Category: PSF Software

• RCA: sparsity-based dimension reduction and super-resolution algorithm.

  Python source code available here: rca.tar.gz

  The method

  RCA (Resolved Components Analysis, [1]) aims at characterizing globally a space-variant PSFs field. Given a set of aliased and noisy images of unresolved objects (stars images from a telescope for instance), RCA estimates well-resolved and noise-free PSFs at the observations positions, in particular, exploiting the spatial correlation of the PSFs across the imaging system field of view (FOV). Let consider a set of p images of unresolved objects $$\mathbf{y}_1,\dots,\mathbf{y}_p$$ in a given instrument FOV and $$\mathbf{x}_1,\dots,\mathbf{x}_p$$ the corresponding ideal PSFs; these images are treated as column vectors and we note $$\mathbf{Y} = [\mathbf{y}_1,\dots,\mathbf{y}_p]$$ and $$\mathbf{X} = [\mathbf{x}_1,\dots,\mathbf{x}_p]$$. RCA minimizes $$ \frac{1}{2}{\|\mathbf{Y}-\mathcal{F}(\mathbf{X})\|_F^2}$$, where $$\mathcal{F}$$ accounts for the observed stars downsampling; this is done subject to several constraints over $$\mathbf{X}$$:

  • Positivity constraint: each PSF $$\mathbf{x}_j$$ should be positive;
  • Low rank constraint: each estimated PSF is forced to be a linear combination of a few number esimated of "eigen" PSFs; $$\mathbf{x}_j = \sum_{i=1}^r a_{ij} \mathbf{s}_i$$, with $$r \ll p$$ or equivalently, $$\mathbf{X} = \mathbf{S}\mathbf{A}$$, $$\mathbf{S}$$ and $$\mathbf{A}$$ being low rank matrices;
  • Piece-wise smoothness constraint: we can assume that the vectors $$\mathbf{x}_j$$ are structured; promoting the sparsity of the "eigen" PSFs $$\mathbf{s}_j$$ in an appropriate dictionary $$\boldsymbol{\Phi}$$ allows to capture the spatial correlations within the PSFs themselves;
  • Proximity constraints: the more two PSFs are close in the FOV, the more they should be similar; the p values relative to the $$i^{th}$$ line of $$\mathbf{A}$$ correspond to the contribution of the $$i^{th}$$ "eigen" PSF across the field of view; therefore the PSFs field regularity is enforced by constraining $$\mathbf{A}$$'s lines; specifically, each line is calculated as a sparse linear combination of spatial frequencies atoms: $$\mathbf{A} = \boldsymbol{\alpha}\mathbf{U}^T$$ (see Fig.1).

  

  Hence RCA solves the following problem:
  \[ \underset{\boldsymbol{\alpha},\mathbf{S}}\min \frac{1}{2}{\|\mathbf{Y}-\mathcal{F}(\mathbf{S}\mathbf{A})\|_F^2} + \sum_{i=1}^r \|\mathbf{w}_i\odot\boldsymbol{\Phi}\odot\mathbf{s}_i\|_1\; \text{s.t.}\; \|\boldsymbol{\alpha}[l,:]\|_0 \leq \eta_l, \; l=1....r \;\text{and } \mathbf{S}\boldsymbol{\alpha}\mathbf{U}^T \geq 0. \]

  The operator $$\mathcal{F}$$ is set according to the resolution enhancement factor specified. Besides, this operator also accounts for the observed images subpixel offsets which are automatically estimated. $$\boldsymbol{\Phi}$$ is a user-chosen redundant dictionary. The frequency dictionary $$\mathbf{U}$$ is automatically built based on the observations $$\mathbf{y}_i$$ locations in the imaging instrument FOV. The sparsity parameters $$(\mathbf{w}_i)_i$$ and $$(\eta_l)_l$$ as well are automatically selected. The number of "eigen" PSFs, r, has to be specified. It might be automatically reduced depending of the signal-to-noise ratio, the observations resolution and the sample size. A principal components analysis should provide the user a good first guess for r.

  Numerical examples

  RCA was tested on set on 500 Euclid telescope simulated PSFs. The simulated observables were downsampled to Euclid telescope resolution and corrupted with a white gaussian noise. Four examples of recovered PSFs are given in Fig.2. In this example, 10 "eigen" PSFs were required. We compare the result to the PSFs obtained from the software PSFEx, using the same observations.

  

  As it can be seen from these reconstructions, RCA handles the undersampling and the PSFs spatial variability, with both noise-robustness and a high visual quality.

  Article

  • • [1] F. M. Ngolè Mboula, J.-L. Starck, K. Okomura, J. Amiaux, P. Hudelot. Constraint matrix factorization for space variant PSFs field restoration, Inverse Problems. Available here.

SPRITE: sparsity-based super-resolution algorithm

The method

SPRITE (Sparse Recovery of InstrumenTal rEsponse, 1) aims at computing a well-resolved compact source image from several undersampled and noisy observations. SPRITE solves the succession of problems of the form
\[ \underset{\Delta}\min \frac{1}{2}\sum_{k=1}^n{\|\mathbf{y}_k-f_k\mathbf{D}\mathbf{H}_k(\Delta+\mathbf{x}^{(0)})\|_2^2}/{\sigma_k^2} +\kappa\|\mathbf{w}^{(l)}\odot\Lambda\odot\Phi\Delta\|_1 ; s.t. \Delta \ge -\mathbf{x}^{(0)},$$ with $$l=1..N \].
The vectors $$\mathbf{y}_k$$ are the low resolution (LR) observations. The scalars $$f_k$$ account for possible luminosity differences between the LR images and $$\sigma_k^2$$ is the noise variance in the LR image $$k^{th}$$. The matrices $$\mathbf{H}_k$$ account for the shifts between the observations and the matrix D is the downsampling operator. The vector $$\mathbf{x}^{(0)}$$ is a rough estimate of the well-resolved image. The final image is given by $$\mathbf{x}=\mathbf{x}^{(0)}+\Delta_N$$, where $$\Delta_N$$ is the minimizer of the $$N^{th}$$ problem solved.

$$\Phi$$ is a user-chosen redundant dictionary. The method relies on the prior that a suitable solution $$\Delta$$ should have a sparse decomposition into the dictionary $$\Phi$$. Finally the inequality constraint (which is element-wise) insures that the final well-resolved image has positive pixels values.

It's important to note that the only parameters to be provided by the user are $Phi$ and $kappa$. The other parameters are automatically calculated.

The source code can be downloaded here: SPRITE package. The data and IDL codes used to perform the benchmark tests presented in 1 are available here: super-resolution benchmark.

Please cite the reference below if you use these codes in a publication. This is a preliminary version of the SPRITE package, which may be imperfect. Please feel free to contact us if you have any problem.

Article

• [1] F. M. Ngolè Mboula, J.-L. Starck, S. Ronayette, K. Okomura, J. Amiaux. Super-resolution method using sparse regularization for point spread function recovery, Astronomy and Astrophysics, 2014. Available here.