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Cone-beam image reconstruction using spherical harmonics

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2001 Phys. Med. Biol. 46 N127 (http://iopscience.iop.org/0031-9155/46/6/401) View the table of contents for this issue, or go to the journal homepage for more

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INSTITUTE OF PHYSICS PUBLISHING

PHYSICS IN MEDICINE AND BIOLOGY

Phys. Med. Biol. 46 (2001) N127–N138

www.iop.org/Journals/pb

PII: S0031-9155(01)16332-3

NOTE

Cone-beam image reconstruction using spherical harmonics Katsuyuki Taguchi1 , Gengsheng L Zeng2 and Grant T Gullberg2 1 CT and Nuclear Medicine Systems Development Department, Research and Development Center, Medical Systems Company, Toshiba Corporation, 1385 Shimoishigami, Otawara, Tochigi 324-8550, Japan 2 Medical Imaging Research Laboratory, Center for Advanced Medical Technologies, Department of Radiology, University of Utah, 729 Arapeen Drive, Salt Lake City, UT 84108-1218, USA

E-mail: [email protected]

Received 14 August 2000, in final form 28 March 2001 Abstract Image reconstruction from cone-beam projections is required for both x-ray computed tomography (CT) and single photon emission computed tomography (SPECT). Grangeat’s algorithm accurately performs cone-beam reconstruction provided that Tuy’s data sufficiency condition is satisfied and projections are complete. The algorithm consists of three stages: (a) Forming weighted plane integrals by calculating the line integrals on the cone-beam detector, and obtaining the first derivative of the plane integrals (3D Radon transform) by taking the derivative of the weighted plane integrals. (b) Rebinning the data and calculating the second derivative with respect to the normal to the plane. (c) Reconstructing the image using the 3D Radon backprojection. A new method for implementing the first stage of Grangeat’s algorithm was developed using spherical harmonics. The method assumes that the detector is large enough to image the whole object without truncation. Computer simulations show that if the trajectory of the cone vertex satisfies Tuy’s data sufficiency condition, the proposed algorithm provides an exact reconstruction.

1. Introduction Recently, multislice x-ray computed tomography (CT) systems equipped with an area detector with a small cone-angle were developed and introduced to the market (Taguchi and Aradate 1998). However, for the last few years image reconstruction from cone-beam projections with large cone-angles has been pursued in single photon emission computed tomography (SPECT) imaging (Jaszczak et al 1986, Gullberg et al 1991, Zeng et al 1997). In the future, as area detectors for x-ray CT systems become larger (Saito et al 2000, 2001), there will be the need to develop accurate cone-beam reconstruction algorithms for x-ray CT imaging with large cone-angles. It is the purpose of this paper to show that spherical harmonic expansion can be 0031-9155/01/060127+12$30.00

© 2001 IOP Publishing Ltd

Printed in the UK

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used in a new implementation of Grangeat’s algorithm to provide accurate reconstruction of cone-beam data that have been acquired using large cone-angles. Cone-beam problems can be categorized into three types: (a) Short-object with oversized (large) detector. (b) Short-object with moderately sized detector. (c) Long-object with moderately sized detector. This paper solves the short-object with oversized-detector problem, reconstructing a whole object bounded in the axial (longitudinal) direction with an oversized (large) area detector, which is capable of obtaining the entire projection data for the object. In order to achieve a good image quality using cone-beam CT with a large cone-angle, exact or quasi-exact image reconstruction algorithms are required. For the short-object with oversized-detector problem, exact image reconstruction methods from cone-beam projections have been developed. Grangeat’s (1991) algorithm computes the radial derivative of the conebeam planar integral of the object from a pair of closely spaced parallel lines on the detector plane, then rebins the data and performs the 3D Radon backprojection to reconstruct the image. Kudo and Saito (1994, 1995) and Defrise and Clack (1994) used shift-variant filtering with cone-beam backprojection instead of rebinning with the 3D Radon backprojection. Recently, You et al (1999) developed a method to obtain the 2D Radon transform (i.e. parallel projection data) from variable-focal-length fan-beam projection data using a circular harmonic expansion. Their results show that implementation of harmonic expansion can be a practical and useful solution in some cases. Also, image reconstruction methods using spherical harmonics have been proposed for the Compton camera (Basko et al 1998a, 1999) and for cone-beam data (Basko et al 1998b, Basko 1998). This paper further develops the theory for a new implementation of Grangeat’s algorithm for cone-beam reconstruction using the harmonic expansion technique and provides computer simulations to demonstrate that an accurate implementation of the proposed technique is feasible.

2. Spherical harmonics for cone-beam reconstruction In this section we outline a new implementation technique for the first stage of Grangeat’s algorithm, obtaining the radial derivative of the plane integral from cone-beam projection data. 2.1. Truncated spherical harmonics Let G(θ, ϕ) be an arbitrary function which can be described by the spherical harmonic expansion using spherical coordinates (θ, ϕ) (figure 1) G(θ, ϕ) = lim

Nn →∞

Nn l

glm Ylm (θ, ϕ)

(1)

l=0 m=−l

or Nn l 1 G(θ, ϕ) = √ glm lm (cos θ)eimϕ lim 2π Nn →∞ l=0 m=−l

(2)


z

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* n

θ t

y

ϕ

* ψ

x

O

The origin of the coordinate Figure 1. Local spherical coordinates for a vertex position at ψ. The centre of the support of the object is system is shifted to the end of the arrow of vector ψ. located at O.

where Ylm (θ, ϕ) is the 3D spherical harmonic function and the function lm (t) is related to the associated Legendre function Plm (t) as follows:   2l + 1 (l − m)! m    P (t) for 0 m  2 (l + m)! l (3) lm (t) =   (l + m)! 2l + 1  −m m  P (t) for m < 0.  (−1) 2 (l − m)! l Interchanging the order of summations in equation (2), we get |m|+N Nm l −1 1 glm lm (cos θ) eimϕ . G(θ, ϕ) = lim lim √ Nm →∞ 2π Nl →∞ l=|m| m=−Nm

(4)

If G(θ, ϕ) is a band-limited function, it can be expressed by a truncated expansion where Nm and Nl are finite. Similarly, once the function G(θ, ϕ) has been sampled with a finite number of data points (corresponding to Nm and Nl for m index in ϕ and l index in θ respectively), the information in those data sets is band-limited according to Nyquist sampling theory. Thus, all of the coefficients of l and m higher than Nl and Nm must be zero. Therefore, the band-limited function can be expressed by a truncated expansion Nm Nm l −1 1 |m|+N glm lm (cos θ) eimϕ = gm (cos θ)eimϕ (5) G(θ, ϕ) = √ 2π l=|m| m=−Nm m=−Nm where 1 gm (cos θ ) = √ 2π

|m|+N l −1

glm lm (cos θ).

(6)

l=|m|

We assume that the function is sampled at equiangular grid points k − 0.75 n − 0.75 π, π , k = 1, . . . , Nl ; n = −Nm + 1, . . . , Nm . Nl Nm

(7)

Here ‘0.75’ (known as a quarter-offset detector in the third generation CT) offsets the sampling positions of all data, otherwise π apart data (in ϕ) sample exactly the same positions and are redundant.

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An efficient method for converting between a function G(θ, ϕ) and its harmonic transform glm was developed by Basko et al (1998a) and is summarized by {G(θ, ϕ)}

IDFT

IDLT

DFT

DLT

−→ −→ {gm (cos θ )} {gml } ←− ←−

(8)

where DFT and DLT refer to the discrete Fourier transform and the discrete Legendre transform, respectively, and IDFT and IDLT refer to their inverse transforms, respectively. 2.2. Description of the cone-beam projection and the first derivative of the 3D Radon transform using spherical harmonics

Using spherical harmonics, the cone-beam projection g ψ (θ, ϕ) and the first derivative of the 3D Radon transform p˜ ψ (θ, ϕ) for a fixed vertex position at ψ are described by Nm l −1 (ψ) 1 |m|+N ψ glm lm (cos θ) eimϕ (9) g (θ, ϕ) = √ 2π m=−Nm l=|m| Nm l −1 (ψ) 1 |m|+N ψ p˜ lm lm (cos θ) eimϕ (10) p˜ (θ, ϕ) = √ 2π l=|m| m=−Nm (ψ)

(ψ)

where glm and p˜ lm are coefficients of spherical harmonics respectively. Note that equation (10) is the derivative of the 3D Radon transform of the object on planes containing the cone vertex, as available from the cone-beam projection using Grangeat’s formula. (ψ) (ψ) As established in the appendix, the relationship between glm and p˜ lm is (ψ)

(ψ)

(11) p˜ lm = −2πlPl−1 (0)glm where Pl (t) is the Legendre polynomial of order l. A smooth window function Wlm can be applied in order to reduce the ringing effect of the truncation, both in the Fourier expansion and in the Legendre expansion: (ψ)

(ψ)

p˜ lm = −2πlPl−1 (0)Wlm glm .

(12)

3. Computer simulations Computer simulations were performed to demonstrate that an accurate implementation of the described method is feasible for addressing the short-object with oversized-detector problem. 3.1. Algorithm The implemented algorithm consists of three stages. 3.1.1. Stage 1: converting the cone-beam projection to the first derivative of the plane integral. The first stage of the algorithm is to obtain the first derivative of the plane integral (3D Radon transform) from cone-beam projections using the harmonic expansion for each cone vertex position. This can be described in three steps: (ψ)

(1a) Calculating the coefficients glm in spherical harmonics for cone-beam projections. (ψ)

(ψ)

(1b) Converting glm into p˜ lm by equation (12). (1c) Obtaining the first derivative of the plane integral p˜ ψ (θ, ϕ) by evaluating equation (10). An efficient method described by equation (8) is used in steps (1a) and (1c).


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Helical orbit of the cone vertex r h R

the support of the object

L

Figure 2. Geometry and parameters for the helical orbit in computer simulations.

3.1.2. Stage 2: rebinning and the second derivative of the plane integral. The second stage of the algorithm rebins the first derivative of the plane integral (in the local spherical coordinates) at each cone vertex into global spherical coordinates. We use a rebinning method (Weng et al 1993) that performs interpolation in the Radon space. The number of samples after this process are Nθ0 in θ0 over π, Nϕ0 in ϕ0 over 2π, and Nt in the radial range [0, r 2 + h2 /4] (figure 2), where (θ0 , ϕ0 , t) represents spherical coordinates in the Radon space. No additional process is employed to fill missing data in the shadow-zone of 3D Radon space. From this, the second derivative of the plane integral is obtained from the rebinned first derivative using a finite difference between closely spaced data points. 3.1.3. Stage 3: backprojection. The third stage of the algorithm uses the 3D Radon backprojection to obtain the reconstructed image. 3.2. Methods 3.2.1. Scanning orbits and geometry. Two vertex paths, a one-circle and a two-turn helix, were chosen for the scanning orbits. The scanning conditions of the helical orbit satisfied the data sufficiency conditions (Tuy 1983). The parameters used in the simulation are illustrated in figure 2. The radius R of the circular and the helical orbit was 60 units. Both the size of the support (a cylinder with radius r and height h) and the pitch of each helix, L/2, depended on which phantom was used, and these are defined in the next section. The number of sampling points at each vertex position was 512 in θ over π and 1024 in ϕ over 2π , respectively (i.e. a sampling pitch of 0.0061 rad), and the number of vertex positions was 256 over 2π . The data were rebinned into a 128 × 128 × 128 (θ0 , ϕ0 , t) matrix and the image matrix size was 1283 . The angular sampling interval at each cone vertex is similar to that used in SPECT. (The length of the arc corresponding to angular sampling interval at the rotation axis, that is, at the centre of the object is a 0.368 detector bin size.) 3.2.2. Phantoms. Two mathematical phantoms, the Defrise phantom and the Shepp–Logan phantom, were chosen for the evaluation. The Defrise phantom consisted of seven stacked discs, each with a radius of 10, a height of 2 and a value of 1.0, which were separated axially

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by dz = 6. The cylindrical support was r = 10 and h = 38. The parameter L in the helical orbit was 52. The Shepp–Logan phantom consisted of 12 ellipsoids. The semiaxes of the largest ellipsoid were 21.0, 15.8, 20.5, along the x, y and z axes, respectively. The support parameters were r = 21, h = 41 and L = 62. 3.2.3. Window function. follows:

The continuous window function in equation (12) was chosen as

Wlm = Wl (Nl ) × Wm (Nm ) where

(13)

1.0 (for l − |m| Cl Nl ) 3x 2 − 2x 3 (for l − |m| > Cl Nl )

1.0 (for |m| Cm Nm ) Wm = 2 3 3x − 2x (for |m| > Cm Nm )  1 − (l − |m|)/N l   (for l) A x = 1 − |m|/N m   (for m) A (for Cl , Cm = 1) 1 − Cl or 1 − Cm A= 1 (for Cl , Cm = 1).

Wl =

(14) (15)

(16)

(17)

We chose Cl = Cm = 1.0 for the Defrise phantom and Cl = Cm = 0.0 for the Shepp–Logan phantom. The window function is one of the key factors affecting the image quality (e.g. Gibbs artefacts). It is possible that other window functions may provide a similar performance. 3.3. Results 3.3.1. Defrise phantom. Figure 3 shows (a) the central coronal slice of two reconstructions of the Defrise phantom and (b) the central vertical profile of each slice. The image for the circular orbit (figure 3(a), left) shows strong oblique shadows and blurring in the axial direction. The profile is also degraded (figure 3(b), dotted curve). These artefacts are similar to the wellknown artefacts that are present in the Feldkamp reconstruction and are caused by ‘missing data’ in 3D Radon space. The helical orbit, which enables us to obtain sufficient 3D Radon data, essentially improves both the image quality (figure 3(a), right) and the profile (figure 3(b), thick full curve) by eliminating the artefacts. The results of the helical scanning orbit verify that the artefacts associated with the circular orbit are caused by insufficient cone-beam data. 3.3.2. Shepp–Logan phantom. Figure 4 shows (a) an off-centre transaxial slice in the reconstructed Shepp–Logan phantom images and (b) the central vertical profile of each slice. The circular scanning resulted in an inaccurate profile (figure 4(b), dotted curve) showing the cone-beam effect of insufficient data, while the helical scanning gave a more accurate result (figure 4(b), full curve). The truncation of the expansion may cause the wide-band circular ringing artefacts in the circular scan and the smaller ripples in the helical scan. A window function can smooth out the truncation effect and reduce the artefacts, but the spatial resolution is also reduced to some degree. Optimizing the window function for better image quality requires further investigation.


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(a) pixel value

1.0 0.8 Circle orbit

0.6

Helical orbit Accurate line

0.4 0.2 0.0 -18.0 -12.0 -6.0 0.0

6.0 12.0 18.0

z coordinate

(b) Figure 3. (a) Reconstructed coronal slice image (x = 0) of the Defrise phantom with the circular orbit (left) and the helical orbit (right). The grey scale range is [−0.5, 0.5]. (b) Central vertical profiles of the images with the circular orbit (dotted curve) and the helical orbit (thick full curve). The thin full curve is the exact profile. Note that the image by the circular orbit shows strong oblique ‘cone-beam’ artefacts which are similar to those obtained by the so-called Feldkamp algorithm. The image by the helical orbit does not show any artefact, which indicates that the ‘cone-beam problem’ has been solved by the proposed approach.

4. Discussion The new implementation of Grangeat’s algorithm using spherical harmonics can solve the short-object cone-beam reconstruction problem with an oversized detector if the orbit satisfies Tuy’s data sufficiency condition. Both the proposed and the classical implementation of Grangeat’s algorithm can, in theory, provide exact reconstruction if an infinite number of data points and an infinite number of terms for the expansion are used. When implementing the proposed method with a finite number of data samples and a finite number of terms in the spherical harmonic expansion, the accuracy depends upon the number of each condition. In order to improve the image quality using spherical harmonics further work must be done on optimizing the window function and selecting optimum sampling strategies. The proposed method allows us to obtain the first derivative data without interpolating the cone-beam projection data. Therefore, no interpolation errors are introduced when converting the cone-beam data into the first derivative of the Radon data. However, the number of terms in the expansion depends upon the sampling bandwidth of the cone-beam projections. The accuracy of performing the second derivative and performing the Radon backprojection to obtain the reconstructed image depends on the number of cone vertex positions. Using more positions decreases the intervals between the rebinned data points and increases the accuracy of the second derivative.

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Figure 4. (a) Reconstructed axial slice image (z = 8.128) of the Shepp–Logan phantom. The grey scale range is [0.98, 1.02] for the circular orbit (left) and [1.0, 1.04] for the helical orbit (right). (b) Central vertical profiles of the images with the circular orbit (dotted curve) and the helical orbit (thick full curve). The thin full curve is the exact profile.

Another method that can be used to obtain accurate first derivative data is the linogram technique (Axelsson and Danielsson 1994), which has been used in other implementations of the cone-beam reconstruction problem (Kudo and Saito 1994, Defrise and Clack 1994). A comparison of the accuracy of the reconstructed images obtained using the proposed method with results obtained using Grangeat’s original approach, should be of interest for future study. Let us briefly discuss it from a theoretical point of view. The accuracy may depend on a data sampling scheme that corresponds to the number of samples and to various detector configurations. Typically there are two basic detector designs for a converging beam. One provides equiangular sampling and the other equispatial sampling, commonly used with flat-panel, cylindrical (Saito et al 2000, 2001) or spherical detectors. The equiangular sampling is advantageous for the method using spherical harmonics because the calculation of the harmonic expansion coefficients requires evaluation of integrals with respect to angular variables θ and ϕ. In such data sampling, the proposed method is more accurate without interpolation, while Grangeat’s method introduces more numerical errors. On the other hand, if equispatial data are used, Grangeat’s method can be more accurate with a linogram approach; the proposed method requires transforming equispatially sampled data to equiangular sampling introducing a Jacobian factor. Since this is a feasibility study the code was not optimized for optimum processing time. Without an efficient method for the discrete Legendre transform (LT), the first step at each cone vertex required 90 s (at an actual wall-clock time) using a Pentium II with 450 MHz and 128 MB RAM. Short processing time is required for practical implementation. We believe that the processing time can be reduced by enacting the following implementations:


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(a) Apply a fast LT of a linear time fast LT in a forward and an inverse discrete LT in equation (8). (b) Use a parallel computing configuration for the fast LT or the linear time fast LT (Corrians 1996, Lucet 1997, Drake et al 1996). (c) Calculate only some angles that contain data (i.e. do not calculate angles whose projection data are zero). Projection data at most angles are zero because the angular range corresponding to the object support is limited. Accessing the hard drive frequently significantly slows the current implementation. Using a PC or workstation with larger memory size (up to 1 GB is now available even in a PC) can also reduce processing time significantly. In order to apply this approach to medical scanners work will need to be done on the truncation problems, such as the short-object problem with a medium sized detector and the long-object problem. For those problems, several exact or quasi-exact algorithms have been proposed (Kudo et al 1998, 1999, 2000, Noo et al 1998a, b, Tam et al 1998, Schaller et al 1999, 2000, Defrise et al 1999, 2000). Using a similar approach in these studies, one can solve both the short-object problem by calculating the first derivative of the plane integral using the spherical harmonics for plural partial planes (Taguchi et al 2000b, 2001) and the long-object problem by using the spherical harmonics, as done in this paper, and by applying the PHI method for backprojection as described by Schaller et al (1999, 2000). In summary, a new implementation of Grangeat’s algorithm for cone-beam image reconstruction using spherical harmonics has been developed. Computer simulations verified the approach by showing improved image quality and accurate profiles. However, the artefact caused by the ringing effect of the truncation remains a problem. Further work will include: optimizing the window functions and sampling parameters; improving the efficiency of calculations; replacing rebinning and 3D Radon backprojection with cone-beam filteredbackprojection; applying the spherical harmonics approach to the cases with truncated data (i.e. short-object problem with moderately sized detector and long-object problem).

Acknowledgments We thank Roman Basko for his original theoretical work, and Dr Vladimir Panin and Dr Chuanyong Bai from the University of Utah (Salt Lake City, UT, USA) for helpful discussions during the study. We would also like to thank one reviewer for pointing out that equations (9)–(11) are a direct result of the Funk–Hecke theorem for special harmonics. The derivation of equation (11) using this result has been included as the appendix. We also thank Sean Webb from the University of Utah for proofreading the manuscript. This work was partially supported by NIH Grants R29 HL51462 and R01 HL39792, and by Medical Systems Company, Toshiba Corporation (Tokyo, Japan).

Appendix Here we derive equation (11), the relationship between the harmonic coefficients for the conebeam projection and those for the first derivative of the plane integral, using Grangeat’s formula and the Funk–Hecke theorem. Another derivation is available in Taguchi et al (2001). First, we will rederive Grangeat’s formula (1991) (equation (A5)). The cone-beam projection data from the cone vertex point ψ can be described using a unit

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vector n = (θ, ϕ):

n) = g ψ (

∞

−∞

f (ψ + r n) dr.

(A1)

Also, the plane integral (3D Radon transform) is described using a unit vector k = (θk , ϕk ) t)|t=ψ· = p(k, f (r )δ(k · r − t) dr |t=ψ ·k . (A2) pψ (k) k = R3

Thus, the first derivative of the plane integral is d ψ f (r )δ (k · r − t) dr |t=ψ· f (r )δ (s) dr |s=k( p˜ (k) = p(k, t)|t=ψ ·k = r −ψ) k = 3 3 dt R R = f (ψ + r )δ (s) d(ψ + r )|s=k·r = f (ψ + r)δ (k · r) dr R3

R3

(∵ s = k · r − t, r = ψ + r , dψ = 0).

(A3)

Changing the variables and using the properties of the delta function, we have 1 n = 2 δ (k · n)r 2 dr d n = δ (k · n) dr d n. δ (k · r) dr = δ (k · r n)r 2 dr d r 1 ∵ r = r n, dr = r 2 dr sin θ dθ dϕ = r 2 dr d n, δ (ax) = 2 δ (x). a From equations (A1), (A3) and (A4) we obtain Grangeat’s formula: ∞ = f (ψ + r n)δ (k · n) dr d n= g ψ ( n)δ (k · n) d n p˜ ψ (k) s2

(A4)

(A5)

s2

0

where S 2 is the unit sphere. Similar to equation (10), the spherical harmonics for cone-beam projections g ψ (θ, ϕ) can be described as follows: l ∞ (ψ) |m| glm Alm Pl (cos θ)eimϕ . (A6) g ψ (θ, ϕ) = l=0 m=−l

Substituting equation (A6) into equation (A5): l ∞ (ψ) |m| ψ ψ imϕ δ (k · n) d g ( n)δ (k · n) d n= glm Alm Pl (cos θ)e n p˜ (k) = s2

=

∞

l

l=0 m=−l

(ψ)

glm

s2

s2

l=0 m=−l

δ (k · n)Ylm ( n) d n

(A7)

where |m|

Ylm ( n) = Ylm (θ, ϕ) = Alm Pl (cos θ)eimϕ . The Funk–Hecke theorem states (Natterer 1986) that for a function h(t), t ∈ [0, 1]    h(k · n)Ylm ( n) d n = Clm Ylm (k)  s2 1   h(t)Pl (t) dt.  Clm = 2π −1

(A8) (A9) (A10)

1 Let h(t) = δ (t), then Clm = 2π −1 δ (t)Pl (t) dt. Using the derivative of the Legendre polynomials defined by 1 d Pl (x) = 2 (lxPl (x) − lP1−l (x)) (A11) dx x −1


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we have Clm = −2πlPl−1 (0).

(A12)

Substituting equations (A9), (A10) and (A12) into equation (A7), we get p˜ ψ (k) =

l ∞ l=0 m=−l

=

l ∞ l=0 m=−l

(ψ)

|m|

[−2πlPl−1 (0)glm ]Alm Pl (cos θ)eimϕ (ψ)

|m|

p˜ lm Alm Pl (cos θ)eimϕ .

(A13)

Thus we get equation (11): (ψ)

(ψ)

p˜ lm = −2πlPl−1 (0)glm . References Arfken G B and Weber H J 1995 Mathematical Methods for Physicists 4th edn (San Diego: Academic) Axelsson C and Danielsson P-E 1994 Three-dimensional reconstruction from cone-beam data in O(N 3 log N ) time Phys. Med. Biol. 39 477–91 Basko R 1998 Personal communication Basko R, Zeng G L and Gullberg G T 1998a Application of spherical harmonics to image reconstruction for the Compton camera Phys. Med. Biol. 43 887–94 ——1998b Application of spherical harmonics to cone-beam image reconstruction Conf. Rec. 1998 IEEE Nuclear Science Symp. and Medical Imaging Conf. (Toronto, ON, Canada) (New York: IEEE) pp 1649–50 ——1999 Image reconstruction for Compton camera including spherical harmonics US Patent 5861 627 (19 January 1999) Cifune N G 1999 Multislice CT: the next step in medical imaging Med. Imaging October 44–9 Clack R and Defrise M 1994 Overview of reconstruction algorithms for exact cone-beam tomography Proc. SPIE 2299 230–41 Clack R, Zeng G L, Weng Y, Christian P E and Gullberg G T 1992 Cone beam single photon emission computed tomography using two orbits Information Processing in Medical Imaging: 12th IPMI Int. Conf. ed A Colchester and D Hawkes (New York: Springer) pp 45–54 Corrians L 1996 Fast Legendre–Fenchel transform and applications to Hamilton–Jacobi equations and conservation laws SIAM J. Numer. Anal. 33 1534–58 Defrise M and Clack R 1994 A cone-beam reconstruction algorithm using shift variant filtering and cone-beam backprojection IEEE Trans. Med. Imaging 13 186–95 Defrise M, Noo F and Kudo H 1999 Quasi-exact region-of-interest reconstruction from helical cone-beam data Conf. Rec. 1999 IEEE Nuclear Science Symp. and Medical Imaging Conf. (Seattle, WA, USA) (New York: IEEE) CD-ROM ——2000 A solution to the long-object problem in helical cone-beam tomography Phys. Med. Biol. 45 623–43 Drake J B et al 1996 Parallel community climate model: description and user’s guide webpage http://www.epm.ornl.gov/chammp/pccm2.1/doc/pccm2.html Feldkamp L A, Davis L C and Kress J W 1984 Practical cone-beam algorithm J. Opt. Soc. Am. A 6 612–19 Grangeat P 1991 Mathematical framework of cone-beam 3D reconstruction via the first derivative of the Radon transform Mathematical Methods in Tomography (Lecture Notes in Mathematics 1497) ed G T Herman, A K Louis and F Natterer (Berlin: Springer) pp 66–97 Gullberg G T, Christian P E, Zeng G L, Datz F L and Morgan H T 1991 Cone beam tomography of the heart using single-photon emission-computed tomography Invest. Radiol. 26 681–8 Gullberg G T and Zeng G L 1992 A cone beam filtered backprojection reconstruction algorithm for cardiac single photon emission computed tomography IEEE Trans. Med. Imaging 11 91–101 Hu H 1999 Multi-slice helical CT: scan and reconstruction Med. Phys. 26 5–18 Hu H, He H D, Foley W D and Fox S H 1999 Helical CT imaging performance of a new multislice scanner Proc. SPIE 3661 450–61 Jaszczak R J, Floyd C E, Manglos S H, Green K L and Coleman R E 1986 Three-dimensional single photon emission computed tomography using cone-beam collimation (CB-SPECT) Proc. SPIE 671 193–9

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