Longitudinal particle motion in circular accelerators is typically monitoring by one dimensional (1-D) profiles. Adiabatic particle motion in two dimensional (2-D) phase space can be reconstructed with tomographic techniques, using 1-D profiles. A computer program RADON has been developed in C++ to process digitized mountain range data and perform the phase space reconstruction for the AGS, and later for Relativistic Heavy Ion Collider (RHIC)

Mane, V.

Peggs, S.

Wei, J.

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1997 Particle Accelerator Conference, Vancouver, B.C., Canada, 5/12-16/97 Abstract BNL-639 26 REOEIVED JUN 2 5 @p Radon Reconstruction in Longitudinal Phase Space * V. Mane, S. Peggs, J. Wei Brookhaven National Laboratory, Upton, New York 11973, USA Longitudinal particle motion in circular accelerators is typ- ically monitored by one dimensional (1-D) profiles. Adia- batic particle motion in two dimensional (2-D) phase space can be reconstructed with tomographic techniques, using 1-D profiles. A computer program W O N  has been d e  veloped in C++ to process digitized mountain range data and perform the phase space reconstruction for the AGS, and later for Relativistic Heavy Ion Collider (RHIC). 1 INTRODUCTION The mathematical problem of 2D image reconstruction from an infinite number of ID projections was formulated and solved by J. Radon in 1917 [l]. In practical algo- rithms, now in use in computerized tomography, the im- age is reconstructed on a 2D grid from a finite number of projections. These algorithms can be broadly classi- fied into Fourier convolution algorithms and iterative al- gorithms. The Algebraic Reconstruction Technique (ART) is a commonly used iterative technique and is described by Gordon [2]. The Fourier convolution algorithm is based on papers by Bracewell and Riddle [3] and Ramachandran and Lakshminarayanan [4]. These algorithms are also pre- sented in texts on computerized tomography [51,[6], [71. Beam tomography in accelerators for longitudinal and transverse phase space has been used by several authors [8], [9], [IO], using the ART algorithm. In longitudinal phase space, the synchrotron osillation period is typically several hundred turns and many bunch profiles are avail- able over half a period. Therefore the Fourier convolution techniques are adequate. In this paper, we present recon- struction for mountain range data shown in Figure 1 and discuss a filtered backprojection algorithm for phase space reconstruction. During the Sextant test of RHIC in early 1997, this program was employed to reconstruct the mo- tion of Au7?+ beam in the AGS. 2 RADONTRANSFORMAND RECONSTRUCTION Let f(z, y) represent a 2D object (Figure 2). Let ( I ~ ,  y,.) be the coordinate system rotated by angle 0 with respect to the (t, y) coordinate system, where t, = I cos 8 + y sin 8 and y, = -zsin 0 + ycosB. The projections P(I,, e) obtained by integrating f ( z ,  y) along all lines of constant 'Work performed under the auspices of the U.S. Department of Energy. I .  I 2.0 2.5 3.0 3.5 4.0 RF phase Figure 1: Mountain Range Data from the AGS, used for reconstruction in figure 5. Figure 2: The profile P(z,, 0) is the Radon Transform of the two dimensional function f(z, y). The filtered backprojection algorithm, a form of Fourier convolution algorithm, is employed in the program RADON. In this algorithm, each projection P(tr,B) is Fourier transformed, multiplied by a high pass filter, in- verse Fourier transformed, backprojected, and all the back- projections are summed up. Note that backprojection is not the inverse of projection, and hence the high pass filter, Let F(u,  v) be the 2D Fourier Transform of f ( z ,  y) and let (w, e) represent the polar coordinates in the frequency space,withu=wcoseandv=wsine ST Since z p  = z cos 8 + y sin 8 and F (  w, 8 + T) = F (  -w, 8), the above equation can be written as f ( z ,y )  = S," J_m_ ~ ( z u , ~ )  I w I ei2awxr dWd6 (3) The Fourier Transform of a projection at angle 8 with re- spect to x gives a slice of the 2D Fourier Transform F(u,v) at angle 6 wrt u. Therefore, F(w, 6) is the Fourier Trans- form of P( z+, 6) and &(Zr, 8) = l, F ( w ,  8) 1 w I eiznwxrdzu (4) gives the filtered projection, where the frequency response of the filter is given by Iwl. The reconstructed image can be obtained by integrating over the filtered backprojections 03 The following two filters are defined in RADON [6] Ramp Filter if1 f I< f c  otherwise where f, is the cutoff frequency Hann Filter 3 RECONSTRUCTION EXAMPLES The program RADON, written in C++ generates simula- tion profiles, reconstructs images, and displays them graph- ically. The program also reads mountain range data, re- constructs in longitudinal phase space, and calculates the longitudinal emittance. This section gives an example of image reconstruction from simulation and compares it with the original image. Reconstruction with data acquired from AGS is described in next section. The signal and back- ground errors, as described below, compare the original im- age with the reconstruction. 3.1 Signal E m r  Let Ii j  and Xij  be the intensity of the original image and the reconstructed image, respectively, in cell (i,j). Let Imaz be the maximum intensity of the original image. Define the cutoff intensity IC to be 10% of the maximum intensity, that is I, = O.lImuz. Define the signal region S such that Iij > I,. Signal Error is defined by I -  C '  where as = Figure 3: Original image (top) and reconstructed image us- ing RADON (bottom). A total of 30 profiles with 64 sam- ples per profile are used for reconstruction. 3.2 Background Error Define the background region B consisting of all the cells (ij) that are not in S. Background Error is defined by r r  3.3 Example Figure 3 compares the reconstructed image with the origi- nal image for an annular distribution, modulated in the az- imuthal direction. A total of 30 profiles equally spaced be- tween 0 and w,  with 64 samples per profile are used for the reconstruction. The signal and background errors are 0.21 and 0.07, respectively. In this case, the signal error is due to the difference in amplitude of the reconstruction. Sensitivity analysis were performed to determine the error in reconstruction due to insufficient profiles, samples and the error in synchrotron period. Figure 4 gives the signal and the background errors, for a gaussian distribution due to the error in synchrotron period. Several examples of re- construction with gaussian and annular distribution, and the associated errors are given in [ 1 11. 0.20 0.10 0.00 -1 00.0 -50.0 0.0 50.0 100.0 % error in synchrotron period Figure 4: Signal and Background error vs. error in syn- chrotron period, for a gauss beam. 4 APPLICATION IN RHIC SEXTANT TEST RADON application was successfully used during the Spring 1997 RHIC Sextant Test [ 121 for longitudinal phase- space reconstruction of Au77+ beams in AGS. Mountain- range beam profile data from the Wall Current Monitor taken at AGS exrtraction flattop was captured into a fast scope, LeCroy Model 9354TM, with a rearm time of 100 ps. The data was transfered to a computer via GPIB using a LabVIEW program. Particle motion was reconstructed from a set of profiles recorded over half a synchrotron period (Figure l), along with basic parameters of the machine and the rf system. Since the bunch only occupied a small fraction of the rf bucket, synchrotron motion was linearized. The rf voltage was calibrated by observing the synchronous phase differ- ence as transition was crossed at a known ramp rate. Sig- nal width broadening caused by finite cable bandwidth 1121 was taken into account. Figure 5 shows the reconstructed phase space diagram at an early stage of the test after two bunches were coa- lesced to increase the beam intensity. The bunch rotated by 110’ in synchrotron motion, from the top figure to the bottom figure. The calibrated rf voltage was 254 kV, the harmonic number was 8, the rf frequency was 2.96 MHz, the relativistic y was 12.1, uansition yt was 8.49, and the synchrotron period was 9.6 ms. Totally, 49 profiles with 100 ps separation sampled at 4 ns were used for the recon- struction. 5 CONCLUSIONS A program RADON has been developed to reconstruct 2D longitudinal phase space plot of the beam from 1D profiles. During the Sextant test of RHIC in early 1997, this program has been successfully employed to reconstruct the motion of Au77+ beam in the AGS. In the future, we plan to gen- eralize the application for both longitudinal and transverse phase space, including nonlinear motion. 6 ACKNOWLEDGEMENTS We would like to thank W. VanAsselt for the LabvDEw MountainRange program, and for his help with obtaining data We would also like to thank S. Tepikian for the XRT 5.59 5.06 4.53 4.00 3.47 2.94 2.42 1.89 1.36 0.83 0.30 -0.23 5.13 4.64 4.16 3.68 3.20 2.71 2.23 1.75 1.27 0.78 0.30 -0.18 2.0 2.5 3.0 3.5 4.0 RF phclse [t-od] Figure 5: This figure shows two distinct globs of charge of rotating in an AGS rf bucket, with the bottom frame rotated by 1 10 degree wrt the top frame. graphics display program, B. Andrews, K. Brown and the AGS Main Control Room crew for their help and support. 7 REFERENCES [ 11 Johann Radon, Ber. Verh. Sachs. Akad. Wiss. Leipzig, Math. [2] R. Gordon, R. Bender, G. T. Herman, J. Theor. Biol., (1970), [3] R. N. Bracewell, A. C. Riddle, Astrophys. J. 150 (2), pp 427- [4] G. N. Ramachandran, A. V. Lakshminarayanan, Proc. Natl. [SI Stanley R. Deans, The Radon Transform and some of its ap- [6] G. T. Herman, Image Reconstruction from Projections, [7] A. Kak, M. Slaney, Computerized Tomographic Imaging, [8] J. S. Fraser, IEEE Trans. Nucl. Sci., Vol. NS-26, Feb 1979, [9] 0. R. Sander et al., BNLSI 134 (1980). 314. [lo] G. Jackson,IEEEPAC, WashmgtonD. C. (1993), 2418. [l l]  V. Mane, RHIUAPB6, June 1996. [I21 J. Wei et al., “RHIC Sextant Test- Accelerator Systems and Performance”, these proceedings. Phys. Kl. 69,262-277 (1917). 29, pp 471-481. 434 (1 967). Acad. Sci. 68 (91, pp 2236-2240 (1971). plications, J. Wiley, 1983. Springer-Verlag, 1979. IEEE Press, 1988. pp 1641-1646. DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their emplcyees, makes any warranty, express or implied, or assumes any legal liability or rcsponsi- bility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Refer- ence herein to any specific commercial product, process. or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, ream- mendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not nectssarily state or reflect those of the United States Government or any agency thereof. 

RADON reconstruction in longitudinal phase space

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RADON reconstruction in longitudinal phase space

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