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LBL-5604
FAN BEAM AND PARALLEL REAM PROJECTION AND BACK-PROJECTION OPERATORS
Grant T. Gull berg
Lawrence Berkeley Laboratory University of California
Berkeley, California 94720
January 19/7
ABSTRACT
Expressions for the fan beam and parallel beam projection and back-
projection operators arc Tiven alonn with an evaluation of the point source
response for the back-projection operators. The tack-projection operator
for the fan beam geometry requires the superposition of projection data
measured over 360". Both the fan beam and parallel beam geometries have
back-projection operators with point source responses which are proportional
to 1/lr-r I and thus two-dimensional Fourier filter techniques can be
used to reconstruct transverse sections from fan beam and parallel beam
projection data. The two-dimensional Fourier filter techniques may have
the speed over other methods for reconstructing fan beam data but the
reconstructed image requires four times the core storage so that the
convolution result of one period does not overlap the convolution result
of the succeeding period when implementing the fast Fourier transform.
TABLE OF CONTENTS
Abstract
0 INTRODUCTION 1 0 FAN BEAM AND PARALLEL BEAM PROJECTION OPERATORS 4
2.1 The Equations for the Fan Beam and Parallel Beam Projection Operators 4
2.2 Projection of a Point Source 9 2.3 Projection of a Circular Diik 10
0 FAN BEAM AND PARALLEL BEAM BACK-PROJECTION OPERATORS 12 3.1 Fan Beam Adjoint Projection Operator 12 3.2 Parallel Beam Adjoint Projectior Operator 21 3.3 Other Back-Projection Operators 22
0 POINT SOURCE RESPONSE FOR BACK-PROJECTION OPERATORS 24 4.1 The Back-Projection Operation Represented as a
Convolution with the Original Density 24 4.2 Adjoint Transform of a Point Source Projection
Function for Fan Beam Geometry 25 4.2.1 The zeros •?* and e* of g(a) 26 4.2.2 The expressions for Idg(f>l/d0] .* and
tdg(e)/d9? 0 = 6* e7 9.' .28 4.2.3 The expressions for 5(0*) and S(6*) 29 4.2.4 Solution to the transformed adjoint transform . . 32
4.3 Adjoint Transform of a Point Source Projection Function for Parallel Beam Geometry 39
4.4 The Back-Projection of a Fan Beam Point Source Projection Function 41
4.5 The Back-Projection of a Parallel Beam Point Source Projection Function 42
0 DISCUSSION 43 Acknowledgments 52 Bibliography 53
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1.0 INTRODUCTION
Fan beam and parallel beam geometries have back-projection operators which accommodate the use of two-dimensional Fourier filter techniques for reconstructing transverse sections from projections. These back-projection operators give an image representing the true image convolved with a blurring function which for the parallel beam geometry equals 1/r. The relationship between the true and the back-projected image is given by the equation
b =(r,«) = f(r,.{,) * 1/r
where b_ is the back-projected image for parallel rays and f is the true
image. Budinger and Gullberg [1] conjectured that a similar result was
also true for fan beam geometry and for beams of arbitrary orientations
such as curved rays and randomly oriented chords from positron annihilation
events. This report shows that this conjecture is true for fan beam
geometry where the blurring function is equal t i 2/r.
Using two-dimensional Fourier f i l t e r techniques the true image is
reconstructed from the back-projected image for parallel beam geometry by
deconvolving the 1/r blurring function using the relationship
f(r.*) • q * 1 ||R| l 2 ( b = ( r , * ) K .
This result was obtained by Bates, Peters and Smith [2,3,4] and used in phantom studies by Budinger and Gullberg [1], The results developed herein show that for the fan beam geometry the true image can be reconstructed from the back-projected image by deconvolving the 2/r blurring function
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using the relationship
where b <(r,$) is the back-projected image for fan beam geometry. The methods of reconstructing fan beam projection data can be divided
into five categories: 1. Two-dimensional filtering of the back-projection.
This is developed in this report. Z. Re-ordering and rebinning the fan beam projection data
so that the data correspond to a parallel beam gpometry [5,6]. This allows the use of iterative methods, convolution methods, or Fourier space methods which are presently developed for reconstructing parallel beam geometry.
3. Special convolution method [7,8,9]. first a convolution function is applied to the fan beam projection data and the result is then back-projected.
4. Radon's integration method [10,11]. The iirojection data are first differentiated, then followed by a Hilbert transform or an approximation using the trapezoidal method of integration.
5. Back-projection of the filtered fan beam projection data [3,12]. For each projection the data ii first Fourier transformed, then filtered, nex<: the inverse Fourier transform is taken and the result is back-projected.
Presently most of the methods being utilized for reconstructing fan beam data either rearrange the projection data so the dat.i correspond to parallel
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rays or use special convolution techniques which are adapted for fan beam geometry.
The text gives a detailed mathematical analysis of fan beam and parallel beam projection and back-projection operators by first developing the expressions for the projection operators, next developing the expressions for the back-projection operators, and then evaluating the point source response for these back-projection operators. The results are summarized in Figure 6, Section 5.
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2.0 FAN BEAM AND PARALLEL BEAM PROJECTION OPERATORS
2.1 The Equations for the Fan Beam and Parallel Beam jYoj^ct_ion_0perators
The coordinate systems for the fan beam and parallel beam geometries are shown in Figure 1 and Figure 2 respectively. The equations for the linear paths are dependent on their orientation relative to the axis of rotation. If vie require that the vertex of the fan beam is left of the center of rotation as illustrated in Figure 1, then the equation of the line emanating from the vertex in the (x,y) coordinate system is
Ri sin C/R2 - x sin (S - r/R:> - y cos (0 - i/Ri) = 0 . (1)
This expression can easily be derived if we first express the eauation of the line in the (x',y') coordinate system,
y' = tan C/R2 x'. (2)
With the coordinate transformation (x,y) -(x',y'),
x' = x cos G - y sin 9 + Ri (3) y' = x sin 0 + y cos £ , (*)
we can substitute equations (3) and (4) into equation (2) giving
x sin 0 + y cos 0 = tan £/R2 (x cos 9 - y sin 0 *• Ri)
x sin e + y cos 9 = •J'*" ffi {x cos 0 - y sin 0 + Ri) COS t,/Ki
x sin 0 cos 5/^2 + y cos 0 cos £/R2 -
x cos 0 sin £/R 2 - y sin 0 sirs 5/R2 + R, sin £/R 2 . (5)
Collecting terms, we can express the above equation as
Rj sin £/R 2 - x (sin 0 cos £/R 2 - cos 0 sin e/rt 2) - y (cos 0 cos 5/R2 + sin 9 sin £/Ri) = 0. (6)
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FAN BEAM GEOMETRY
R|»ln({ /R 2 ) -» i ln;e-£/R 2 ) -yco«(«-f /R 2 )«0
XBLT67-9100
Figure 1. The projection data p^C.g) for a fan beam geometry represent line integrals for the lines Rj sin(£/Rj) -x sin(0-£/R ) - y cos(0-S/R 5) = 0.
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PARALLEL BEAM GEOMETRY
XM 7i)'. '-9099
Figure 2. The projection data D_{f',"') for a parallel beam geometry represent line integrals for the'lines r'-xsinO' - y cos t* = 0.
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Usinc. the identities for the sine and cosine of the difference of two angles, equation (6) can be simplified giving the result in equation (I),
R, sin \/R. - x sin(0-r./R2) - y cos(n - r/R.) = 0 . (1)
The equations of the colinear lines for parallel beam geometry can be derived in a similar fashion. Another approach would be to keep |R2 -R,| fixed and allow R ( and R, to approa. , » in equation (1) which in the limit gives the equation of a line for the parallel beam geometry,
C - x sin 0' - y cor 0' = 0 . (7) The coordinate system for the parallel bean geometry is illustrated in Fig. 2.
Using equation (1), the fan beam projection operator, P , is defined as P :f • p^ where
pjr.,0) * if f(x.y) «(R1 sin r/R2 - x sin(0-C/R 2) - y cos(0 - C/R2))dxdy
The integral is taken over R' and it is assumed that f(x,y) =0 outside the cone spanned by the fan beam. The limits of the cone are a function of the length of the curved detector and a function of R 2 which is the distance between the vertex of the fan beam and the detector. Throughout this report we assume that the detector spans an arc between -irR /2 and TTR /2. Using equation (7) the parallel beam projection operator, P_, is defined as P_:f * p_, where
//ft.* pJt',0') = II f(x,y)<5(r - x sir,©' - y COS0') dxdy . (9)
R 2
The parallel beam projection operator is linear and continuous when f decreases rapidly at the boundary [13], and the projection function satisfies [14]
P =(C,0') * P=(-5',0- + u) •
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Likewise the fan beam projection operator is linear and continuous; however, the fan beam projection operator satisfies
P<(?.0) = P^-5, e + n - 25/Rj) • (10)
The result given by equation (10) will become important when we consider back projection operators and the appropriate limits of integration.
The coordinates (5',0*) for the parallel beam projection function represent polar coordinates in the x-y plane and are related to the coordinates (x,y) through the transformation
x = 5' sin 6' (11) y = 5' cos 0' .
However, the coordinates (5,9) for the fan beam projection function do not represent polar coordinates in the x-y plane but are related to the coordinates (x,y) through the transformation
x = R sin 5/R sin(0 - £/R ) (12)
y = Rj sin £/R2 cos(0 - 5/R2) and are related to the parallel beam projection coordinates (5',0') through the transformation
5" = R. sin 5/R (13)
0- = e - C/R 2
It is clear that the relationship between either the fan beam coordinates (5,0) or the parallel beam coordinates (5',0') and (x,y) is not one-to-one since for any 0 or 0', (0,6) and (0,0') map onto (x,y)= (0,0). The transformation given by equation (13) is useH to reorganize the fan beam data into parallel beam data so that reconstruction algorithms applicable to parallel beam geometry can be used [5,6].
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2,2 Projection of a Point Source The impulse response for the fan beam projection ooerator can be
evaluated for a point source at the position (x ,y ) by substituting the two dimensional delta function f(x,y) = 6(x - x0)6(y -y o) into equation (8) giving
pJS.e) = JJ 6 ( x -x o )6 (y -y o ) « ( R , sin S/R2 - x s in(0-5/R 2 ) - y cos(0-e/R 2))dxdy
S?2
= S(RI s in 5/R2 - x Q s in{0-£/R 2 ) - y o cos(0-(-/R.,)) .
Substituting the expressions for x a and y 0 in terms of polar coordinates, x o = r
0
c o s * o a n d ^ o = r o s 1 n * o ' 9 i v e s
P<(e,9) = «(R, sin £7R2 - r 0 sin(4.0 + 0 - e/R2)) (14)
Similarly evaluating the projection of a point source for the parallel beam
geometry gives
P=(S',0") = « ( r - r 0 sin(<t>o + e'}) . (15)
Later we will see that we can rewrite equation (14) as
r r0 sin (0+9) ~|\
„ , , „ , . _ _ ^ [R1 + r o c o s U o + 0 ) j ; R, <5[ C - R tan " 1 '
V R * + rj; + Zr0R2 cos(<j,„ + e)
I f we integrate p U,0) over £ we find that
/ P_(?>0)de = K V R 2 + r* + 2 ^ , cos(* o+0)
Distance from fan source to detector Distance from fan source to (x , y )
Whereas for the parallel beam geometry
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/p =(5'.o-)de-IR
1 .
2.3 Projection of a Circular Disk For the fan beam projection operator we consider the projection of
a circular disk, f(x,y), defined as
f(x,y) i: x z + y 2 < R < R, otherwise
Since the projection of a circular disk is independent of angle, the projection of f(x,y) is just the length of a cord going through the circle as illustrated in Figure 3.
Figure 3. The length of the cord traversing the circular disk is equal to 2 V R 7 - x 2 .
The length of the cord subtending the circle is
. / o 2 J 2 2 v R 2 - x 2 (16) where x is the perpendicular distance from the center of rotation to the
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line. Substituting the expression of x in terms of 5, x = R, sin f'R2
into equation (16) gives the projection of the circular disk, f(x,y), for the fan beam geometry,
I 2%/R2 - R 2 sin e/R |sin ?/R, | < R/R, ' 0 otherwise .
Using the transformation given by equation (13) gives
i 27R 2 - r 2 ' i n < R P =(e' .e ') * (i?)
' 0 otherwise .
The expression given in equation (17) is equal to the parallel beam projection function where £' is identical to x and represents the closest distance between the line and the center of rotation.
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3.0 FAN BEAM AND PARALLEL BEAM BACK-PROJECTION OPERATORS
3.1 Fan Beam Adjoint Projection Operator The adjoint operator P of the fan beam projection operator P_. is by
definition the operator which satisfies the relationship <P <f, P < > = (f, P* P <> (18)
where ( , > denotes an inner product defined as
<f.9> II f(x,y) g(x,y) dxdy . IR*
We will show in this section that the adjoint operator P,. :p,, * p^ is defined by the equation
+ f " * R ' P J S * . 9 ) >"sin{0+,i>)[R1 + rcosU + 0)] P*(r.«) - - ^ - 575 do J-i, [ r + R i + 2 Ri rcosU + 0)Yu
where
-1 I rsin(0 + <t) 1 L Rj + rcos(0-*-$) J £* = R tan
The kernel of the adjoint transformation satisfies the equation
R 2 rsin(0 + i(>)[R, + rcos(<|> + 0)] K(R,,r,*,0) = - i — 1 — — (19)
1 [r 2 + R 2 + 2Rj rcos(<i> + 0 ) ] 3 / 2
If we eliminate the kernel K(R1(r,<j>,0) from the expression for the adjoint operator and extend the limits of integration for 0 to be from 0 to 2ir, then we obtain the fan beam back-projection operator, B <:p < •* b <, defined by the expression
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b..(r,:) = I P,(r»,0)dO J The adjoint operator of the projection operator and the back-projection operator are members of a class of "back-projection" operators which give the superposition of the projection line integrals weighted by a particular kernel which for the back-projection operator is equal to the identity function.
In Section 3.2 we will develop the adjoint operator for the parallel beam projection operator [13] which satisfies the relationship
•n
p!(r,<.) = I P.(rsin(* + '),0') Vsin(« + 0')! dO' .
Removing the kernel |rsin(:+. ') from the integrand gives the expression for the parallel beam back-projection operator, B_:p. - b_, defined by the equation
bjr,*) = I p.(rsin(* + O'),0')dO' . •'o
Thp back-projection operator for parallel beam geometry has been defined by various authors - Peters [3] identified the back-projection operator with the layergram, Gilbert [15] called it back-projection, and Budinger and Gullberg [16] identified it with linear superposition.
For the digital implementation of the projection and back-projection operation we use a projection operator represented by the matrix (y_ which maps the transverse section vector A into the projection vector P according to the relationship
<$_ A = P
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The adjoint operator of this projection operator is equal to the transpose of the projection operator.
The back-projection operator used for the digital application of the back-projection operation is equal to the adjoint operator as indicated by the expression
This is in contrast to the continuous situation where the back-projection and adjoint operator are not identical. The back-projection approach to estimating the original density from its projection, as given by the equation
B = ;g= P = ©I P
where B is the back-projection vector, is the most rapid method of reconstruction but gives an image which is equal to the original density convolved with 1/r.
To develop the expression for the adjoint operator of the fan beam projection operator we first evaluate the inner product in equation (18)
P<f'P<' = II P< f ( x'' y' ) p< ( x"' y' ) dx'' < p< f'P<> = II P^(x',y')p,(x',y-) dx'dy- (20>
Substituting the expression for the projection operator of equation (8) into equation (20) gives the relationship
2ir R 2it/2 <f'P<> = I f f| f(x'-y) p<(5'0) * 0 0 R* ( 2 1
S(R ( sin ?/R 2-x sin(0-5/R2)-y cos(0-S/R2)) dxdy|0|d?dO
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where d.t'dy' is replaced by |J(df;dO and J is the Jacobian for the transformation relating the coordinates (x',y") to the coordinates (C,Q) given by equation (12),
x' = Rt sin '/R? sin(0 - r,/R.,) y' = R, sin K/R: cos(G - K/K2) .
In equation (21) the limits of integration for £ and for 0 were chosen so that the space 5?J is properly sampled. Figure 4 gives the trace of the locus of points representing the projected coordinates of a point source as it is rotated from 0 to 360 degrees. The trace for the parallel beam geometry illustrates that p_(C",0') = p_(-£\ n+O'); therefore, we can always properly sample the projection space if we integrate £' between -•» and ^'o, and o' between 0 and n. However, for the fan beam geometry the trace illustrates that p<(f,,Q) = P <(- f", "+" - 2f,/R ) and in order to properly sample the space IR' we need to choose the limits of integration for r,
between 0 and R 2JI/2 and for 9 between 0 and 2TT. In choosing the limits for r., we assume that the detector for the fan beam spans an arc between -R,TT/2 and R 2n/2-
The Jacobian for the transformation given above is the determinant
3(x',y") 3(C,0)
I ax' K
3?
3x 30
30 (22)
where
3x 3C -isin(0 - 25/R ) ,
3x 30 Ri sin c/R2 cos(0 - t/R2) ,
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SINQGROM of a POINT SOURCE FOR FAN BEOn PNO PQRPLLEt BEOM GEOMETRY
TMEOTO
PQINT SOURCE COORDINATES - 1 3 0 . 3 0 1
fdU BEPH UERTE* COOROINflTES - R1=B0 . R 2 = 1 2 0
Figure 4. The locus of points representing the coordinates (£,9) and (£ ' ,8 ' ) for a projected point source for fan beam and parallel beam geometry, respectively.
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& f = !l cos(0 - 2C./R,) .
^ = -R, sin c/R2 sin(6 - <;/R2) .
Evaluating tht determinant in equation (22) gives
J ' IT % - W fr = - R - s i n e / R
2
c o s e / R > ' ( Z 3 )
Returning to equation (21), we can rewrite the inner product by interchanging the integrals
< P < f ' P < > = II f< x'V) I J P..U.0) «(Ri s i n £ / R ? " xsin(0-i/R 2) IR
- ycos(0 - Z/R2)) !J|d£d0 dxdy , (24)
provided the functions f and p_, satisfy the conditions of Fubini's theorem [17,p.323]> i.e. f and p< are Lebesque integrable on a closed subset of !R*. This is always satisfied in real problems since f and p are continuous functions almost everywhere and bounded functions on a closed subset of K r. In order to agree with equation (18), we take the inner double integral in equation (24) and define the adjoint operator of the fan beam projection operator as _ ,„
2* V / 2
P X = P*(x,y) = 1 I p<(e,0) S(R, sin 5/R, - xsin(0-e/R2)
- ycos(0 - f./R2)) |J| df,dO (25)
Equation (25) can be simplified by int—jrating over the variable £,. In order to do this we use the relationship [18,p.38]
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irR,/2 ^ V 2
/ p<{€,e)«(h(e)) UU)|d? = I SP. IE.8) —HL-21L- u ^ ) ^ J. i I M i l l
1 a t- ' =C - (26)
where J{£) indicates that the Jacobian is a function of £ and E* are the zeros of h,
h(£) = RjSin £/R2 - xsin(6 - Wt) - ycos(e - </R ?) . (27)
The function h has a unique zero E* which satisfies
E* = R. tan"
= R, tan
•1 I xsinG •*• ycose 1 \_Rl + xcose - ysinej
• l {" rsin(e + $) "1 (_R1 + rcos(e + c|>) J
(28)
and sin K*/R2 and cos E*/R2 satisfy
sin £*/R = xsine + ycose rsin(<ti + e) [ R 2 + x 2 + y z + 2R I(xcose-ysin6)]' i [ r 2 + R2 + 2Rircos(iJ> + e ) ] ! i
(29)
„ R, + xcose - ysine Rj t rcos(* + e) cos £ /R, = r - —; r
[Rz + x 2 + y 2 + 2 R (xcose - ysine)P [r*+R 2+ 2R1rcos(<}, + e ) p (30)
where the poiar coordinates x =rcos$, y = rsin$ are substituted for x and y. Integrating the expression In equation (26) gives
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r-?.,/Z \ i dK
— i f e* > o
I dh(g) 1 w
P„«,0) «(h(c))|0{t)!d5 = i "O 1 0 otherwise
(31) From equation (28) we see that if R, is greater than r, then the condition r * > 0 can be replaced by jo + ijl < it.
In order to evaluate the denominator in equation (31), we take the derivative of h giving
I R ! ^ " | . ..* = |"T " S C/RJ + ^ - C O S ( 6 - 5/R2) - -sin(e - C/R2) £=£*
Expanding and collecting the terms which are factors of sin 5/R and cos £/R2
gives
dhU) = ^ R, + xcose - ysine) (xsine + ycose) cos £/R, + s sin C/R 5 df, R 2 "" v n i R 2
and in polar coordinates gives
dhfrl T R i + r cos((t + 6 )1 mLL B J _ J c o s e / R j + s 1 n ( ^ + e ) S i n C / R 2
Substituting the value of sin £*/R, and cos £*/R 2 given by equations (29) and (30) gives
-,2 « .„• [R, + rcos(<j) + e)] r 2 sin2(<t> + 8) R j[r 2+R| + 2Ri rcos^ + e)]'"4 R,[r2 + R 2 + 2R, rcos(* + 6J]1*
[r2 + R 2 + 2Ri rcos(* + e)]'s / R 2 . (32)
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Substituting equation (32) into equation (31) and replacing the condition t* > 0 with -$ < 0 < TT - (j> gives the expression
R2 P < ( 5 * , 0 ) | O ( £ ; * ) | i f -$ <0< IT
" V 2 \ [ r 2 + R ? + 2Rl rcosU + G)]
| P<.{C,0)«(n(C))lJ(d|d5 =
otherwise (33)
We can now give an expression for the adjoint operator P by substituting equation (33) and the expression for the Jacobian [equation (23)] into equation (25) giving
+ + 7* R2IP.^*'9) s i n '*/R2 cos ''*/R2 P-P< = P.(r,*) = J — ••- . — dfi J [r 2 + R 2 + 2R, rcosU + e ) ] 1 ' 2
-0
The absolute value sign is removed since both sin C*/R2 and cos ?*/R, >0 for -$ < 0 < ir-(f> and Rj > r. If we substitute for sin £*/R2 and cos 5*/R? given in equations (29) and (30), the adjoint operator is
, r R; p,(£*,e) rsin(6 + <t»)[R + rcos(* + B)] P +p = p+(r,<J.) = I ~ L ± - : w do (34) ' •= < J [r 2 + R + 2R, rcos(* + 6 ) ] 3 / 2
where K(R ,r,<)>,0) is given by equation (19) and the expression for £* is given by equation (28).
The adjoint operator given by equation (34) represents the integration over all projection values p,,!?*,©) which pass through the point (r,<|>) weighted by the kernel K(R ,r,$,0). Therefore this adjoint operator is a
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merofcer of the class of "back-projection" operators. It will be proved later that if the kernel is removed, we can define a back-projection operator which gives a response proportional to 1/r for a point source projection function.
3.2 Parallel Beam Adjoint Projection Operator The adjoint projection operator for the parallel beam geometry can be
derived using the same methods developed in evaluating the fan beam adjoint projection operator. From Fig. 2 we see that the coordinate transformation relating (x,y) to (£',0') 1 s
x = S'COS(TT/2 - 0')
y = 5'sin(Tr/2 - 0') and the Jacobian for this transformation is J = -£*. Therefore the adjoint projection operator defined in equation (25) for fan beam geometry can also be defined for parallel beam geometry using the equation
P*P= = p!(x,y) = I I P=(e'.0')«(e' - xsine- - ycose')|-r|drd0' . J0 -oo
or in polar coordinates using the equation
P ! P = = p!(r,<f>) = I I P =(C',e')6(r - rsin($ + e'))]r|drtie' • •0 -co (35)
If we integrate over £' in equation (35) then the adjoint projection operator for parallel beam geometry is
P*P= = P*(r,<(>) = I p =(rsinU + e'),e')|rsin((|) + 0')|de' . (36) 0
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3.3 Other Back-Projection Operators Ein-Gal [13] defines two other transforms for parallel beam geometry
which are "back-projection" operators similar to the adjoint transform. One of these operators is the unweighted integration of p_(£',0') along the circle C shown in Fig. 5,
it c(r,$) = f P =(rsin(e' + <)>),0') rde'
0 (37)
Figure 5. The value for the back-projection at the point (r,$) represents integration along the circle C.
Ein-Gal also defines another back-projection operator which is the transform
ation of the scaled projection function,
ir s(r,<|>) = f p (rsin(d) + e J) ,0-) ^
0 " |sin(<f> + 0-)| (33)
Here again the integration is performed along the circle C in Fig. 5 but this time the projection values are scaled by the factor l/|sin(<|> + 0')|.
Another back-projection operator for parallel beam geometry is
B =P = = bjr,j>) = f P_(rsin(<f> + 0-),g') d0' 0
(39)
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Peters [3] has identified this as the layergram, others have defined this as linear superposition [16J or just back-projection [15]. Notice that in this definition the differential arc length, rdG', for the circle C in equation (37) is replaced by do'. The result of this transform applied to a point source projection function gives
EL 6(5' - r sin(<b+0')) = b_(r,<|>) = ^ . (40) — U Q — i i
l<"->-„l For fan beam geometry, if we define a back-projection operator
B <P < = b <(r,*) = f P <(5*,0) d0 (41)
where £* is given by equation (28), then b<{r,(J>) has a similar response for
a point source projection function as does the parallel beam geometry
[equation (40)],
B< fi(R, sin 5 /R 2 - r s in(+ 0 +e-5/R )) = b<(r,<j.) = — . (42) l r- rol
The kernel ^(Rj.r.ilj.O) in the definition of the fan beam adjoint projection operator given in equation (34) does not appear in the integrand for the definition of the fan beam back-projection operator [equation (41)]. The proof of the result in equation (42) will be given in the next Section. The results in equations (40) and (42) are desirable since it means there exists back-projection operators both for the fan beam and parallel beam geometries which represent the original density convolved with a function proportional to 1/r. Therefore, two-dimensional Fourier filter techniques can be applied for both fan beam and parallel beam geometries to obtain a reconstructed image from its projections.
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4.0 POINT SOURCE RESPONSE FOR BACK-PROJECTION OPERATORS
4.1 The Back-Projection Operation Represented as a Convolution with the Original Density In order to prove that a back-projection operation represents a
convolution with the original density f(x,y) we need only to evaluate the back-projection of a point source projection function. To illustrate why this is so, let us consider writing the original density f(x,y) as
«,y) = l / f ( x ' , y f(x,y) = J I f ( x ' . y ' ) s ( x - x ' ) s(y-y') dx'dy .
Then applying a projection operator, P:f -+p, we have
P{£.0) = l l l l f ( x ' , y ' )6 (x -x ' )6 (y -y ' )dx 'dy " 6(g(x,y,e,o)) dxdy
where g(x,y,£,Q) represent some curvilinear path. Then assuming that the conditions are met which allow us to interchange the integrals, we can integrate over x and y, giving
P(C.e) = II f(x'.y-) 6(g(x',y',c,e)) dx-dy-
The delta function S(g(x',y',?,0)) represents the projection of a point source. Then applying a back-projection operator, B:p->-b, gives
b(x,y) = \h f(x',y) 6(g(x'!y',5*,0)) dx'dy d0
where C* is a function of x, y and 0. Interchanging the order of integration gives
b(x,y) = ij f(x',y') / 6(g(x\y',C*(x,y,0),e))de dx'dy
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where
J«(g(x',y',c*(x,y,Q),e))de
represents the point source response to the projection - back-projection operation. Therefore if b(x,y) is to be represented as a convolution, the point source response must be a function that can be represented as h(x-x'.y-y'). We will first show that such a functional representation does not exist for the adjoint transform of a projected point source both for the fan beam and parallel beam geometry. Then we will show that the back-projection operators given by equations (39) and (41) represent a convolution of the original density with a function proportional to 1/r.
4.2 Adjoint Transform of a Point Source Projection Function for Fan Beam Geometry The adjoint transform of a point source projection function is
evaluated by substituting the fan beam projection function for a point source given in equation (14) into equation (34) giving
1T-(|> R*S(R,sin£*/R, -r„sin(ij)„ + 0-£*/Rj)rsin(*+e)rR,+rcos{<t>+0)]
de. P*S • P*(r.<,) = J R,6(R1sinC*/R2-rosin(ij)o + 0-e*/R2))rsin(*+e)[R!+rcos(iti+0)]
O 2 + R, + 2R : rcosU + 0)]° (43)
I f we l e t g(e) denote the argument in the del ta r i nc t i on in equation (43),
then we can subst i tu te for £ * using equations (29) and (30) ,
g(0) = ( R , + r o c o s U o + 0 ) ) s in£* /R 2 - r 0 s i n U 0 + 0)cos5*/R 2
R ] ( rcosi) i - r o cosi t i | ) )s in0 + R ^ r s i n * - r ^ i n i t J cosQ + r o r s i ' n ( * - <fl0)
[ r 2 + R* + 2Rj rcos($ + 0 ) ] 1 / 2
-26-
Rewriting g(e),
g(e) _ a cosO + bsinO * c cl/2 (44)
where
a = R^rsin* - r^sin^) = R ((y - y j
b = R^rcosifr - rocos« ) = R,(x - x o)
c = r„rsin((j, - $„) = x oy - xyo
S = r 2 + R2 + 2R: rcosU + 9)
and substituting into equation (43) gives
+ V R* 6(g(e))rsin(e + $)[R. + rcosU + 9)] P <(r,») = J -! -J72 d9
(45)
(46)
To evaluate equation (46) we need to determine the zeros o* of g(9) as we did in equation (28) for h(,r) and evaluate |dg(9)/d9L_„*. This then allows us
i to rewrite equation (46)
P*(r.*) - ( Rj rsinO + ^JCRj+ rcos($ + e)]
Tsn £ 6(9-9*) i |dg(9) I
de do
4.2.1. The zeros 9* and 9* of g(9) Let us determine the zeros of g('0 by first setting u = sin" and writing
g(-i) as
g(u) iVl - U ? + bu + c cl/Z
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Qctermining the zeros of g{u) is equivalent to evaluating the solution to the equation
• as'l -u' + bu + c = 0 . (47)
The solution of equation (47) is outlined in the following steps:
• av'l - u r ' -(bu + c)
a'(l - u ;) = b V + 2bcu • c;
(a: + b :)u : +2bcu + c ; - a ; = 0
which gives
. -be • a\ a" + b" - c a r + b :
Substituting u back into equation (47) we see that there are two roots and •* such that
-be + a\ a' • b' - c" a' + b'"
(48)
/>'- 4. h J . r» coso* = - a c - b V a + b - c { w )
a * b"
Sinn* = -be • a v V • b ? _ ^ c L ( 5 0 )
a- + b -
C 0 S P » = ,-ac^bVa-*b'_-_cl. ( 5 1 )
a : + b*
-28-
The expressions for a2+b2 and a 2+b 2-c 2 can be simplified using equations (45),
giving the vector equations
a 2 + b 2 = R 2(r 2 + r 2 -2rr 0 cos { * - * 0 ) ) = R 2 | r - r J 2 (52)
a 2 + b 2 - c 2 • R 2 l ! l - i ; 0 | Z - r 2 r 2 s i n 2 U - * o ) = R 2 | r - r j 2 - |r x r „ | 2 .
(53)
Therefore g(9) has two roots 9* and 8 which satisfy equations (48)-(51).
4.2.2. The expressions for |dg(e)/d8|Q_0„ and |dg(e)/del3_Q*
Next we want to evaluate |dg(e)/d8L_ 0* and jdg(e)/deL * where the
derivative of g(e) [equation (44)] with respect to 9 is
dg(6) = (-asin9 +bcos9)S1 / 2 - (acos8 + bsine + c)dS 1 | / 2 /de d9 s
Me know fro.i equation (47) that the expression acosfl+ bsinf)+ c is equal to
zero fur '>=o* and 0=9*. Therefore
jdg(8) I . I -asinB* + bcos8* .
Substituting the expression for sine* and cos8* given in equation (48) and equation (49) and using the expression in equation (53), gives
1 dg(9) I . V a ' + b ' - c ' - V»;ir.-r,l°-|£»rol 2
I d e U=e* s 1 / z(e*) s 1 / z(9*) (54)
-29-
Lifcewise for o=e* we have
|dg(o) ' I -asine* + bcoso* ! d 0 '-3=n* " S 1 / 2 ( 0 * >
V a 2 + _ ' - . * . V"
4.2.3. The expressions for S(8*) and 5(9*) The denominators in equation (54) and equation (55) are evaluated by
substituting sin9. and cosO. given by equations (48)-(51) into the expression for S given in equation (45). First, for S(8,) we have
S(y*) = r' + R'j' + ZR rcos<J> cos"* - 2R, rsinj sino*
(r 2 + R 2)(a 2 + b 2) - 2R,c(xa-yb) - 2Rj(xb+ya) V a 2 + b 2 - c 2
_____
where the expressions,
2R tc(xa-yb) = W^y - xyJCxR^y - y o ) - y R t ( x - x 0 ) ]
,2
(56)
2 R > . * r |
and
2R1(xb + y a ) v
/ a 2 + b 2 - c 2 = 2RJXRJX - x.) +yR,(y -y„))y/a'- +b 2 - c 2
2 R 2 [ r . ( r - r J ] V R ] T r - r J 2 - | rxr_
-30-
Substituting these expressions along with equations (52) and (53) into equation (56) gives
(r2 + ^ ) R 5 | r - r 0 | 2 - 2 ^ | r 0 x r | 2 - 2 R ; [ r . ( r - r o ) ] V R ' | r - r 0 | 2 - | r > r o | 2 S(9 I - -,
We can rewrite S(©*) as
R | i£-i j i i 2 - i£. K d 2 - 2 tr ( !:-£. ) : i V R l i^£.i 2 - iO'£.i *^i*\z-\u*ii Sfflj) - —
in - Coi 2
(57)
n 2 2 7
Since [ r - ( r - r ) ] = r | r - r | - | r 0 * r | , the numerator in equation (57) is a
perfect square and we can write S(9*) as
s ( < ) . {«\^-\u^-i^Jr . ( 5 8 )
For S(e*) we obtain
S(9*) = r 2 + R2 + 2Rj rcos* cose* - 2Rt rsin* sine*
(r 2+R 2)(a 2+b 2)-2R 1c(ax-by)+2R l(xb + ya )y a
? +b 2 -c 2
(" r 2*»?)i!i-i,i 2- 2ir.' <i:i 2 + 2fc-{£-E„)WRIit-£oi 2-ir?«r 0i 2
\L-lf { V R>- r. l 2 - I v l 2 + Z• (£- r»>} 2
72 : (59) \Z - £o'
-31-
Substituting the expression for S(e*) into equation (54) gives the expression for |dg(e)/de| „*,
dg(B) d e io-e!
i ^ V f e ! 2 - \™.i2
m
V R t in-rioi 2- ino^i:)2 - c{ciJ
and substituting the expression for S(e*) into equation (55) gives the expression for |dg(0)/de|„_„* ,
V«!iF IdgMj t = jp^-yrr^irrJ2 _ ( 6 1 )
9 = e* VR' i r . - : 0 i 2 - iv ! i i 2 + c^-i,)
Notice that the expressions for these derivations differ only in the sign of the second term in the denominator. The denominator in equation (60) would normally be expressed as the absolute value since it comes from the square root of a perfect square. However, if we assume that R is greater than r o
and r then «Rj|r-£ 0l 2- |r„xr|2 > £"(j;-^0) - V r 2 | r " r o I 2 " l r o x r l 2 a n d t h e
denominator expressed in equation (60) is precisely equal to its absolute value.
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4.2.4. Solution to the transformed adjoint transform
We can now rewrite the adjoint transform given in equation (46) as
TT—4>
P<(|-,*) = I R* rsinJe+^JER,+rcos($+9)] s(n-op
+ 6(a-op
S 3 / 2 ( D ) i |<ig(o) 1 +
ldq(o) 1 1 d" U* I i "*-• in -n" ' u " IG
(62) where the expressions for |dg(e)/de|0_Q* and |dg(9)/do|0=[)* are given by equations (54) and (55). Integrating over 6 the adjoint transform of the point source projection function can be expressed as
. 2 R' rsin(et + «)[R +rcos(* + e*)] P*(r,*) - £ ~ , D[9*|(-*.K-«)] (63)
1 = 1 S(8*)VR;|r-r o| 2- |r<r D| 2
where 0£e^](-4«.TT-4>)] = 1 if 9 e(-<t>,ii-iti) and 0 otherwise. From the informati we have about 9* and e* given in equations (48)-(51), it is not immediately obvious whether 9* and 6 2 are both in the interval (-<t>,ir-<t>) or just one or neither. Since the terms of equation (63) must be positive for any 8c(-<fi,TT-<f>) we can determine under what conditions the roots 6* and 9* will be in the interval (-$,*-•) by evaluating equation (63). It turns out that oniy one of the two roots will be in the interval (-4>,ir-<f) for any projected point source.
For i=l, we have
rsinld^tp) = rcos* sin8* + rsin<J> cos8*
. r c o s , / - b c + a 7 a W ^ l \ r s . Y - a c - b y ^ b X F X V a'+b 2 ) \ a2 + b2 j
on
-(xb + ya)c + (xa-yb)s/a 2+b 2-c 2
-33-
Substituting the values for a,b,c, a 2+b 2, and V a z + b ! - c 2 from equations
(15), (52) and (53) gives
. , * , -R 1 (x (x -x ( ) )+y (y -y ( 1 ) ) (x ( , y -xy ( ) )+R 1 (x (y -y ( ) ) -y (x -x 0 ) )yR 2 ! r - r o | Z - | rKr o | 2
rsin(n +i) = 5 R > i r - r » i 2
-C"<C-CoHx0y-xy0) + { x 0 y - x y 0 ) ^ R 2 | r - r a | Z - | r * r 0 | 2
R I*" " r n l 2
-vizrt*)+ V R f in-Ho) z - I ^ L , (x„y - xy0) — (64)
R. r - r „ p
Likewise for rcos(6* + <(i), we have
rcos(ii* + .j>) = rcos* cosO* - rsin$ sinfi*
/ -ac - b y V + b 2 - c 2 \ . / -be + a y V + b ' - c 2 \ \ a2 + b2 / " \ a 2 + b2 /
- -(xa-yb)c - (xb+ya)y / a T +b 2 - c 2
a 2 + b 2
- R ] ( x ( y - y „ ) - y ( x - x o ) ) ( x ( , y - x y 0 ) - R 1 ( x ( x - x 0 ) + y ( y - y 0 ) ) ^ R 2 | r - r o | 2 - | r x r 0 | 2
R 2|r - r I 2
-iio^i2 -r<rr.WRtir:c«i z-irioi z , . (65) "Jr-rJ2
I f we express rsin(6* + <|>) and rcos(e* + <(.) as
-34-
rsin(8* + *) = (x 0y - xy 0) A + s ' R i !*:-J: 0 I
r c o s ( 9 * + , ) = » + » -i " i - - •- | 2
where
R, k - r\
A = - £ - ( r - r 0 )
B = - | r * r „ | Z , (66)
r- r | z - |r x r r-s- = V R |
then tha f i r s t term in equation (63) gives
R2 rsinie^ + JCRj + rcos{6* + * ) ]
S(6*)VR^|r-r 0 | 2 - I r ^ l 2
R2[R, rsin(8*+<)>) + rsin(9*+()>) rcos(e*+i{i)]
S(e*)VR*|r-r,|2 - |r*rj2~
V ' " Mr-£.l2 'VM^clvWil^l 2 / . s ( e * ) - s '
« ' (x o y-xy | ) ) [R 1
2 ! r - r 0 | 2 (A + S')+(ft + S-)(B + ASJ)]
S(9*)-S'R=|r-r o | 4
( x o y -xy o ) [R ; | r - r 0 l 2 A + R ^ r - r J 2 S' + AB + S'B + A V + AS'2]
(equation cont.)
SCeTJ 'S ' l r - r , ! *
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( x 0 y - x y „ ) [ ^ | r - r 0 | 2 A + AB+AS'2 +(R*|r-rJ +B + A 2)S']
S {e * ) .S ' | r - r , | *
(x o y-xy o ) [ (R^]r-r 0 | 2 4B)A-^AS' 2 +(S' 2 - i -A 2 )S-] SK> s >- : , i 4
(x 0 y-xy 0 ) [S ' 2 A^AS J 2 + (S^+A 2 )S ' ]
S(e*) S ' | r - r o | 4
. , [S' 2+ 2AS-+A ]
( v " x y o l iKn^7" I f we substitute S(9 ) from equation (58) into the above equation, we have
Rj rsinfe^ + JCRj + rcos(e* +<j>)]
(S'+ A) 2 | r - r 0 | 2
= (x„y-xy j ° ' r : * ' ° '2 ,_ _ ,4
( y - xy„) J&-r*f" _ (x0y - xy0) k - r o | 2 4 S ^ T r ^ " | r - r „ | 2
r-r„
(67)
Equation (67) is the expression for the first term in equation (63) and immediately one sees that this is positive only if (x 0y-xy 0) > 0. Ther 6* is in the interval (-<(>,ir-<|>) only if (x.y-xy ) > 0.
-36-
For i = 2 , we have
-be - aVa' + b ' - c 2 -ac + bv/a 2 + b 2 - c 2
rsin(8* + <t>) = x + y a 2 + b 2
-(xb+ya)c + (-xa + y b J v V + b 2 - c a
2 .o'
-R i (x (x -x o )+y(y -y o ) ) (x o y-xy a )+R 1 ( -x (y -y [ , )+y(x -x 0 ) ) N /R 2 i r - r 0 | 2 - | r*r _ _ _ _ _
- " " ' r f r . - I i>Hx 0 y-x y o ) - R / x ^ - x y ^ y ^ - r J 2 - | r x r 0 | 2
R 2|r - r | 2
- r - ( r - r 0 ) - ^ I r - r J 2 - | r x r „ | 2
= (x „y -xy 0 ) - = - — " ~ ~ (68) R r - r
-ac + bVa 2 + b 2 - c 2 -be - aVa 2 + b 2 - c 2
rcos(9 +*) = x - y a 2 + b 2 a 2 + b 2
(-xa + yb)c + (xb + y a ) \ / a 2 + b 2 - c 2
a 2 + b 2
R 2!r - r I 2
1 '-s, ^ 0 '
| 2 + r - ( r - r ) \ /R 2 | r - r I 2 - Irxr I 2
I ~ _ ^ V 1 ____J _ _ _ _ _ (eg) R,|r - r I '
-37-
If we express rsin(fl2+ <!>) and rcos(e* + <ji) as
rsin(0* + o>) = (x 0y - xy 0) A ~ S
rcos(0* + <t,) - B " A S ' R , l £ - ! ^
where A, B, and S'are defined in equation (66), then the second fraction of equation (63) becomes
R* rsin(9* + <j,)[R] + rcos(((,+ e*)]
R [Rj rsin(e*+<t>) + rsin(e*+<|>) rcosU+e*)]
S (e t )VR 2 | r - r o | 2 - | r * r o | 2
Rlk(x,y-xy 0)—A^l_ + (x )—Azll B - AS- J
s(ep s-
R^V^NlC-Co l 2 ( f t " s ' 5 + ( A " S ' ' ( B - AS")] S ( e * ) S ' R 2 | r - r /
(x 0 y-xy 0 ) [R^ | r - r 0 | 2 A - R 2 | r - r „ | 2 S'+ AB - S'B - A2S" + AS'2]
Steps' | r - r g | ?
(x o y-xy 0 ) [R; | r - r 0 | 2 A + AB + AS' 2- ( R ' | r - r 0 | 2 + B + A2)S"]
S(e*)S- | r - r . | 4
(equation continued)
-38-
( x 0 y - x y 0 ) [ 2 S ' 2 - ( S J 2 + A s )S J ]
SO:>S" I r - r J 4
(x v - x y J S ' [ 2 S - - S - 2 - A J ]
S(0*)S- | r - r | 4
-(x 0y - xy 0 ) (S ' -A ) Z
• I 4 S(9*) | r - r o
I f we substitute S{92) from equation (59) into the above equation, we get
R( rsintej+itOCRj + rcos(9*+*)]
S ( e * ) v ^ r - r j 2 - | r * r j Z
, (S'- A)2 | r - r 0 [ 2
= -(x 0y-xy 0)
-(x0y - xy0) SH^Ttf _ -(x„y - xy0)
2 XS^^TO* ' - - - ' 2 (70)
I r - r j 1 - i&-^k)- |r - r,
from equation (70) we see that the second term in equation (63) is positive only
i f ( x „ y - x y 0 ) < 0. Therefore 6* is in the interval (-<!>,IT-*) only i f
{ x „ y - x y 0 ) < 0.
Comparing equations (67) and (70), we see that conditions for both
9* and 6* being in the interval (-*,T-<J>) are mutually exclusive. Therefore
using the results from equations (67) and (70) we can express the adjoint
transform of a projected point source for the fan beam geometry given in
equation (63) as
-39-
(x0y-xy,,) - j — if {x 0y-xy 0) > 0
P<(r,>f) = '1-C'
-(x oy-xy 0)
'£"£<> 2~ i f (xoy-^) < °
The expression x oy - xy 0 represents the z-component of the vector cross product r *r, therefore we can express p<(r,<f.) as
P*(r,«) - - - — - x (71) ir-rJ
4.3 Adjoint Transform of a Point Source Projection Function for Parallel Beam Geometry The adjoint transform of a point source projection function given by
equation (15) for parallel beam geometry is equivalent to the result obtained for the fan beam. If we substitute equation (15) into the expression for the adjoint transform given by equation (36) we have
pV,<t>) = 1 6(rsin(9' + <t>) - rosin(<j>0+9')) |rsin(e'+ <j>)|de" . (72)
Again, in order to evaluate the integral we need to determine the zeros of
f(e') = rsin(6' + J>) - rosin(<f0 + e')
which are 8* and 9* such that
-40-
rcos'.'- - r.cos« , * O n
C O S " . = !r . -r' ir.-ri and
si no
'ro-ri ic,-ri r cos} - rcos<J> x - x ,* 0 0 0 COS0? = — - = — - — — k-H I'Vl
The derivative of f with resDect to 9" evaluated at 8* and (i* aives
lltiHI . iP . ri
Therefore equation (72) can be expressed as 17
pV.oi) = f Irslne- cos» + rs1n» cose-| [ a ( r . 9«) + a ( e - - e*)]de' •
Since the roots are 180° out of phase the adjoint transform can be expressed as
+/ I* l y x ° - "y. ! P.(r,i)>) = ,—
ll-Iol Using the expression r *r = yx 0 -xy 0, we can express the adjoint transform as
p!(r,<M - -=!—=--- . (73) u ~o'
Therefore the adjoint transform of a point source projection function for the parallel beam geometry given in equation (73) is identical to the result given in equation (71) for the fan beam geometry.
-41-
2*
4.4 The Back-Projection of a Fan 8eam Point Source Projection Function We have already seen the definition of the fan beam back-projection
operator given in equation (41). Notice that the adjoint operator given in equation (34) gives a weighted integral of the projection function p (E;*,e) whereas the fan beam back-projection operator defined as
„2n ,e) de (41)
where £* is given by equation (28), is an unweighted integral of the projection
function. I f we substitute the equation for the fan beam point source
projection function given in equation (14) into equation (41), then
2TT
b<(r,<j,) = ( e ^ s i n £*/Rj - t ^s in f ^+e -eVRjHde .
0
The back-project ion function can be rewritten as
f «(e-e*) 6(e-e*) Mr,*) s I !— + —: de (7<> n idg(9) I |dg(e)
de „_ n* de
where 6* and 8* satisfy equations (48)-(51) and [dg(e)/de| e = Q* and |dg(e)/de| e = e* are given by equations (60) and (61). Substituting these equations into equation (74) and integrating over 6 gives
b.(r,*) = V^ln-rJ2- lr*rJz- r(r-r-) '~ ~0 '
(75)
-42-
4.5 The Back-Projection of a Parallel Beam Point Source Projection Function The equation for the parallel beam back-projection operator is given
in equation (39),
Mr,*) = J p=(rsin(<j> + 9'),e') de' . (39) 0
Substituting the equation for the parallel beam point source projection function given in equation (15) into equation (39) gives
b = ( r , * ) = | 6(rsin(* + e') - r0sin(<)>0+9')) de' '0
We can express b_(r,<t>) as
r «(e--e*) >.•> - -
h jdf(e-) |
6(9'-e*) + 1 d e-'o l d f< r> I |df(e') I I de' L- = 0* I de' L, „
where 8* and 6* are the roots of
f(6) = rsin(<j> + 9') - r ^ i n ^ + e')
given in Section 4.4 and
L d f & 2 | = |r - r| . 1 d e 'e'=e*,e*
Since 8* and 9* are 180° out of phase, then the result for the back-projection of a point source projection function is
b =(r,*) = - ]
i:-r,i The back-projection result for the parallel beam geometry differ from the
result for a fan beam geometry only in a factor of 2.
-43-
5.0 DISCUSSION The adjoint, and back-projection operators for the fan beam and parallel
beam geometries are summarized in Fig. 6. These operators are members of a class of operators which give the superposition of the projection line integrals weighted by a particular kernel which for the simple back-projection is equal to the identity function. The back-projection operator for the parallel beam geometry requires the summation of line integrals over a range of 180° whereas the fan beam back-projection operator requires the summation of line integrals over a range of 360°. Therefore, the fan beam geometry requires a double sampling of projection data which is reflected in the 2/|r-r | response for the back-projection of a point source projection function as opposed to a l/|r-r | response for the parallel beam back-projection operator.
The results of the fan beam back-projection operation for various point source orientations and rotations of 180° and 360° are shown in Fig. 7. The projection data represent a sampling at 1° increments. The middle column of figures are images of the back-projection multiplied by |r-r | which has uniform gray only if the back-projection is proportional to 1/lr-rJ. The back-projection operation for a rotation of 360° is approximately proportional to l/|r-r0J, but due to the digitalization, the image of |r-rJ-B(r,<f>) represents a 10% variation between the maximum and minimum values. The contrast for the case of 360° rotation has been turned up to illustrate this variation.
The results shown in Fig. 7 for a 180° sampling can be expressed explicitly as
PARALLEL BEAM FAN BEAM
P.(£,'e')=//f(x,y)Sie1xsine-ycose')dxdy R(£,e)=//f (x ,y )8(R|S in f /R 2 -xs in<e-^ /R 2 ) -ycos( f r f /R 2 ) )dxdy 1R2 IR Z
where € * - R a ton ' [ p
f s i n ( » * f ' 1 z L R,+rcos(^+S) J
bir,<£) ^ " p U s i n U + e ' l . f l ' l df l ' ^ ( r , ^ ) = / " & ( £ * « ) dfl o °
For a point source at the point (r 0 ,<£ 0)
p,(f',e') = 8 ( C ' - r 0 s i n ( ^ 0 + e ' ) ) 0 ^ , 9 ) = 8(R,sin C /R z - r o s in (c£ o +0 - £ / R 2 ) )
. , r - , . l loxr| I-Toxxl P J r ^"M z ^ F ^
bjr,*) - ^ L j - W r , * ) . - j ^ j j -
XBL768-9234
Figure 6. Expressions for the projection, adjoint and back-projection operations for the fan beam and parallel beam geometry and the results for a point source with coordinates (r0,<)>0).
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POINT SOURCE RESPONSE AS A FUNCTION OF POSITION
Roiated 180°
Back-Projection Back-Projection. t£-£0| Reconstruction
Rotated 360°
X B B 760-9638 Figure 7. Comparison of the point source response as a function of position and rotation angle. The fan beam vertex coordinates are R, = 80 and R = 120.
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b"ir,4) =
1 ^ l t - E . I 2 - | r > r 4 | 2 - r - ( r - r p )
in-c-i M^-:f - ir:» | 2
i VR'ii:-£oi2 - irtoi2 + r<i:-£o)
z ^"I'rroi 2 - i n d 2
if 9* and 6* e(-n,0)
\ otherwise
if 0*e(-Ti,0)
if 0*e(-n,O)
(76)
where 9* and 9* satisfy equations (48)-(51). Equation (76) follows immediately from equation (74) where the limits of integration are taken to be from 0 to -it instead of from 0 to 2TT. A similar result was given by Peters [3] for the impulse response of the fan beam layergram. For a centrally positioned source (r =0) we see from equations (48) and (50)
sin 9,
sin 9* = -^_ IrJ
Therefore for points in the lower half of the plane 6*e(-Tr,0) and equation
(76) reduces to
M |r| b»(r.*) = 1 - f -
For points in the upper half of the plane 9*e(-ir,0) and equation (76) reduces to
|r| b*(r,*) • 1 +-f-
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Therefore for a centrally positioned point source the upper and lower planes show a contrast about the abscissa as the result of the discontinuity at y = 0.
Fiqures 8 and 9 give a comparison of the results from filtering the parallel beam and fan beam back-projection for two simulation phantoms. Artifacts are generated when the fan beam back-projection operation is integrated only over 180°. For an infinite number of projections, the method of filtering the back-projection for parallel beam geometry can be expressed as
f(r,$) = g - 1 {iRr.j2{b=(r,*m
and for the fan beam geometry as
f(r,*) = \ 3J 1 j|R|^{b <(r.«)>l .
However, the Parzen filter (Fig. 10) was used instead of the ramp filter in order to eliminate the ringing which occurs with a sharp cut-off. Other filters such as the Hamming, Hann, and Lutterworth have been studied and give results comparable to that of the t'arzen filter. Of these filters, the Hann filter appears to give the best results.
Three key points are derived from the analysis presented: 1. There exists a fan beam back-projection operator which
gives a result equal to 2/|r-r 0| for a point source projection function.
2. This operator involves the integration of projection data taken over 360' and not just 180° which is sufficient for a parallel beam geometry.
J**&*-*
Original Parallel 28Proj.6.4° Fan Beam 28 Proj.6.4° Fan Beam 56 Proj. 6.4° XEB 763-2175
Figure 8. Comparison of results from f i l te r ing the back-projection for parallel beam and fan beam geometries.
Original Parallel 9 0 Proj. 2° Fan Beam 9 0 Proj. 2° Fan Beam ISO Droj. 2° XBB 762-1243
Figure 9. Comparison of resuUs from filtering the back-projection for parallel beam and fan beam geometries.
- 5 C -
0 16 32 Frequency
XBL7611-9463
Figure 10. Plot in frequency space of the Simple Ramp and Parzen filters.
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3. The adjoint transform of the point source projection function for both the parallel beam and fan beam geometry cannot be represented as a function with the form h(x-x'.y-y').
Points 1. and 3. imply that only the back-projection operation gives an image which can be represented as a convolution and thus apdicable for the filter of the back-projection approach to transverse section reconstruction. A disadvantage of using this approach over other fan beam reconstruction methods which first filter and then back-project, is that a matrix which is four times the size of the reconstructed image is required so that the convolution result of one period does not overlap the convolution result of the succeeding period when implementing the fast Fourier transform [19, Chapter 7]. This method may be faster; however, we have not yet compared it with other methods.
The results using the filter of the back-projection approach to reconstructing fan beam data hints at a general approach to reconstructing projection data generated from nonparallel beams such as curved rays. It is required that a back-projection operator exists which gives a point source response which is either proportional to l/|r-r | or is a function which depends only on the radial distance from the point source. It is also conceivable that there are back-projection operators for three-dimensional leometries such as the cone beam which give a back-projection image that can be deconvolved using the filter of the back-projection approach to reconstruction.
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ACKNOWIEOGMENTS
This work was supported by the U.S. Energy Research and Development Administration. The basis of this paper was developed out of the work of Or. Thomas Budinger. His encouragement and suggestions are greatly appreciated. I would also like to thank Dr. Ronald Huesman whose suggestions led to some of the key results presented.
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BIBLIOGRAPHY
1. T. F. Budinger and G. T. Gullberg, Reconstruction by Two-Dimensional Filtering of Simple Superposition Transverse-Section Image, in Imaging Processing for 2-D and 3-D Reconstruction from Projections, Stanford, August 4-7, 1975 (Opt. Soc. Am.) pp. ThA9-l to 4.
2. R. H. T. Bates and T. M. Peters, Towards Improvements in Tomography, New Zealand J. Sci. T4 (1971), pp. 883-896.
3. T. M. Peters, Image Reconstruction from Projection, Ph.D. Thesis, University of Canterbury, New Zealand (1973).
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Polychromatic Degradation, to appear. 10. P. Stonestrom and A. Hacovski, The Scatter Problem in Fan Beam
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18. A. Papoulis, Systems and Transforms with Applications in Optics, McGraw-Hill, New York, 1968.
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