This repository implements MENT, an algorithm to reconstruct a distribution from its projections using the method of maximum entropy. The primary application of this algorithm is to phase space tomography in particle accelerators.
A probability distribution is constrained, but not determined, by a finite set of its projections. MENT finds the distribution
where
MENT uses the method of Lagrange Multipliers combined with a nonlinear Gauss-Seidel relaxation method to solve the constrained optimization problem. There are two equivalent ways to run the algorithm. The first, called "reverse mode", uses numerical integration; the second, called "forward mode" uses particle sampling, i.e., MCMC. Numerical integration is the best choice in low-dimensional problems, while particle sampling is the better choice in high-dimensional problems.
This repository contains both a forward-mode and reverse-mode implementation of MENT. In forward mode, one must sample particles from an unnormalized distribution function. An accurate grid-based sampler is included for problems of dimension
Each projection is defined as a sum over one or more axes after a transformation of the coordinates. The only requirement on the transformations is that they must be deterministic and one-to-one. The code is set up to take arbitrary transformation functions as inputs. This allows straightforward integration with particle tracking codes.
We also include routines to fit an
git clone https://github.com/austin-hoover/ment.git
cd ment
pip install -e .
To run examples us 576E ing built-in plotting functions:
pip install -e '.[test]'
Several examples are included in the examples folder. These examples demonstrate convergence on a variety of 2D, 4D, and 6D distributions, for both synthetic and real data from particle accelerators.
[1] G. Minerbo, MENT: A Maximum Entropy Algorithm for Reconstructing a Source from Projection Data, Computer Graphics and Image Processing 10, 48 (1979).
[2] G. N. Minerbo, O. R. Sander, and R. A. Jameson, Four-Dimensional Beam Tomography, IEEE Transactions on Nuclear Science 28, 2231 (1981).
[3] J. C. Wong, A. Shishlo, A. Aleksandrov, Y. Liu, and C. Long, 4D Transverse Phase Space Tomography of an Operational Hydrogen Ion Beam via Noninvasive 2D Measurements Using Laser Wires, Phys. Rev. Accel. Beams 25, 042801 (2022).
[4] A. Hoover, Four-dimensional phase space tomography from one-dimensional measurements of a hadron beam, Physical Review Accelerators and Beams 27, 122802 (2024).
[5] A. Hoover and J. Wong, High-dimensional maximum-entropy phase space tomography using normalizing flows, Physical Review Research 6.3, 033163 (2024).
[6] A. Hoover, N-dimensional maximum-entropy tomography via particle sampling, arXiv preprint arXiv:2409.17915 (2024).