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BinaryTomo

This MATLAB toolbox solves the reconstruction of binary images from their tomographic projections. The challenge with this reconstruction problem is that the number of tomographic projections are much smaller than the size of the image. By exploiting the binary nature of the image, it is possible to solve the problem. This framework is based on the convex programming approach and can scale up fairly easily for large-scale tomographic problems.

Problem description

The least-squares formulation of a binary tomography problem is

equation

where A is a tomography matrix of size m times n, b is the tomographic data of size m times 1, and x is a binary image that has grey levels 0 and 1. This problem is NP-hard to solve. We propose to solve the following convex program instead:

equation

This convex program is a Lagrangian dual of the main problem. The binary image is retrieved from a dual solution using

equation

Authors

  • Ajinkya Kadu ([email protected])
  • Tristan van Leeuwen
    Mathematical Institute, Utrecht University, The Netherlands

License

You can distribute the software as you wish.

Dependencies

This framework has been tested on Matlab 2019a.

Usage

The examples scripts are

  1. test_tomo : classic discrete tomography problem with no regularization.
  2. test_tomo_gen : discrete tomography problem with option for grey values to be other than -1 and 1.
  3. test_tomo_cvx : small-scale discrete tomography problem with dual problem solved using CVX toolbox.
  4. test_tomo_TV : Total-variation regularized discrete tomography problem
  5. test_tomo_TVmin : Minimum total-variation discrete solution to noisy tomography problem

image

Citation

If you use this code, please use the following citation

@ARTICLE{8637779,
author={A. {Kadu} and T. {van Leeuwen}},
journal={IEEE Transactions on Computational Imaging},
title={A Convex Formulation for Binary Tomography},
year={2020},
volume={6},
number={},
pages={1-11},
keywords={Tomography;Mathematical model;Optimization;Iterative methods;Noise measurement;Phantoms;Binary tomography;inverse problems;duality;LASSO},
doi={10.1109/TCI.2019.2898333},
ISSN={2573-0436},
month={},
}

A preprint of the article can be found here

Reporting Bugs

In case you experience any problems, please contact Ajinkya Kadu