vphantom{{{G}_i\otimes{G}^*_j}}\right] \left[\vphantom{{{D}_i\otimes{D}^*_j}}\right. \otimes \left.\vphantom{{{D}_i\otimes{D}^*_j}}\right] \left[\vphantom{{{C}_i\otimes{C}^*_j}}\right. \otimes \left.\vphantom{{{C}_i\otimes{C}^*_j}}\right]$$

$$\left[\vphantom{{{E}_i {\left(\underline{\rho}_k\right)} \otimes {E}^*_j{\left(\underline{\rho}_k\right)}}}\right. \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \otimes \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \left.\vphantom{{{E}_i {\left(\underline{\rho}_k\right)} \otimes {E}^*_j{\left(\underline{\rho}_k\right)}}}\right] \left[\vphantom{{{P}_i {\left(\underline{\rho}_k\right)} \otimes {P}^*_j{\left(\underline{\rho}_k\right)}}}\right. \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \otimes \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \left.\vphantom{{{P}_i {\left(\underline{\rho}_k\right)} \otimes {P}^*_j{\left(\underline{\rho}_k\right)}}}\right] \left[\vphantom{{{F}_i {\left(\underline{\rho}_k\right)} \otimes {F}^*_j{\left(\underline{\rho}_k\right)}}}\right. \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \otimes \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \left.\vphantom{{{F}_i {\left(\underline{\rho}_k\right)} \otimes {F}^*_j{\left(\underline{\rho}_k\right)}}}\right]$$

S (3)

It is not immediately obvious how to construct dirty and residual images for this measurement equation, since the Fourier inverse does not apply. Actually it is not clear just what a dirty image is supposed to be. I argued previously that a good definition of a generalized dirty image $\vec{\cal I}^{D}_{k}$ is:

$$\vec{\cal I}^{D}_{k} \left[\vphantom{{\partial^2 \chi^2\over\partial \vec{\cal I}_k\partial \vec{\cal I}^T_k}}\right. {\partial^2 \chi^2\over\partial \vec{\cal I}_k\partial \vec{\cal I}^T_k} \left.\vphantom{{\partial^2 \chi^2\over\partial \vec{\cal I}_k\partial \vec{\cal I}^T_k}}\right]^{-1}_{} {{\partial \chi^2\over\partial \vec{\cal I}_k}} \mid_{\vec{\cal I}_k=0}^{}$$ (4)

Note that the first term on the RHS of this equation is the inverse of a 4 by 4 matrix. By inverting this matrix, we are correcting for the coupling of different polarizations in the interferometer. By ignoring the non-diagonal terms of the Hessian, we are ignoring the coupling between different pixels in the final image. This is reasonable since, first, the coupling is singular, and, second, it is the role of an Image Solver to correct for this coupling.

A generalized residual image can be similarly defined as the update direction for a given estimate of the sky brightness, $\vec{\cal I}\,$. Thus, as is reasonable, the residual image tends towards zero as the model image reproduces the observed visibility data. Note that since the residual image is defined in terms of the gradients and Hessian, these are the primary interface to the measurement equation. In fact, it makes good sense to think of residual image as an image solver of status no different from any other image solver.

In addition, there is a PSF B($\underline{\rho}$,$\underline{\rho}{^\prime}$) associated with the generalized dirty image. Conceptually, it is calculated by propagating an appropriately centered point source, at position $\underline{\rho}{^\prime}$ through the measurement equation to obtain the predicted visibility and then back into the image plane, to position $\underline{\rho}$, via the equation for the dirty image. Thus the PSF is not necessarily shift-invariant. Note also that the PSF is a (4 by 4) matrix for each point in ($\underline{\rho}$,$\underline{\rho}{^\prime}$) space (!) while the dirty and residual images are images in which each pixel is a 4-vector. This means that any image solver must estimate a 4-vector at every pixel, and that finding the PSF requires evaluating the response separately for the 4 basis vectors (e.g. once for each of I, Q, U, V). The Stokes parameters, I, Q, U, V are therefore coupled together both in the calibration and in the solution for the image. Note that the ``normal'' approach to imaging is to correct coupling between Stokes parameters in the measurements and then to deconvolve separately. Here the decoupling and deconvolution are done simultaneously. At this point, it may be a good idea to reassure the reader that conventional imaging can be acheived using this formalization by making the appropriate approximations: e.g. the PSF matrix is diagonal and shift-invariant, in which case one is justified in using a straightforward application of CLEAN to each of I, Q, U, V.

Let us now consider how to use a priori information that applies to the 4-vector $\vec{\cal I}\,$.