PSWF-Radon approach for 2D/3D

PSWF-Radon approach allows to recover unknown function from its Fourier transform in 2D/3D being truncated to a finite euclidean ball. In comparison with conventional method where unknown part of Fourier spectrum is extrapolated with zeros, we recover this part (up to some limit) using theory of one-dimensional Prolate Spheroidal Wave Functions.

Recovery of high frequencies of the signal allows to achieve super-resolution while the choice of tuning parameters is fully automatic.

   

This repository contains an implementation of PSWF-Radon algorithm and related numerical experiments from

1. PSWF-Radon approach to reconstruction from band-limited Hankel transform by Fedor Goncharov, Mikhail Isaev, Roman Novikov, Rodion Zaytsev, 2024 HAL ArXiv

2. Super-resolution reconstruction from truncated Hankel trasnform by Fedor Goncharov, Mikhail Isaev, Roman Novikov, Rodion Zaytsev, 2024 HAL

Original papers introducing PSWF-Radon approach are:

3. Reconstruction from the Fourier transform on the ball via prolate spheroidal wave functions Mikhail Isaev, Roman Novikov, 2022 journal

4. Numerical reconstruction from the Fourier transform on the ball using prolate spheroidal wave functions Mikhail Isaev, Roman Novikov, Sabinin Grigory, 2024 journal

One word before

Code for PSWF-Radon approach is developed for Octave on basis of Scatter/Spheroidal library originally designed for Matlab (current repository is a fork from this library). We use only part of this libary to pre-compute Prolate Spheroidal Wavefunctions (PSWFs) which are later used in PSWF-Radon approach for reconstructions.

Requirements & Installation

Required libraries/software

• Linux-kernel OS with Octave 8.4.0

• installed NFFT library - for installation read the corresponding guidline from here; we also advise you to build it with flags

  • --enable-all

  • --enable-openmp

  • --with-matlab=<path-to-save Matlab's MEX and .m files> - compile NFFT into Matlab's MEX binaries, Octave will make calls to these functions

• GNU MPFR libary

In principle, it is possible to adapt our implementation for Windows if, for example, you

• use Matlab instead of Octave (this requires minimal changes of our codes due to close syntaxes)

• build NFFT for Windows with the above flag to enable support of Matlab

Building binaries

1. copy this repository on your local machine (e.g. via git clone or download ZIP archive and unpack it)

2. go to your home .octaverc and add path to compiled NFFT library for Matlab by adding line

    addpath(<path to folder with NFFT>)
   

   Note: folder <path to folder with NFFT> should contain nfftmex.mex, nfft_get_*.m, nfft_set_*.m and other Matlab's scripts for NFFT

3. build binaries for computation of PSWFs

    cd <path-to-folder with project>/spheroidal/sphwv; make
   

Pre-computation of PSWFs

1. in Octave go to folder of this repo

    cd <path-to-folder of this repo>
   

2. add paths

    cd spheroidal; sandbox; 
   

3. to sample PSWFs up to order n_max for frequency parameter c with equispaced n_points on [-1,1] run

    generate_data_for_pswfs_m0_only(c, n_max, n_points)
   

   this command may take some time and will take some hard-drive memory as it samples PSWFS and stores them in files in /data folder.
   To run codes from our papers run the above command for c=10 and c=15 with n_max=29, n_points=1024.

Running scripts for Papers 1, 2

1. run Octave

2. go to folder of this repo

    cd <path-to-folder of this repo>
   

3. add all necessary paths

    sandbox; cd prolate-2d-3d; sandbox; cd ../spheroidal; sandbox
   

4. check that current pwd should be <path-to-folder of this repo>/spheroidal

    pwd
   

5. in Octave's script editor open files

   1. <path-to-folder of this repo>/prolate-2d-3d/generate-scripts/generateArticleImages.m - codes for Paper 1

   2. <path-to-folder of this repo>/prolate-2d-3d/generate-scripts/generateMultipoleReconstructions.m - codes for Paper 2

6. select pieces of code with your cursor and run them (in Octave to run selected text is F9 key; logical pieces of code are separated with multiline comments)

Important note: simulations for 3D case in Paper 1 may require significant amount of RAM as one of the routines being called is the inverse Radon transform implemented using NFFT (up to 80GB in our experiments).

Playground

All codes related to PSWF-Radon approach are stored in <path-to-folder of this repo>/prolate-2d-3d/.
Check source codes there if you want to develop your own simulations.

The scattering library

Routines for computing the analytical solutions to acoustic scattering problems involving prolate spheroids, oblate spheroids, and disks.

The software in this library is described in the following paper: http://scitation.aip.org/content/asa/journal/jasa/136/6/10.1121/1.4901318.

Using the scattering library

1. Clone this repository to your local machine. Alternatively, you can download this repository as a ZIP by clicking on the link to the right. This will give you the entire repository, including not only the scattering library, but also the spheroidal library.

2. Open up MATLAB, and run sandbox.m. This will set up all of the paths.

3. Run generate_all_test_figures.m. This repository comes with the precomputed spheroidal wave functions necessary to generate these test figures. It will generate four figures, saving each one as a PDF in the images directory.

4. To run generate_figures.m, you'll need to download more precomputed spheroidal wave functions at https://www.dropbox.com/s/tmqr3x05mpskinw/saved.zip?dl=0. This ZIP is approximately 800 MB, so make sure you have enough disk space. Unzip the contents of saved.zip into the saved directory. Once done, run generate_all_figures.m, which will generate 11 figures, saving each one as a PDF in the images directory.

5. For those interested in lower-frequency stuff, you can download https://www.dropbox.com/s/sor9szivwbk7v5f/saved_small_c.zip?dl=0, which contains precomputed spheroidal wave functions for c = 0.1, 0.5, 1.0, and 5.0.

6. You can use the the spheroidal library to compute the spheroidal wave functions for values of c, m, and/or n that aren't included in either of these ZIPs.

License

The scattering library is Copyright (c) 2014, Ross Adelman, Nail A. Gumerov, and Ramani Duraiswami, and is released under the BSD 2-Clause License (http://opensource.org/licenses/BSD-2-Clause).

The spheroidal library

The spheroidal library includes routines for computing the spheroidal wave functions, and is located in the spheroidal directory.

Third-party software

This repository contains four libraries from the MATLAB Central File Exchange. They are:

1. export_fig by Oliver Woodford (http://www.mathworks.com/matlabcentral/fileexchange/23629-export-fig)

2. cubehelix by Stephen Cobeldick (http://www.mathworks.com/matlabcentral/fileexchange/43700-cubehelix-colormaps)

3. legendflex by Kelly Kearney (http://www.mathworks.com/matlabcentral/fileexchange/31092-legendflex--a-more-flexible-legend)

4. spaceplots by Aditya (http://www.mathworks.com/matlabcentral/fileexchange/35464-spaceplots)

These four libraries were released under their own licenses, which can be found along with their code in their respective directories in this repository.