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  • MAXIMUM-ENTROPY-BASED TOMOGRAPHIC RECONSTRUCTIONOF BEAM DENSITY DISTRIBUTION

    Y.-N. Rao, R. Baartman, TRIUMF, Vancouver, CanadaI. Tashev, UBC, Vancouver, Canada

    G. Goh, SFU, Burnaby, Canada

    Abstract

    For ISAC at TRIUMF, radioactive isotopes are gener-ated with a 500 MeV proton beam. The beam power isup to 40 kW and can easily melt the delicate target if tootightly focused. We protect this target by closely monitor-ing the distribution of the incident proton beam. There is a3-wire scanner monitor installed near the target; these givethe vertical profile and the +45 and 45 profiles. Ourobjective is to use these 3 measured projections to find the2-D density distribution. By implementing the maximumentropy (MENT) algorithm, we have developed a computerprogram to realize tomographic reconstruction of the beamdensity distribution. Of particular concern is to make thecalculation sufficiently efficient such that an operator canobtain the distribution within a few seconds of the scan. Aswell, we have developed the technique to perform phasespace reconstruction, using many wire scans and the cal-culated transfer matrices between them. In this paper wepresent details of the computer code and the techniquesused to improve noise tolerance and compute efficiency, af-ter reviewing the MENT algorithm.

    MAXIMUM ENTROPY TOMOGRAPHY

    Tomographic reconstruction algorithms offer the possi-bility to reconstruct higher dimensional density distributionfrom a series of projections measured in a lower dimen-sional subspace. In the absence of a large number of pro-jections, the Maximum Entropy (MENT) algorithm [1] canreconstruct a distribution that maximizes the entropy andsimultaneously reproduces all the measured projections ex-actly. In other words, MENT can find a most probable solu-tion with minimized artifacts to describe the observed data.This has proven to be superior in the case where there areonly a few projections available.

    We will only be dealing with 2-D distribution here. Letf(x, y) be the source distribution. It satisfies

    f(x, y) 0 and

    f(x, y)dxdy = 1 (1)

    The projection P (x) of this distribution on the x-axis isdefined by

    P (x) =

    +

    f(x, y)dy (2)

    The input data for tomographic reconstruction is a set ofsuch projections onto N different s-axes defined by a set

    TRIUMF receives funding via a contribution agreement through theNational Research Council of Canada.

    of transformation matrices Ri (i = 1, 2, ..., N ):(st

    )= Ri

    (xy

    )=

    (a bc d

    )i

    (xy

    )(3)

    The transformation matrixRi can be a rotation matrix, usedfor real space reconstruction, or the beam transport matri-ces for reconstruction of phase space density. It conservesthe area of the source distribution because det(Ri) = 1.Using the inverse transformation from the ith projectioncoordinates (s, t) back to the (x, y) source plane, the ith

    projection is represented as

    Pi(s) =

    +

    f [xi(s, t), yi(s, t)]dt (4)

    The goal is to invert Eq. 4 and determine the functionf(x, y). However, the inversion is not unique unless thenumber of projections N is infinite. For a finite number ofmeasurements, many different distributions exist that canreproduce all the measured projections. Out of these distri-butions, the one that maximizes the entropy

    E(f) = +

    +

    f(x, y) ln f(x, y)dxdy (5)

    and satisfies the boundary conditions of Eq. 4 is the mostappropriate one, because it contains the least information.

    The extended entropy function, with boundary condi-tions, can be written as

    (f, ) = E(f)Ni=1

    +

    i(s)[f(xi, yi)dtPi]ds (6)

    where xi and yi are functions of s, t, and where the i(s)denotes the Lagrange multiplier functions. The conditionsfor the stationary solution are

    (f, )

    i= 0 and

    (f, )

    f= 0 (7)

    The first condition in Eq. 7 is in fact equivalent to the con-straints defined by Eq. 4, whereas the second one gives

    ln[f(x, y)] =

    Ni=1

    i 1 or f(x, y) =Ni=1

    Hi (8)

    where the unknown Lagrange multipliers i have been re-placed by the equally unknown functions Hi = exp(i 1/N). The arguments of these functions are si = aix+biy,completely determined by the projection. So, the task ismerely to find these H-values for the equation.

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  • Since the measured projections are received as discreterather than continuous distributions, its natural to formu-late a binned projection as following

    Gij =

    si(j+1)sij

    Pi(s)ds =

    +

    +

    f(x, y)ijdxdy

    (9)where ij denotes a characteristic function

    ij(s) =

    {1 sij s si(j+1)0 otherwise

    (10)

    Therefore, Eq. 8 can be written as

    f(x, y) =Ni=1

    Mij=1

    Hijij (11)

    Substituting Eq. 11 in Eq. 9 gives an iteration relation forthe factors Hij

    Hij =Gij

    dxdyij{N

    k =iMk

    l=1 Hklkl}(12)

    After the H-factors are computed, they can be substi-tuted back into Eq. 8 to compute the distribution functionf(x, y).

    COMPUTER PROGRAM

    We implemented the MENT algorithm and developeda computer program [2][3] in C/C++ at TRIUMF fromscratch to accomplish tomographic reconstruction of beamdensity distribution. Our program is stand-alone, indepen-dent of any external libraries. Of particular concern is tomake the program robust and sufficiently efficient such thatone can obtain the distribution within a few seconds of thescan. First of all, the conditions defined in Eq. 1 imply thatthe input projection profiles must have non-negative back-ground and must be normalized to one, namely, Gij 0and

    Mij=1 Gij = 1. However, the input data are usually

    noisy, and even spiky and also inconsistent in terms of theintensities from different projections. Therefore, its cru-cial to pre-process the data before feeding them into theMENT routine.

    Data Pre-processing

    Data pre-processing is composed of two major steps inour program: statistics calculation and FFT smoothing. Indetails:

    Calculate the 2rms size T and centroid C so as to de-termine a proper window for efficient smoothing withFFT. Only the data within a window between C 2Tand C + 2T are retained for next use; everything out-side is discarded. C and T are calculated by samplingan area with no signal to determine offset and 2rmsnoise, subtracting the offset, cutting off the 2rms noise

    level in the background and then removing isolatedspikes. But in this step, we actually do not make anychanges (subtraction or cut) to the data in the retainedwindow.

    Do FFT to smooth the retained data using a low-passfilter. The cut-off frequency of the filter is one of theinput parameters that is allowed to change by the user.The profile after being smoothed may still have non-zero offset and wiggles in the background. So, nextwe subtract this offset and then make a 2rms cut tozero off everything in the background. This resultsin a clean profile with zero background to pass on tothe MENT function for tomographic reconstruction.However, the noisier the data, the more of the distri-bution tails will be unavoidably deleted.

    Test Runs

    One of the test runs [3] we made was specifically re-lated to the profile monitor near the ISAC target, whichmeasures only three projections: 45 and vertical. Thisexample deals with a distribution of a Gaussian core plusan asymmetric halo, moreover, this Gaussian core is talland narrow: the vertical rms size is 3 times the horizon-tal one. The reconstruction from the 3 profiles gives dis-tribution function f(x, y) correct to within 15% (standarddeviation). This is shown in Fig. 1.

    Another test was for phase space reconstruction, relatedto the measurements in the TRIUMF injection line. In thistest, we used a crescent shape to represent the initial phasespace of beam, instead of a regular elliptical shape. Thisphase space was transformed through a quadrupole to a lo-cation downstream. For a specified quad setting, a profilewas obtained from projection on the x-axis. There were12 profiles simulated; these, along with the correspond-ing transfer matrices, were fed into the MENT program toreconstruct the initial phase space. Fig. 2 shows the re-sult. The crescent shape is very well reproduced. Also, animportant conclusion from this exercise is that in order toreveal the crescent result, the phase advance between thesource point and the observation point must be made largeenough, i.e. 90, and over this 90 phase advance, themore projections one uses, the more precise reconstructionresult one can achieve.

    APPLICATIONS

    Our program has been incorporated into the cyclotronand beamline central control computer to accomplish on-line monitoring of the beam distribution near the ISAC tar-get. Within 5 seconds of a scan, an operator can ob-tain the result. Fig. 3 is an example snapshot, showing the3 projections measured and the reconstructed 2-D contourplot and H,V distributions.

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    Figure 1: Illustration of the first test made to the tomog-raphy program for the real space reconstruction, using anarrow Gaussian core plus an asymmetric halo as is shownon top as 3-D surface plot (left) and contour plot (right).The reconstructed result is shown in the middle