Journal of X-Ray Science and Technology 16 (2008) 95–117 IOS Press
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Application of projection simulation based on physical imaging model to the evaluation of beam hardening corrections in X-ray transmission tomography Shaojie Tanga , Xuanqin Moua,∗, Ying Yanga , Qiong Xua and Hengyong Yub a Institute
of Image Processing & Pattern Recognition, Xi’an Jiaotong University, Xi’an, Shaanxi, 710049, P.R. China b Biomedical Imaging Division, VT-WFU School of Biomedical Engineering and Science, Virginia Tech., Blacksburg, VA 240601, USA Received 12 December 2007 Revised 16 March 2008 Accepted 31 March 2008 Abstract. The polychromatic spectrum of photons emitted by an X-ray tube in medical diagnostic CT system brings beam hardening artifact. Although many correction methods have been developed, it is still the major artifact in clinical CT imaging. Motivated by CT reconstruction researches benefiting from mathematical phantoms, one so-called physical mathematical phantom and its projection simulation are presented in this paper, which are used in comprehensive evaluation of beam hardening corrections. Due to the inherent merits as compared to discrete and stochastic simulations, an analytic method is proposed to simulate the polychromatic projections of FORBILD head phantom. Firstly, CT-numbers are mapped to material compositions under the precondition sub-regions represent different human tissues. Second, boundary parameters of subregions are determined by the presented algorithms. Finally, projections are calculated according to X-ray energy spectrum. For validating the usefulness of our simulation, several typical correction methods are analyzed using simulated projections. The performance of every correction is exhibited distinctly. The analysis results show that there are still the spaces to improve those correction methods. Keywords: Polychromatic spectrum, beam hardening, projection simulation, FORBILD
1. Introduction A polychromatic spectrum of photons emitted by an X-ray tube can cause beam hardening effect in a CT system. It results in beam hardening artifact (e.g., cupping, streak, spill-over, and pseudo cortex) in the reconstructed images, which are relevant to not only the imaging chain of CT device, but also the ∗ Corresponding author. Prof. Xuanqin Mou, Institute of Image Processing & Pattern Recognition, Xi’an Jiaotong University, Xi’an, Shaanxi, 710049, P.R. China. E-mail:
[email protected].
0895-3996/08/$17.00 2008 – IOS Press and the authors. All rights reserved
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imaged object. It is well known that beam hardening artifact is one of the most important artifacts in the CT field, and it is difficult to be completely corrected. The researches on beam hardening correction have been launched since the invention of the modern medical X-ray CT (XCT) system by Hounsfield in EMI at 1970’s. Up to now, several correction methods have been developed. The typical methods include water correction [1], bone correction [2], ECC [3], and so on. In these literatures, most correction methods were evaluated by the data acquired from the real CT scanner, which have only partially confirmed their validity. During the comprehensive evaluations, manufactured phantoms are lack of design complexity and incapable of completely isolating the acquired data from other error sources. Additionally, results from clinic regime also disclosed that the beam hardening artifact is still one of artifacts in current CT devices [4–6]. Hence, the existing beam hardening corrections have still not been performed ideally according to the clinical requirements. The CT reconstruction algorithms mostly benefit from the various specially designed mathematical phantoms. Those phantoms are employed to validate the developed algorithms, evaluate their performances, and compare different algorithms, which consequently lead to provide us more knowledge about the perfect reconstruction algorithm. By the same idea, to develop a sophisticated physical mathematical phantom and its projection calculation algorithm, which can exhibit the physical characteristics of the projection of real polychromatic spectrum, can help us to develop, evaluate and choose the correction method, and therefore reach the perfect solution finally while researching the beam hardening effect. Experiments in this paper show that the beam hardening effects not only induce artifacts in reconstructed images, but also deteriorate modulate transfer function (MTF), which with spatial resolution and contrast resolution are also the major indexes to score the CT reconstruction algorithms. This phenomenon hints that it is necessary to unify the evaluation phantoms for both beam hardening correction and CT reconstruction. On the other hand, we have shown that the beam hardening can be corrected based on the HelgassonLudwig consistency in references [7,8], which is a basis to construct the CT reconstruction algorithms. It also indicated that the research of beam hardening correction and CT reconstruction could be unified under an identical mathematical framework. It further induced the requirement to design a physical mathematical phantom applicable to evaluate both beam hardening correction and CT reconstruction. Because beam hardening effects are relevant to both CT device and imaged objects, several aspects should be considered during choosing a mathematical phantom, e.g., comparability between sub-regions of the phantom and anatomical structures of human body, and capability of evaluating more performance indexes. In the current theoretical studies of XCT, there are already some well-known mathematical phantoms, e.g., FORBILD head phantom [9], Shepp-Logan phantom, and Defrise Disk phantom. By comparison, we find that both Shepp-Logan and Defrise Disk phantoms are lack of capability of conveniently evaluating spatial resolution, contrast resolution, and so on. In contrast to Shepp-Logan and Defrise Disk phantoms, almost all sub-regions in FORBILD head phantom are the anatomical analogues of human brain. Therefore, FORBILD head phantom is a good choice for our destination. However, the first problem still left is how to determine the corresponding material compositions while FORBILD only presents the CT-numbers of all sub-regions of the FORBILD head phantom. In this paper, CT-numbers are converted to the material compositions, under the constraint that only human tissues are considered [10]. The second problem is which simulation method should be used during projections computation. Nowadays, available methods can be classified into three categories: discrete simulation [11–14], stochastic simulation [15,16], and analytic simulation [17–19]. Discrete simulation can be applied to a phantom with larger number of irregular geometrical shapes and requires less computation, but it necessitates massive storage and the computation precision is limited. Therefore, it is not very compatible
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for the studies of CT reconstruction and especially for beam hardening correction. Monte Carlo methods are essentially stochastic simulation, which may successfully reflect the realistic physical process of X-ray imaging, however, is time-consuming and not accessible to use [15,16]. Analytic simulation is of merits requiring less storage, shorter computation, and generating projections accurately, but it becomes somewhat hard to calculate projections when larger number of irregular geometrical shapes in a phantom. Here, shortcomings of analytic simulation become uncritical due to the moderate complexity of the geometrical shapes in the FORBILD head phantom. Thereby, we evaluate beam hardening corrections on the basis of this analytic simulation, although the functionality of our previously improved discrete simulation has been successful in some degree [14]. The last is how to calculate polychromatic projections. In general, linear integral model is utilized to acquire the projections from a mathematical phantom [11,13,17,18]. However, there is a discrepancy between real and linear integral projections, because the monochromatic spectrum, a hypothesis in linear integral model, is dissimilar to the polychromatic spectrum of real X-ray source. Hence, it is necessary to calculate projections, according to the real polychromatic spectrum of X-ray source, the material compositions of the phantom, and the knowledge of X-ray imaging physics [12,14,19]. The final obtained projections will be more consistent with those acquired by a real data acquisition system. Here, we also mention that there have already existed several references about the polychromatic projection simulation [20–23,35]. However, there are still some differences between our present job and others. Firstly, in our job, only CT-numbers are needed to map to mass densities and element weights, which is useful when sub-regions of a phantom are not of information of anatomy analogues, such as the two spheres of 47.5HU and 52.5HU in FORBILD head phantom. Second, although analytic simulation method, extensively adopted to calculate polychromatic projections by many references, is reported in this paper in detail due to its foundation merit, we focus our main attention on how to analyze, evaluate, and compare beam hardening correction methods in a virtual and unified framework. To the best of our knowledge, there is still no previous works along this line. This paper is organized as follows. In Section 2, we introduce an analytic method to simulate polychromatic projections by geometrical models. In Section 3, three typical beam hardening corrections are applied to projections data and their correction performances are analyzed and compared. In Section 4, we conclude the paper. 2. Method In the simulation of XCT, researchers mainly use the mathematical phantom rather than digitalized images of human being body, because utilization of a deliberately designed mathematical phantom may make the assessments of the important characteristics more convenient. In the following, we will give a general polychromatic projection simulation method for mathematical phantom demonstrated by the FORBILD head phantom. 2.1. Mapping CT-numbers to mass densities and element weights Although the detailed information of all sub-regions in the FORBILD head phantom is available [9], the corresponding material compositions and mass densities are still unknown. These should be determined as a precondition to calculate the energy-dependent mass attenuation coefficients. However, it is impossible to determine material compositions and mass density only from the CT-number since there are multiple solutions. Fortunately, we can determine one solution under the constraint that only
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Fig. 1. One cross section of the FORBILD head phantom at x3 = 0 mm and the labels of the principal sub-regions.
the human tissues are considered. Here, the CT-numbers firstly are mapped into tissue mass densities and essential element (i.e., H, C, N, O, P, and Ca) weights [10]. Then, the XCOM software of NIST is utilized to obtain the energy-dependent mass attenuation coefficients [24]. The same procedure had been described in references [14]. 2.2. Geometrical models 2.2.1. Coordinate system As shown in Fig. 2, a local coordinate system is set up for an X-ray source point S on the scanning trajectory, where the v -axis is parallel to natural x3 -axis and perpendicular to the u-axis. Let point A be the projection position of detector pixel B on the u-axis along the direction of v -axis. The rectangular triangle SOd A is completely inside the middle plane of the X-ray cone beam. The trajectory of X-ray source can be expressed as below in nature coordinate system, if it is smooth enough. Γ : = y = y(λ) ∈ R3 : y1 = y1 (λ), y2 = y2 (λ), y3 = y3 (λ), λ ∈ R (1) The space coordinates of S and B are denoted by y = (y 1 , y2 , y3 ) and z = (z1 , z2 , z3 ), then a unit vector β = (β1 , β2 , β3 ) pointing from S to B by β = (z − y)/||z − y||2 can be determined. Thereby, the X-ray SB can be parameterized as: x = y + tβ, (t 0). 2.2.2. Parameters calculation It is known that the geometrical shapes in mathematical phantom generally include sphere, ellipsoid, elliptical cylinder, elliptical cone, and so on. Here, any sphere is seen as a special case of ellipsoid. For any general geometrical shape, we first define its inside-outside function F (x) = F (x 1 , x2 , x3 ) as described in reference [18], which can be used to judge a point x = (x 1 , x2 , x3 ) inside, outside, or on the geometrical shape. For simplifying the simulation algorithm, we further define a standard geometrical shape F (x ) = F (x1 , x2 , x3 ) for any general shape, which can be used to judge if a point x = (x1 , x2 , x3 ) inside, outside, or on the geometrical shape.
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Fig. 2. Local coordinate system in a planar X-ray detector.
In order to transform a general geometrical shape to its corresponding standard shape, an affine transform with nine parameters is necessary. This transform can be expressed by a sequential multiplication of multiple matrices: 1 0 0 x01 cos φ − sin φ 0 0 x1 x2 0 1 0 x02 sin φ cos φ 0 0 = x3 0 0 1 x03 0 0 1 0 1 0 0 0 1 0 0 0 1 (2) cos ω 0 sin ω 0 1 0 0 0 a 0 0 0 x1 0 1 0 0 0 cos θ − sin θ 0 0 b 0 0 x2 − sin ω 0 cos ω 0 0 sin θ cos θ 0 0 0 c 0 x3 1 0 0 0 1 0 0 0 1 0 0 0 1 where x = (x1 , x2 , x3 ) is a point in R3 , x0 = (x01 , x02 , x03 ) is the coordinate position of geometrical center; θ , ω , and φ are the rotation angles around the three orthogonal semi-axes of geometrical shapes, whose lengths are a, b, and c, respectively. Formula (2) can be further compactly expressed as:
x x . (3) =Θ 1 1 Using Θ−1 , the inverse matrix of Θ, y , the transformed coordinates of X-ray source point S is obtained by
y −1 y =Θ (4) . 1 1
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Likewise, β can be determined by Θ −1 with x0 = 0. Therefore, X-ray S B , the transformed X-ray SB , can be parameterized as: x = y + t β , (t 0). Here we further emphasis that x, y , β , x 0 , θ , ω , φ, a, b, and c are related to the general geometrical shape, while x , y , and β correspond to the standard shape. According to the theory of differential geometry, all scalar quantities are invariants under affine transform. Therefore, t is equal to t and can be calculated according to the method described in appendix. This makes the whole simulation procedure more convenient and highly efficient. 2.2.3. Parameters sorting As for the ith X-ray, there is a number of parameter t, which are sorted in ascending order as Ti := {tm : tn tn+1 ; m, n, n + 1 ∈ N; m, n, n + 1 Ni } ,
(5)
where Ni is the intersection number between the ith X-ray and the surfaces of all geometrical models in mathematical phantom. By this way, the semi-infinite interval [0, ∞) has been divided into several subintervals. For every subinterval, its length and the parameter of its middle point are calculated and denoted by |tn+1 − tn | and tn,n+1 = (tn + tn+1 )/2, respectively. According to x = y + tn,n+1 β , we can judge if this point belongs to the interior of a sub-region in phantom, and then determine the corresponding CT-number, tissue mass densities and element weights, by the method introduced in Section 2.1. 2.3. Projections computation Generally speaking, if we have known the incident energy spectrum of X-ray photons, P i (tn , E), of point tn on the ith X-ray, the transmitted energy spectrum, P i (tn+1 , E), of next point tn+1 can be conveniently calculated using the distance, mass attenuation coefficients and mass density of the corresponding tissue between these two points, by the formula as follows, Pi (tn+1 , E) = Pi (tn , E) exp {−µ(tn,n+1 , E)ρ(tn,n+1 )|tn+1 − tn |} ,
(6)
where E represents the photon energy in keV; µ(t n,n+1 , E) and ρ(tn,n+1 ) are the energy-dependent mass attenuation coefficient and the mass density of the tissue at point t n,n+1 , respectively. In fact, both µ(t, E) and ρ(t) are constants within the subinterval [t n , tn+1 ). As long as all parameters t in T i of the ith X-ray have been traversed, the following expression can be utilized to calculate the final projection data [29]:
E max E min Pi (t1 , E)Q(E)dE , fi = log E max (7) P (t , E)Q(E)dE i N i E min where t1 and tNi are the position parameters of the first and final boundary points on the ith Xray, respectively. Emin and Emax denote the minimum and maximum photon energy in X-ray energy spectrum, respectively. P i (t1 , E) and Pi (tNi , E) are the energy spectrum distributions of a polychromatic X-ray before and after passing through the mathematical phantom, respectively. P i (t1 , E) is shown in Fig. 3(a). Pi (tNi , E) is calculated according to formula (6). Q(E) is the absorption energy spectrum distribution of the X-ray detector as shown in Fig. 3(b).
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Fig. 3. (a) The energy spectrum distributions of the polychromatic X-ray of the potential 120 kVp with and without aluminum filtration. The values of solid and dashed lines have been magnified 5 times. The horizontal axis is photon energy with 1keV interval and the vertical axis is the relative number of photons with different energy. (b) The energy spectrum distribution of the Na:CsI X-ray detector. The horizontal axis is photon energy with 1 keV interval and the vertical axis is energy spectrum distribution.
When quantum noise of X-ray photons are considered, there is
E max Poisson {P (t , E)} Q(E)dE i 1 E min , fi = log E max E min Poisson {Pi (tNi , E)} Q(E)dE
(8)
where Poisson{} is used to generate a random number obeying Poisson distribution.
3. Numerical simulation and analysis 3.1. Mechanical parameters The generation of projections depends on the mechanical configuration of XCT imaging system. In our developed software, several parameters can be configured such as rotation axis orientation, scanning trajectory (circular or helical), the distance from rotation centre to detector centre, the distance from rotation centre to X-ray source point, detector sampling method (equi-spatial or equi-angular), the numbers of row and column of detector cell, the size of detector cell, and projection number per rotation circle. For the numerical simulation below, we simulate a simple circular scanning trajectory with a radius 50 cm to demonstrate our proposed method. The locus is located in the x 1 x2 plane whose centre is the origin O of the natural coordinate system. Therefore, the X-ray source will rotate around the x 3 -axis. The imaged object, i.e., FORBILD head phantom, is placed such that the center of its circumcircle with a radius 12.8 cm is the origin O. An equi-spatial planar detector is positioned opposite to the X-ray source about the origin O, and the distance between O and O d is 50 cm. The detector includes 850 × 80 apertures, each of which covers a region of 1 mm × 1 mm. When the X-ray source is rotated a circular turn, 1080 cone-beam projections are equi-angularly acquired.
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Fig. 4. The polychromatic projections of the water cylinder phantom (a) and the water cylinder phantom with a bone shell (b). Note that one million photons penetrate through every X-ray path.
3.2. X-ray energy spectrum The primary energy spectrum of X-ray source is shown in Fig. 3(a) [25], and the absorption energy spectrum of Na:CsI X-ray detector is shown in Fig. 3(b) [26]. It is assumed that no scattered photon reaches the surface of X-ray detector. Energy interval 1keV is used to divide the beam into numerous energy levels. As compared to the polychromatic simulation, a monochromatic simulation is carried out simultaneously, which represents the ideal situation of linear integral projection. In our developed software, the tube potential of polychromatic X-ray source, the photon energy of monochromatic Xray source, and the thickness of aluminum filtration can be configured conveniently. In the numerical simulation below, the tube potential of X-ray source is 120 kVp, and the incident polychromatic X-ray is filtered by a 35 mm aluminum slab, which is similar to the situation of the real CT equipment (Fig. 3(a)). The photon energy of monochromatic X-ray source is 66 keV, because this photon energy is the effective energy of the polychromatic X-ray source. 3.3. Simulation validation In order to validate our simulation method, a water cylinder phantom and a water cylinder phantom with a bone shell are projected by our simulation method and MCNP-4C [16]. The radius of the water cylinder is 8 cm, the thickness of bone shell is 2cm. Only one view of projection is calculated for each phantom as shown in Fig. 4, where analytic and Poisson simulations have been described at formulas (7) and (8) in Section 2.3. From Fig. 4, we can notice that our simulation projections are of very higher consistency compared with the projection results of MCNP-4C [16], which can be seen as a realistic data as no real X-ray CT system can be operated. In this paper, one million photons penetrate through every X-ray path, no matter what simulation method is considered. 3.4. Generalized Feldkamp reconstruction Generalized Feldkamp algorithm has been published, adopted, or discussed in references [27,30– 34], and all these previous works are accomplished only considering line integral projections, i.e., monochromatic projections. In this paper, the algorithm is to reconstruct the mid-plane image from
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simulated polychromatic projections. The reconstructed image is a 512 × 512 matrix and the size of each pixel is 0.5 mm × 0.5 mm. The reconstruction is theoretically exact since the reconstructed plane is completely inside the mid-plane of cone beam. In order to meet convention, we further transform reconstructions by a simplified “HU correction” formula as follows, I(i, j) =
µ (i, j) − µw × 1000, µw
(9)
where µ is an untransformed reconstructed image (i.e., effective linear attenuation coefficients) in 1/mm. µw is the effective linear attenuation coefficient of pure water under the same imaging condition, 0.0184 1/mm by experience. I is the transformed reconstructed image in HU. The reconstructed images of the FORBILD head phantom are shown in Fig. 5. In these images, beam hardening effects are visible. That is, there are some evident dark streak artifacts along the extending direction of sub-region 14, which can be seen from polychromatic and difference images in x 1 = 0 mm (first row); similarly, along the extending directions of sub-regions 10 and 11, we also can see evident dark streak artifacts from images in x3 = 0 mm (third row) and x3 = 5 mm (last row). From all difference images (right column), we can clearly find that there are anticipated cupping artifacts. Cupping artifacts and streak artifacts are the two representative characteristics of beam hardening effects in polychromatic XCT. 3.5. Beam hardening correction Based on our projection simulation method, several typical beam hardening corrections, such as water correction [1], bone correction [2], and ECC [3], are operated. In bone correction, 1.39 is chosen as a typical empirical value of λ0 , which is a scaling factor and adopted to evaluate the mass density ρ of bone material from a pre-reconstruction image µ as follows, ρ(i, j) = µ (i, j) λ0 , µ (i, j) > µT , (10) where µT is an empirical threshold between the effective linear attenuation coefficients of bone material and soft tissue material. For these three corrections, the form of linearization polynomial is as follows, f c = c1 f + c2 f 2 + c1/2 f 1/2 + c3 f 3 + c1/3 f 1/3 = c · f,
(11)
which is slightly different from its usual form but more efficient in correction performances. Where f and f c are uncorrected and corrected polychromatic projections, respectively. For water and bone corrections, c = c1 c2 c1/2 c3 c1/3 (12) = 0.9791 0.0037 −0.0076 −0.0001 0.0049 , is calculated by a sequential of simulated water slabs with area density from 1 g/cm 2 to 50 g/cm2 with 1 g/cm2 interval. Note that water correction is used as a pre correction in bone correction. As for ECC, c = c1 c2 c1/2 c3 c1/3 (13) = 0.9868 −0.0022 −0.0033 0.0002 −0.0273 , is calculated by a simulated water ellipse with two orthogonal semi-axes lengths of 96 mm and 256 mm, respectively.
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Fig. 5. The reconstructed images of the FORBILD head phantom. From the first row to the last row are in x1 = 0 mm, x1 = − 67 mm, x3 = 0 mm and x3 = 5 mm, respectively. The left and central columns, displayed with the window [0 167]HU, are the monochromatic and polychromatic images. And the right column, displayed with the window [−17 17]HU, is the difference images between polychromatic and monochromatic images. The white lines in the first column are for marking the positions of profiles. The positions of the lines in the left low image are at the 215th, 256th and 385th from up to low.
3.6. Results analysis The corrected reconstructions of the section x 1 = 0 mm in the FORBILD head phantom are shown in Fig. 6, where in the first and second rows are analytic simulations without noise, in the third and last rows are analytic simulations with Poisson noise. This phantom section is chosen for its characteristic of a long strip of bone regions. In the first and third rows of the figure, the evident dark streak artifacts along the extending direction of sub-region 14 have been perfectly compensated by bone correction. On the contrary, water correction and ECC can not combat with streak artifacts. From the second and last rows of the figure, we notice that all corrections can overcome cupping artifact. And especially, bone correction can deal with both cupping and streak artifact effectively. Water correction and ECC have a residue cupping artifact uncorrected, which is proved by the slightly lower values in the central regions than in the outer soft tissue regions. The corrected reconstructions of the sections x 1 = − 67 mm, x3 = 0 mm, and x3 = 5 mm are shown in Figs 7, 8, and 9, respectively. Their layouts are as in Fig. 6. The phantom section in Fig. 7 is selected
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Fig. 6. The corrected reconstructions of the FORBILD head phantom at x1 = 0 mm. The left, central, and last columns are water correction, bone correction, and ECC, respectively. And the images in the first and third rows, displayed with the window [0 167]HU, are corrected reconstructions without and with Poisson noise, respectively; the images in the second and last rows, displayed with the window [−17 17]HU, are the difference images between the corresponding corrected images and monochromatic reconstructions.
for its characteristic of large block of soft tissue region surrounded by an annulus of bone. Figure 8 is for its high spatial resolution in right ear. Figure 9 is for its large block of bone in right ear and the subdural hematoma in sub-region No. 13. In the second and last rows of Fig. 7, the effectiveness of water correction and ECC are almost the same. However, the capability of bone correction is quite different from water correction and ECC. The second and last rows of the figure also indicate that water correction and ECC have a residue cupping artifact uncorrected. Both Figs 6 and 7 say that water correction and ECC will not be very effective if there are plural kinds of materials in the imaged objects. In Figs 8 and 9, more attentions should be paid to the region from sub-region No. 14 to right ear. The reason that this region is infected with dark streak artifact is the collective contributions of sub-regions No. 9, 11, 14 and right ear. Bone correction is validated again that it is capable of decreasing dark streak artifact in polychromatic reconstructions. And other arguments are similar with in Figs 6 and 7. In the legends of Figs 10, 11 and 12, polychromatic reconstruction, monochromatic reconstruction, bone correction, water correction and empirical cupping correction are denoted by Poly, Mono, Bone, Water and ECC, respectively. The curves in Fig. 10 are the profiles of the first and second rows in
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Fig. 7. Same as Fig. 6 but at x1 = − 67 mm.
Figs 6–9, at the corresponding positions of the white lines in the first column of Fig. 5. Figure 10(a) corresponds to Fig. 6. In the inset of Fig. 10(a), the reconstructed values of monochromatic situation and bone correction are nearest. The reconstructed values of water correction and ECC stay with that of polychromatic situation, and present a discrepancy around 60HU as compared to monochromatic reconstruction. Figure 10(b), corresponding to Fig. 7, shows the reconstructed values in a soft tissue region near the inner side of a bone wall, where a so-called “pseudo cortex artifact” arise if an improper correction or no correction is performed. As for bone correction, the corresponding profile is the closest with monochromatic situation, however, shows an over correction as compared to monochromatic result. Figure 10(c), corresponding to Fig. 8, is dedicated to illustrate the correction characteristic of bone correction in a region with higher density contrast and higher spatial resolution such as the cochlea of human being. In this situation, polychromatic reconstruction appears a difference quantity of CTnumber near 80HU as compared to monochromatic result. As illustrated by the curves in Fig. 10(c), bone correction can reduce this difference quantity from 80HU to 30HU. It is already an outstanding performance, whereas may be further improved. On the other hand, these difference quantities degrade MTF of XCT system in some degree, and the degradation extent can be expressed by a degradation factor as follows, Df = (80HU − 30HU)/860HU × 100% ≈ 6%,
(14)
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Fig. 8. Same as Fig. 6 but at x3 = 0 mm.
where 80HU and 30HU are the difference quantities mentioned above, and 860HU is the CT-number of monochromatic reconstruction in the region of right ear. We notice that this degradation factor is only for a fixed region and spatial resolution. As for other region or spatial resolution, corresponding degradation factor can be determined similarly by designing a slightly different phantom. Figures 10(d), (e) and (f) correspond to the 256th, 215th and 385th lines of the first row of Fig. 9, respectively. The beam hardening artifacts in Figs 10(b) and 10(d) are also called “spill-over” phenomenon in earlier literatures. Due to the large block of bone region in right ear, the difference quantity of polychromatic and monochromatic reconstructions is large near 50HU in Fig. 10(d). Result of bone correction is near monochromatic value, whereas those of water correction and ECC are close to polychromatic value. Water correction and ECC are of less effect on “spill-over”. Figure 10(e) is to illustrate the correction effectiveness for dark streak artifact, as shown by curves from pixel 280 to 380. In this interval, the maximum fluctuation of bone correction is less than 10HU, however, those of water correction, ECC and polychromatic result are almost near 15HU. Although these profiles show that bone correction can reduce the streak artifact in some degree, the over correction is also observed. It seems that there is still possibility to improve the bone correction. Figure 10(f) is for the subdural hematoma in sub-region No.13 as shown by curves from pixel 370 to 400. In the neighborhood of pixel 400, because of the negative influence of “pseudo cortex artifact”, water correction, ECC, and polychromatic result can’t distinguish the subdural hematoma from
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Fig. 9. Same as Fig. 6 but at x3 = 5 mm.
surrounding background successfully. Here, bone correction is superior to water correction and ECC, however, also shows an over correction as compared to monochromatic result. The curves in Fig. 11 are the profiles of the third row in Fig. 9, corresponding to the positions of the central horizontal lines. Although there are visible noises on the curves of Fig. 11 as compared to Fig. 10(d), their tendencies are similar to each other, except that the profile amplitude of bone correction is influenced by the addition of Poisson noise. The curves in Fig. 12 are the correlation coefficients in bone region (Fig. 12(a)) and soft tissue region (Fig. 12(b)) at section x3 = 0 mm in the FORBILD head phantom. The horizontal axis is λ 0 in bone correction and the vertical axis is the correlation coefficients between respective noiseless correction and monochromatic reconstruction. Here, bone and soft tissue regions refer to the regions of CT numbers 800HU and 50HU in original definition of FORBILD head phantom, respectively. In both regions, the performances of water correction and ECC are superior to polychromatic reconstruction. For bone correction, there is an obvious maximum of correlation coefficient in both regions, where the maximum positions are slightly different and the maximum values are near 100%. We also notice that the correlation coefficients are more superior in both regions, when the incident polychromatic X-ray is filtered by a 35 mm aluminum slab than by a 25 mm aluminum.
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Fig. 10. The profiles of reconstructions at the positions of white lines. (a) is from the first row of Fig. 6; (b) is from the first row of Fig. 7; (c) is from the second row of Fig. 8; (d), (e), and (f) are from the first row of Fig. 9, corresponding to the positions of 256th, 215th, and 385th horizontal lines, respectively.
4. Conclusion Designing a sophisticated physical mathematical phantom can help to precisely evaluate beam hardening correction methods and to develop more perfect correction algorithm. In this paper, we have presented an analytic method on simulating the realistic X-ray projections of a well-known mathematical phantom. During simulation, three general geometrical models are converted into their corresponding standard models by an affine transformation. The mathematical phantom is mapped to its physical counterpart, and its polychromatic projections are calculated. And thus, simulated projections will be
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Pixel Fig. 11. The profiles of reconstructions with and without correction, at the positions of white lines. The curves are from the third row of Fig. 9, corresponding to the positions of the central horizontal lines.
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0
(b)
Fig. 12. The correlation coefficients in bone region (a) and soft tissue region (b) at x3 = 0 mm of FORBILD head phantom. The horizontal axis is λ0 in bone correction and the vertical axis is the correlation coefficients between respective noiseless correction and monochromatic reconstruction. Note that 120 kVp + 25 mmAl and 120 kVp + 35 mmAl in the legends denote corresponding energy spectrum distributions in Fig. 3(a).
more consistent with those acquired by a real data acquisition system. The whole method is concise and compact, and it is very convenient to rapidly develop a dedicated software program. This is definitely meaningful for creating other more sophisticated mathematical phantoms. On the basis of this simulation method, several typical beam hardening corrections have been analyzed and compared quantitatively. From all reconstructed images and profiles, firstly, we can see that water correction and ECC are of a similar performance, bone correction is superior to them. The underlying causality is that bone correction distinguishes imaged object into two kinds of materials, i.e., bone and soft tissue, rather than water correction and ECC treat imaged object only as a water-like or soft tissue-like material. As we know, pre-correction, pre-segmentation and re-projection are necessary in bone correction for its excellent capabilities. Therefore, much more computation cost must be paid.
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By this way, bone method is applicable to the corrections of “cupping artifact”, “streaks artifact”, and “spill-over”, rather than water correction and ECC can only cope with “cupping artifact”. Second, it should be noticed that bone correction is prone to be infected with over correction due to the empirical choice of λ0 , because the fixed λ 0 can not completely adapt to the attenuation content change of imaged object. There is still a considerable space to improve the capability of bone correction as compared to monochromatic reconstruction. Third, all these beam hardening corrections mentioned above are stable when Poisson noise is added to the polychromatic simulation projections before logarithm operation. Finally, the polychromatic reconstruction degrades MTF of XCT system in some degree, and bone correction can improve this MTF by a factor of near 6%.
Acknowledgements The project is partially supported by National Science Fund of China (N0. 60551003 and No. 60472004), and the program of NCET (No. NCET-05-0828).
Appendix A.1. Ellipsoid model For a general ellipsoid, its inside-outside function is defined by Fe (x) = Fe (x1 , x2 , x3 ) := [(x1 − x01 ) cos ω cos φ + (x2 − x02 ) cos ω sin φ + (x3 − x03 ) sin ω]2 a2 +
[(x1 −x01 )(sin θ sin ω cos φ−cos θ sin φ)+(x2 −x02 )(sin θ sin ω sin φ+cos θ cos φ)+(x3 −x03 ) sin θ cos ω]2 (A.1) b2
+
[(x1 −x01 )(cos θ sin ω cos φ+sin θ sin φ)+(x2 −x02 )(cos θ sin ω sin φ−sin θ cos φ)+(x3 −x03 ) cos θ cos ω]2 , c2
as described in reference [18]. If F e (x) < 1, x is inside the ellipsoid; if Fe (x) = 1, x is on the surface of the ellipsoid; if Fe (x) > 1, x is outside the ellipsoid. The inside-outside function of a standard ellipsoid, in fact a unit sphere, is defined as follows, 2
2
2
Fe (x ) = Fe (x1 , x2 , x3 ) := x 1 + x 2 + x 3 ,
(A.2)
where if Fe (x ) < 1, x is inside the unit sphere; if Fe (x ) = 1, x is on the surface of the unit sphere; if Fe (x ) > 1, x is outside the unit sphere. After the affine transform Θ−1 has been executed, there is a quadric equation with respect to t: (y 1 + tβ 1 )2 + (y 2 + tβ 2 )2 + (y 3 + tβ 3 )2 = 1.
(A.3)
After the above equation is solved, the length of the line segment between the boundaries of ellipsoid can be calculated. Here, using the similar method described in reference [28], we have ψt2 + 2ξt + η = 1,
(A.4)
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Fig. A1. The ellipsoid body and x-ray SB. S is the x-ray source point, Sa and Sb are two intersecting points of the x-ray with the ellipsoid body. And the left part of figure is the transformed result.
where ψ = β · β ξ = β · y , η = y · y and sign “·” denotes cross multiplication. The roots of Eq. (A.4) are √ −ξ ± ξ 2 − ψ(η − 1) −ξ ± ∆ = . ta,b = ψ ψ
(A.5)
(A.6)
If ∆ > 0, Eq. (A.4) yields two intersections between X-ray S B and the unit sphere as shown in Fig. A1, and then we record these two parameters t a and tb corresponding to intersections S a and Sb , respectively.
A.2. Elliptical cylinder model Similarly, given a general elliptical cylinder, we define its inside-outside function by Fecy (x) = Fecy (x1 , x2 , x3 ) := [(x1 − x01 ) cos ω cos φ + (x2 − x02 ) cos ω sin φ + x3 sin ω]2 a2 +
(A.7)
[(x1 − x01 )(sin θ sin ω cos φ − cos θ sin φ) + (x2 − x02 )(sin θ sin ω sin φ + cos θ cos φ) + x3 sin θ cos ω]2 , b2
where if Fecy (x) < 1, x is inside the elliptical cylinder; if Fecy (x) = 1, x is on the surface of the elliptical cylinder; if Fecy (x) > 1, x is outside the elliptical cylinder. However in mathematical phantom, elliptical cylinders are always clipped by two parallel planes. And in general situation, these two planes are perpendicular to the longitudinal axis of elliptical cylinder. Although there is no conceptions of inside and outside for a plane in R 3 , we still define an inside-outside function for one parallel plane by F±1 (x) = F±1 (x1 , x2 , x3 ) :=
(A.8)
(x1 − x01 )(cos θ sin ω cos φ + sin θ sin φ) + (x2 − x02 )(cos θ sin ω sin φ − sin θ cos φ) + (±x3 − x03 ) cos θ cos ω , c
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where if F±1 (x) < 1, x is inside the plane; if F±1 (x) = 1, x is on the surface of the plane; if F±1 (x) > 1, x is outside the plane. And we notice that the interior of a general elliptical cylinder in mathematical phantom may be expressed as {x ∈ R3 : Fecy (x) < 1} ∩ {x ∈ R3 : F±1 (x) < 1}.
(A.9)
Just as in above subsection, we need to further define a standard elliptical cylinder, in fact a cylinder clipped by two parallel planes. The inside-outside function of cylinder is as follows, 2
2
F ecy (x ) := x 1 + x 2 ,
(A.10)
(x ) < 1, x is inside the cylinder; if F (x ) = 1, x is on the surface of the cylinder; if where if Fecy ecy Fecy (x ) > 1, x is outside the cylinder. And inside-outside function of two parallel planes is as follows, F±1 (x ) = F±1 (x1 , x2 , x3 ) := ±x3 ,
(A.11)
(x ) < 1, x is inside the plane; if F (x ) = 1, x is on the surface of the plane; if where if F±1 ±1 F±1 (x ) > 1, x is outside the plane. Simultaneously, we define the interior of a cylinder clipped by two parallel planes by
{x ∈ R3 : F ecy (x ) < 1} ∩ {x ∈ R3 : F ±1 (x ) < 1} 2
2
= {x ∈ R3 : x 1 + x 2 < 1, −1 x3 1}.
(A.12)
In order to successfully convert a general elliptical cylinder to a standard one, also an affine transform Θ−1 is required. After accomplishing this transform, we determine four intersections between X-ray S B and cylinder as shown in Fig. A2, and record these parameters t a , tb , tc , and td corresponding to intersections Sa , Sb , Sc , and Sd , respectively. (x ) = 1, there is a quadric equation with respect to t: If X-ray S B intersects Fecy (y 1 + tβ 1 )2 + (y 2 + tβ 2 )2 = 1.
(A.13)
We have ψt2 + 2ξt + η = 1,
where ψ = [β 1 , β 2 , 0] · [β 1 , β 2 , 0] ξ = [β 1 , β 2 , 0] · [y 1 , y 2 , 0] . η = [y 1 , y 2 , 0] · [y 1 , y 2 , 0] The roots of Eq. (A.14) are √ −ξ ± ξ 2 − ψ(η − 1) −ξ ± ∆ = . ta,b = ψ ψ
(A.14)
(A.15)
(A.16)
(x ) = 1, there is a simple equation with respect to t: If X-ray SB intersects F±1
y3 + tβ3 = ±1.
(A.17)
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Fig. A2. The elliptical cylinder and x-ray SB. S is the X-ray source point. Sa and Sb are two intersecting points of the X-ray with the elliptical cylinder; Sc and Sd are two intersecting points of the X-ray with the two parallel planes clipping the elliptical cylinder. And the left part of figure is the transformed result.
We have tc,d =
−y3 ± 1 . β3
(A.18)
A.3. Elliptical cone model Given a general elliptical cone, its inside-outside function is defined by Fec (x) = Fec (x1 , x2 , x3 ) := [(x1 − x01 ) cos ω cos φ + (x2 − x02 ) cos ω sin φ + (x3 − x03 ) sin ω]2 a2 [(x1 −x01 )(sin θ sin ω cos φ−cos θ sin φ)+(x2 −x02 )(sin θ sin ω sin φ+cos θ cos φ)+(x3 −x03 ) sin θ cos ω−b]2 − b2 (A.19) [(x1 −x01 )(cos θ sin ω cos φ+sin θ sin φ)+(x2 −x02 )(cos θ sin ω sin φ−sin θ cos φ)+(x3 −x03 ) cos θ cos ω]2 , + c2
where if Fec (x) < 0, x is inside the elliptical cone; if Fec (x) = 0, x is on the surface of the elliptical cone; if Fec (x) > 0, x is outside the elliptical cone. Correspondingly, we define a standard elliptical cone, in fact a cone, as follows, 2
2
F ec (x ) = F ec (x 1 , x 2 , x 3 ) := x 1 − (x 2 − 1)2 + x 3 , F
(x )
x
F
(x )
(A.20) x
where if ec < 0, is inside the cone; if ec = 0, is on the surface of the cone; if F ec (x ) > 0, x is outside the cone. Actually in mathematical phantom, a general elliptical cone is generally clipped by its vertex. The interior of a standard elliptical cone may be expressed as {x ∈ R3 : F ec (x ) < 0} ∩ {x ∈ R3 : x 2 < 1} 2
2
= {x ∈ R3 : x 1 − (x 2 − 1)2 + x 3 < 0, x 2 < 1}.
(A.21)
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Fig. A3. The elliptical cone and X-ray SB. S is the X-ray source point. Sa and Sb are two intersecting points of the X-ray with the elliptical cone. And the left part of figure is the transformed result.
In order to convert a general elliptical cone to a standard one, also an affine transform Θ −1 is needed. After the operation of this transform, the two intersections between X-ray S B and cone are determined as shown in Fig. A3, and we record the parameters t a and tb corresponding to intersections S a and S b , respectively. If X-ray S B intersects F ec (x ) = 0, there is a quadric equation with respect to t: (y 1 + tβ 1 )2 − (y 2 + tβ 2 − 1)2 + (y 3 + tβ 3 )2 = 0.
(A.22)
We have ψt2 + 2ξt + η = 0,
where ψ = [β 1 , β 2 , β 3 ] · [β 1 , −β 2 , β 3 ] ξ = [β 1 , β 2 , β 3 ] · [y 1 , 1 − y 2 , y 3 ] . η = [y 1 , y 2 − 1, y 3 ] · [y 1 , 1 − y 2 , y 3 ] The roots of Eq. (A.23) are √ −ξ ± ξ 2 − ψη −ξ ± ∆ = . ta,b = ψ ψ
(A.23)
(A.24)
(A.25)
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