In many applications, the sampling frequency is limited by the physical characteristics of the components: the pixel pitch, the rate of the A/D converter, etc. A low-pass filter is then often applied before the sampling operation to avoid aliasing. However, when multiple copies are available, it is possible to use the information that is inherently present in the aliasing to reconstruct a higher resolution signal. If the different copies have unknown relative offsets, this is a non-linear problem in the offsets and the signal coefficients. They are not easily separable in the set of equations describing the super-resolution problem. Thus, we perform joint registration and reconstruction from multiple unregistered sets of samples. We give a mathematical formulation for the problem when there are M sets of N samples of a signal that is described by L expansion coefficients. We prove that the solution of the registration and reconstruction problem is generically unique if MN>= L+M-1. We describe two subspace-based methods to compute this solution. Their complexity is analyzed, and some heuristic methods are proposed. Finally, some numerical simulation results on one and two-dimensional signals are given to show the performance of these methods.

This repository contains all the code and data to reproduce the results of the paper
*Super-Resolution from Unregistered and Totally Aliased Signals Using Subspace Methods*.

First download both code and data provided with the paper. Then, for each figure i, simply execute

```
run figure_i.m
```

Note that some of the figures might take a bit long to run. Here are some abbreviations that might help in estimating the execution time:

- ER: easily reproducible, taking less than 15 minutes
- CR: complexly reproducible, taking more than 15 minutes
- NR: not reproducible (images of original data)

With the above notations in mind, here are the estimated times for each figure:

- Figure 1: ER (can be reproduced using figure_1.m)
- Figure 2: ER (can be reproduced using figure_2.m)
- Figure 3: NR (manually drawn figure)
- Figure 4: left: ER (can be reproduced using figure_4.m), right: NR (manually drawn figure)
- Figure 5: ER (can be reproduced using figure_5.m)
- Figure 6: NR (manually drawn figure)
- Figure 7: NR (manually drawn figure)
- Figure 8: ER (can be reproduced using figure_8.m)
- Figure 9: NR (manually drawn figure)
- Figure 10: ER (can be reproduced using figure_10.m)
- Figure 11: ER (can be reproduced using figure_11.m)
- Figure 12: NR (manually drawn figure)
- Figure 13: ER (can be reproduced using figure_13.m)
- Figure 14: ER (can be reproduced using figure_14.m)
- Figure 15: CR (can be reproduced using figure_15.m)
- note that because of the long duration of these simulations, it might be better to compute the results using each method separately, saving the data, and then plotting the results

- Figure 16: CR (can be reproduced using figure_16.m)
- Note that because of the long duration of these simulations, it might be better to compute the results using each method separately, saving the data, and then plotting the results

- Figure 17: CR (can be reproduced using figure_17.m)
- Note that because of the long duration of these simulations, it might be better to compute the results using each method separately, saving the data, and then plotting the results

Figure 18: CR (can be reproduced using figure_18.m) *Note that because of the long duration of these simulations, it might be better to compute the results for each of the images separately

Table 1: NR (complexity analysis of the different algorithms)

- Microsoft Windows 8, MATLAB R2015b

Copyright (C) 2006 Laboratory of Audiovisual Communications (LCAV), Ecole Polytechnique Federale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland.

This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This software is distributed in the hope that it will be useful, but without any warranty; without even the implied warranty of merchantability or fitness for a particular purpose. See the GNU General Public License for more details (enclosed in the file GPL).

Patrick Vandewalle, Luciano Sbaiz, Joos Vandewalle and Martin Vetterli Laboratory for Audiovisual Communications (LCAV) at EPFL.

Comment | Rating | Date |

5 | 2015-06-15 |

2007

Easy to reproduce figures

★★★★★

★★★★★

Well-commented code

★★★★★

★★★★★

Patrick Vandewalle; Luciano Sbaiz; Joos Vandewalle; Martin Vetterli, "Super-Resolution from Unregistered and Totally Aliased Signals Using Subspace Methods", IEEE Transactions on Signal Processing, Vol. 55, No. 7, Part 2, pp. 3687-3703