I have been recently working on the deconvolution of spectropolarimetric data observed with space-borne instruments like Hinode. The advantage of such telescopes is that the observations are not corrupted with the atmosphere point spread function (PSF). As a consequence, the observations are really stable. In spite of this advantage, the instrumental PSF (telescope+spectropolarimeter) always affects the observations. This PSF induces a smearing of the profiles, with the consequence that the information obtained at a certain pixel is a combination of the Stokes profiles of many surrounding pixels.
Inferring the physical conditions from such kind of observations have been traditionally carried out using the filling factor approach. A fraction of the pixel is assumed to contain information from the magnetic region of the pixel itself and the remaining is just contamination from the surrounding pixels and of the non-magnetic part of the pixel you observe. Correcting for the unresolved structure of the pixel is quite difficult and I guess that it can only be done statistically from the observation of many pixels. However, correcting for the smearing can be done if the PSF of the instrument is known. Given that the PSF of Hinode is quite well known, there are recent efforts to deconvolve the data from it.
Deconvolution is an ill-posed problem. Under some simplifying conditions, the effect of a PSF (that we represent as the image $\mathbf{P}$) on an image $\mathbf{O}$ is given by: $$\mathbf{I}=\mathbf{O}*\mathbf{P}$$ where $\mathbf{I}$ is the degraded image and $*$ is the convolution operator. In the Fourier domain, the image degradation process transforms into a simple product:
$$\mathcal{I}=\mathcal{O}\mathcal{P}$$
where the calligraphic fonts refer to the Fourier transforms. Even with a good knowledge of the PSF, the deconvolution process is an ill-posed problem in the noisy case, given that the deconvolved image is obtained as a division of the Fourier transforms of the degraded images and the PSF. If noise is present, the information about the high frequencies disappear and the resulting deconvolution becomes a mess.
This can be solved using regularization. There is a recent effort by van Noort (2012) in which he carries out the inversion of the whole observed map while simultaneously taking into account the convolution process. That's a very neat idea but extremely time-consuming.
I have come with an idea that is much simpler and also gives good results at a fraction of the computing time. Additionally, the deconvolution and the inversion of the Stokes profiles is uncoupled, so that one can apply the desired inversion scheme once the observations have been deconvolved. The idea is to introduce a regularization on the Stokes profiles and not on the images, as usually carried out in the deconvolution of natural images. Before reaching the telescope, the Stokes profiles at each pixel can be quite generally written as a series expansion in terms of a set of orthonormal eigenfunctions. If these eigenfunctions are, for instance, the principal components obtained from the observations, this expansion can be very compact and it can be truncated with a reduced number of eigenfunctions:
$$\mathbf{O}(\lambda)=\sum_{i=1}^N \mathbf{w}_i \phi_i(\lambda)$$
The resulting convolved image is then given by:
$$\mathbf{I}(\lambda)=\sum_{i=1}^N (\mathbf{w}_i * \mathbf{P}) \phi_i(\lambda) + \mathbf{N}$$
where $\mathbf{N}$ is a Gaussian noise image. The advantage of using orthonormal eigenfunctions is that we can project the previous equation along the eigenfunctions and obtain:
$$\langle \mathbf{I}(\lambda), \phi_k(\lambda) \rangle=\mathbf{w}_k * \mathbf{P} + \mathbf{N}$$
where the noise is still Gaussian with zero mean and the same variance.
The previous equation tells us to deconvolve each of the images obtained by projecting the observed Stokes profiles onto the first $N$ eigenfunctions and recover the final Stokes profiles by using the deconvolved images of coefficients and the eigenfunctions. The deconvolution is done with a standard Richardson-Lucy deconvolution scheme with the advantage that the images of coefficients have a very small noise level. The reason is that the first $N$ projections of the Stokes profiles onto the principal components only capture the real signal, while the remaining capture the noise. This way, noise is efficiently filtered. Finally, one ends up with almost noiseless Stokes profiles to inject to the inversion code of choice.
Inferring the physical conditions from such kind of observations have been traditionally carried out using the filling factor approach. A fraction of the pixel is assumed to contain information from the magnetic region of the pixel itself and the remaining is just contamination from the surrounding pixels and of the non-magnetic part of the pixel you observe. Correcting for the unresolved structure of the pixel is quite difficult and I guess that it can only be done statistically from the observation of many pixels. However, correcting for the smearing can be done if the PSF of the instrument is known. Given that the PSF of Hinode is quite well known, there are recent efforts to deconvolve the data from it.
Deconvolution is an ill-posed problem. Under some simplifying conditions, the effect of a PSF (that we represent as the image $\mathbf{P}$) on an image $\mathbf{O}$ is given by: $$\mathbf{I}=\mathbf{O}*\mathbf{P}$$ where $\mathbf{I}$ is the degraded image and $*$ is the convolution operator. In the Fourier domain, the image degradation process transforms into a simple product:
$$\mathcal{I}=\mathcal{O}\mathcal{P}$$
where the calligraphic fonts refer to the Fourier transforms. Even with a good knowledge of the PSF, the deconvolution process is an ill-posed problem in the noisy case, given that the deconvolved image is obtained as a division of the Fourier transforms of the degraded images and the PSF. If noise is present, the information about the high frequencies disappear and the resulting deconvolution becomes a mess.
This can be solved using regularization. There is a recent effort by van Noort (2012) in which he carries out the inversion of the whole observed map while simultaneously taking into account the convolution process. That's a very neat idea but extremely time-consuming.
I have come with an idea that is much simpler and also gives good results at a fraction of the computing time. Additionally, the deconvolution and the inversion of the Stokes profiles is uncoupled, so that one can apply the desired inversion scheme once the observations have been deconvolved. The idea is to introduce a regularization on the Stokes profiles and not on the images, as usually carried out in the deconvolution of natural images. Before reaching the telescope, the Stokes profiles at each pixel can be quite generally written as a series expansion in terms of a set of orthonormal eigenfunctions. If these eigenfunctions are, for instance, the principal components obtained from the observations, this expansion can be very compact and it can be truncated with a reduced number of eigenfunctions:
$$\mathbf{O}(\lambda)=\sum_{i=1}^N \mathbf{w}_i \phi_i(\lambda)$$
The resulting convolved image is then given by:
$$\mathbf{I}(\lambda)=\sum_{i=1}^N (\mathbf{w}_i * \mathbf{P}) \phi_i(\lambda) + \mathbf{N}$$
where $\mathbf{N}$ is a Gaussian noise image. The advantage of using orthonormal eigenfunctions is that we can project the previous equation along the eigenfunctions and obtain:
$$\langle \mathbf{I}(\lambda), \phi_k(\lambda) \rangle=\mathbf{w}_k * \mathbf{P} + \mathbf{N}$$
where the noise is still Gaussian with zero mean and the same variance.
The previous equation tells us to deconvolve each of the images obtained by projecting the observed Stokes profiles onto the first $N$ eigenfunctions and recover the final Stokes profiles by using the deconvolved images of coefficients and the eigenfunctions. The deconvolution is done with a standard Richardson-Lucy deconvolution scheme with the advantage that the images of coefficients have a very small noise level. The reason is that the first $N$ projections of the Stokes profiles onto the principal components only capture the real signal, while the remaining capture the noise. This way, noise is efficiently filtered. Finally, one ends up with almost noiseless Stokes profiles to inject to the inversion code of choice.
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