Authors: | F. Ngolè-Mboula |

Language: | C++ |

Download: | sprite_v1.tgz |

Description: | SPRITE: sparsity-based super-resolution algorithm |

Notes: |

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.