Published on January 3, 2008
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) Optimal reconstruction of Gaussian: Optimal reconstruction of Gaussian Error = 5.5% K=150 Error = 7.6 % K=160 J= 8, N=13, rep= .7, Viewing angle ~ 45 degrees Gaussian representation Pillbox representation Reconstruction with 5% noise added: Reconstruction with 5% noise added Error =11.6% K=120 Error = 10.8 % K=150 Gaussian representation Pillbox representation J= 8, N=13, rep= .7, Viewing angle ~ 45 degrees Bottom end: J = 2? 3?: Bottom end: J = 2? 3? Viewing angle=18o K=50 Error =31% J= 3, N=13, M=400, rep= 1 Gaussian representation Gaussian representation Viewing angle=45o K=65 Error =30% The data: The data Synchrotron period = 24.3 us, 26.4 projections per 180o I II III 1.61 ns 500 us Region 1 - sample with 184.09o view: J=27, N = 12, no smoothing, Gaussian representation Region 1 - sample with 184.09o view Region 1: comparison: Pillbox representation Region 1: comparison Gaussian representations J=27 J=8 J=10 error 25% error 31% Gaussian representation Region 2: 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.