Direct calculation of the collimator transfer function for single bore and multichannel focusing collimators

International Journal of Applied Radiation and Isotopes, 1974, Vol. 25, pp. 193-202. Pergamon Pre~. Printed in Northern Ireland

Direct Calculation of the Collimator Transfer Function for Single Bore and Multichannel Focusing Collimators D. C. B A R B E R Regional Medical Physics Department, Weston Park Hospital, Whitham Road, Sheffield SI0 2SJ, England

(Received 2 August 1973; in revisedform 16 November 1973) A method of directly calculating the collimator transfer function (CTF) of both single bore and multicharmel focusing collimators is presented. Since the point spread function (PSF) of a single hole collimator is obtained by the cross correlation of the exit pupil and the projection of the entrance pupil onto the exit pupil plane, the C T F may be obtained directly from the cross correlation theorem, by a product of the Fourier Transform of the exit pupil and a suitably scaled Fourier Transform of the entrance pupil. The method is extended to multichannel focusing collimators and examples are given. C A L C U L D I R E C T DE LA F O N C T I O N DR T R A N S F E R T DR C O L L I M A T E U R P O U R LES C O L L I M A T E U R S A CANAL U N I C E T M U L T I C A N A L A F O Y E R AJUSTABLE On p r ~ e n t e une m6thode de calculer directement la fonction de transfert de collimatenr (CTF) pour les collimateurs e t / t canal unic et multicanaux/l foyer ajustable. Puisque la fonction d'extemion de point (PSF) d'un collimateur/t canal unic se trouve par la corr61ation mutudle de la pupille de sortie et de la projection de la pupRle d'entr~e sur le plan de la pupliie de sortie, on peut obtenir la C T F directement du thdor6me de corr61ation mutuelle, par un produit de la Transforme Fourier de la pupiUe de sortie et uric Trensforme Fourier convenablement proportionnde de la pupille d'entr~e. On ~tent la m6thode aux collimatcurs multicanaux/t foyer ajustable et on donne des exemples. H E I I O C P E ~ C T B E H H O E BbItIHCJIEHHE I I E P E ~ A T O t I H O F I ( I ) Y H I ~ I 4 I I HOJIJIHMATOPA ~ J I H ¢OH~CIdP]FIOIIII4X I~OJIJDIMATOPOB O ~ H O P A C T O q H O F O H MHOFOHAHAJIBHOFO THIIOB IIpe~cTaB~eH meTo~ B ~ n e a e H n a .oam~MaTop-o~ nepe~aToqHo~ ~ym~=n~ (I~II¢) qSoKyc~pym~ax ~0aanMaTOpOB O~HopacToqHoro ~ M~oro.a~a~sHoro ~,nom Ta~ ~ a . ~08MOmHO no~y~ qSy~n~m pa~0poca To~e~ ~o~ammavopa c O ~ M owepc~e~e~ ( ¢ P T ) np~ np~Men e m ~ ~Sa~MHO~ ~ o p p e a . ~ . ~ m~xo~Horo 8 p a ~ a ~ ~ p o e . n ~ Bxo~umro ~paqsa Ha ~OCSOC~b ~ x o ~ . o r o apaq~a ~oa~om~o ~oayq~vb IKH¢ ~enocpe~cv~e~o ov ~eopeM~ o ~sa~MHO~ . o p p e a a ~ . a , ~euoms~ya n p o ~ e ~ e ~ e ~ p e o 6 p a a o ~ a ~ a ~ypbe ~m'xo~oro 8pa~sa eOOTBeVcTBeH~O ~aBeme~Horo npeo6pasoBaH.a qSypbe BXO~HOrO a p a ~ a . P a s t r y MeTo~ BmU~O.~S ~,~oro~a~a~n~e ¢o~ye~pymn~ae ~oaaaMa~op,~ a Ta~me n a ~ np~Mep~,. DIREKTE BERECHNUNG DER KOLLIMATOR-OBERTRAGUNGSFUNKTION F~R KOLLIMATOREN MIT EINZELBOHRUNG-UND MEHRKANAL-FOKUSSIERUNG Eine direkte Berechnungsmethode ftir die Kollimator-]3bertragtmgsfunktion (KOF) yon Kollimatoren m/t Einzelbohrung- und Mchrkanal-Fokussiertmg wird vorgelegt. Da die Punkt-Ausbreitungsfunktion (PAF) eines Einzdloch-Kollimators durch die Krenzkorrdation der AustrittspupiUe und der Projektion der Eintrittspupille auf die Ebene dcr Austrittspupille erhalten wird, kann die KC~F direkt aus dem Kreuzkorrelations-Theorem erhahen werden, durch ein Produkt der Fourier-Bildfunktion der Austrittspupi[le mad eine Fourier-Bildfunktlon der Eintrittspupille in einem passenden Massstabe. Die Methode wird a u f Kollimatoren mit Mehrkanal-Fokussierung ausgedehnt und Beispicle werden angefiihrt. 193

194

D. C. Barber

INTRODUCTION M O S T radioisotope imaging devices ("scanners" and g a m m a ray cameras) need to collimate the g a m m a radiation incident on the detector in order to form an image. This is because in the absence of a gamma ray lens information about the direction in which the gamma rays are travelling may only be obtained (except for positron coincidence systems) by eliminating those gamma rays from the incident flux which are not travelling along or at a small angle to a fixed axis. Practical collimators are usually made up of one or more holes (channels) in a gamma ray opaque medium, typically lead. I n the ease of a mnitichannel parallel hole collimator for a gamma camera these are usually simple straight holes with parallel axes arranged in some close packed array. In the case of a "focusing" collimator the field of vlew of several holes is arranged to exactly overlap at a given distance (in the "focal plane") from the collimator by inclining some of the holes; increased sensitivity is achieved for a given resolution in the focal plane at the expense of reduced resolution (compared to the performance of one hole of this arrangement) in other planes. T h e overall response of the imaging device depends on other factors besides the collimator. I n the case of a gamma ray camera the response depends largely on the image transfer system (i.e. the crystal/photomultiplier/position computer arrangement). T h e overall image quality also depends on the display associated with the device. I n this paper, however, we shall be interested only in the collimating section of the imaging device. T h e imaging properties of a collimator for a radioisotope imaging device can be described, assuming uniformity of response across the field of view, by either the point spread function (PSF) or equivalently its Fourier transform, the collimator transfer function (CTF) which is mathematically identical to the optical transfer function in optics (GooDMAN{I)). Strictlyspeaking the more familiar Modulation Transfer Function (MTF) is the modulus of the CTF. However, it has become common usage in collimator work to refer to the real part of the C T F as the modulation transfer function (e.g. MACIN~RE et al32)). I f the PSF

(called S(x,y) in this paper) is at all asymmetric this is not really correct since the M T F defined in this way does not correctly describe the way the spatial frequency components of the object are modified by the imaging device. However, most PSF's exhibit a high degree of centro-symmetric symmetry (in the sense that S(x,y) = S(--x, --y)) and in this case little error is involved. Nevertheless we ought to be careful on this point, and in this paper we shall refer to the transform of the PSF as the CTF.

THE COLLIMATOR TRANSFER FUNCTION I n practice the CTF is computed by taking the Fourier transform of a measured line spread function (LSF). It might appear that this C T F is different to the C T F computed from the PSF. In fact the two are directly related. The C T F is given by P(u, v) oo

ff s(x,y) exp --2vti(ux + P(u, v) =

-oo

vy) dx dy

oo

ff s(x,y) dx dy --0o

(I) where u and v arc spatialfrequencies, that is,the reverse transformation

S(x,y) = (I P(u, v) exp 2~ri(ux

+

vy) du dv

--oO

(2)

represents a decomposition of S(x,y) into sinusoidal components of frequency u, v. I f we set v = 0 we have:

e(u, 0) ="

f f S(x,y) dxdy --oo

(3) N o w the line spread function

L(x) is given

L(x) = fo~ S(x,y) dy d-oo

by (4)

Direct calculation of the collimator transfirfunction for a line source positioned along the y axis. Therefore V(u, o) =

j-®

L(x) exp --2~ux dx

(5)

~ s(,,y) d~dy and the CTF calculated from a line source is seen to be a profile (in this case along v = 0) through the complete CTF calculated from the PSF. The Collimator Transfer Functions to be computed in this paper are therefore identical in concept to those computed from appropriate LSF data. Both the PSF and the CTF allow calculation of the image from the object; however, interest has been focused on the CTF because of the mathematical simplicity of the imaging equation in frequency space. The imaging equation is given by

G(u, v) -~- O(u, v) . P(u, v)

(6)

where G(u, v) is the Fourier transform of the image and O(u, v) is the Fourier transform of the object. From the normalization of equations (1) and (5), we see that P(0, 0) = 1. Since P(u, v) is generally less than P(O, 0) it follows that G(u, v) < O(u, v). Information about the presence of abnormal features in the scan is generally contained in the low spatial frequencies, which have a large ratio of signal to noise. However, information about the detail in the scan is concentrated in the higher spatial frequencies. Empirical methods of collimator design seek to optimize simple parameters such as sensitivity and a resolution parameter (e.g. FWHM). These methods control reasonably well the behaviour of the GTF at lower spatial frequencies but in general not the higher spatial frequencies. This is not unreasonable i f a simple scan is being performed. However, methods exist (e.g. BARBER,(a) BROWN et al. t4)) for enhancing various parts of the spatial frequency spectrum in order to improve the presentation of the information contained in these regions of the frequency spectrum. The success or otherwise of these techniques is rather more dependent on the detailed form of the CITF at the higher spatial frequencies than is suggested by simple op-

195

timization techniques. It may be useful therefore to be able to obtain information, both experimental and theoretical, about the detailed shape of the CTF. In real space the response of the collimator to a point source, given by the PSF, may be considered to be made up of three components. The first component, the geometric component, is the response of the system to radiation passing through both the entrance and exit pupils of the collimator. The second component is that due to penetration of the collimator material by radiation. The third component is that due to the effect of scattered radiation from the collimator material, and following the concept proposed by BECKel, al., c5~ from scattering in the source medium as well. We may therefore write

S(x,y) = S(x,y)~oo... + S(x,y)~..

+ S(x,y),~.t..

(7)

Taking the Fourier transform of this gives

(So + 8,, + S,)P(u, V)total = S a . P(u, v),

+ s../'(u, v)~ + s0. P(u, v)0 (a) where 8o, 8s, S" are the plane sensitivities due to the geometric, penetration and scatter effects respectively and as before P(0, 0)t.t.l , P(0,0)~, P(0,0)~ and P(0,0), have been normalized to unity. In general the PSF components from penetration and scatter will be broader than the geometric component and therefore in frequency space their transforms should fall to zero faster than P(u, v)o. The shape of the G~FF at the higher spatial frequencies should therefore be dominated by the geometric component. For the purposes outlined above, the failure of the method to be presented to deal with the effects of scattering and penetration is less serious than might at first appear. In this paper we shall present a method of directly computing P(u, v)o from the physical dimensions of the collimator. We shall show that if either the CTF or the LSF are required this method is in fact more effective than direct computation in real space. THEORY Let us first consider a general collimator hole (Fig. 1) consisting of an entrance and

196

D. G. Barber yw

\

l IV

l/' n

• l--

o~

.....

pu

i~ po;.tSou.fplo.°

Fxo. 1. The entrance pupil F,~(x',y') is projected onto the exit pupil plane by the point source at P.

exit pupil, each of arbitrary shape (F,,(x',y'), F,(x",y") respectively, where these functions are equal to unity within the pupil aperture and zero elsewhere). The only conditions we shall specify are (a) that there is no penetration or scatter, i.e. we are considering geometric response only, (b) that the plane of the entrance pupil is parallel to the plane of the exit pupil and the plane in which the CTF is to be calculated, (c) that the solid angles involved are small enough for a point source to produce a constant flux density on the detector surface through the exit pupil as it is moved in the point source plane. Before embarking on a mathematical analysis it is worth reminding ourselves what is happening at the collimator. Gamma rays emitted from a point source in front of the collimator will pass through the entrance pupil of the collimator. Only a fraction of these gamma rays will succeed in escaping through the exit pupil, this fraction depending on the position of the point source relative to the entrance and exit pupils and their relationship to each other. This fraction, as a function of the position of the point source is proportional

to the point spread function, and the Fourier transform of this function is the result we require here. For any position of the point source the point spread function is clearly proportional to the area of overlap of the p r o j e c ~ f i of the entrance pupil onto the exit pupil since this area defines the solid angle into which gamma rays must be emitted by the point source for them to pass through both pupils. As the point source is moved, the projection of the entrance pupil onto the exit pupil moves relative to the exit pupil; the corresponding change of overlap area reflects the changing response of the collimator to the point source. Two points need to be made. The size of the projection of the entrance pupil onto the exit pupil plane clearly depends on the distance of the point source plane from the entrance pupil plane. If the point source plane is close to the entrance pupil plane, the projection of the entrance pupil will be greatly magnified relative to the actual dimensions of the pupil. Conversely for a point source removed to a great distance virtually no magnification will take place. The relative magnification is given, from simple geometry, by (d + t)[d where t is the collimator thickness and d is the normal distance from the entrance pupil plane to the point source plane. The second point is that if the point source plane is dose to the entrance pupil plane a small movement of the point source will cause a large movement of the projection of the entrance pupil in the plane of the exit pupil, whereas the converse is true for a point source removed to a great distance from the entrance pupil plane. As well as this, moving the point source in one direction causes the projection of the entrance pupil to move in the opposite direction in the exit pupil plane. The relative movement of the projection of the entrance pupil in the exit pupil plane is given from simple geometry by --(rid) where the negative sign indicates the reverse direction effect described above. In constructing the collimator transfer function from the geometric properties of the collimator we must take into account both the scaling effects outlined in the above paragraph. For convenience we shall assume that the origins in each plane (0, 0', 0") lie on a common line as shown in Fig. 1. As noted above a point

197

Direct calculationof the collimatortransferfunction source a t any point in front of the entrance pupil projects the entrance pupil onto the plane of the exit pupil with a magnification of (d -b t)/d where t is the normal distance between entrance pupil and exit pupil planes (the collimator thickness) and d is the normal distance between the entrance plane and the parallel plane containing the point source. Let the gamma ray flux density through the entrance pupil from a point source at P(coordinates x,y) be D,r Then we can describe the distribution of flux passing through the entrance pupil plane by the function DnFa(x',y'). The projection of this flux distribution onto the exit pupil plane is given, remembering the magnification effect, by (d/(d + t)) s D,,Fn'(x" -- p, y" -- q), where F . ' is the function F,, magnified into the exit plane and p, q are the coordinates of the origin point O' projected into the exit pupil planeby thepoint source (Fig. 1). Fn'is ofcourse, the same shape as Fn but expanded dimensionally; a moments thought will show that in the plane 0" we can write F.'(x" -

p,y" - q)

The flux distribution leaving the exit pupil plane is given by d 2 (-~-~t) D,F,'(x" -- p,y" -- q)F®(x",y") (9)

of the collimator, i.e. to the response to the point source as it is moved in the 0 plane. It is clear from inspection of Fig. 1 that as the point source is moved in the point source plane p, q are related to x, y by the relationships p ~ (--x. t)/dand q ---- (--y. t)/d. Substituting these into Q(p, q) gives a function T(x, y) = Q ( ( - x . t)ld, (--y . t)]d) which represents the response of the collimator measured in the point source plane to moving the point in this plane. T(x, y) is not the point spread function since this must be measured by keeping the point source still and moving the collimator (this being the way images are formed). The point spread function is given by reversing T(x, y) i.e. by S(x, y) "- T(--x, --y). In terms of Q the point spread function is given by

(ll) Equations (10) and (11) express the point spread function in terms of the functions describing the shape of the entrance and exit pupils. Taking the Fourier transform of Q(.p, 9) first and then of S(x,y), the normalized Fourier transform of S(x, y) is shown in the Appendix to be given by

P(u' v)" = F"*( u(d t+ t) , v(d ~ t).) x

and it follows therefore that the total flux passing through the exit pupil is given by the integral of this function over the exit pupil plane. Making the substitution for F.' this is given by d

s

-(<"- p)s,o"_= q-)¢i

×

Y.\

a+t

a+t]

--00

X F,(x", y") dx" dy".

(I0)

{ ud

' t!

(12)

where F,,(u, v) and F®(u, v) are the Fourier transforn~ of the entrance and exit pupils respectively and F,,* means the complex conjugate ofF,,. The G'TF has been expressed in terms of the geometric parameters of the collimator hole. Since P(u, v)a will by convention be normalized to unity at 0, 0 we need only consider the normalized forms of F,~ and F,, in what follows.

SPECIAL CASES The value of the integral is seen to be proFor a square pupil of side 2a (Goovm~m), portional to the area of overlap of the exit F(u, v) is real and when normalized is given by pupil and the projection of the entrance pupil sin (21rag) . sin (2~rav) onto the exit pupil plane. The function Q(p, q) 03) F ( u , v) - 2~rau . 21ray is clearly related to the point spread function

D. C. Barber

198 Writing sin (2~'z) S(Z) = " 2~'z

(14)

P(u, v) becomes, for a square section tapered hole with an entrance pupil of side 2b and an exit pupil of side 2a.

~,tl

\t/ (15)

I f b is less than a the collimator tapers to a point at a distance f given by ( f + t)b = af. T h e n P(u, v) becomes (16) T h e C T F is therefore always positive at the focal point for this type of hole. For a circular pupil of radius a, the norrealized F(u, v) is given by Y ( u , v) =

Jt(21r(u ~ + v~)ll2a) Ja(21rwa) ,~(u2 + v~)ma -,~wa

(17)

MULTICHANNEL FOCUSING COLLIMATORS By far the majority of collimators in use on scanning instruments are mnltichannel focusing collimators, in which several collimator channels are arranged so that their fields of view exactly overlap at a distance (the focal plane) in front of the collimator. T h e advantage of increased sensitivity this confers over a single channel of the same shape is partly offset by the loss in resolution at depths other than that of the focal plane; however this type of collimator generally gives adequate performance. Since the total response of the collimator is a sum of the responses of the individual holes it follows that we should be able to calculate the G T F by a suitable summation over the G T F of the individual holes. I f we only consider hole shapes which are even (in the sense thatF(x,y) = F(--x, --y)) then P(u, v) for each hole has a real part only when taken about the axis of the hole. I f there are N holes in the multichannel collimator the coordinates of the ith hole on the front face are m~, n~ taken about the central axis of the collimator (Fig. 2). I f the axis of the ith hole intercepts the axis of the collimator atf~, then the coordinates of the axis in the plane at a

where J l is a Bessel function of the first kind order one (C~ooDMAN{1)) and w is the radial frequency (uz + v~).1/2 Writing Jl(2~z) B(z)

-

-

-

~z

(18)

P(w) for a circular section tapered hole with an entrance pupil of radius b and an exit pupil of radius a becomes, again suitably normalized (19) Provided that the dimensions t, a, b are constant the C T F of the hole is independent of the angle between the line joining the origins of the entrance and exit planes and the normal between these planes. For a general hole shape the transforms of the entrance and exit pupils could easily be computed and stored. With this data the collimator transfer function may be computed for any distance d from equation (I 2).

FIo. 2. Dimensions of a collimator hole.

Direct calculation of the collimatortransferfumtion distance d from the front face of the collimator are given by m~ t, ?1~t =

m~

~,~n,

,

( f 'f g- d)

199

For simple regular hole shapes it is shown in the Appendix that the sensitivity of the ith hole is proportional to

y8

Z, = (j~ + m** + n,8)31* T h e transfer function of the ith hole taken about the central axis of the collimator is given by the shift theorem, et~

CR,(u, v) = P,,(u, v) . cos

((

2~" m,

f, (20)

and

2,~ m,

f,

× u+n, (A-d))) ~

(21)

where the subscripts R, I refer to the real and imaginary terms respectively, and Pa~(u, v) is the C T F of the ith hole at a distance d from the front face of the collimator. ~ I f the plane sensitivity of the ith bole is Z, the C T F of the whole collimator becomes N

CR(u, v) "=c~Z~CR,(u, v) N

q(u, v) =~ z , G , ( u ,

v).

Pat(u, v) has been calculated above.

and therefore the C T F can be expressed completely in terms of the geometric parameters of the collimator.

RESULTS

x u+~, ( A - d ) ) )~ C.(u, v) = P.,(~, ~) sin

(25)

(22)

(23)

Figure 3 shows the real part of the C T F of a square section taper hole collimator o f dimensions a - - - - 0 " 2 5 c m b = 0 . 1 2 5 c m , t = 10 cm evaluated in a direction parallel to one side of the entrance pupil, for various values of d. Figure 4 shows the real part of the C T F of an imaginary but not atypical multichannel focusing collimator of square holes of dimensions identical to the previous example and arranged as in the inset. As expected the shape of the collimator transfer function shows maximum response at the focal plane of the collimator (d = 10 cm in this example) and gets worse for values of d greater than or less than this value.

I n both these examples the imaginary part of the C T F is zero for all frequencies.

DISCUSSION

T h e C T F can only be calculated analytically In practice, this summation can be simplified. for fairly simple shapes. T h e hexagonal shape I f the placing of the holes is even, i.e. for a of hole is a fairly common variant and as far as hole of coordinates m, n, there is one of co- the author is aware does not have a simple ordinates --m, --n then the imaginary part analytic transform. However, the transform El(U, v) is zero everywhere, j~ is usually the could be computed once by numerical methods same for all holes and is equal to the taper and then stored for further calculations with length of each hole (i.e. to f ) . Finally, if the holes of this shape. For more general shapes holes are circular in cross section or all have the and orientations the calculations required are a same orientation (e.g. hexagonal holes in a hex- little more complex, but there is in principle agonal array or square holes in a close packed no reason why they cannot be performed. square array) then P~(u, v) is the same for I t is easily shown that this method gives each hole. improved speed of computation over real space T h e C T F becomes for all these simplifications calculations. Let us suppose we wish to calculate the full two-dimensional C T F of a CTF = P,(u, v) collimator. In real space we must compute the N PSF at say n x n points (where the spacing × between points is adequate for complete N representation of the CTF). I f the PSF of each hole is available in analytic form (for every (24) value of d) then each PSF point requires a

200

D. C. Barber 18 cm Single Topered Hole " t - I 0 cm. o= 0-25¢m. b- 0"125 cm.

I t0 cm

6¢m ¢TF

O!

\

2 cm

0.$

X\ 1;0

2:0

"

clcm

Fro. 3. The real part of the CTF of a single tapered hole calculated for various distances (2-18 cm) from the front face of the collimator (i.e. the entrance pupil plane). The dimensions of the hole arc I -----10 cm, a ----0.25 cm, b == 0.125 crn. s u m m a t i o n of the appropriate values over the m holes of the collimator. T h u s m X n X n calculations arc required. T h e C T F m a y then bc computed by a fast Fourier transformation involving 4n * log s n operations. O n the other h a n d using the present direct m e t h o d the C T F may bc c o m p u t e d (if the transforms are available in analytic form) by m × n × n

calculations only, assuming the C T F is required

over an n X n array. Problems associated with inadequate sampling of the PSF are avoided. T h e time taken for the F F T is therefore eliminated. I f the Fourier transform of the LSF is required, then for the real space calculation n × n points must still be calculated and then the array summed along one axis to produce the LSF, which m a y then be transformed. By the current method only m x n 18 cm

¢TF

I-0

l©m

/

I

0"$

I

I ~ J I:0-"

2:0

3:0

Clcm

FIo. 4. The real part of the C T F of a multichannel focusing collimator calculated for various distances (2-18 cm) from the front face of the collimator. The dimemiom of each hole are t = 10 cm, a ~ 0"25 cm, b = 0.125 c m with a septal thickness at the back face of 0.I c m (0.05 c m at the front face). The arrange.mcnt of holes is as shown in the inset.

201

Direct calculation of the collimator transfir funaion

calculations are required (rather than ra × n × n) since we are only interested in the C T F along one axis in frequency space. Since n will be at least 20, it follows that a substantial reduction in computing time m a y be achieved. This function m a y then be transformed to give the LSF and it is clear that calculation of the LSF by this method is faster than by direct calculation in real space. Even greater savings m a y be achieved in more difficult cases. For example, GENNA et al. ~e~ have published details of a collimator consisting of two rings of segmented holes with a circular central hole. T h e r e are therefore, three types of holes in this collimator. Since the channel shapes are not simple the PSF for each hole shape must be calculated by a numerical method, in fact a digital convolution. This must be done not only for each hole shape but for each value of d. Calculation of the total PSF of the whole collimator for different values of d is therefore seen to be a very laborious task. I n the present method only three data arrays need be computed, i.e. the Fourier transform of each shape of hole (the F F T needed for this calculation is m u c h faster than the digital convolution calculation). All the relevant data for the calculation of the C T F for all values of d m a y then be extracted by interpolation of these values. I t is almost certain that the computational advantages of the present method would enable the PSF of this collimator to be calculated faster by transformation of a calculated C T F than b y direct calculation. A relatively daunting task in real space becomes quite feasible when tackled by the present method. CONCLUSION A method has been described and illustrated for computing the C T F of a collimator directly from the geometric parameters of the collimator. This method has been shown to be faster than the Fourier transformation of a calculated PSF and m a y be used to calculate the LSF of a collimator faster than direct computation in real space. For collimators with holes of nonsimple shape it permits calculation of the C T F (and even the PSF) in situations where the real space calculation would be excessively tedious.

Acknowledgcffwnt--The bulk of this work was carried

out while the author was a member of the Department of Medical Physics, University of Aberdeen. The author wishes to th~nk Prof. J. R. MALLARD for his encouragement during the course of this work. REFERENCES 1. GOODMAN J. W. Introduaion to Fourier Optics. McGraw-Hill, New York (1968). 2. MACINTYRE W. J., FEDORUK S. O., HARRIS C. C., KUHL D. E. and M A ~ J. R. Medical Radioisotope Sdntigraphy, Vol. 1, pp. 391--435. I.A.E.A., Salzburg (1969). 3. BARBER D. C. Ph.D. thesis,UniversityofAberdccn (1972). 4. BROWN D. W., KmcH D. L., RYEsO~ T. W., THROCKMORTONA. J., K n ~ O U R N A. L., BRENNER N. M. d. nud. Med. 12, 287 (1971). 5. BECK R. N., SC.HUX-XM. W., COH~,~ T. D. and ~¢RZS

N.

Medical Radioisotope Scintigraphy,

Vol. 1, pp. 595--616. I.A.E.A., Salzburg (1969). 6. GENNA S., FARMEX~rr M. H. and BURROWS B. A. Medical Radioisotope Scintigraphy, Vol. I, pp. 561574. I.A.E.A., Salzburg (1969). APPENDIX Derivation of equation (12)

The Fourier transform of Q~, q) (equation (10)) is given by OO

Q(u,v) =ffQ(p,q).exp-2~/(up

+ w) @ dq.

--oO

(A1) Writing d[(d + t) = % x" -- p = r,y" -- q = s and therefore for xn, y , constant dp = --dr and dq -----ds we have from equations (I0) and (AI)

Q(u,v) F.(r~,, s~) exp +2~/(ru + sv) dr

=

× F~(x',f) exp

-2~(x~u +y"v) dx"dy". (A2)

The integral in brackets now becomes

GO

~O0

(AS)

D. C. Barber

202

a plane source of specific activity p photons/sec/ cm i is placed against the entrance pupil. The flux from a small area dA on the plane source within the entrance pupil reaching the exit pupil is given by

Then 0o

=

,

F,,(:,:") --OD

× exp --2ui(x"u + y%) dx" dy" 1

u

(A4)

v

By a derivation similar to that in equation (A3) if P(u, v) is the Fourier tramform of S(x, y) (equation O f ) ) a n d t/d = 3.

P(u, v) = ( I Q(x3'y3) exp -2u/(ux + vy) dx dy --00

(A6) which gives

P(u,~)=~Q

,

.

(A7)

From (A5) and (A7) and normalizlng to unity at 0, 0 we obtain equation (12), i.e. '

,

02) Derivation of equaIion (26) Let the area of the entrance pupil in Fig. I be A n and the area of the exit pupil be Az. Suppose

d F = ~ p d~

CA8)

where ~ is the solld angle subtended by A~ at dA. £1 is given by _ A®. cos 0 4mr,~ (A9) where 0 is the angle between the line joining the centres of the entrance and exit pupils (i.e. 0'0 u) and the normal between the entrance and exit planes and t ' is the distance 0 ' 0 " . But t ' = t/cos 0 and therefore d F = Ax cos s 0 dA p4~t ~ (AIO) Integration over the area of the entrance pupil gives F =

A=A,, cos a 0

(All)

/~rt s For comtant An, A~ and t the sensitivity of a hole inclined at an angle to the normal between the entrance and exit pupil planes is proportional to cosa 0, i.e. from Fig. 2 to

fs (fs + rots + n s)a/l"

(26)