New relationship between the divergent beam projection and the Radon transform

Transcription

1 Journal of X-Ray Science and Technology 9 ( DOI.333/XST--3 IOS Press New relationship between the divergent beam projection and the Radon transform Yuchuan Wei a, and Ge Wang a,b a Biomedical Imaging Division, VT-WFU School of Biomedical Engineering and Science, Wake Forest University, Winston-Salem, NC, USA b Biomedical Imaging Division, VT-WFU School of Biomedical Engineering and Science, Virginia Tech, Blacksburg, VA, USA Received 7 August Revised 6 June Accepted 9 June Abstract. In this paper, we present some new relationship between the divergent beam projection and the Radon transform. An application of our results is to implement the Radon inversion from divergent beam projection. Keywords: CT, Generalized function, homogeneous function, Radon formula, Grangeat formula, divergent beam, data consistency. Introduction Since the parallel beam reconstruction is usually easier than the divergent beam reconstruction in x-ray CT, many formulae for divergent beam reconstruction were derived based on the relationship between parallel and divergent beam projections [,], including Grangeat s [3] and Smith s [4] cone-beam formulae and Noo s [5] fan-beam formulae, etc. Here we first recall the existing relationships between the divergent beam projection and the Radon transform, then present a new theorem, and finally apply it for the Radon inversion from divergent beam projections.. Existing relationships between the divergent beam projection and the Radon transform As shown in Fig., a function f( r describes an object in the n-dimensional space R n (n =, 3, whose divergent beam projection or x-ray transform at the origin O is defined as P O ( n = f(r ndr, where n is a vector on the n-dimensional unit sphere Ω n [,6,7]. We assume that f( r is a good function [8], which means that it is very smooth and decreases rapidly. Note that f( r may be not compactly supported. The n-dimensional vector r =(x,x,...,x n becomes r =(x,x =(x, y in Corresponding author. yuchuanwei@gmail.com //$7.5 IOS Press and the authors. All rights reserved

2 386 Y. Wei and G. Wang / New relationship between the divergent beam projection and the Radon transform Fig.. An object in the D (a and 3D spaces (b. R and r =(x,x,x 3 =(x, y, z in R 3.Also,(r, φ with r [,,φ [, π] are polar coordinates in R,and(r, θ, φ with r [,,θ [,π],φ [, π] or r [,,θ [, π],φ [,π] are spherical coordinates in R 3. A general way to study the relationships among the divergent beam projection, the parallel beam projection and the Radon transform is to calculate the weighted integral R n f( rh( rd n r in different coordinate systems, where h( r is a function with a certain symmetry [7]. If a weight function h( r is a homogeneous function of order n in R n [9,], i.e., h(r n =r n h( n forr>, ( the integral of the object weighted by h( r can be expressed as an integral of its divergent beam projection [6,7]: f( rh( rd n r = P O ( nh( ndω, ( R n Ω n where d n r = dx dx dx n = r n drdω is the differential element in the n-dimensional space R n and dω is the differential element on the unit sphere Ω n of R n. If a weight function h( r satisfies that h(x,...,x n =h(x,...,x n,, the weighted integral can be expressed as f( rh( rd n r = p(x,...,x n h(x,...,x n, d n x, (3 R n R n where p(x,...,x n = f(x,...,x n dx n is called a parallel projection, and dx n = dx dx...dx n is the differential element in the space R n. If a homogeneous function h ( r of order n satisfies that h (x,...,x n =h (x,...,x n,, one can set up a relationship between divergent and parallel beam projections [7] P o ( nh ( ndω = p(x,...,x n h (x,...,x n, d n x. (4 Ω n R n

3 Y. Wei and G. Wang / New relationship between the divergent beam projection and the Radon transform 387 If a weight function h( r satisfies that h(x,...,x n =h(x,,...,, the weighted integral can be expressed as f( rh( rd n r = R(x h(x,,...,dx, (5 R n where R(x = f(x,...,x n dx...dx n R n is called a Radon transform of the object. If a homogeneous function h ( r of order n satisfies that h (x,...,x n = h (x,,...,, one can set up a relationship between the divergent beam projection and the Radon transform P o ( n h ( ndω = R(x h (x,,...,dx. (4 Ω n The initial form of Formula (4 appeared in Hamaker early paper [6], which is sometimes called Hamaker s formula []. Different from Hamaker s work, we emphasize that in this note the function f( r is supposed to be a good function [8], and the function h( r (h ( r, h ( r, etc. is a generalized function [9,] instead of an ordinary function [6]. In the D space R, h ( r are identical to h ( r, a parallel beam projection coincides with the Radon transform, and Hamaker s formula is identical to Eq. (4. In the 3D space R 3, h ( r is a special case of h ( r, andeq.(4 is a special case of Eq. (4. Givenanexampleofh ( r (or h ( r, one can set up a relationship between a divergent beam projection and a parallel beam projection (or the Radon transform. The important homogenous function of order in the D space R include [7]. h ( r =δ(x for the opposite ray condition in fan-beam CT [7,];. h ( r = x for Noo s super short scan formula [5,]; 3. h 3 ( r = exp(iπωxdω = δ(x iπ x for a fan-beam version of the Tuy formula [3]. The important homogenous function of order in the 3D space R 3 include. h 4 ( r =δ (x for the Grangeat formula [3,];. h 5 ( r = for the Smith formula [4]; x 3. h 6 ( r = ω exp(iπωxdω = δ (x 4πi for the Tuy formula [,4,5]; 4π x 4. h 7 ( r = xδ(y for the Katsevich formula [6] and the Ye-Wang formula [7]; 5. h 8 ( r =δ(xδ(y for the opposite ray condition in cone-beam CT [7,]. 3. New relationships between the divergent beam projection and the Radon transform From any point S in the space R n (n =, 3, the divergent beam projection of an object is defined as P S ( n = f( OS + r n dr where n is a unit vector in the space R n.

4 388 Y. Wei and G. Wang / New relationship between the divergent beam projection and the Radon transform Theorem. In the space R n, if a homogeneous function h ( r is of the form h (x,...,x n = g(x,...,x n x n,whereg(x,...,x n is a homogeneous function of order n, the divergent beam projection and the parallel beam projection of an object f( r are related by P o ( nh ( ndω P o ( nh ( ndω = D p(x,...,x n g(x,...,x n d n x, (6 Ω n Ω n R n where O (,...,,D is a point in R n, and D is a real number. Proof. This relationship can be directly verified as follows: P o ( nh ( ndω P o ( nh ( ndω Ω n Ω n = f( rh ( rd n r f( OO + rh ( rd n r R n R n = f( rh ( rd n r f( rh ( r OO d n r R n R n = f( rg(x,...,x n x n d n r f( rg(x,...,x n (x n Dd n r R n R n = D f( rg(x,...,x n d n r R n = D p(x,...,x n g(x,...,x n d n x. R n Theorem. In the space R n, if a homogeneous function h ( r is of the form h (x,...,x n = g(x x n,where g(x is a homogeneous function of order n, the divergent beam projection and the Radon transform of an object f( r are related by P o ( n h ( ndω P o ( n h ( ndω = D g(x R(x dx, (6 Ω n Ω n R where O (,...,,D is a point in R n, and D is a real number. The proof is similar to that for Theorem, and we omit it here. Clearly, Theorem is a special case of Theorem. Now let us discuss some examples useful for Radon inversion in the D and 3D space. 3.. h ( r = x y in the space R Now, Formula (6 becomes π P o (φ sin φ π cos φ dφ P o (φ sin φ cos dφ =D R(xdx (7 φ x with P o (φ =P o ( n(φ = P o ( i cos φ + j sin φ.

5 Y. Wei and G. Wang / New relationship between the divergent beam projection and the Radon transform h ( r =δ (xy in the space R Since f(x, yδ (xydxdy = f(r, φδ (r cos φr sin φrdrdφ R R π = f(r, φδ (cos φsinφdrdφ = P o (φδ (cos φsinφdφ R = π = P o ( π P o (φdδ(cos φ = ( 3π + P o, π δ(cos φp o (φdφ Formula (6 becomes ( ( π ( 3π P o + P o ( P o ( π + P o ( 3π = DR (, (8 which says that in the D space R the derivative of the Radon transform can be calculated from divergent projections at two points. Here is an explanation on the case r =. In the above deduction, the claim δ (r cos φ =δ (cos φ/r is not strict since δ (ax =δ (x/(a a holds for a only. Clearly, at the origin of the coordinate system, we have r =. To avoid this problem, we can remove a small neighborhood of the origin in the space R first, and then let the neighborhood tend to zero. A strict deduction goes as follows: f(x, yδ (xydxdy = R x x= f(x, yydy = lim ε y >ε f(x, yδ (xydxdy = lim ε = lim f(x, yδ ε {(x,y:x (xydxdy +y >ε } π = lim f(r, φδ (r cos φr sin φrdrdφ ε = lim ε = lim ε π π ε = lim ε = P o ( P o ε ( π + P o f(r, φdrδ (cos φsinφdφ P o (φ, εδ (cos φsinφdφ ( π ( 3π,ε + P o ( 3π,ε {(x,y: x >ε,and y >ε} f(x, yδ (xydxdy

6 39 Y. Wei and G. Wang / New relationship between the divergent beam projection and the Radon transform where P o (φ, ε = ε f(r, φdr. Since the result does not change, we will not use the limiting process in the rest deduction work h 3 ( r =δ (xz in the space R 3 In this case Formula (6 becomes P o ( nh 3 ( ndω P o ( nh 3 ( ndω = D δ (xr(xdx. (9 Ω 3 Ω 3 R As shown in the Appendix, Formula (9 becomes π cos θ π φ φ= π P o(θ, φ sin θ dθ cos θ φ φ= π P o (θ, φ sin θ dθ = DR (x x=, ( which says that in the 3D space R 3 the second derivative of the Radon transform on a plane can be calculated from the divergent projections at two points on the plane, where P o (θ, φ =P o ( n(θ, φ and n(θ, φ =(sinθcos φ, sin θ sin φ, cos θ. Recall that the Grangeat formula provides a link between the first derivative of the Radon transform and the divergent projection at one point. 4. Radon inversion formula for the divergent beam projection 4.. A new form of the D reconstruction formula A equivalent form of the D Radon inversion formula [8,9] can be expressed as π f(x, y= p θ (t t =x cos θ+y sin θ dθ = π π p θ (t (t t dt t =x cos θ+y sin θ dθ, ( where p θ (t is the parallel beam projection (Radon transform and p θ (t the filtered projection, as shown in Fig.. The fan beam geometry is shown in Fig. 3. An x-ray source S moves along a circle of radius R, parameterized by the view angle β. The divergent beam projection at S is denoted as P (β,γ, whereγ is called ray angle. γ is the ray angle of the ray through a point T to be reconstructed. We suppose that T (x, y is inside the scanning circle with L = ST. As illustrated in Fig. 4, a line through the point T, parameterized by (θ, t in the parallel beam geometry, intersects the scanning circle at two positions S and S respectively. To apply Formula (7, we set up a coordinate system x y with S as the origin and S S as the y axis. Note that φ = 3π + γ γ for S and φ = π + γ γ for S. According to Formula (7, we have ( π p θ (t = π R cos γ π P (β,γ cos(γ γ π sin (γ γ dγ + P (β,γ cos(γ γ π sin (γ γ dγ. (

7 Y. Wei and G. Wang / New relationship between the divergent beam projection and the Radon transform 39 Fig.. The geometry for parallel beam reconstruction. Fig. 3. The basic geometry for fan beam reconstruction. In Fig. 5, when θ changes from to π, S moves from S to S, S moves from S to S, β changes from β to β,andβ changes from β to β. Based on Fig. 5, or Fig. 3 in [], we have the relationship dθ = R cos γ dβ = R cos γ dβ, (3 L L where L and L are the length of S T and S T respectively.

8 39 Y. Wei and G. Wang / New relationship between the divergent beam projection and the Radon transform Fig. 4. Fan beam reconstruction based on the new relationship. Fig. 5. When θ changes from to π, S moves from S to S, S moves from S to S, β changes from β to β, β changes from β to β +π. Putting in Eqs (3 and ( in (, we have f(x, y= β π 4π β π cos(γ γ P (β,γ L sin (γ γ dγ dβ

9 Y. Wei and G. Wang / New relationship between the divergent beam projection and the Radon transform 393 β +π π cos(γ γ 4π P (β,γ β π L sin (γ γ dγ dβ (4 = π π 4π P (β,γ cos(γ γ L sin (γ γ dγdβ π If the object is inside the circular locus, the inner integral can be expressed as π/ π/ : f(x, y = π π/ 4π P (β,γ cos(γ γ π/ L sin (γ γ dγdβ (4 Comparing with the standard form of the full scan formula in the text book [9,] π π/ f(x, y = R cos γ 4π P (β,γ π/ L sin dγdβ, (5 (γ γ it is found that a main difference lies in the power of L and the weight factor. Based on the observation in [], the decrement in the power ofl could result in a more uniform noise distribution in a reconstructed image. There are two alternative ways to obtain Formula (4. The first one is to put a weight function w(β,γ = L cos(γ γ R cos γ into the general full scan formula [,3] π π/ f(x, y = R cos γ π w(β,γp (β,γ π/ L sin dγdβ. (7 (γ γ As shown in Fig. 6, the weight function is normalized for opposite rays w(β,γ +w(β + π +γ, γ =. (8 As far as we know, it is the first time for this weight function Eq. (6 is ever suggested. Another way is to start with the Radon inversion formula for divergent beam data, which is Eq. ( in [4]: f(x, y = 4π π π/ π/ (6 ( L sin(γ γ γ P (β,γdγdβ. (9 β Noting that the integral of the second term in the Radon inversion formula for divergent beams is zero, i.e., π π/ ( π/ L sin(γ γ β P (β,γ dγdβ =, ( since ( ( L sin(γ γ β P (β,γ = β=β,γ=γ L sin(γ γ β P (β,γ, β=β +π+γ,γ= γ

10 394 Y. Wei and G. Wang / New relationship between the divergent beam projection and the Radon transform Fig. 6. The normality of the weight function w(β,γ. Fig. 7. The explanation for Eq. (. (a indicates P (β,γ = P (β,γ (b indicates β β=β,γ=γ, β β=β +π+γ,γ= γ, = L sin(γ γ L sin(γ γ, (c indicates that the positive value and negative value appear symmetrically. one can obtain Formula (4 from the first term in Eq. (9 via integration by parts. An intuitive explanation of Eq. ( is shown in Fig. 7. Finally, we mention a fan-beam reconstruction formula for a smooth closed convex locus (Fig. 8. Based on the general relationship between dθ and dβ in [] dθ = R(βcosγ R (βsinγ dβ, ( L we have f(x, y = π π R(βcosγ R (βsinγ LD π π P (β,γ cos(γ γ sin dγdβ, ( (γ γ

11 Y. Wei and G. Wang / New relationship between the divergent beam projection and the Radon transform 395 Fig. 8. A smooth closed convex locus described by R(β. where R(β describes the scanning locus, and D the length of the chord through the source S and the point T. 4.. A new form for the full-scan Hilbert image By Noo s work [5], another equivalent form of the D Radon inversion formula is f(x, y = π Hb(x, y = π H π p θ (t t =x cos θ+y sin θ dθ (3 where H is the Hilbert transform along the y direction, and b(x, y is a Hilbert image. Applying Formula (8 in the local system x y in Fig. 4, we have [( p θ (t = γ =γ R cos γ γ P (β,γ + γ =γ γ +π P (β,γ ( (4 γ =γ γ P (β,γ + ] γ =γ γ +π P (β,γ Inserting Eqs (3 and (4 into Eq. (3, we have b(x, y= β γ =γ β γ P (β,γ + β +π γ =γ γ P (β,γ + = β π γ γ =γ +π P (β,γ γ γ=γ P (β,γ+ γ γ=γ +π P (β,γ γ =γ γ +π P (β,γ dβ L dβ (5 L sgn (sin(β + γ L dβ

12 396 Y. Wei and G. Wang / New relationship between the divergent beam projection and the Radon transform For an object inside a scanning circle, we have b(x, y = π ( γ γ=γ P (β,γ sgn (sin(β + γ L dβ (5 For comparison, we rewrite Noo s expression Eq. (37 in [5] b(x, y= π R L γ (P (β,γcosγ γ=γ sgn (sin(β + γ dβ + P (β,γ P (β,γ, L L where the meanings of the parameters L,L,β,β,γ,andγ are illustrated in Fig. 5. Since Eq. (5 is short, it is easier for programming. There two alternative ways to obtain Eq. (5. In fact, if one put the new weight function (6 in the general form for b(x, y [5] (6 b(x, y= π R L γ (w(β,γp(β,γcosγ γ=γ sgn (sin(β + γ dβ + L w(β,γ P (β,γ L w(β,γ P (β,γ, one can arrive at Formula (5. Alternatively, we have b(x, y= = = = π π π π p θ (t t =x cos θ+y sin θ dθ = p θ (t t =x cos θ+y sin θ ( R cos γ β + p(β,γ γ=γ sgn (sin(θdθ γ R cos γ γ p(β,γ γ=γ sgn (sin(θdθ L γ p(β,γ γ=γ sgn (sin(β + γ dβ. π Also, we have π R cos γ β p(β,γ γ=γ sgn (sin(θdθ =, since ( sgn (sin(θ at S = R cos γ β p(β,γ γ=γ ( R cos γ β p(β,γ γ=γ Please see Fig. 4 and Fig. 7a for more explanations. sgn (sin(θ at S. sgn (sin(θdθ (7 (8

13 Y. Wei and G. Wang / New relationship between the divergent beam projection and the Radon transform 397 Fig. 9. Cone beam reconstruction for circular locus. Furthermore, for a closed smooth convex locus we have f(x, y= π H π sgn (sin(β + γ dβ, R(βcosγ R (βsinγ DL where R(β describes the scanning locus (Fig D Radon formula for cone beam reconstruction with a circular locus The 3D Radon inversion formula can be expressed as [8] γ γ=γ P (β,γ+ γ γ=γ +π P (β,γ f( r = Ω3 d 8π dt R m(t t= r m dω, (3 where m =(sinϑcos ϕ, sin ϑ sin ϕ, cos ϑ runs over the 3D unit sphere Ω 3. For simplicity, we first considera circular scanninglocus (Fig.9. An existingpractical reconstruction formula for circular scanning is the well-known Feldkamp formula [6]. Let us reconstruct a point T (x, y, on the middle plane z =and inside the locus. We denote the plane through T (x, y, and orthogonal to m as Π( m. The plane Π( m intersects the locus at two points S and S with D = S S. We set up a new local system x y z with S as the origin, S S as the z axis, the x axis opposite to the vector m, andthey axis inside the plane Π( m. Applying Formula ( in the local system, we have R m (t t= r. m = D π cos θ sin θ dθ, φ cos θ φ= π P S (θ, φ sin θ dθ D π φ (9 φ= π P S (θ, φ (3

14 398 Y. Wei and G. Wang / New relationship between the divergent beam projection and the Radon transform Fig.. The points can be reconstructed exactly or approximately for circular locus. where P S (θ, φ =P S ( n with a unit vector n =(sinθ cos φ, sin θ sin φ, cos θ defined in the local coordinate system x y z. Putting Eq. (3 in Formula (3, any point T (x, y, inside the locus can be reconstructed. This is a direct implementation of the Radon formula, while in the Grangeat scheme the first derivative is calculated first. We cannot exactly reconstruct the object function at a point T (x, y, outside the circular locus or a point T (x, y, z with z since in either case the second derivative of the Radon transform cannot be calculated for all the Radon planes, see Fig.. However, if for a point missing data constitutes only a small fraction of the needed information, and we can use some interpolation methods to compensate for the missing data, and reconstruct the point approximately. Thus, an approximate algorithm can be designed for cone beam reconstruction with a circular locus. For a closed convex smooth locus in a plane, the above approach is also valid for construction of an approximate cone-beam reconstruction formula. For a saddle locus [7,8], more points of the object can be exactly reconstructed. For a general locus, a new condition for a point to be exactly reconstructable using this approach is that every plane through the point must have two intersection points with the locus, even if the size of the object is larger than the locus. Certainly, if some Radon planes do not contain two source points but the second derivative of the Radon transform can be obtained from other method such as the Grangeat method, the exact reconstruction can be achievable as well. In conclusion, we have presented some new results on relationships between a divergent beam projection and the Radon transform. Based on these results, various Radon inversion schemes have been discussed that directly use divergent beam data. Since our new reconstruction formulas are based on divergent projections at two points instead of one point, it would seem reasonable to apply this new approach for multiple source CT systems [9]. Interested readers may read [3 34] and the references therein for more information.

15 Y. Wei and G. Wang / New relationship between the divergent beam projection and the Radon transform 399 Appendix A. The deduction of the Grangeat s relation Consider the weighted integral R f(x, y, zδ (xdxdydz in two coordinate systems. 3 In the xyz coordinate system, we have f(x, y, zδ (xdxdydz = f(x, y, zdydzδ (xdx = R(xδ (xdx = R (x x=. R 3 R 3 R In the spherical coordinate system with r [, ],θ [, π],φ [,π], wehave f(x, y, zδ (xdxdydz = R 3 = = = = π π π π π π π π = = π π π π = φ φ= π f(r, θ, φδ (r sin θ cos φr sin θ drdθdφ f(r, θ, φdrδ (sin θ cos φ sin θ dθdφ P (θ, φδ (sin θ cos φ sin θ dθdφ P (θ, φδ (cos φ sin θ dθdφ π P (θ, φδ ( sin ( P (θ, φδ φ π P (θ, φ sin θ dθ. ( φ π sin θ dθdφ sin θ dθdφ Thus, we arrive at the classic Grangeat relation R (x x= = π P (θ, φ φ sin θ dθ. φ= π In the above, we have used δ (ax =δ (x/(a a with a such as δ ( x = δ (x Appendix B. The deduction From Eq. (9 to Eq. ( We have f(x, y, zδ (xzdxdydz R 3 = f(r, θ, φδ (r sin θ cos φr cos θr sin θ drdφdθ R 3

16 4 Y. Wei and G. Wang / New relationship between the divergent beam projection and the Radon transform = P o (θ, φδ (sin θ cos φ sin θ cos θdφdθ Ω 3 π π = P o (θ, φδ (cos φdφ cos θ sin θ dθ π π ( ( = P o (θ, φδ sin φ π dφ cos θ sin θ dθ π π ( = P o (θ, φδ φ π dφ cos θ sin θ dθ π cos θ = φ φ= π P o(θ, φ sin θ dθ wherewehaveusedδ (ax =δ (x/ a 3 with a such as δ ( x =δ (x. Then, one can complete the rest part of the deduction without difficulty. References [] F. Natterer, Recent developments in x-ray tomography, Lectures in Appl Math 3 (994, [] R. Clack and M. Defrise, Cone-beam reconstruction by the use of Radon-transform intermediate functions, J Opt Soc Am A (994, [3] P. Grangeat, Mathematical framework of cone beam 3d reconstruction via the st derivative of the Radon-transform, Lecture notes in Mathematics 497 (99, [4] B. Smith, Image reconstruction from cone-beam projections, IEEE Trans Med Imaging 4 (985, 4 5. [5] F. Noo et al., Image reconstruction from fan-beam projections on less than a short scan, Phys Med Biol 47 (, [6] C. Hamaker et al., The divergent beam x-ray transform, Rocky Mountain J Math (98, [7] Y.C. Wei, H. Yu and G. Wang, Integral Invariants for Computed Tomography, IEEE Signal Process Lett 3 (6, [8] M.J. Lighthill, Introduction to Fourier Analysis and Generalized Functions, (Cambridge U.K.: Cambridge Univ. Press, 958. [9] I.M. Gel Fand and G.E. Shilov, Generalized Functions: Properties and Operations, vol New York: Academic, 964. [] I. Richards and H. Youn, Theory of Distributions: A Non-Technical Introduction, Cambridge U.K.: Cambridge Univ. Press, 99. [] Y.C. Wei et al., Scheme of computed tomography, J X-ray Sci Tech 5 (7, [] G. Chen, A new data consistency condition for fan beam projection, Med Phys 3 (5, [3] G. Chen, A new framework of image reconstruction from fan beam projections, Med Phys 3 (3, 5 6. [4] H.K. Tuy, An inversion formula for cone-beam reconstruction, SIAM J Appl Math 43 (983, [5] G. Chen, An alternative derivation of Katsevich s cone-beam reconstruction formula, Med Phys 3 (3, [6] A. Katsevich, Improved exact FBP algorithm for spiral CT, Adv Appl Math 3 (4, [7] Y.B. Ye and G. Wang, Filtered backprojection formula for exact image reconstruction from cone-beam data along a general scanning curve, Med Phys 3 (5, [8] J. Radon, Uber die Bestimmung von Funktionnen durch ihre Integralwerte l angs gewisser Mannigfaltigkeiten, Ber Verh Sachs Akad Wiss Leipzig- Math-Natur 69 (97, [9] A.C. Kak et al., Principles of Computerized Tomography Imaging (Philadelphia:SIAM,. [] Y. Wei, J. Hsieh and G. Wang, General formula for fan-beam computed tomography, Phys Rev Lett 95 (5, 58. [] B.K.P. Horn, Fan-beam reconstruction methods, Proceedings of the IEEE 67 (979, [] X.C. Pan and L.F. Yu, Image reconstruction with shift-variant filtration and its implication for noise and resolution properties in fan-beam computed tomography, Med Phys 3 (3, [3] D.L. Parker, Optimal short scan convolution reconstruction for fan beam CT, Med Phys 9 (98, [4] G. Herman and Naparstek, A fast image reconstruction based on a Radon inversion formula appropriate for rapidly collected data, SIAM J. Applied Mathematics 33 (977,

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