Improve spatial resolution by Modeling Finite Focal Spot (MFFS) for industrial CT reconstruction

Ming Chang; Yongshun Xiao; Zhiqiang Chen

doi:10.1364/OE.22.030641

1. Introduction

Since the x-ray CT technology was employed, it has been widely developed in both the medical diagnostic imaging field and the industrial Non-Destructive Testing (NDT) area. It has achieved great success in diagnosis of diseased tissues and defect detections. However most of these applications are concentrated on qualitative detection. Recently, an increasing requirement for high-resolution quantitative measurement has arisen with the development of advanced technologies, especially in the industrial NDT area. For example, it has become a new task to perform dimensional metrology by CT [1–3]. High spatial resolution CT images can not only help us to inspect small-sized defects but also provide more accurate geometric information for further comparisons and analysis [4, 5]. Therefore the high spatial resolution is always pursued as one of our goals [6, 7].

In general, there are many factors that influence the spatial resolution of industrial CT systems, including the focal spot size of an x-ray source a, detector unit size d, the magnification of the scanning system M, mechanical precision, reconstruction algorithm and the material of the scanned object [8–10]. As shown in Fig. 1 the first three factors determine the system’s physical resolution and Eq. (1) is commonly used to describe the scanner’s maximum ability to resolve closely placed objects [11].

BW = \sqrt{{U_{F}}^{2} + {U_{D}}^{2}} = \sqrt{{[(1 - \frac{1}{M}) a]}^{2} + {(\frac{d}{M})}^{2}}

Fig. 1 Edge blurs caused by the finite focal spot size a and detector unit size d.

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As the magnification M can be adjusted appropriately, the first two factors become the major physical limitations of the scanning systems. To reduce the detector’s limitations, a lot of work has been carried out to decrease the effective detector unit size [12] or increase the sampling frequency by focal spot wobbling [7, 13]. However these methods become invalid when the detector unit size is relatively smaller than the focal spot size. It can be explained by Eq. (1). For example, the typical focal spot size of the 600keV industrial CT x-ray tube is about 1mm at the power of 1.5kW and the corresponding detector unit size is about 0.2mm. The physical resolution is calculated by Eq. (1) with the magnification of the scanning system set to 2.0. To make the problem clearer, the physical resolution BW is calculated with focal spots and detector units set to different sizes respectively. The result is shown in Fig. 2. The solid line shows that the BW value changes with the detector unit size d ranging from 0.02mm to 0.2mm and the focal spot size a is fixed to 1.0mm. The dotted line indicates the BW value as a function of the focal spot size a ranging from 0.1mm to 1.0mm and the detector unit size d is fixed to 1.0mm.

Fig. 2 Illustration of the different influences on the physical resolution BW as a function of focal spot size a (dotted line) and detector bin size d (solid line) respectively..

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It can be found from Fig. 2 that the decreasing of the detector unit size has little impact on the physical resolution. On the contrary, the physical resolution is obviously enhanced with the decrease of the focal spot size. These results indicate that the finite focal spot size is one of the most important factor that restrict the spatial resolutions of the high-energy industrial CT system. Therefore, to improve the spatial resolution of the high-energy CT system with a macro-focus x-ray source, more efforts should be devoted to reduce the effective focal spot sizes.

However, due to the requirements of high energy and high power to the industrial CT system, there are still some other factors in the construction of an x-ray tube, such as the heat dissipation within the target, which limits the minimum size of the target of the x-ray tube. Therefore the practical projection model of an industrial CT doesn’t meet the assumption that the projections are produced by the point x-ray sources, which is commonly used in conventional reconstruction algorithms. As a result, the spatial resolution of a reconstructed image is limited by the blurring effect due to the finite focal spot size of the real CT scanning system. To overcome this limitation, many methods have been proposed in the literatures [14–18]. They can be divided into two categories. The first one is finite focal spot discretization based methods. The main idea behind such methods is to divide the focal spot into several finer components. Then the model-based iterative reconstruction methods [15–18], which take the finite spot size into consideration, are adopted for achieving a high-spatial resolution. The finer the discretization of the focal spot is, the more exact the projection model is. This makes it possible to achieve a higher spatial resolution. However it also requires a heavier computational cost due to the reduplicative forward and backward projection operations with a more complex projection model [16]. Therefore the tradeoff between the image quality and the time spent must be considered when it is applied to practical applications. The second one is to use the aperture collimator to refine the large focus of industrial x-ray source [6]. In this method, a specially designed aperture collimator is needed and it is used to reduce the coupling between different parts of the attenuation coefficient distribution in the projection data. However, to block the high-energy x-ray, a fairly thick collimator with several uniformly distributed tiny holes should be manufactured, which is difficult to achieve. Moreover the improvement of the spatial resolution is at the expense of about 35% radiation utilization efficiency [6]. Furthermore, to determine the parameters of the finite focal spot model, many efforts have been devoted to measuring the focal spot size and the intensity distributions [19–21]. Its influences to the spatial resolution and the accuracy of the CT number are also widely developed [15, 22].

It should be noted it is hard to handle the reconstruction problem under such a complex projection model by the analytical methods directly. When all the aforementioned methods are implemented by iterative reconstructions, the time spent or the computational cost is the major obstacles to the industrial applications. In this work, an efficient reconstruction framework by modeling finite focal spot (MFFS) is presented to improve the spatial resolution. The unique advantages of the proposed method are the high efficiency and great improvements in the spatial resolution without any hardware changes. The key is to build the approximate linear equivalence relationship between the finite focus model and the ideal point source model. Then, under this relationship, the projection of the finer focal spot can be recovered efficiently from measured projections of a relatively large focus before the reconstructions. Finally the standard analytical algorithms, such as the FBP method, can be applied to achieve a high spatial resolution image using the recovered projections. In the following section, the details about the proposed method are described. In the third section, the numerical simulations and experiments are reported to show the significant improvements in the spatial resolution. Finally, some discussions and conclusions are drawn at the end of this paper.

2. Methodology

First, a finite focal spot size based projection model is presented. Then the approximate linear equivalence relationship between the actual finite focus model and the ideal point source model is established according to the data completeness. The projection recovery method based on such an equivalence relationship, is presented. However, the recovery process by the proposed method will cause the high noise in projections and reconstructed images. Therefore a difference image based fusion method is proposed in the end of this section to further improve the reconstructed image quality.

2.1. Finite focal spot size based projection model

In the ideal CT imaging model, the x-ray focal size can be ignored or it is regarded as an ideal point source. The projection model under ’ideal’ conditions can be expressed as Eq. (2) according to the Beer-Lambert law.

I (a, t) = I_{0} (a, t) \exp (- p (a, t)) where p (a, t) = \int_{L} μ (l) d l

For a given attenuating path determined by the source and detector unit, denote the incident and the penetrated x-ray intensity as I₀(a, t) and I(a, t), respectively. The total attenuation of the x-ray caused by the object is calculated by p(a, t). The μ(l) is the linear attenuation coefficient distribution of the scanned object and L is the path x-ray passing through the object. The a and t is the source coordinate and the detector coordinate respectively.

However the measurements obtained on any industrial CT scanner rarely satisfy these conditions. As mentioned before, due to the heat dissipation within the target of an x-ray tube, the x-ray source of a high-energy industrial CT is far from the ideal point source. To make the projection model more realistic, the focal spot size must be taken into consideration. The measured intensity of the x-ray before and after attenuated by the object under the finite focal spot size based projection model should be expressed as Eq. (3) and Eq. (4).

I_{measure} (t) = \int_{Ω} I (a, t) d a = \int_{Ω} I_{0} (a, t) \exp (- p (a, t)) d a

I_{air} (t) = \int_{Ω} I_{0} (a, t) d a

As the Eq. (3) shown, the measured data I_measure(t) is the accumulation of the ideal projections I(a, t) produced by point source over the finite focal spot support Ω. The negative logarithm of the quantity should be taken before the measured data is used for reconstruction.

\begin{array}{l} q (t) & = - ln [I_{measure} (t) / I_{air} (t)] \\ = - ln {\exp (- p (a_{0}, t)) [\int_{Ω} I_{0} (a, t) \exp (- (p (a, t) - p (a_{0}, t))) d a]} + ln [I_{air} (t)] \\ = p (a_{0}, t) - ln (\int_{Ω} I_{0} (a, t) \exp (- p (a, t) - p (a_{0}, t)) d a) + ln [I_{air} (t)] \\ where a_{0} \in Ω \end{array}

We assume that the difference p(a, t) − p(a₀, t) between the projections produced by two different point focuses in the support is minimal. This assumption is satisfied in most cases because the focal spot support is always small. Then Eq. (5) can be approximate by Eq. (6) using the Taylor expansion.

\begin{array}{l} q (t) & = p (a_{0}, t) - ln (\int_{Ω} I_{0} (a, t) [1 - (p (a, t) - p (a_{0}, t)] d a) + ln [I_{air} (t)] \\ = p (a_{0}, t) - ln {1 - \int_{Ω} I_{0} (a, t) / I_{air} (t) [(p (a, t) - p (a_{0}, t)] d a} \\ = \int_{Ω} \frac{I_{0} (a, t)}{I_{air} (t)} p (a, t) d a \end{array}

If the intensity angular distribution of the x-ray source can be ignored, the attenuated projections based on the finite focal spot size can be simplified as Eq. (7). It is a weighted integral of ideal projections of a given point source over the finite focal spot support Ω. The weighted factor w(a) only depends on the source coordinate a and it can be determined by the normalized intensity contributions of a given point source to the whole finite focal spot support Ω.

q (t) = \int_{Ω} w (a) p (a, t) d a, where w (a) = I_{0} (a) / \int_{Ω} I_{0} (a) d a

By Eq. (7), the corresponding discrete model can be expressed as Eq. (8), where the finite focal spot size is divided into M virtual point sources.

q (t) = \sum_{i = 1}^{M} w (a_{i}) p (a_{i}, t) where w (a_{i}) = I_{0} (a_{i}) / \sum_{i = 1}^{M} I_{0} (a_{i})

2.2. Projection recovery method and CT reconstruction

Compared to the ideal point source projection model, (M − 1) times more forward-projections and backward-projections are necessary in Eq. (8). This means high computational costs when the iterative reconstruction methods are applied, which is usually unacceptable in practice. In this work, a more efficient reconstruction framework by modeling finite focal spot (MFFS) is proposed. The key is to recover the projections of the finer focal spot from the measured projections, which is produced by larger focal spot source, before reconstructions. Then the highly efficient FBP algorithm can be applied for reconstructions using the recovered projections.

To recover the projections, the equivalence relationships between the projections produced by the virtual point source and by the ideal point source are deduced. To make it clear, the geometric relationships are shown in Fig. 3. The explanations of corresponding symbols in it are given in Table 1. In the ideal projection model, the detector unit Det₁ is irradiated by the point source S₁. However, in the finite focal spot size model, the x-ray fluxes from other point sources besides the S₁ in the focal spot support, such as the virtual point source S₂_i, should also be considered. Taking the completeness of the projection data into consideration, for any x-ray attenuating path in the scanned objects, there is always a equivalent x-ray path, which can be found in the standard complete projection data. Specifically, as shown in Fig. 3, the x-ray path determined by the point source S_2i and the detector unit Det₁ is equivalent to the path determined by the point source S_2i′ on the circular orbit and the virtual detector unit Det_2i. This equivalence between the projection p(a_i, t₁) produced by the virtual point source S_2i and the projection from a standard circular orbit projection p_cir(β_i, t_2i) can be calculated as Eq. (9) shown. The β₁ is the projection view angle and it is marked in Fig. 3.

\begin{array}{l} {\begin{array}{c} p (a_{i}, t_{1}) = p_{cir} (β_{i}, t_{2 i}) \\ p (0, t_{1}) = p_{cir} (β_{1}, t_{1}) \end{array} & where {\begin{array}{l} θ_{i} = {tan}^{- 1} (a_{i} / R) \\ β_{i} = β_{1} + θ_{i} \\ t_{2 i} = \frac{D [t_{1} cos θ_{i} + (D - R) sin θ_{i}]}{[R + (D - R) cos θ_{i} - t_{1} sin θ_{i}]} \end{array} \end{array}

Table 1. The explanations of corresponding symbols in Fig. 3.

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Fig. 3 Geometric relationships between the virtual focus projections and the ideal projections.

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As shown in Eq. (10), the linear equivalence relationship between the finite focal spot size based projection model and the ideal point source projection model is built when the Eq. (9) is substituted into Eq. (8).

q (β_{1}, t_{1}) = \sum_{i = 1}^{M} w (a_{i}) p_{cir} (β_{i}, t_{2 i})

Then the projections of a finer focus can be recovered from the measured projection data of a relatively large focal spot size by solving the linear problem (10). The scale of this problem is much less than the reconstruction problem. Therefore it can be solved efficiently even though the iterative method is used in current work. Specifically, the linear problem is rewrote as Eq. (11). The Q is a column vector and its elements are the measured projections. The elements of each row in the weighting matrix W are calculated according to Eq. (8) and Eq. (9). They depend on the virtual focus intensity, projection view angles and the geometric relationships. However the matrix is not stable for directly inverting Eq. (11). Therefore ART-like iterative method is adopted in this work to address it. The exact update formula in the iteration is expressed as Eq. (12). Furthermore, the measured projection is used as the initial guess P₀. It will be terminated when it reaches the maximum iteration number, which is set in advance. Finally, the analytical reconstruction methods, such as the FBP algorithm, can be applied using the recovered projections.

Q = WP

{p_{j}}^{(n + 1)} = {p_{j}}^{(n)} + \frac{q_{i} - \sum_{m = 1}^{M} w_{im} {p_{m}}^{(n)}}{\sum_{m = 1}^{M} {w_{im}}^{2}} w_{i j}

2.3. Difference image based reconstructed image fusion method

In this section, we will consider the influence of the projection recovery method on the projection noise. In general, there are two types of noise to be considered: the additive noise and shot noise in x-ray tomography [23]. To make it simple, we assume that the projections consist only of zero-mean random noise and its values are uncorrelated for any two rays in the system. Therefore, the variance of the measurement noise can be expressed as follows.

{σ_{q}}^{2} (β_{1}, t_{1}) = \sum_{i = 1}^{M} w^{2} (a_{i}) {σ_{p}}^{2} (β_{i}, t_{2 i})

If the variance of noise for each adjacent ray is the same, the Eq. (13) can be simplified.

{σ_{q}}^{2} (β_{1}, t_{1}) = {σ_{1}}^{2} (β_{1}, t_{1}) \sum_{i = 1}^{M} w^{2} (a_{i}) where σ_{1} (β_{1}, t_{1}) = σ_{p} (β_{i}, t_{2 i})

It implies that, compared to the variance of the measurement noise σ_q, the variance of the recovered projection noise σ_p increases, which obviously results in the reduction of the SNR of the reconstructed image. Also, in general, the more virtual sources used, the more noise will be introduced.

On one hand, the image reconstructed using the measured projection has a high SNR but a low spatial resolution. On the other hand, the reconstruction using the recovered projection achieves a great improvement in the spatial resolution at the expense of the SNR. To combine the two advantages, the image fusion of the two reconstructed images is presented. Many image fusion methods have been developed to incorporate more complementary information into the fused image [24, 25]. The weight strategy is the key to various fusion methods. In this paper, the difference between the reconstructed images using the measured projections and recovered projections gives a good reference for the determination of the weighting factor. As shown in Fig. 4, the significant differences are mainly concentrated on the edges of the object, which has a great effect on the spatial resolution, and other minor differences should be contributed to the different noise level of the projections used for reconstruction. In other words, the image reconstructed from the recovered projections should have a higher weight where the difference is significant, and the high SNR image reconstructed from the original measurements should account for more percent in the final fused image. Given such weighting, a linear fusion method, as shown in Eq. (15), is adopted in this paper. The weight strategy is designed based on the difference image of the two reconstructed images and the image noise level σ. The image noise level can either be estimated from the measured data or be chosen by experience. Specifically, in our experiment, the I_m can be reconstructed with the measured data by the conventional method. Then the standard deviation σ_m of a flat ROI in I_m can be calculated. Finally the σ can be estimated based on such value. The fused image keeps the characteristics of high spatial resolution at the object’s edges on one hand. The SNR of the flat regions has been improved on the other hand.

\begin{matrix} I_{f} (x, y) = α (x, y) I_{c} (x, y) + (1 - α (x, y)) I_{m} (x, y) \\ where α (x, y) = exp (- \frac{| I_{m} (x, y) - I_{c} (x, y) |}{σ}) \end{matrix}

Fig. 4 (c) The difference image between the reconstructed images using (a) the measured projections and (b) recovered projections.

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The entire reconstruction framework of the MFFS method can be summarized as follows. First, under the finite focal spot size based projection model, the projection of a finer focal spot is recovered from the measurements by solving Eq. (10). Then a high SNR image and a high spatial resolution image are reconstructed by FBP methods with different projections respectively. It should be noted that the FBP kernel used in the following experiments is the Ramp filter. At the end, the final image is fused based on the difference of two such reconstructed images, as shown in Eq. (15). A complete flowchart of the proposed reconstruction framework is given in Fig. 5.

Fig. 5 A complete flowchart of the proposed efficient reconstruction framework.

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3. Experiments and results

Both numerical simulations and experiments were carried out to evaluate the proposed reconstruction framework. In the numerical simulations, special phantoms are designed for the quantitative evaluation of the spatial resolution. Furthermore, the experimental results using the measured projection from a 200keV industrial CT verified the effectiveness and the feasibility of the proposed reconstruction framework.

3.1. Numerical simulations

In the first numerical simulation, a line-pair phantom was designed to evaluate the improvements in spatial resolution of the proposed reconstruction framework. As shown in Fig. 6(a), the spatial resolutions of the five line-pairs ranged from 2.0 LP/mm to 3.6 LP/mm and they were equally spaced around a circle. Their attenuation coefficients are set to 0.02 mm⁻¹. The diameter of each line-pair is 14mm. The number of the simulating photos produced by the x-ray source was set to 1.0 × 10⁶. As shown in Fig. 6(b), to approximate the finite focal spot size (about 1.0mm), 21 virtual point sources equally spaced in the interval of [−1.0, 1.0] were used to simulate the projections. The intensity has a Gaussian distribution. The mean value and the standard variance of the distribution were set to 0 and 0.2 respectively. The detector unit size was set to 0.13mm according to the parameters of the real industrial CT systems. Other geometrical configurations in the simulations are listed in Table 2.

Fig. 6 The line-pair phantom and the Gaussian distributed focus spot model.

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Table 2. The geometrical configurations in the experiments.

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As shown in Fig. 7, due to the finite focus spot size, the reconstructed images using the ideal projection model was blurred. It was hard to distinguish the line-pair of 3.6 LP/mm and the contrasts of other line-pairs were decreased. By contrast, the spatial resolutions of our results were obviously enhanced and all the line-pairs could be easily distinguished. To make a clear comparison, the profile lines of 2.8 LP/mm, 3.2 LP/mm and 3.6 LP/mm were plotted in Fig. 8. The system resolution is to quote the frequency where the MTF is reduced to 50%. The spatial resolution of the ideal projection model was about 2.4 LP/mm while the spatial resolutions of 5 virtual focuses and 11 virtual focuses projection models were 3.1 LP/mm and 3.6 LP/mm. The noise in the reconstructed images can be attributed to the projection model errors, the discretization errors and quantum noises. As aforementioned, the projection errors will be slightly magnified by the proposed projection analysis method. However we focused on the evaluations of the spatial resolution in this simulation and the image fusion based method to suppress the noise will be applied to the following experimental data.

Fig. 7 Simulation results of the line-pair phantom based on (a) ideal projection model, (b) 5 virtual focuses projection model and (c) 11 virtual focuses projection model respectively. The gray windows of all images are set to [0.00 0.03].

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Fig. 8 Comparisons of the profiles along the lines in Fig. 7.

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In the second numerical simulation, a low-contrast phantom was designed to evaluate the density resolution of the reconstructed image using the proposed framework. As shown in Fig. 9(a), there were four low-contrast disks placed in the phantom. They were made of the same material with different densities (about 0.5% and 1.0% difference). The specific attenuation coefficients were labeled in Fig. 9(a). The number of the simulating photos produced by the x-ray source was set to 5.0 × 10⁸ to distinguish the low-contrast disks. Other simulating parameters and geometrical configurations are similar to the previous numerical simulation.

Fig. 9 (a)The low-contrast phantom and the simulation results of the low-contrast phantom based on (b) ideal projection model, (c) 5 virtual focuses projection model and (d) 11 virtual focuses projection model respectively. The gray windows of all images are set to [0.0095 0.0105].

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From Fig. 9 and Fig. 10, it can be found that the density resolution is almost the same but the edges of the finite focus spot projection model results are sharper. It should be noted that the ring artifacts in the Figs. 9(c) and 9(d) are caused by the discretization errors in the projection analysis methods. The more virtual focus points used, the fewer produced artifacts. The artifacts are actually not as obvious as it seems because the gray window is limited to a very small range for a clear display of the low-contrast disks. This simulating result demonstrates that the proposed method can improve the spatial resolution, while maintaining sufficient density resolution.

Fig. 10 Comparisons of the profiles along the lines in Fig. 9.

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3.2. Experiments

To further demonstrate the effectiveness and the feasibility of the proposed method, experiments were done on a 200keV industrial CT. In the experiment, we set the energy of the x-ray source to 200keV and the current was set to 3.0mA. A line-pair phantom was used to demonstrate the superior spatial resolution of the proposed method. The spatial resolution of the phantom ranged from 2.4 LP/mm to 3.0LP/mm. The geometrical configurations were the same with the simulation study.

As shown in Fig. 11, in order to determine the parameters of the finite focal spot size model, the intensity profile of the x-ray source was measured by the pinhole imaging method before scanning (ASTM E1165 [26]). Taking the magnification factor of the image into consideration, the actual focus spot size was about 1.0mm in the horizontal axis and about 0.4mm in the vertical axis. A significant side lobe can be found in the focus profile along the horizontal line, which leads to the decline of the spatial resolution. The weight factor w(a_i) of each virtual point source in the focal spot support was determined by the horizontal profile. The numbers of the virtual source points were set to 3, 5 and 11 respectively in this experiment and they were uniformly distributed along the horizontal line.

Fig. 11 The measurement of the focal spot size by the pinhole imaging method. (a) The image of the focus and (b) the profile of the focus along the horizontal and vertical lines in (a) respectively. The magnification factor of the image was 2.4×.

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The results without image fusions are shown in Fig. 12. The reconstructed image I₁ by the conventional FBP method without projection modeling is seriously blurred due to the finite focal spot size of the x-ray tube. By contrast, visible improvements are seen in the spatial resolution results of I₃, I₅ and I₁₁ and the line-pair of 3.0 LP/mm is easily identified. To give a quantitative comparison, the profiles along the line in the images are plotted in Fig. 13. It can be found that the contrasts of our results have been enhanced by about 70% and 100% respectively. Compared to the result of the 3 virtual focus based model, the contrast was a little better in the result using 5 virtual focuses and 11 virtual focuses based model because more virtual sources provide a better description. However the noise level of 11 virtual focuses is a little higher than others. Therefore the balance between the noise level and the spatial resolution should be considered when the virtual focuses number is decided.

Fig. 12 Experimental reconstruction results without image fusions using the measured projections from the 200keV industrial CT. (a) ideal projection model; (b) 3 virtual focus projection model; (c) 5 virtual focuses projection models and (d) 11 virtual focuses projection models. All of the gray windows are set to [0.00 0.07].

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Fig. 13 Comparisons of the profiles along the lines in Fig. 12.

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However, as mentioned before, the noise level was increased in our results due to the projection analysis process. The image fusion based on the difference image was applied to further improve the SNR of our results. As shown in Fig. 14, this method maintains the sharpness of the object’s edges and the noise in the flat region is suppressed. To make a quantitative comparison the small flat region (which is marked by the red box in Fig. 14) in the line-pair phantom was chosen as the region of interest (ROI). The definitions of SNR and CNR of the ROI given in Eq. (16) and the Eq. (17) and they were used to evaluate the effectiveness of the proposed fusion method, where N is the total number of pixels in the ROI. The corresponding results with and without the image fusion are shown in Table 3. The SNR and CNR of the blurred images using the ideal projection model were also calculated as a reference. After the image fusions, the SNR and CNR results are close to the ideal projection model without sacrificing spatial resolution.

\begin{array}{l} SNR = 10 {log}_{10} (\frac{\bar{I_{ROI}}}{σ_{ROI}}) \\ where \bar{I_{ROI}} = \frac{1}{N} \sum_{i \in ROI} I_{i} σ_{ROI} = {[\frac{1}{N} \sum_{i \in ROI} {(I_{i} - \bar{I_{ROI}})}^{2}]}^{1 / 2} \end{array}

\begin{array}{l} CNR = 10 {log}_{10} (\frac{\bar{I_{{ROI}_{1}}} - \bar{I_{{ROI}_{2}}}}{σ_{{ROI}_{BG}}}) \\ where \bar{I_{{ROI}_{k}}} = \frac{1}{N} \sum_{i \in {ROI}_{k}} I_{i} σ_{{ROI}_{BG}} = {[\frac{1}{N} \sum_{i \in {ROI}_{BG}} {(I_{i} - \bar{I_{{ROI}_{BG}}})}^{2}]}^{1 / 2} \end{array}

Fig. 14 Comparisons of reconstructed results with and without the image fusion process. The gray window of the difference image is set to [0, 0.01] and the windows of other images are set to [0, 0.07].

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Table 3. Results of SNR and CNR using MFFS method with and without image fusion.

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In the end, the computational cost results of MFFS method are provided. The MFFS method including the FBP reconstruction was implemented by MATLAB programming without any special accelerations. The total execution time of the MFFS method at 3 virtual focuses, 5 virtual focuses and 11 virtual focuses were 758s, 872s and 1082s respectively including about 480s for the FBP reconstructions. More virtual focuses used in the projection model, the higher the computational burden was. However, the 3 virtual focuses model based iterative reconstruction method took about 6332s for each iteration. Much more time was required for more virtual focuses and iterations. The outstanding performance on the efficiency was achieved by the proposed method. It should be noted that it might not be the most optimal implementation though we tried our best to give a fair comparison. As it is a verification experiment, the programming efficiency has not been improved. These results only provids a qualitative comparison. The potential of accelerating the implementation of the MFFS method is discussed in the following section.

4. Discussions and conclusions

The results show the proposed finite focal spot size based reconstruction method has achieved significant improvements in the spatial resolution and keeps the capacity to distinguish the low-contrast structures. Furthermore it should not be ignored that the low computational cost is another unique advantage of the proposed framework. Under the established equivalence relationship between the finite-focus projection model and the ideal model, the standard projections of the finer focal spot can be recovered from the measured projections by the MFFS method. Then a high-resolution image can be reconstructed efficiently by the conventional analytical algorithm with the recovered projections. Compared to the reduplicative projection operations of the iterative method in the image domain, the computational costs are greatly reduced using the MFFS method with some simple calculations in the projection domain. Moreover the method is a local processing method in the projection domain because, according to the Eq. (9) and Eq. (10), any recovered projection is only related to the measured data from adjacent views and detector units. Therefore the projection analysis process can be performed during the projection measurements and takes no additional time than the currently used standard. The MFFS method is highly parallelized because projections of different views and units can be recovered at the same time, which makes it possible to further speed it up using GPUs.

In conclusion, we have proposed an efficient reconstruction framework for the high spatial resolution industrial CT system by modeling finite focal spot. The equivalence relationships between the ideal projection model and this model are deduced and then used to recover the projections of the finer focus from the measured projections. This method avoids the reduplicative projection operations in the iterative algorithm and it is highly efficient due to the low computational cost and its high parallelism. The feasibility and effectiveness of the proposed method have been demonstrated by numerical and experimental results. It can be applied in most existing high-energy industrial CT systems with macro-focal spots without any hardware modifications. In the same spirit of 2D fan-beam scanning, our method can be extend for enhancing the in-plane spatial resolutions of the 3D cone-beam geometry.

Acknowledgments

This work was partly supported by the grants from the Transportation Construction Science and Technology Project 2013 328 49A090.

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Symbols	Explanations
S₁	The ideal point source on the circular orbit
S_2i	The virtual point source belongs to the finite focal spot support
S′_2i	The corresponding virtual source on the circular orbit
a_i	The distance between the ideal point S₁ and the virtual point S_2i
R	The distance between the source and the rotating axis
D	The distance between the source and the detector array
Det₁, Det_2i, t₁, t_2i	The detector units with the positions t₁ and t_2i on the detector array

Scanning configuration parameters	Values
Trajectory radius (mm)	678.75
Object radius (mm)	40.00
Source-to-detector distance (mm)	1501.75
Projection number per circle	8192
Linear array detector size (mm)	169.00
Detector unit number	1300
Reconstructed image dimensions	4096 × 4096

Projection model	SNR1 (without fusion)	SNR2 (with fusion)	CNR1 (without fusion)	CNR2 (with fusion)
Ideal	10.56		10.74
3 Virtual focuses	8.26	10.40	10.10	10.72
5 Virtual focuses	7.18	10.26	9.63	10.70
11Virtual focuses	5.14	10.11	8.06	10.48

Symbols	Explanations
S₁	The ideal point source on the circular orbit
S_2i	The virtual point source belongs to the finite focal spot support
S′_2i	The corresponding virtual source on the circular orbit
a_i	The distance between the ideal point S₁ and the virtual point S_2i
R	The distance between the source and the rotating axis
D	The distance between the source and the detector array
Det₁, Det_2i, t₁, t_2i	The detector units with the positions t₁ and t_2i on the detector array

Scanning configuration parameters	Values
Trajectory radius (mm)	678.75
Object radius (mm)	40.00
Source-to-detector distance (mm)	1501.75
Projection number per circle	8192
Linear array detector size (mm)	169.00
Detector unit number	1300
Reconstructed image dimensions	4096 × 4096

Improve spatial resolution by Modeling Finite Focal Spot (MFFS) for industrial CT reconstruction

Abstract

1. Introduction

2. Methodology

2.1. Finite focal spot size based projection model

2.2. Projection recovery method and CT reconstruction

2.3. Difference image based reconstructed image fusion method

3. Experiments and results

3.1. Numerical simulations

3.2. Experiments

4. Discussions and conclusions

Acknowledgments

References and links

Cited By

Figures (14)

Tables (3)

Equations (17)

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