PSF field learning based on Optimal Transport Distances


Authors: F. Ngolè Mboula, J-L. Starck
Journal: arXiv
Year: 2017
Download: ADS | arXiv



Context: in astronomy, observing large fractions of the sky within a reasonable amount of time implies using large field-of-view (fov) optical instruments that typically have a spatially varying Point Spread Function (PSF). Depending on the scientific goals, galaxies images need to be corrected for the PSF whereas no direct measurement of the PSF is available. Aims: given a set of PSFs observed at random locations, we want to estimate the PSFs at galaxies locations for shapes measurements correction. Contributions: we propose an interpolation framework based on Sliced Optimal Transport. A non-linear dimension reduction is first performed based on local pairwise approximated Wasserstein distances. A low dimensional representation of the unknown PSFs is then estimated, which in turn is used to derive representations of those PSFs in the Wasserstein metric. Finally, the interpolated PSFs are calculated as approximated Wasserstein barycenters. Results: the proposed method was tested on simulated monochromatic PSFs of the Euclid space mission telescope (to be launched in 2020). It achieves a remarkable accuracy in terms of pixels values and shape compared to standard methods such as Inverse Distance Weighting or Radial Basis Function based interpolation methods.


Space variant deconvolution of galaxy survey images


Authors: S. Farrens, J-L. Starck, F. Ngolè Mboula
Journal: A&A
Year: 2017
Download: ADS | arXiv


Removing the aberrations introduced by the Point Spread Function (PSF) is a fundamental aspect of astronomical image processing. The presence of noise in observed images makes deconvolution a nontrivial task that necessitates the use of regularisation. This task is particularly difficult when the PSF varies spatially as is the case for the Euclid telescope. New surveys will provide images containing thousand of galaxies and the deconvolution regularisation problem can be considered from a completely new perspective. In fact, one can assume that galaxies belong to a low-rank dimensional space. This work introduces the use of the low-rank matrix approximation as a regularisation prior for galaxy image deconvolution and compares its performance with a standard sparse regularisation technique. This new approach leads to a natural way to handle a space variant PSF. Deconvolution is performed using a Python code that implements a primal-dual splitting algorithm. The data set considered is a sample of 10 000 space-based galaxy images convolved with a known spatially varying Euclid-like PSF and including various levels of Gaussian additive noise. Performance is assessed by examining the deconvolved galaxy image pixels and shapes. The results demonstrate that for small samples of galaxies sparsity performs better in terms of pixel and shape recovery, while for larger samples of galaxies it is possible to obtain more accurate estimates of the galaxy shapes using the low-rank approximation.


Point Spread Function

The Point Spread Function or PSF of an imaging system (also referred to as the impulse response) describes how the system responds to a point (unextended) source. In astrophysics, stars or quasars are often used to measure the PSF of an instrument as in ideal conditions their light would occupy a single pixel on a CCD. Telescopes, however, diffract the incoming photons which limits the maximum resolution achievable. In reality, the images obtained from telescopes include aberrations from various sources such as:

  • The atmosphere (for ground based instruments)
  • Jitter (for space based instruments)
  • Imperfections in the optical system
  • Charge spread of the detectors


In order to recover the true image properties it is necessary to remove PSF effects from observations. If the PSF is known (which is certainly not trivial) one can attempt to deconvolve the PSF from the image. In the absence of noise this is simple. We can model the observed image \mathbf{y} as follows


where \mathbf{x} is the true image and \mathbf{H} is an operator that represents the convolution with the PSF. Thus, to recover the true image, one would simply invert \mathbf{H} as follows


Unfortunately, the images we observe also contain noise (e.g. from the CCD readout) and this complicates the problem.

\mathbf{y}=\mathbf{Hx} + \mathbf{n}

This problem is ill-posed as even the tiniest amount of noise will have a large impact on the result of the operation. Therefore, to obtain a stable and unique solution, it is necessary to regularise the problem by adding additional prior knowledge of the true images.


One way to regularise the problem is using sparsity. The concept of sparsity is quite simple. If we know that there is a representation of \mathbf{x} that is sparse (i.e. most of the coefficients are zeros) then we can force our deconvolved observation \mathbf{\hat{x}} to be sparse in the same domain. In practice we aim to minimise a problem of the following form

\begin{aligned} & \underset{\mathbf{x}}{\text{argmin}} & \frac{1}{2}\|\mathbf{y}-\mathbf{H}\mathbf{x}\|_2^2 + \lambda\|\Phi(\mathbf{x})\|_1 & & \text{s.t.} & & \mathbf{x} \ge 0 \end{aligned}

where \Phi is a matrix that transforms \mathbf{x} to the sparse domain and \lambda is a regularisation control parameter.

Low-Rank Approximation

Another way to regularise the problem is assume that all of the images one aims to deconvolve live on a underlying low-rank manifold. In other words, if we have a sample of galaxy images we wish to deconvolve then we can construct a matrix X X where each column is a vector of galaxy pixel coefficients. If many of these galaxies have similar properties then we know that X X will have a smaller rank than if images were all very different. We can use this knowledge to regularise the deconvolution problem in the following way

\begin{aligned} & \underset{\mathbf{X}}{\text{argmin}} & \frac{1}{2}\|\mathbf{Y}-\mathcal{H}(\mathbf{X})\|_2^2 + \lambda|\mathbf{X}\|_* & & \text{s.t.} & & \mathbf{X} \ge 0 \end{aligned}


In the paper I implement both of these regularisation techniques and compare how well they perform at deconvolving a sample of 10,000 Euclid-like galaxy images. The results show that, for the data used, sparsity does a better job at recovering the image pixels, while the low-rank approximation does a better job a recovering the galaxy shapes (provided enough galaxies are used).


SF_DECONVOLVE is a Python code designed for PSF deconvolution using a low-rank approximation and sparsity. The code can handle a fixed PSF for the entire field or a stack of PSFs for each galaxy position.



Constraint matrix factorization for space variant PSFs field restoration


Authors: F. Ngolè Mboula, J-L. Starck, K. Okumura, J. Amiaux, P. Hudelot
Journal: SIAM
Year: 2016
Download: ADS | arXiv



Context: in large-scale spatial surveys, the Point Spread Function (PSF) varies across the instrument field of view (FOV). Local measurements of the PSFs are given by the isolated stars images. Yet, these estimates may not be directly usable for post-processings because of the observational noise and potentially the aliasing. Aims: given a set of aliased and noisy stars images from a telescope, we want to estimate well-resolved and noise-free PSFs at the observed stars positions, in particular, exploiting the spatial correlation of the PSFs across the FOV. Contributions: we introduce RCA (Resolved Components Analysis) which is a noise-robust dimension reduction and super-resolution method based on matrix factorization. We propose an original way of using the PSFs spatial correlation in the restoration process through sparsity. The introduced formalism can be applied to correlated data sets with respect to any euclidean parametric space. Results: we tested our method on simulated monochromatic PSFs of Euclid telescope (launch planned for 2020). The proposed method outperforms existing PSFs restoration and dimension reduction methods. We show that a coupled sparsity constraint on individual PSFs and their spatial distribution yields a significant improvement on both the restored PSFs shapes and the PSFs subspace identification, in presence of aliasing. Perspectives: RCA can be naturally extended to account for the wavelength dependency of the PSFs.


Super-resolution method using sparse regularization for point-spread function recovery


Authors: F. Ngolè Mboula, J-L. Starck, S. Ronayette, K. Okumura, J. Amiaux
Journal: A&A
Year: 2015
Download: ADS | arXiv



In large-scale spatial surveys, such as the forthcoming ESA Euclid mission, images may be undersampled due to the optical sensors sizes. Therefore, one may consider using a super-resolution (SR) method to recover aliased frequencies, prior to further analysis. This is particularly relevant for point-source images, which provide direct measurements of the instrument point-spread function (PSF). We introduce SPRITE, SParse Recovery of InsTrumental rEsponse, which is an SR algorithm using a sparse analysis prior. We show that such a prior provides significant improvements over existing methods, especially on low SNR PSFs.