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https://github.com/lcav/surface-reconstruction

Code for papers 'Sampling at unknown locations: Uniqueness and reconstruction under constraints' and 'Sampling at unknown locations, with an application in surface retrieval'
https://github.com/lcav/surface-reconstruction

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Code for papers 'Sampling at unknown locations: Uniqueness and reconstruction under constraints' and 'Sampling at unknown locations, with an application in surface retrieval'

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# Sampling at unknown locations: Uniqueness and reconstruction under constraints



This repository contains all the code to reproduce the 

results of the paper

**Sampling at unknown locations: Uniqueness and reconstruction under 

constraints**

 by G. Elhami, M. Pacholska, A. Scholefield, B. Bejar and M. Vetterli.



The part of the code related to polynomials with rational warping builds on top 

of the code for paper

**Sampling at Unknown Locations**, by M. Pacholska, A. Scholefield, B. Bejar and 

M. Vetterli.



Code that generated the figures form the previous paper can be found under fist 

version `v1.0`



## Abstract



Traditional sampling results assume that the sample locations are known. 

Motivated by simultaneous localization and mapping (SLAM) and structure from 

motion (SfM), we investigate sampling at unknown locations. Without further 

constraints, the problem is often hopeless. For example, we recently showed 

that, for polynomial and bandlimited signals, it is possible to find two 

signals, arbitrarily far from each other, that fit the measurements. However, we 

also showed that this can be overcome by adding constraints to the sample 

positions.



In this paper, we show that these constraints lead to a uniform sampling of 

a composite of functions. Furthermore, the formulation retains the key aspects 

of the SLAM and SfM problems, whilst providing uniqueness, in many cases.



We demonstrate this by studying two simple examples of constrained sampling 

at unknown locations. In the first, we consider sampling a periodic bandlimited 

signal composite with an unknown linear function. We derive the sampling 

requirements for uniqueness and present an algorithm that recovers both the 

bandlimited signal and the linear warping. Furthermore, we prove that, when the 

requirements for uniqueness are not met, the cases of multiple solutions have 

measure zero.



For our second example, we consider polynomials sampled such that the 

sampling positions are constrained by a rational function. We previously proved 

that, if a specific sampling requirement is met, uniqueness is achieved. In 

addition, we present an alternate minimization scheme for solving the resulting 

non-convex optimization problem.



Finally, simulation results are provided to support our theoretical 

analysis.



## Authors



Michalina Pacholska, EPFL



Golnoosh Elhami, EPFL






#### Contact



Michalina Pacholska, michalina.pacholska at epfl.ch



Golnoosh Elhami, golnoosh.elhami at epfl.ch



## About



### Polynomial simulations

In order to recreate figures used in the paper related to polynomial based simulations,

one has to first run:



    python surface-tests.py

    

or, in the python console:



    `exec(open("surface-tests.py").read())`

   

Note that this script takes several hours to run on four Intel i7 cores.

After data generation all figures can be generated by Jupyter Notebook `generate_figures.ipynb`.



If you want to just have a preview how the code works, you can use Notebook `examples.ipynb`.

This notebook contains an example how to use ALS solver and how to use the whole pipeline 

(with few tests, which compute fast). 



### Unwarping simulations

In order to recreate figures used in the paper related to unwarping of periodic bandlimitted simulations,

one has to first run:



    python simulate_alpha_equal_2pi_over_2Kplus1.py

    python simulate_alpha_less_than_alpha_c.py

    python simulate_alpha_less_than_pi_over_K.py

    python simulate_alpha_more_than_pi_over_K.py

    python simulate_change_b.py

    

Note that these scripts take several hours to run. The simulation results are however, saved in folder `unwarping_simulation_results/`.

After data generation all figures can be generated by Jupyter Notebook `generate_figures_unwarping.ipynb`.



## Requirements



This project uses Python 3. It requires:



    scipy

    matblotlib

    jupyter

    sortedcontainers

   

You can install all of them by running:



    pip install -r requirements.txt

    

or in the `conda` environment:



    conda install --file requirements.txt

    

Specific version of all packages used are in the file `all_requirements.txt`,

which can also be used with `pip` and `conda`. 



## License



```

Copyright (c) 2018, Michalina Pacholska



Permission is hereby granted, free of charge, to any person obtaining a copy

of this software and associated documentation files (the "Software"), to deal

in the Software without restriction, including without limitation the rights

to use, copy, modify, merge, publish, distribute, sublicense, and/or sell

copies of the Software, and to permit persons to whom the Software is

furnished to do so, subject to the following conditions:



The above copyright notice and this permission notice shall be included in all

copies or substantial portions of the Software.



THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR

IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE

AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER

LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,

OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE

SOFTWARE.

```