Phase-Space Tomography of Fast Processes using Single Value Decomposition Method:

Phase-Space Tomography of Fast Processes using Single Value Decomposition Method Kevin Chalut, Duke University
Vladimir Litvinenko, BNL

Content:

Content Traditional method of tomography
Needs for additional methods
Description of the SVD approach to tomography
Results of mathematical modeling
Application to real electron beam data
Conclusions and prospects

Traditional method of tomographyeven angle rotations :

Traditional method of tomography even angle rotations J evenly spaced (180o/J) projections
Radon transform
Drawbacks
Works only with J projections of object rotated in steps of =180o/J
Applicable only to rotations - hence limiting the use of arbitrary linear projections
Susceptible to errors in the angles =-180o/J f

Longitudinal Phase-Space:

Longitudinal Phase-Space x/ xo e/eo Synchrotron
Oscillations Strip-line for nsec, streak camera for psec

Motivations:

Motivations Ability of using a limited number of arbitrary , but known, linear projections to restore full N-D information
Make the restoration process less sensitive to errors and uncertainties about the projection operators (i.e. in angles, beam-line settings, oscillation frequencies, etc.)
Make the restoration process less sensitive to errors and noise in the image detectors
Studies of fast processes in storage ring, i.e. those faster than one oscillation

Slide6:

Data from OK-4 storage ring FEL I II III 1.61 ns 500 s One synchrotron period All physics is here

Transverse phase space:

Transverse phase space x´ x z x´ x x´ x x y L - variable

Transverse phase space:

Transverse phase space x´ x z x´ x x y Quadrupole y´ y y´ y x´ x y´ y L - fixed F

Linear projections:

Linear projections Object P…. Projected
Images

Using SVD for tomography:

Using SVD for tomography I - full set of projected images
P - projection matrix
F - array representing the distribution F(X,P)
M - number of pixels in one image
J - number of projections
N - number of grid divisions in the phase space
D - number of the phase space dimensions Informatics says for unique reconstruction we need
M·J > N2D plus non-degenerative projection matrix.

What is degenerated projection matrix?:

What is degenerated projection matrix?
M·J > N2D

Method: SVD:

Method: SVD Singular value decomposition SVD is linear process which satisfies criteria of least squares
minimum of the error function:
U = eigenvectors of A, orthogonal matrix
V = eigenvectors of B, orthogonal matrix
D = matrix with zero non-diagonal elements 1 > 2… > N2D
are eigenvalues of matrix A

Pseudoinverse Projection Matrix:

Pseudoinverse Projection Matrix

The data:

The data Synchrotron period = 24.3 us, 26.4 projections per 180o I II III 1.61
ns 500 us

The 2D-Model:

The 2D-Model

Projection matrix forpillbox representation:

Projection matrix for pillbox representation Pillbox representation
Exact intersection of bins and grid
Advantage
Exact projections
Disadvantage
Sharp corners

Projection matrixGaussian representation:

Projection matrix Gaussian representation Projection of 2-D Gaussians onto pixels
Advantage
No sharp corners
Disadvantage
Not exact representation

Toy model: Theoretical Gaussians=2:

Toy model: Theoretical Gaussian s=2 Projections with 5% noise Gaussian, s=2,
centered at (1,1)

Region 2 srep=.8, K=90 Gaussian representation J= 8
0.152 of synchrotron period J=20
0.38 of synchrotron period

Interesting phase-space pictures:

Interesting phase-space pictures Code and theory
Developed by
V. Litvinenko “hot spot”

Region 3 – after lasing:

Region 3 – after lasing Gaussian representation N=12
srep=1
J=8 J= 8 error 14% Gaussian representation J= 27

Memory limitations:

Memory limitations Memory = N4D + J2 M2 + J M N2D
1D
N=100 D=1 -> N4D = 100M
J=25 M=400 -> J2 M2 = 100M
2D
N=100 D=2 -> N4D = 1016
10,000,000 G x #bytes But most of P matrix are Zeros!!!
New SVD decomposition methods needed…

Conclusions:

Conclusions This method works remarkably well at performing phase-space reconstruction with limited number projections
We explored two representations
Resolution (N and D) is limited by the RAM needed to implement SVD for inverting projection matrix.
We have seen that the method is very robust
A lot of interesting new representations and new method of using SVD
A lot of applications in science, medicine, industry and military

Acknowledgements:

Acknowledgements Thanks to Igor Pinayev and Samadrita Roychowdhury for all the support.

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