Fast reconstruction with uniform noise properties in half-scan tomography

[0002] This application is a continuation-in-part of U.S. patent application Ser. No. 09/289,297, filed on Apr. 9, 1999 and assigned to the assignee of the present invention.

[0004] The field of the invention relates to computer tomography and more particularly to the reconstruction of images from fan-beam sinograms.

[0005] In tomographic imaging, a finite set of imaging samples are obtained of the underlying multi-dimensional function of interest. However, because of various physical restrictions of the sampling system, these samples are often obtained on nonuniform grids, thereby preventing the direct use and meaningful interpretation of these data. For example, in medical tomographic imaging such as the two-dimensional (2D) fan-beam computed tomography (CT), single-photon emission computed tomography (SPECT), positron emission tomography (PET), spiral (or helical) CT, diffraction tomography (DT),and magnetic resonance imaging (MRI), the acquired data are often sampled on nonuniform grids in the sinogram space, thus preventing the direct use of existing methods that are computationally efficient and numerically stable for reconstruction of tomographic images. In these situations, one can always use various multi-dimensional interpolation (MDI) methods to convert the samples that lie on nonuniform grids into samples that lie on uniform grids so that they can be processed directly and be presented meaningfully.

[0006] A wide variety of MDI methods have previously been developed. The methods that are based upon the Whittaker-Shannon sampling (WST) theorem can potentially provide accurate interpolation results. Unfortunately, these methods generally possesses the shortcoming of great computational burden, which increases drastically as the number of interpolation dimensions increases (“the curse of the dimensionality”). Attempts have been made to alleviate the computational burden by developing efficient interpolation methods. However, these methods are all associated with certain approximations. Virtually all of the previously developed methods calculate the desired uniform samples directly from the measured nonuniform samples, which generally requires the use of computationally burdensome algorithms if accuracy is to be preserved.

[0007] A method and apparatus are provided for reconstructing a tomographic image from a halfscan fan-beam sinogram using the parallel beam reconstruction algorithms. The method includes the steps of weighting at least some elements of the halfscan fan-beam sinogram to forming a weighted fan-beam sinogram and expanding the weighted fan-beam sinogram into a Fourier series. The method further includes the steps of linearly interpolating at least some elements of the Fourier series to form a parallel beam sinogram of linearly interpolated data and reconstructing the image from the parallel beam sinogram using the parallel beam reconstruction algorithm.

[0008] FIG. 1 depicts an system for reconstructing images in accordance with an illustrated embodiment of the invention;

[0009] FIG. 2 depicts a flow chart of method steps that may be used by the system of FIG. 1;

[0010] FIG. 3 depicts exemplary images produced by the system of FIG. 1;

[0011] FIG. 4 depicts further exemplary images produced by the system of FIG. 1;

[0012] FIG. 5 depicts a surface plot of the image variance obtained with the halfscan algorithm of the system of FIG. 1;

[0013] FIG. 6 depicts the image variance obtained with the halfscan-FFBP algorithm; and

[0014] FIG. 7 depicts image variance along a horizontal line passing through a center of the image space of the system of FIG. 1.

[0015] FIG. 1 is a half-scan imaging system 10, generally in accordance with an illustrated embodiment of the invention. FIG. 2 is a flow chart of processing steps that may be used by the structure of FIG. 1 to generate images from half-scan fan-beam sinograms.

[0016] Under the illustrated embodiment, a sampling device 12 acquires samples 100 under an appropriate format (e.g., halfscan fan-beam CT) under the control of a central processing unit (CPU) 14. The CPU 14 may store the samples in a memory unit 16 or process the samples directly.

[0017] In general, conventional interpolation techniques calculate a function by directly using its measured samples. However, because the values of a function can readily be calculated from its Fourier transform (FT), the interpolation task could be compared to estimating the FT of a function from its measured samples. Based upon this observation, an approach is provided for reducing the interpolation dimensions in MDI problems.

[0018] Let f({right arrow over (x)}₁, {right arrow over (x)}₂) and g({right arrow over (y)}₁, {right arrow over (y)}₂) be K-dimensional (KD) band-limited functions, where {right arrow over (x)}₁=(x₁, x₂. . . x_k)^Tand {right arrow over (y)}₁=(y₁, y₂. . . y_k)^Tare kD vectors, {right arrow over (x)}₂=(x_k+1, x_k+2. . . x_k)^Tand {right arrow over (y)}₂=(y_k+1, y_k+2. . . y_K)^Tare MD vectors, and K=M+k and K>k. Suppose that f({right arrow over (x)}₁, {right arrow over (x)}₂)=g({right arrow over (y)}₁, {right arrow over (y)}₂) mathematically. Because the transformation between the {{right arrow over (x)}₁, {right arrow over (x)}₂} and {{right arrow over (y)}₁, {right arrow over (y)}₂} spaces can be generally non-linear, uniform samples of g({right arrow over (y)}₁, {right arrow over (y)}₂) in the {{right arrow over (y)}₁, {right arrow over (y)}₂} space produce generally non-uniform samples of f({right arrow over (x)}₁, {right arrow over (x)}₂) in the {{right arrow over (x)}₁, {right arrow over (x)}₂} space. The MDI task is to estimate uniform samples of f({right arrow over (x)}₁, {right arrow over (x)}₂) in the {{right arrow over (x)}₁, {right arrow over (x)}₂} space from a set of measured uniform samples of g({right arrow over (y)}₁, {right arrow over (y)}₂) in the {{right arrow over (y)}₁, {right arrow over (x)}₂} space, where the sampling rate satisfies the Nyquist conditions. In a wide class of MDI problems such as those that arise in medical imaging, the transformation between the {{right arrow over (x)}₁, {right arrow over (x)}₂} and {{right arrow over (y)}₁, {right arrow over (y)}₂} spaces has a general form, given by

[0019] {right arrow over (x)}₁={right arrow over (h)}₁({right arrow over (y)}₁)  (1a)

[0020] {right arrow over (x)}₂={right arrow over (h)}₂({right arrow over (y)}₁, {right arrow over (y)}₂)=a{right arrow over (y)}₂+{right arrow over (b)}({right arrow over (y)}₁),  (1b)

[0021] where {right arrow over (h)}₂denotes a general linear transformation (LT) between the {{right arrow over (x)}₂} and {{right arrow over (y)}₂} subspaces, whereas {right arrow over (h)}₁denotes a general non-LT between the {{right arrow over (x)}₁} and {{right arrow over (y)}₁} subspaces. The known matrix a is independent of {right arrow over (y)}₁and {right arrow over (y)}₂, and the known vector {right arrow over (b)} can be any real or complex function of {right arrow over (y)}₁. The known matrix a and vector {right arrow over (b)} represent transfer functions (i.e., a set of relationships) which relate the first set of samples f({right arrow over (x)}₁, {right arrow over (x)}₂) to the second set of samples g({right arrow over (y)}₁, {right arrow over (y)}₂) (i.e., the two sampling spaces) in a known manner.

[0022] Let F({right arrow over (x)}₁, {right arrow over (ν)}_{{right arrow over (x)}}) and G({right arrow over (y)}₁, {right arrow over (ν)}_{{right arrow over (y)}}) be the partial FTs of f({right arrow over (x)}₁, {right arrow over (x)}₂) and g({right arrow over (y)}₁, {right arrow over (y)}₂) with respect to {right arrow over (x)}₂and {right arrow over (y)}₂, respectively, where {right arrow over (ν)}_{{right arrow over (x)}}and {right arrow over (ν)}_{{right arrow over (y)}}denote the corresponding frequencies of {right arrow over (x)}₂and {right arrow over (y)}₂. It can be shown that $\begin{array}{cc}F\left({\overrightarrow{x}}_1,{\overrightarrow{v}}_{{\overrightarrow{x}}_2}\right)=\Vert a\Vert G\left({\overrightarrow{y}}_1,a^T{\overrightarrow{v}}_{{\overrightarrow{x}}_2}\right)e^{-j2\pi {\overrightarrow{v}}_{{\overrightarrow{x}}_2}·\overrightarrow{b}\left({\overrightarrow{y}}_1\right)},& \left(2\right)\end{array}$

[0023] where ∥a∥ and a^Tare the determinant and transpose of a. Because G({right arrow over (y)}₁, {right arrow over (ν)}_{{right arrow over (y)}}) can readily be obtained with the fast FT (FFT) from the measured samples of g({right arrow over (y)}₁, {right arrow over (y)}₂) in {{right arrow over (y)}₂} subspace, one can calculate F({right arrow over (x)}₁, {right arrow over (ν)}_{{right arrow over (x)}}) by using Eqn. (2) and then f({right arrow over (x)}₁, {right arrow over (x)}₂) in the {{right arrow over (x)}₂} subspace by invoking the inverse FFT. Therefore, the CPU 14 accomplishes the interpolation task in the {{right arrow over (x)}₂} subspace and effectively reduces the original KD interpolation task of Eqn. (1) to a kD one between the {{right arrow over (x)}₁} and {{right arrow over (y)}₁} subspaces.

[0024] For another class of MDI problems, the transformation between the {{right arrow over (x)}₁} and {{right arrow over (y)}₁} subspaces is described by Eqn. (1a), whereas the transformation between the {{right arrow over (x)}₂} and {{right arrow over (y)}₂} subspaces is given by

[0025] x_k+i=c_iy_k+i+b_i({right arrow over (y)}₁, y_k+1, y_k+2. . . y_k+i−1),  (3)

[0026] where x_k+i, y_k+i, b_iand c_iare the components of {right arrow over (x)}₂, {right arrow over (y)}₂, {right arrow over (b)} and {right arrow over (c)}, i=1, 2 . . . M. We assume that the vectors {right arrow over (b)} and {right arrow over (c)} are known. It should be noted that {right arrow over (b)} in Eqn. (1b) is completely independent of {right arrow over (y)}₂, whereas {right arrow over (b)} in Eqn. (3) can be a general non-linear function of the components of {right arrow over (y)}₂. One can show that F({right arrow over (x)}₁, {right arrow over (ν)}_{{right arrow over (x)}}) can be expressed as

[0027] F({right arrow over (x)}₁, ν_k+1, ν_k+2. . . ν_k+M)=c₁e^{−j2πνb({right arrow over (y)})}×∫dy_k+1e^−j2πcνyG^(M−1)({right arrow over (y)}₁, y_k+1, ν_k+2. . . ν_k+M)  (4)

[0028] where {right arrow over (ν)}_{{right arrow over (x)}}=(ν_k+1, ν_k+2. . . ν_k+M)^T, and

[0029] G⁽ⁱ⁾({right arrow over (y)}₁, y_k+1. . . y_k+M−i, ν_k+M−i+1. . . ν_k+M)

[0030] =c_M−i+1e^{−j2πνb({right arrow over (y)}, y. . . y)}

[0031] ×∫dy_k+M−i+e^−j2πcνyG⁽ⁱ⁻¹⁾({right arrow over (y)}₁,

[0032] y_k+1. . . y_k+M−i, y_k+M−i+1, ν_k+M−i+2. . . ν_k+M)  (5)

[0033] with G⁽⁰⁾({right arrow over (y)}₁, y_k+1. . . y_k+M)=g({right arrow over (y)}₁, {right arrow over (y)}₂) and i=1, 2 . . . M−1. The integrals in Eqns. (4) and (5) denote 1D FTs that can be calculated by invoking the 1D FFT. Using Eqns. (4) and (5), the CPU 14 can calculate F(({right arrow over (x)}₁, {right arrow over (ν)}_{{right arrow over (x)}}) from knowledge of g({right arrow over (y)}₁, {right arrow over (y)}₂) in the {{right arrow over (y)}₂} subspace. From F(({right arrow over (x)}₁, {right arrow over (ν)}_{{right arrow over (x)}}), the CPU 14 can obtain f({right arrow over (x)}₁, {right arrow over (x)}₂) by invoking the inverse FFT and thus accomplish the interpolation task in the {{right arrow over (x)}₂} subspace. Therefore, the original KD interpolation task of Eqns. (1a) and (3) is thus reduced effectively to a kD one between the {{right arrow over (x)}₁} and {{right arrow over (y)}₁} subspaces.

[0034] In fan-beam CT, one can measure uniform samples of g(α,β) in the {α,β} space. Reconstruction of a CT image requires uniform samples of 𝑓(ξ,φ) in the {ξ,φ} space. One can show that 𝑓(ξ,φ)=g(α,β), provided

[0035] ξ=F sinα  (6a)

[0036] φ=β+α,  (6b)

[0037] where F is the focal length of the fan-beam system. Because the transformation in Eqn. (6) is non-linear, a 2D interpolation may be used to calculate uniform samples of 𝑓(ξ,φ) in the {ξ,φ} space from that of g(α,β) in the {α,β} space. However, comparison of Eqns. (6b) and (1b) indicates that φ is a 1D LT of β with a=1 and b=α and that the interpolation between the {φ} and {β} subspaces can thus be accomplished by using Eqn. (2). Hence, the 2D interpolation task in 2D fan-beam CT is reduced effectively to a 1D interpolation task between the {ξ} and {α} subspaces.

[0038] The use of the multidimensional interpolation discussed above allows for the development and use of hybrid algorithms for the reconstruction of CT images. As used herein a hybrid algorithm is a method of reconstruction that relies upon the multidimensional interpolation method set forth above as a method of substantially simplifying image reconstruction.

[0039] Hybrid algorithms inherently possess computational and noise properties superior to those of the fan-beam filtered backprojection (FFBP) algorithm. However, the hybrid algorithms cannot be applied directly to a halfscan fan-beam sinogram because they require knowledge of a fullscan fan-beam sinogram. In the following description, halfscan-hybrid algorithms are described which can be used for image reconstruction in halfscan CT.

[0040] Numerical evaluation indicates that the proposed halfscan-hybrid algorithms are computationally more efficient than are the widely used halfscan-FFBP algorithms. More important, the results of quantitative studies demonstrate clearly that the noise levels in images reconstructed by use of the halfscan-hybrid algorithm are generally lower and spatially much more uniform than are those in images reconstructed by use of the halfscan-FFBP algorithm. Such reduced and uniform image noise levels may be translated into improvement of the accuracy and precision of lesion detection and parameter estimation in noisy CT images without increasing the radiation dose to the patient. Therefore, the halfscan-hybrid algorithms have significant implication for image reconstruction in conventional CT and, in particular, helical CT.

[0041] The widely used fan-beam filtered backprojection (FFBP) algorithm in computed tomography (CT) reconstructs an image directly from the fan-beam sinogram. Alternatively, the hybrid algorithms estimate the parallel-beam sinogram from the fan-beam sinogram without an explicit interpolation between the fan-beam and parallel-beam projection angles and then reconstruct the image by use of parallel-beam reconstruction algorithms such as the filtered backprojection (FBP) algorithm. The hybrid algorithms possess several potential advantages over the FFBP algorithm: the former are computationally more efficient than is the latter; and the noise levels in images reconstructed by use of the former are significantly lower and much more uniform than those in images reconstructed by use of the latter. Low and uniform noise levels in CT images are always desirable because they may improve the accuracy of estimation of parameters of interest and the detection of subtle signals in noisy images

[0042] It is possible to reduce the scanning time and radiation dose in CT imaging by use of the so-called halfscan approach, in which one acquires the halfscan fan-beam sinogram at projection angles from zero to r plus the fan angle. Other investigators have proposed the halfscan-FFBP algorithms, which first weighted the halfscan fan-beam sinogram and then applied the FFBP algorithm to reconstruct the image. However, because of their use of the FFBP algorithm, the halfscan-FFBP algorithms suffer from computational and noise problems similar to those that plague the FFBP algorithm. Additionally, although the noise properties in parallel-beam and fullscan fan-beam CT have been investigated previously, there is a lack of analysis of the noise and numerical properties of image reconstruction in halfscan CT.

[0043] The hybrid algorithms cannot be applied directly to the halfscan fan-beam sinogram because they require knowledge of the full scan sinogram at projection angles from zero to 2π. Because of the potential advantages of the hybrid algorithms, however, it is theoretically interesting and practically important to generalize them to the halfscan situation. In this work, halfscan-hybrid algorithms are developed for image reconstruction in halfscan CT. The noise and numerical properties of the halfscan-hybrid and the widely used halfscan-FFBP algorithms are also developed and compared, demonstrating that the halfscan-hybrid algorithms possess properties similar to those of fullscan hybrid algorithms, and, more importantly, possess computational and noise properties superior to those of the halfscan-FFBP algorithms.

[0044] The description in the following sections is organized as follows. In the following section, the necessary notation and equations are introduced by summarizing the hybrid algorithms and the halfscan-FFBP algorithms. In the section after that, the halfscan-hybrid algorithms are developed. In the next following section, we describe numerical studies for evaluation and comparison of the halfscan-FFBP and halfscan-hybrid algorithms. In the final section, we present a discussion of the work.

[0045] We first consider the theoretical background. Let a(r,θ) denote a two dimensional (2D) image, where r and θ are the polar coordinates. Also, let p(ξ,φ) denote the parallel-beam sinogram of a(r,θ), where ξ and φ represent the detector-bin indices and projection angles, respectively. From knowledge of p(ξ,φ), one can reconstruct a(r,θ) by use of a variety of reconstruction algorithms such as the FBP algorithm, which can be expressed as $\begin{array}{cc}a\left(r,\theta \right)={\int }_{\phi =0}^{2\pi }{\int }_{\xi =0}^{\infty }\sum _{k=-\infty }^{\infty }{P}_k\left(\xi \right)e^{jk\phi }h\left(r\cos \left(\phi -\theta \right)-\xi \right)d\xi d\phi ,& \left(7\right)\end{array}$

[0046] where h(ξ) is the inverse Fourier transform of the ramp filter, P_k(ξ) is the Fourier series expansion of p(ξ,φ) with respect to φ, and k=0, ±1, ±2 . . . is the angular frequency index.

[0047] The Fan-Beam Filtered Backprojection(FFBP) Algorithm will be considered next. Let q(α,β) denote the fan-beam sinogram of the image a(r,θ), where α and β represent the detector-bin indices and projection angles in the fan-beam projection configuration. In a fullscan situation, one acquires the fullscan fan-beam q(α,β) in the complete data space W=[|α|≦α_m,0≦β≦2π], where α_m>0 is the maximum open angle of the curved detector array in a CT system such that q(α,β)=0 for α>α_m. A well-known relationship between the parallel-beam sinogram p(ξ,φ) and the fan-beam sinogram q(α,β) is given by

[0048] p(ξ,φ)=q(α,β)  (8)

[0049] where

[0050] ξ=F sinα and φ=β+α  (9)

[0051] and F is the focal length of the fan-beam system.

[0052] Using Eqs. (8) and (9) in Eq. (7), one can readily derive the FFBP algorithm that reconstructs the image directly from the fullscan fan-beam sinogram q(α,β). Mathematically, the FFBP algorithm can be expressed as $\begin{array}{cc}a\left(r,\theta \right)=\frac{1}{2}{\int }_{\beta =0}^{2\pi }\frac{F}{L^2}{\int }_{\alpha =-{\alpha }_m}^{{\alpha }_m}{q\left(\alpha ,\beta \right)\left[\frac{\left({\alpha }_0-\alpha \right)}{\sin \left({\alpha }_0-\alpha \right)}\right]}^2\cos \alpha h\left({\alpha }_0-\alpha \right)d\alpha d\beta ,\mathrm{where}L={\left\{{\left[F+r\sin \left(\beta -\theta \right)\right]}^2+{\left[r\cos \left(\beta -\theta \right)\right]}^2\right\}}^{\frac{1}{2}}\mathrm{and}{\alpha }_0=\arctan \left[\frac{r\cos \left(\beta -\theta \right)}{F+r\sin \left(\beta -\theta \right)}\right].& \left(10\right)\end{array}$

[0053] Hybrid algorithms will be considered next. As an alternative to the FFBP algorithm, hybrid algorithms have been proposed for image reconstruction from a fullscan fan-beam sinogram. Let Q_k(α) denote the Fourier series expansion of the fullscan fan-beam sinogram q(α,β) with respect to β, i.e., $\begin{array}{cc}{Q}_k\left(\alpha \right)=\frac{1}{2\pi }{\int }_{\beta =0}^{2\pi }q\left(\alpha ,\beta \right)e^{-jk\beta }d\beta ,& \left(11\right)\end{array}$

[0054] where |α|≦α_m, and the angular frequency index k=0, ±1, ±2 . . . Using Eq. (11), one can readily obtain Q_k(α) from the fullscan fan-beam sinogram q(α,β).

[0055] The hybrid algorithm first estimates P_k(ξ) (i.e., the parallel-beam sinogram) from Q_k(α) (i.e., the fullscan fan-beam sinogram):

[0056] P_k(ξ)=ω(α,k)γ^kQ_k(α)+(1−ω(α,k))(−1)^kγ^−kQ_k(−α)  (12)

[0057] where ξ≧0, k=0, ±1, ±2 . . . , γ=e^−jα, and ω(α,k) is the combination coefficient that can be a general complex function of α and k. From the estimated P_k(ξ), the hybrid algorithm subsequently reconstructs the image a(r,θ) by use of computationally efficient and numerically stable algorithms such as the FBP algorithm in Eq. (7). Different choices of ω(α,k) specify different hybrid algorithms. It was observed theoretically and demonstrated quantitatively that, because the ratio F/L in the FFBP algorithm (see Eq. (10)) is spatially variant (i.e., a function of r, θ, and β) and can be larger than 1, the FFBP algorithm is computationally less efficient and more susceptible to the effects of data noise and sample aliasing than are the hybrid algorithms.

[0058] The halfscan algorithms will be considered next. One can divide the complete dataspace W into four subspaces A, B, C, and D, where A=[|α|≦α_m,0≦β<2α_m−2α], B=[|α|≦α_m, 2α_m−2α<β≦π−2α], C=[|α|≦α_m,π−2α≦β<π+2α_m], and D=[|α|≦α_m,π+2α_m≦β<2π]. In a halfscan situation, q(α,β) is obtained only in the subspace H=A∪B∪C=[|α|≦α_m,0≦β≦π+2α_m], which is referred to as a halfscan fan-beam sinogram. Although this halfscan fan-beam sinogram contains all of the information necessary for reconstruction of the image, it also includes partially redundant information in subspaces A and C which, without being adequately normalized, can result in image artifacts. The halfscan-FFBP algorithm first normalizes such partially redundant information by generating an appropriately weighted fan-beam sinogram q′(α,β) and then uses the FFBP algorithm of Eq. (10) for exact reconstruction of the image. The appropriately weighted fan-beam sinogram q′(α,β) is given by

[0059] q′(α,β)=q(α,β)ω(α,β)  (13)

[0060] where the weighting function ω(α,β) satisfies

[0061] ω(α,β)+ω(−α,β+2α+π)=1  (14)

[0062] for (α,β) in the complete space M,ω(α,β)=1 for (α,β) in subspace B, and ω(α,β)=0 for (α,β) in subspace D. Although the forms of ω(α,β) for (α,β) in subspaces B and D are completely specified, explicit forms of ω(α,β) for (α,β) in subspaces A and C are unspecified so far. In principle, one can design different halfscan-FFBP algorithms by choosing different ω(α,β) for (α,β) in subspaces A and C as long as they satisfy Eq. (14). The reconstructions obtained using different choices of ω(α,β) will respond differently to the effects of data noise and sample aliasing

[0063] Halfscan-hybrid algorithms will be discussed next. Because the halfscan-FFBP algorithms use the FFBP algorithm for image reconstruction from the weighted sinograms q′(α,β), they are expected to be less computationally efficient and more susceptible to the effects of data noise and sample aliasing than are the hybrid algorithms described above. However, these hybrid algorithms cannot be applied directly to the halfscan fan-beam sinogram because they estimate P_k(ξ) from Q_k(α) in Eq. (12), the calculation of which in Eq. (11) requires knowledge of the fullscan fan-beam sinogram for 0≦β<2π.

[0064] The derivation of the halfscan-hybrid algorithms will be considered next. In the discussion below, we develop halfscan-hybrid algorithms that can be applied to the halfscan fan-beam sinogram and that possess computational and noise properties similar to those of the hybrid algorithms. Using q′(α,β) of Eq. (13) in Eq. (10), the halfscan-FFBP algorithm can be rewritten as $\begin{array}{cc}a\left(r,\theta \right)={\int }_{\beta =0}^{2\pi }\frac{F}{L^2}{\int }_{\alpha =0}^{{\alpha }_m}{\left[q^{\prime }\left(\alpha ,\beta \right)+q^{\prime }\left(-\alpha ,\beta +\pi \right)\right]\left[\frac{\left({\alpha }_0-\alpha \right)}{\sin \left({\alpha }_0-\alpha \right)}\right]}^2\cos \alpha h\left({\alpha }_0-\alpha \right)d\alpha d\beta .& \left(15\right)\end{array}$

[0065] Using Eq. (9) to replace the variables (α,β) by the variables (ξ,φ) in Eq. (15), one obtains $\begin{array}{cc}a\left(r,\theta \right)={\int }_{\phi =0}^{2\pi }{\int }_{\xi =0}^{\infty }\left[q^{\prime }\left(\alpha ,\phi -\alpha \right)+q^{\prime }\left(-\alpha ,\phi +\alpha +\pi \right)\right]h\left(r\cos \left(\phi -\theta \right)-\xi \right)d\xi d\phi .& \left(16\right)\end{array}$

[0066] Let Q′_k(α) denote the Fourier series expansion of the weighted fan-beam sinogram q′(α,β). Then $\begin{array}{cc}{Q}_k^{\prime }\left(\alpha \right)=\frac{1}{2\pi }{\int }_{\beta =0}^{2\pi }q\left(\alpha ,\beta \right)\omega \left(\alpha ,\beta \right)e^{-jk\beta }d\beta \mathrm{and}{q}^{\prime }\left(\alpha ,\beta \right)=\sum _{k=-\infty }^{\infty }{Q}_k^{\prime }\left(\alpha \right)e^{-jk\beta }.& \left(17\right)\end{array}$

[0067] Substituting Eq. (17) into Eq. (16) yields $\begin{array}{cc}a\left(r,\theta \right)=\frac{1}{2}{\int }_{\phi =0}^{2\pi }{\int }_{\xi =0}^{\infty }\sum _{k=-\infty }^{\infty }\left[{\gamma }^k{Q}_k^{\prime }\left(\alpha \right)+{\left(-1\right)}^k{\gamma }^{-k}{Q}_k^{\prime }\left(-\alpha \right)\right]e^{jk\phi }h\left(r\cos \left(\phi -\theta \right)-\xi \right)d\xi d\phi .& \left(18\right)\end{array}$

[0068] Comparison of Eqs. (7) and (18) indicates that

[0069] P_k(ξ)=½[γ^kQ′_k(α)+(−1)^kγ^−kQ′_k(−α)].  (19)

[0070] This is an important relationship that can be used to estimate P_k(ξ) from the halfscan fan-beam sinogram. From this estimated P_k(ξ), one can subsequently reconstruct the image by use of the FBP algorithm. Therefore, the combination of Eqs. (19) and (7) forms a halfscan-hybrid algorithm. Moreover, different choices of the weighting function ω(α,β) yield different halfscan-hybrid algorithms that respond differently to the effects of data noise and sample aliasing.

[0071] The procedure of using the halfscan hybrid algorithms will be discussed in more detail next. As a first step a weighted fan-beam sinogram q′(α,β) is determined. Using the halfscan fan-beam sinogram q(α,β) and the selected weighting function ω(α,β) in Eq. (13), one can readily obtain the weighted fan-beam sinogram q′(α,β).

[0072] Next a partial Fourier transform Q′_k(α) of the weighted sinogram q′(α,β) with respect to β is determined. One can obtain Q′_k(α) from q′(α,β), which is calculated in step (1), by performing a Fourier series expansion with respect to β (see Eq. (17)).

[0073] Next a Fourier series expansion of the parallel beam sinogram P_k(ξ) is determined. Using the calculated Q′_k(α) and Q′_k(−α) in Eq. (19), one can obtain P_k(ξ). Due to the non-linear relationship between ξ and α (see Eq. (9)), the desired points α at which P_k(ξ) is to be calculated do not coincide, in general, with the values of α at which Q′_k(α) and Q′_k(−α) are sampled. However, P_k(ξ) can be obtained by interpolation of the right side of Eq. (17) for each desired ξ value. Because the sample density of q(α,β) in α is generally high, linear interpolation is usually sufficiently accurate.

[0074] Finally the image may be reconstructed. Applying the FBP algorithm in Eq. (7) to P_k(ξ), which is estimated in step (3), one can reconstruct the image a(r,θ).

[0075] As with the halfscan-FFBP algorithms, the halfscan-hybrid algorithms use step (1) to generate the weighted fan-beam sinogram q′(α,β). However, the implementation steps (2)-(4) described above for the halfscan-hybrid algorithms are unique.

[0076] The results of using halfscan hybrid algorithms will be discussed next. A 128×128 numerical phantom, as shown in FIG. 3a, was used in the simulation studies. This phantom consists of a set of square and circular structures with different sizes, orientations, and locations that are embedded in a circular disk background. A noiseless halfscan fan-beam sinogram that contains 92 projections was generated by use of this numerical phantom and an equiangular fan-beam configuration with a focal length F=91 pixels. These projections are uniformly sampled over 260°, each containing 128 bins that have the same size as that of the image pixels. The weighting function proposed by Crawford was used in the simulation studies and is given by

[0077] ω(α,β)=3x²(α,β)−2x³(α,β).  (20)

[0078] where $x\left(\alpha ,\beta \right)=\frac{\beta }{2{\alpha }_m-2\alpha }$

[0079] for (α,β) in subspace A, and $x\left(\alpha ,\beta \right)=\frac{\pi +2\alpha -\beta }{2{\alpha }_m+2\alpha }$

[0080] for (α,β) in subspace B. In an attempt to evaluate the noise level and uniformity in images reconstructed by use of the halfscan-FFBP and halfscan-hybrid algorithms, we generated noisy halfscan fan-beam sinograms that contain stationary white Gaussian noise, using the noiseless halfscan fan-beam sinogram as the mean. From these noisy sinograms noisy images were reconstructed by use of the halfscan-hybrid and halfscan-FFBP algorithms and subsequently empirical image variance was calculated.

[0081] Reconstructed images in Halfscan CT will now be considered. The halfscan-hybrid and halfscan-FFBP algorithms were used to reconstruct images, which are shown in FIGS. 3band 3c, respectively, from the noiseless halfscan sinogram. The evident artifacts in the outer region of the image in FIG. 3csuggest that the halfscan-FFBP algorithm is more susceptible to the effect of sample aliasing than is the halfscan-hybrid algorithm (see FIG. 3b). Similarly, from the simulated noisy sinogram, these two algorithms were used to reconstruct images that are shown in FIGS. 4band 4c, respectively. Clearly, the noise levels in the outer regions of the image in FIG. 4cappear to be higher than are those in the outer regions of the image in FIG. 4b, confirming qualitatively that the halfscan-FFBP algorithm amplifies the effects of data noise and sample aliasing inherent in the noisy and discrete data more seriously than does the halfscan-hybrid algorithm.

[0082] The noise properties of reconstructed images in halfscan CT will now be discussed. The noise properties of the images reconstructed by use of the halfscan-hybrid and halfscan-FFBP algorithms were quantitatively considered. Using 500 sets of simulated noisy halfscan sinograms, each of these two algorithms were employed to reconstruct 500 images and then the image variance was calculated empirically from these noisy images. In FIGS. 5 and 6, we show the surface plots of the image variance obtained by use of the halfscan-hybrid and halfscan-FFBP algorithms, respectively. These results indicate that the noise levels in images reconstructed by use of the halfscan-hybrid algorithm are much more uniform than are those in images reconstructed by use of the halfscan-FFBP algorithm. In an attempt to provide a quantitatively detailed comparison, we plot in FIGS. 7aand 7bthe image variance along a horizontal line and a vertical line, respectively, that pass through the center of the image space. These results indicate quantitatively that the image variance obtained with the halfscan-hybrid algorithm is generally lower and more uniform than that obtained with the halfscan-FFBP algorithm. This result is consistent with those observed in the fullscan situation.

[0083] The non-symmetric noise properties of reconstructed images in Halfscan CT were considered. As analyzed by Pan in “OPTIMAL NOISE CONTROL AND FAST RECONSTRUCTION OF FAN-BEAM COMPUTED TOMOGRAPHIC IMAGE” (Med. Phys., 26, 689-697 (1999)), the high and non-uniform noise levels in images obtained with the FFBP algorithm are caused by the ratio of F/L, which can be larger than 1 for spatial locations in the outer region of the image and, thus, can amplify significantly the effects of data noise and sample aliasing inherent in the noisy and discrete data. However, the noise levels in images reconstructed by use of the FFBP algorithm from fullscan fan-beam sinograms appear qualitatively circularly symmetric around the center of the image space, whereas the noise levels in images reconstructed by use of the halfscan-FFBP algorithm from the halfscan fan-beam sinogram are highly non-symmetric, as shown in FIGS. 4c, 6, and 7. For example, in FIGS. 4cand 6, the noise levels in the upper left regions are much lower than are those in other outer regions of the image obtained with the halfscan-FFBP algorithm.

[0084] These observations can be understood as follows: As shown in Eq. (10), the filtered projections (i.e., the integration over α in Eq. (10)) are multiplied by the ratio of (F/L)²before being backprojected into the image space. For a given projection angle β, the ratio of F/L (see its expression below Eq. (10)) can be smaller or larger than 1, depending upon whether the locations (r,θ) in the image space are close to or far away from the detector array that acquires the projections at angle β. Therefore, when the filtered projections are backprojected into the image space, the factor L⁻²suppresses or amplifies the effects of data noise and sample aliasing, depending on whether the locations in the image space that are close to or far away from the detector array at angle β. In a fullscan situation, the projections and, thus, the corresponding filtered projections can be obtained for 0≦β≦2π. When the products of (F/L)²and the filtered projections are backprojected into the image space for all projection angles between 0 and π, the image noise levels are non-uniform (e.g., they increase as r increase), but are qualitatively circularly symmetric about the center of the image space. However, in a halfscan situation, projections are not acquired at certain angles β. For example, our simulated halfscan sinogram contains no projections for angles around the lower right corner the image space in FIG. 4c. These zero-projection views contribute no values to the reconstructed image, and, therefore, the image regions which are close to and far away from the zero-projection views contain contributions mainly from projection views conjugate to the zero-projection views. Therefore, noise levels in these image regions are either higher or lower than those in the center of the image space, respectively. In our simulation studies, the zero-projection views are around the lower right corner of the image space (see FIG. 4c), and their conjugate projection views are around the upper left corner of the image space. Therefore, the noise levels in the lower right and upper left regions in the image space are higher and lower, respectively, than those in the center of the image space. This observation can readily be confirmed by the result in FIG. 6.

[0085] The hybrid algorithms discussed herein are superior to the FFBP algorithm in terms of computational efficiency and noise amplification. However, these hybrid algorithms cannot be applied directly to the halfscan fan-beam sinogram because they involve the calculation of the Fourier series expansion of the fullscan fan-beam sinogram, which is not available in the halfscan situation. In this work, the halfscan-hybrid algorithms are developed for image reconstruction in halfscan CT. The key is that the weighted q′(α,β) can be treated as a “fullscan fan-beam sinogram,” and one can use its Fourier series expansion to estimate the parallel-beam sinogram without invoking an explicit interpolation between the parallel-beam and fan-beam projection angles. Different hybrid algorithms exploit the redundant information in a fullscan fan-beam sinogram by using different combination coefficients ω in Eq. (12). However, as shown in Eq. (19), different halfscan-hybrid algorithms use the same combination coefficient ω=½, and, instead, they exploit the partially redundant information in a halfscan fan-beam sinogram by using different weighting functions, as shown in Eq. (13).

[0086] Numerical evaluation indicates that the halfscan-hybrid algorithms are computationally more efficient than is the currently used halfscan-FFBP algorithm. Such a computational advantage may become more significant for the reconstruction task in single- and multi-slice helical CT, which generally reconstructs a large number of images of large size. More importantly, the results of our simulation studies demonstrated clearly that the noise levels in images reconstructed by use of the halfscan-hybrid algorithm are generally lower and spatially much more uniform than are those in images reconstructed by use of the halfscan-FFBP algorithm. Such a reduction and uniformity of image noise levels may be translated into improvement of the accuracy and precision of lesion detection and parameter estimation in noisy CT images without increasing the radiation dose to the patient. On the other hand, because the noise levels (i.e., image variance) in quantum limited CT are generally related to the reciprocals of radiation exposures to the patient, to reconstruct images with comparable noise levels, the halfscan-hybrid algorithm requires a lower radiation dose (i.e., a smaller number of X-rays) than does the halfscan-FFBP algorithm. Again, this advantage of the halfscan-hybrid algorithms over the halfscan-FFBP algorithm can become cumulatively significant in single- and multi-slice helical CT, in which the imaging of multiple slices generally requires high radiation dose. Finally, the results in this work can be generalized to other fan-beam imaging modalities such as single-photonemission computed tomography with different fan-beam projection configurations.

[0087] A specific embodiment of a method and apparatus for creating CT images according to the present invention has been described for the purpose of illustrating the manner in which the invention is made and used. It should be understood that the implementation of other variations and modifications of the invention and its various aspects will be apparent to one skilled in the art, and that the invention is not limited by the specific embodiments described. Therefore, it is contemplated to cover the present invention and any and all modifications, variations, or equivalents that fall within the true spirit and scope of the basic underlying principles disclosed and claimed herein.