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Abstract
The accuracy of measuring the effective focal spot of the X-ray source directly affects the spatial resolution of computed tomography (CT) reconstructed images. This study proposes what we believe to be a novel approach to measure the effective focal spot based on the dynamic translation of light barrier using an accessible measuring device. This method discretizes the effective focal spot of the X-ray source into multiple subfocal spots with varying intensities and establishes a nonlinear model between the effective focal spot and measurement data. Measurement data are obtained by moving the light barrier to different positions using the electric displacement stage. The shape, size, and intensity distribution of the effective focal spot are determined by calculating the normalized weighting coefficients for each subfocal spot from measurement data. The measurement device is simple and easy to operate. Additionally, the obtained effective focal spot exhibits high accuracy, and a higher spatial resolution can be realized by reconstructing the CT images using the measured focal spot information. Numerical and real experiments validate the proposed method.
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1. Introduction
Combining the information derived from the X-ray source focal spot is an effective approach for improving the spatial resolution of computed tomography (CT) images [1]. The focal spot of an X-ray source can be conceptually divided into the actual and effective focal spots. The actual focal spot refers to the area where the high-speed electron beam impacts the metal target, whereas the effective focal spot refers to the projection area of the actual focal spot of the X-ray tube in a direction perpendicular to the tube axis [2]. The focal spot affects the resolution of the imaging equipment. Further, its size is affected by various factors, including the geometric shape, material, focal spot position, beam focusing system, tube voltage, and tube current. Although the X-ray manufacturer calibrates the effective focal spot of the equipment according to the IEC60336 [3], EN12543 [4], or ASTME1165 [5] standards, the anode target of the X-ray source may be damaged after prolonged use. This may further result in an increase in diffuse reflectance, which, in turn, results in a change in the effective focal spot (including the size, shape, and normalized intensity distribution). To enhance the spatial resolution of CT reconstructed images, the characteristics of the effective focal spot must be remeasured [6–9].
Existing measurement methods for the effective focal spot of X-ray sources can be classified into three categories [10]—scanning, imaging, and calculation. Scanning method employs mechanical scanning to obtain accurate focal spot information, and yields accurate measurement results. However, it requires numerous devices, such as scintillation counters, double-slit collimators, and pre-collimators, making the measurement process relatively complex. By contrast, the pinhole and slit camera radiographic methods are representative examples of imaging methods. The former works on the pinhole imaging principle, and the main device required for this method is a pinhole plate. Although this is a commonly adopted method, its use is complicated because the aperture of the pinhole plate is required to be small. For instance, when the effective focal spot size is between 0.2 and 1 mm, the required aperture is $30\pm 5\;\mathrm{\mu}$m. Processing such a small aperture on tungsten plate with good ray attenuation properties is a challenging feat. When using a lightweight material that is easy to process, a significant photon transmission may occur under high voltage conditions, and this may potentially lead to inaccurate measurement outcomes [11]. The slit camera method is similar to the pinhole camera method in terms of operational principle. However, this method primarily utilizes a slit baffle, and the characteristics of effective focal spot are determined using the two X-ray photos that are obtained through two perpendicular directions. This method performs well for focal spot with nominal values of less than 0.3 mm [12]. However, it demands high precision in slit manufacturing and alignment, owing to which its implementation is challenging. The calculation methods include edge and star card methods. the edge method indirectly measures the effective focal spot by measuring the geometric unsharpness. It primarily uses a steel cylinder covered with a lead plate with a thickness of more than 1 mm. The measurement process in this method is relatively simple, and the device can be easily manufactured. Thus, this method is suitable for measuring the effective focal spot in field conditions. However, the accuracy of the measured effective focal spot is limited; thus, it is unsuitable for absolute focal spot measurement [13]. The star card method obtains the characteristics of the effective focal spot by testing and calculating the star card X-ray image. It primarily utilizes a star card made of high-absorbing material, and it is commonly used in the medical field [14]. Among the aforementioned methods, only the pinhole, slit, and edge methods are suitable for on-site measurement of effective focal spot in the industrial field. Furthermore, although other methods have been reported in the literature [15–18], their use is constrained by factors such as difficulties in device acquisition or complexity in the measurement process.
To conveniently measure the effective focal spot of the X-ray source, this study proposes a measurement method based on the dynamic translation of the light barrier (DTM), which can be used for on-site measurement of focal spot. The innovation of this method is that we proposed a new type of focal spot measurement device and established a nonlinear model about the effective focal spot and measurement data. The measuring device for the proposed method primarily comprises a light barrier and electric displacement stage. During measurement, the light barrier is placed close to the light exit-window of the X-ray source, and a series of measurement data at different positions is obtained by moving the light barrier. Thus, the effective focal spot of X-ray source is logically discretized into multiple subfocal spots with different intensity, and a nonlinear model for the effective focal spot and the measurement data is established. The size, shape and intensity distribution of the effective focal spot are obtained by calculating the normalized weighting coefficient of each subfocal spot. The introduction of the innovative method involving the dynamic translation of the light barrier not only enhances the convenience and efficiency of the measurement but also achieves groundbreaking results in the design of the focal spot measurement device. This method possesses the capability for on-site measurements, and the focal spot obtained through this method can improve the quality of reconstructed images.
The remainder of this paper is organized as follows. Section 2 describes the design of the measurement device, establishes the relevant model of the measurement data, and presents the solution in detail. Section 3 presents the experimental verification of the proposed method, along with the numerical experiments, real experiments, and the display and analysis of the reconstructed images based on the measured results. Section 4 summarizes the proposed method and concludes the study.
2. Method
An accurate measurement of the effective focal spot is the basis for improving the spatial resolution of CT reconstructed images. In this method, the effective focal plane is logically discretized, and a new measuring device has been proposed accordingly. The effective focal spot is measured using the method of the dynamic translation of the light barrier. This approach includes data acquisition, data modeling, and solving. The schematic diagram is shown in Fig. 1.
Fig. 1. Dynamic translation of light barrier scheme. (a) New measurement device. (b) Dynamically translating to obtain the measurement data. (c) Establishing a nonlinear model between the effective focal spot and measurement data. (d) Solving the model to recover effective focal spot.
2.1 Measurement device
The measurement device primarily comprises an electric displacement stage, light barrier, and detector [19,20]. Further, the electric displacement stage is composed of a translation stage, a lifting stage, and motors, as shown in Fig. 2. It regulates the position and displacement of the light barrier by executing movement command from the motion controller or software. The light barrier is composed of a high-ray-absorbing material and possesses a uniform thickness, which is determined according to the tube voltage and tube current of the X-ray tube. The primary role of the light barrier is to obstruct photons that are emitted from the focal spot of the X-ray source. To ensure the complete shielding of photons, and prevent photon diffraction and scattering at the edge of the light barrier, the length and width of the light barrier should be slightly greater than those of the light exit window of the X-ray source. In addition, to address the issue of inaccurate data collection caused by detector blurring, a detector with a small unit size should be used for data collection [21]. If the size of the detector unit is large, the two-dimensional Modulation Transfer Function (MTF) measured using the noise response (NR) method can be used to correct the detector blur [22].
Fig. 2. (a) Front view of electric displacement stage. (b) End view of electric displacement stage. (c) Measuring device.
2.2 Data collection
A schematic of the dynamic translation of the light barrier for data collection is shown in Fig. 3. First, the tube voltage and tube current of the X-ray tube are set. Next, the X-ray source is preheated until it reaches a stable state, and then the data without a light barrier, that is, the bright field data, are collected. Subsequently, the light barrier is fixed on the electric displacement stage, and its position is adjusted such that it is parallel to the light exit window. Next, the light barrier is moved from one side of the light exit-window to the other side in increments according to the set step value, using an electric displacement stage. The data are collected using the detector at different positions of the light barrier. Data collection is terminated when the light barrier completely blocks the light exit-window. To minimize errors, the aforementioned steps are repeated multiple times, and the average value of the collected data is obtained. Moreover, measurement data are collected in two directions (along the horizontal and vertical directions of the light barrier), and both datasets are used during calculation to obtain more accurate measurement results. Figure 3 shows a schematic of the dynamic translation data collection of the light barrier. Figures 3(a) and 3(b) show the process of the light barrier translational data collection in the vertical and horizontal directions, respectively.
Fig. 3. Schematic of data collection. (a) Along the vertical direction, and (b) along the horizontal direction translation.
During data collection, the parameters of the measurement device, including the number of moves of the light barrier $M$ and step value $dh$, must be appropriately set. When the length or width of the light exit-window (depending on the step direction) is $H$, $dh=H/M$. Here, $M$ must consider the effective focal spot size to be measured, number of discrete subfocal spots, and detector unit size. Focal spot object distance $\left (fod\right )$ and focal spot detector distance $\left (fdd\right )$ can be determined based on the detector unit size $d$ and effective focal spot size $L$. Further, $M$ should not be less than $N\cdot d\cdot fod/\left (L\cdot (fdd-fod)\right )$, where $N$ is the number of discrete subfocal spots, and $dh$ should not be greater than $H/\left (N\cdot d\cdot fod/\left (L\cdot \left (fdd-fod\right )\right )\right )$. The reason for selecting the value of $M$ is explained in Section 3.
2.3 Modeling and solving
This section establishes a model for the effective focal spot and measurement data, and subsequently solves the model to obtain the normalized weight coefficients of each subfocal spot. Thus, the size, shape, and intensity distribution of the effective focal spot are obtained.
2.3.1 Effective focal spot plane discretization
In practical applications, photons can be considered to be emitted from the plane of the effective focal spot of the X-ray source [23]. To measure the effective focal spot, a square containing the focal spot is selected in the plane where the effective focal spot is located; subsequently, it is discretized, as shown in Fig. 4(c). Each unit after discretization is referred to as a subeffective focal spot (note: for some subfocal spots, the intensity may be 0). Assuming that the number of discretized units is $N=P\times Q$, where $P$ and $Q$ represent the number of rows and columns of the subfocal spots, respectively, the total luminous intensity of the effective focal spot can be expressed as $I=\sum _{p=1}^{P}\sum _{q=1}^{Q} I_{p,q}$, where $I_{p,q}$ is the luminous intensity of the subfocal spot whose row and column indices are $p$ and $q$, respectively. Further, the normalized weighing intensity of each subfocal spot is $\omega _{p,q}=I_{p,q}/ I$.
Fig. 4. Schematic of X-ray tube and focal spot. (a) Schematic of structure of X-ray tube. (b) Magnification of focal spot. (c) Effective focal spot and its discretization.
2.3.2 Measurement model based on effective focal spot of X-ray source
Without considering the polychrome of photons, scattering of photons, poisson noise and noise of detectors, at the $m$-th translation of the light barrier, the number of photons that remain after passing through the light barrier measured by the $j$-th detector unit can be expressed using the following mathematical model.
where ${I_{0,j}}$ is the photon data measured when the $j$-th detector unit detects the absence of the light barrier, i.e., the bright field data; ${L_{m,j}}$ is the ray direction at which the photon reaches the $j$-th detector unit, when the light barrier is translated at the $m$-th time; $\mu$ is the linear attenuation coefficient; $S$ is the effective focal spot plane; and $\omega _{s}$ is the intensity weighting coefficient of the effective focal spot plane.In this method, Eq. (1) is discretized into Eq. (2), as follows
where $N$ is the number of subeffective focal spots; $\omega _{n}$ is the normalized weighting coefficients of $n$-th subfocal spot occupying the entire effective focal spot; and $\omega _{n}$ is the variable to be solved. Once this variable is solved, the intensity distribution of the effective focal spot can be obtained. The light barrier is made of a single material. Here, $\mu$ is the line attenuation coefficient per unit length of the light barrier at standard density. Therefore, $\mu$ is a known constant, and its value can be obtained from a website [24]. Further, $\ell _{m,j,n}$ is the straight-line length vector between the $n$-th subfocal spot and the $j$-th detector unit that the ray pass through the light barrier at the $m$-th translation. Further, $\delta$ is the thickness of the light barrier, and when it is known, $\ell _{m,j,n}$ can be calculated as the straight-line distance.2.3.3 Solving $\ell _{m,j,n}$
In the above model, the core problem of working out $\omega _{n}$ involves solving $\ell _{m,j,n}$. In this subsection, the light barrier is translated along the vertical direction (i.e., the positive direction of the y-axis), and the approach to obtain $\ell _{m,j,n}$ is described in detail.
Figure 5(a) shows the coordinate system established in this study, with the center of the X-ray source exit-window acting as the coordinate origin. Here, $fod$ and $fdd$ can be obtained through measurement. The center coordinates of each subfocal spot of the effective focal plane are represented as $S\left (0,y_n,z_n\right )$. The size of the detector is known, and they are denoted as $L_d\times H_d$. The center coordinates of each detector unit are represented as $D\left (fdd,y_j,z_j\right )$, where $0\leq y_j\leq H_d$, and $0\leq z_j\leq L_d$. The X-ray path from a subfocal spot sport center of the effective focal spot plane to the center of a unit of the detector can be represented as ${x}/{fdd}={\left (y-y_n\right )}/{\left (y_j-y_n\right )}={\left (z-z_n\right )}/{\left (z_j-z_n\right )}$.
Fig. 5. Vertical translation of light barrier as an example to calculate $\ell _{m,j,n}$. (a) Coordinate system for mode. (b) Four situations where X-rays intersect with light barrier.
For different situations where the X-ray intersects with the light barrier, the distance $\ell _{m,j,n}$ between the X-ray and light barrier can be calculated accordingly. For this, the height of the top edge of the light barrier must be equal to the bottom edge of the light exit window, denoted as $y_0$, according to the initial position for the translation of the light barrier. After $m$ translations along the positive direction of the y-axis, the distance moved is $y_m=m\times \nabla y$, where $\nabla y$ is the step. In Fig. 5(b), four positional relationships between the X-ray and light barrier are shown. Here, the blue line segment represents the intersection distance between the X-ray and light barrier, whereas the red dashed line represents the initial plane for the translation of the light barrier. In Fig. 5(b), A and B represent the intersection points of the X-ray with the front and back planes of the light barrier, respectively. Further, $y_A$ and $y_B$ represent the vertical distances from A and B, respectively, to $y_0$, such that $y_A={fod}(y_j-y_n)/{fdd}+y_n-y_0$ and $y_B=\left (fod+\delta \right )(y_j-y_n)/{fdd}+y_n-y_0$. $\ell _{m,j,n}$ can be calculated as shown in Eq. (3), where the four subequations correspond to the four intersecting situations in Fig. 5(b). These four situations are: 1. the X-ray does not intersect with the light barrier; 2. the X-ray intersects with the plane where the top edge of the light barrier is located, and the back plane of the light barrier; 3. the X-ray intersects with the plane at which the top edge of the light barrier is located, and the front plane of the light barrier; 4. the X-ray intersects with the front and back planes of the light barrier. Eq. (3) is as follows.
2.3.4 Solving model
When $\mu$ and $\ell _{m,j,n}$ in Eq. (2) are known, the solution for $\omega _{n}$ can be transformed into a linear solution, as follows.
Equation (5) can be solved using various methods, including algebra reconstruction technique (ART) [25], gradient descent method (GD) [26], and expectation maximization (EM) [27]. However, owing to the non-negative nature of the EM algorithm, which can ensure $\omega _{n}\geq 0$, the EM algorithm was employed herein to solve the problem; its iterative form is:
According to the idea of discretizing the effective focal spot plane, when the normalized weight coefficient $\omega _n$ of each subfocal spot is known, the luminous intensity of each subfocal spot can be obtained according to the corresponding relationship between $n$ and $p$, $q$, which further helps obtain the shape and size of the effective focal spot.
3. Experiments
In this chapter, both numerical and real experiments are described. In the numerical experiments, the effective focal spot is recovered based on the simulated data, and the impact of the step value on the measurement results is analyzed. Whereas in the real experiment, different methods are used to measure the effective focal spot of the actual system equipment, and the same algorithm is used for reconstruction based on the measurement results. The effectiveness of the algorithm is verified based on the comparison of the reconstructed images.
3.1 Impact of $M$ and $\nabla h$ on measurement results
Note that $M$ determines the number of equations. In the window area that is completely blocked by the light barrier, the number of photons detected by the corresponding detector unit on the detector is zero when the quantum noise and electronic noise of the detector are not considered. Further, the number of photons detected by the corresponding detector unit on the detector is almost equal to the number of photons of the collected bright field data when the window area is not blocked by the light barrier. This indicates that when $\ell _{m,j,n}=0$, we obtain $e^{-\mu \ell _{m,j,n}}=1$ and $\omega _{1}+\cdots +\omega _{n}+\cdots +\omega _{N}=1$. The two aforementioned cases are invalid equations, excluding the above two cases, and the number of effective equations in a single translation of the light barrier is $\left (L\cdot \left (fdd-fod\right )\right )/\left (d\cdot fod\right )$. To ensure the number of valid equations, the total number of valid equations of $M$ times translation the light barrier should not be less than the number of discrete focal spot $N$; that is, $\left (\left (L\cdot \left (fdd-fod\right )\right )/(d\cdot fod)\right )\cdot M>N$. This implies that $M$ is not less than $N\cdot d\cdot fod/\left (L\cdot \left (fdd-fod\right )\right )$.
In the numerical experiments, the effective focal spot size of the X-ray source was set as 1 mm; the intensity distribution of the effective focal spot is shown in Fig. 6(a). The effective focal spot was dispersed into 1000 subfocal spots to simulate the continuous effective focal spot. The light barrier was a tungsten plate of uniform material, with a length and width of 120 mm each, and thickness of 2 mm. The initial photon number was $I_0=10^6$, and the length and width of the light window were 100 mm each. The geometric parameters of the CT system were $fod$ = 100 mm and $fdd$ = 500 mm, number of detector units was 1000, and detector unit size was 0.2 mm. In $M$ and the range of step value, the impact of $M$ and stepping precision on measurement results was analyzed.
Fig. 6. Effect of $M$ on measurement result. (a) Intensity curve of recovery effective focal spot. (b) Mean square error curve.
First, the effect of different values of $M$ on the measurement of the effective focal spot of the X-ray source was analyzed. The measurement process was simulated at $N$ = 100. The recovery degree of effective focal spot when $M$ is 20, 50, 100, and 500 is shown in Fig. 6(a). Evidently, as $M$ increased, the fitting degree of the curve improved, and the curve more closely resembled the reference curve. For a detailed analysis of the impact of $M$ on the recovery of effective focal spot, the mean-square error (MSE) of the corresponding curves of different $M$ and reference values were calculated, and the MSE curve was drawn, as shown in Fig. 6(b). Evidently, when $M = 200$, the MSE was already small, and when $M > 200$, the MSE did not change significantly. As $M$ increased, the amount of computation and the calculation time both increased. Therefore, when utilizing the algorithm outlined in this study to recover effective focal spot, precision and computational efficiency must be balanced, while accounting for the impact of parameter $M$. Thus, a suitable value of $M$ should be adopted.
Next, the impact of $\nabla h$ on measurement results was analyzed. Stochastic perturbation was used to simulate the translation errors that may occur when the light barrier moves. When $N$ = 100, random fluctuations of different degrees increased for the step values of 0.01 and 0.1. The recovery curves of effective focal spot are shown in Figs. 7(a) and 7(c). The correspondence between the amplitude of random disturbance and the resulting MSE is shown in Figs. 7(b) and 7(d). The reference curves in Figs. 7(a) and 7(c) indicate the measurement results recovered without disturbance. A comparison of the two figures reveals that different degrees of random disturbance had different impacts on the recovery results under different step values. When $\nabla h$ was small, the corresponding curves under different disturbances differed slightly. This suggests that when $\nabla h$ is small, a certain degree of random disturbance does not have a significant impact on the measurement results. However, when $\nabla h$ was large, as the disturbance range increased, the curve of recovery result became farther away from the reference curve, thus indicating that the greater the random fluctuation, the greater the impact on the recovery result. For example, when $\nabla h$ was 0.1 mm and the random fluctuation range was less than the range between −0.05 and 0.05 mm, the disturbance had little effect on the result and could be neglected. However, when the random fluctuation range was between −0.1 and 0.1 mm, the recovered strength curve was at a farther distance from the reference curve.
Fig. 7. Impact of $\nabla h$ on the measurement results. (a) Intensity distribution of effective focal spot curves recovered under different disturbances when step value is 0.1 mm. (c) Intensity distribution of effective focal spot curves recovered under different disturbances when step value is 0.01 mm. (b) and (d) correspond to (a) and (c), respectively, and are mean square errors under different perturbations.
In summary, $M$ affects the recovery degree of the effective focal spot. When considering the accuracy and computational efficiency of the effective focal spot, the number of moves must be set reasonably. Moreover, when the movement error of the light barrier is large, it affects the recovery of effective focal spot.
3.2 Real experiments
The real experiments are divided into two parts. The first part is aimed at measuring the effective focal spot of the actual system equipment, whereas the second part is aimed at reconstructing the same data using the same reconstruction algorithm using the results of measurement effective focal spot. The experiments in this study were performed on a CT imaging device consisting of an X-ray source with model YXLON Y.TU/450-D09, and detector with model Varian PerkinElmer XRD 1620 xN CS.
3.2.1 Effective focal spot measurement
Given that the effective focal spot is different under different tube voltages and tube currents [28–30], the effective focal spots under different tube voltages and tube currents were measured herein. The parameters of the CT equipment used herein are presented in Table 1. The measured X-ray source had factory-specified focal spot sizes of 0.4 and 1 mm. The change in effective focal spots at different tube voltages and tube currents was analyzed when the effective focal spot size was 1 mm, and the edge method (EDM) and DTM method were compared at 160 kV/2.6 mA. When the factory-specified effective focal spot size was 0.4 mm, because EDM is applicable to the focal spot measurement only for $d\geq 0.5$ mm, the measurement using other methods was difficult. Thus, only the proposed method was used for the measurement. The real measuring device is shown in Fig. 8. The device comprises an electric displacement stage along with a controller, light barrier, and flat panel detector. The repeatable positioning accuracy of the electric displacement stage was 0.005 mm, and the absolute positioning accuracy was 0.015 mm.
Table 1. System parameters for effective focal spots measurement.
A. Effective focal spots measurement
The results measured using this method in multiple groups of tube voltages and tube currents are shown in Fig. 9. Evidently, under the same tube voltage, the shape of the effective focal spots under different tube currents did not change significantly, essentially resembling a rectangle. Additionally, the brightness of the image of the effective focal spots increased with the tube current, and concurrently, the horizontal effective focal spot size also increased with the tube current. Furthermore, the images exhibited faint stripes of reduced luminance in the horizontal orientation, located approximately in the middle of the multiple images.
Fig. 9. DTM effective focal spot images measured at different tube voltages and tube currents. (a), (b) and (c) are effective focal spot images of different tube currents at 60, 80, and 100 kV.
The effective focal size is often measured using the full width at tenth maximum (FWTM) of the focal intensity curve. Thus, herein, the intensity curves of multiple positions in the horizontal and vertical directions of the effective focal spot image were plotted, and the corresponding effective focal spot size at one-tenth of the height and width was analyzed. Next, the average value of the size in the two directions was calculated. Figures 10(a) and 10(b) show the change in the curve of effective focal size with tube current when the tube voltage is constant. Evidently, the effective focal spot size in the horizontal direction gradually increased with the tube current, whereas the effective focal spot size in the vertical direction tends to decrease. Figures 10(c) and 10(d) show the change in the curve of effective focal size with tube voltage when tube current is constant. Evidently, the effective focal size in the horizontal direction progressively increased with the tube voltage, whereas the effective focal spot size in the vertical direction initially decreases and then gradually increases. In Fig. 10(c), the curve corresponding to 1.0 mA is above the line corresponding to 0.5 mA, whereas the curve in Fig. 10(d) exhibits a contrasting phenomenon. In addition, a comparison of the intensity curve changes in the horizontal direction of Figs. 10(a) and 10(c) reveals that when the tube voltage or tube current was constant, the effective focal spot size in the horizontal direction progressively increased with the tube current or tube voltage. In particular, this increase was greater with the increase in tube current than with the increase in tube voltage.
Fig. 10. Changes in effective focal spot size in the (a) horizontal direction and (b) vertical direction with increasing tube voltage under different tube currents. Changes in effective focal spot size in the (c) horizontal direction and (d) vertical direction with increasing tube current under different tube voltages.
To further analyze the impact of tube voltage and tube current on intensity distribution of the effective focal spot, the horizontal and vertical profile lines at the center of the effective focal spot images were drawn, as shown in Figs. 11 and 12, respectively. Evidently from Fig. 11, under the same tube voltage, the increase in tube current enhanced the intensity of X-rays, and the intensity distribution in the horizontal and vertical directions of the effective focal spot images were different. Under the same tube voltage, the horizontal intensity distribution of the effective focal image approximately followed a Gaussian distribution, as shown in Figs. 11(a), 11(c), and 11(e), whereas the intensity distribution in the vertical direction exhibited a double peak at low tube currents, as shown in Figs. 11(b), 11(d), and 11(f). In particular, the double-peak phenomenon of the effective focal spots weakened when the tube current was increased while keeping the tube voltage as constant. This phenomenon occurs because when the tube voltage is constant, more electrons are emitted with similar kinetic energy and velocity, and as the tube current increases, more concentrated electron beams are easily formed, that is, the aggregation of electron beams, which may lead to the weakening of the double-peak phenomenon [31]. In addition, a comparison of the curves at of 60 kV/0.5 mA, 80 kV/0.5 mA, and 100 kV/0.5 mA in Figs. 11(b), 11(d), and 11(f), respectively, reveals that when the tube voltage was varied while maintaining the tube current as constant, the degree of double-peak attenuation also varied. The effective focal spots changes corresponding to different tube voltages under the same tube current are shown in Fig. 12. Evidently, under the same tube current, increasing the tube voltage increased the intensity of X-rays. Moreover, the curve of the effective focal spot image in the vertical direction became increasingly pronounced as the tube voltage increased while the tube current was constant, thus leading to a more distinct double-peak phenomenon.
Fig. 11. Normalized intensity changes in the effective focal spot at same tube voltage in the (a),(c),(e) horizontal direction and (b),(d),(f) vertical direction.
Fig. 12. Normalized intensity changes in the effective focal spot at same tube current In the (a),(c) horizontal direction and (b),(d) vertical direction.
B. Comparison experiment with effective focal spot measured using EDM
Under the conditions of a 160 kV tube voltage and 2.6 mA tube current, the effective focal spot was measured using both EDM and DTM methods for an X-ray source with a factory-specified effective focal spot size of 1 mm. Furthermore, the effective focal spot measurement was performed using the DTM method for an X-ray source with a factory-specified effective focal spot size of 0.4 mm. The measurement results are illustrated in Figs. 13(a)–(c). Figures 13(d) and 13(e) show the intensity curves of the measured results in the horizontal and vertical directions, respectively; here, the length of the effective focal spot measured at one-tenth of the height and width is measured in mm. Figure 13(f) shows the intensity curves in horizontal and vertical directions for the measurement effective focal spot size of 0.4 mm using the DTM method. For the X-ray source with a factory-specified effective focal spot size of 1 mm, the horizontal direction and vertical direction of the results measured using the EDM method were 1.150 mm and 1.330 mm, whereas those measured using the DTM method were 1.222 mm and 1.462 mm, respectively. For the X-ray source with a factory-specified effective focal spot size of 0.4 mm, the horizontal direction and vertical direction of the results measured using the DTM method were 0.522 mm and 1.394 mm, respectively.
Fig. 13. Effective focal spot images measured using different methods and normalized intensities. (a) 1mm: EDM, (b) 1mm: DTM, and (c) 0.04mm: DTM. (d), (e), and (f) represent intensity curves corresponding to (a), (b), and (c) in both directions, respectively.
3.2.2 Reconstructed CT images experiment based on effective focal spots measurement results
The CT system used in the experiment is shown in Fig. 14(a), and the scanning parameters of the CT system are presented in Table 2. The standard CT performance detection module catphan600 was used in the experiment, as shown in Fig. 14(b). The CTP528 is a high-resolution module used to measure spatial resolution. Further, it contains pairs of lines from 1 to 21 per cm, and the number of stripes at 5–21 lp/cm is five. Filters and collimators were used in the scanning process to reduce the effects of noise and scattering. The ultimate resolution of the reconstructed image was calculated using Eq. (7) [32,33], where $d$ is the width of the detector unit and $L$ is the effective focal spot size. With regard to the aforementioned parameters, the maximum resolution of the reconstructed images based on focal spot sizes of 1 and 0.4 mm were 19 and 44 lp/cm, respectively.
Table 2. CT system parameters.
A. Reconstructed CT images experiment of real data when the effective focal spot size is 1 mm
The projection data obtained from the CTP528 module at an X-ray source with a factory-specified effective focal spot of 1 mm were scanned. The effective focal spot size was measured using the EDM and DTM methods, and the projected data was reconstructed using virtual focal spot reconstruction (VFS) [34]. The number of discrete virtual focal spot was determined using the sampling theorem. Thus, under the above system parameters, the highest frequency was $f_{\text {max}}={1}/{21}$ and the number of samples was ensured to be greater than ${2\left (fod-fdd\right )}/{f_{\text {max}}\cdot fdd}$, considering the accuracy and computational efficiency of the reconstructed image. The number of the discrete virtual focal spot was seven.
Figure 15(a1) shows the results of reconstruction using the ART method, Fig. 15(b1) shows the results of reconstruction using the virtual focal spot method of effective focal spot measured using the edge method (EDM-VFS), and Fig. 15(c1) shows the result of reconstruction using the virtual focal spot method of effective focal spot measured using the proposed method (DTM-VFS). Figures 15(a2), (b2), and (c2) are the enlarged views of 14–18 lp/cm line pairs reconstructed using three methods, respectively. Evidently from the local magnification in Figs. 15(a2), (b2), and (c2), compared with the other two methods, the boundary of image fringes reconstructed by ART method was more blurred. At 15 lp/cm, all the three methods can correctly reconstructed the five fringes. Further, compared with the EDM-VFS method, the ART method, and DTM-VFS method reconstructed the fringes with higher clarity and contrast. At 16–18 lp/cm, the fringes at 16 lp/cm can be seen in the reconstructed image using the ART method, the fringes at 16–18 lp/cm can be seen in the reconstructed image using DTM-VFS method, and the fringes 16–18 lp/cm are blurred in the image reconstructed using EDM-VFS method. In addition, at the green arrow in the locally enlarged image, the DTM-VFS method reconstructed the image with fewer artifacts.
Fig. 15. Reconstructed images based on re-measured effective focal spot and their zoomed-in views when factory-specified effective focal spot size is 1 mm and window width is set to [0,0.7], (a1) ART, (a2) zoomed-in view of ART. (b1) EDM-VFS, (b2) zoomed-in view of EDM-VFS. (c1) DTM-VFS, (c2) zoomed-in view of DTM-VFS.
When the line pair was larger than 15 lp/cm, the number of reconstructed stripes cannot be visually determined from the reconstructed image. Therefore, the gray values of the middle line pairs in the reconstructed image at different positions (the yellow curve position at 2 lp/cm in Fig. 15(a1) were extracted, and the profile line was drawn on average. When the line pairs were greater than or equal to 18 lp/cm, the number of stripes reconstructed using the three methods were different. Figure 16 shows the profile of the reconstructed resulting image at 14–17 lp/cm line pairs, along with partial enlargements. Evidently, at 16 lp/cm, the number of stripes in the reconstructed image was four, whereas the other two methods correctly reconstructed the number of stripes. At 17 lp/cm, the reconstructed image using the DTM-VFS method contained five strips, and the stripes were well distinguished.
Fig. 16. (a) Line diagrams for factory-specified effective focal spot size of 1 mm for 14–17 lp/cm line pairs. (b)–(e) are enlarged views.
Figure 17 shows the modulation contrast (MC) curve of line pairs between 1 and 17 lp/cm. The formula for calculating MC is shown in Eq. (8), where P is the pixel value of the image. In Fig. 17, the difficulty in demodulating the fringes increased with the number of line pairs. The MC curves for the DTM-VFS and EDM-VFS methods were greater than that of ART. At 1–9 lp/cm line pairs and 15–17 lp/cm line pairs, the curve value of DTM-VFS was greater than that of EDM-VFS. The MC values and accuracy improvement rates of 14–17 lp/cm line pairs were calculated and are presented in Table 3. The improvement rates of DTM-VFS 14-17 lp/cm line pairs were 4.94${\%}$, 32.05${\%}$, 58.21${\%}$, and 38.40${\%}$ compared with ART, and 2.55${\%}$, 29.48${\%}$, 40.74${\%}$, and 80.87${\%}$ compared with EDM-VFS.
Table 3. Improvement rates of modulation contrast.
B. Reconstructed CT images experiment of real data when the factory-specified effective focal spot size is 0.4 mm
Projection data were obtained by scanning the CTP528 module with a factory-specified effective focal spot size of 0.4 mm. Subsequently, the ART method and DTM-VFS method were used for reconstruction.
Figures 18(a1) and (b1) show the ART and DTM-VFS methods used to reconstruct the image. Figures 18(a1) and (b1) are enlarged views at 17–21 lp/cm line pairs, respectively. Evidently, the reconstructed image using the ART algorithm was smoother than that using DTM-VFS method. Additionally, the reconstructed image using the DTM-VFS method was clearer than that reconstructed using the ART method. Figure 19(a) shows plots of 17–21 lp/cm line pairs of ART method and DTM-VFS method. Figure 19(b) shows the MC image of line pairs at 10, 13, 16, 18, 19, 20, and 21 lp/cm. In Fig. 19, the degree of stripe differentiation in DTM-VFS reconstructed images was greater than that in ART reconstructed images. In Fig. 19(b), the MC value of different line pairs of DTM-VFS was greater than the MC value of the corresponding line pairs of ART. Table 4 presents the MC values at 10, 13, 16, 18, 19, 20, and 21 lp/cm line pairs of ART method and DTM-VFS method and their corresponding MC improvement rates. Evidently, compared with the ART method, the improvement rates of the MC values at 10, 13, 16, 18, 19, 20, and 21 lp/cm line pairs were 28.83${\%}$, 16.47${\%}$, 72.93${\%}$, 57.04${\%}$, 62.81${\%}$, 61.55${\%}$, and 69.51${\%}$, respectively.
Fig. 18. Reconstructed images based on re-measured effective focal spot and their enlarged views when factory-specified effective focal spot size is 0.4 mm and the window width is set to [0,1], (a1) ART, (b1) DTM-VFS. (a2) and (b2) are enlarged views of ART and DTM-VFS at 17–21 lp/cm line pairs.
Fig. 19. (a) is profile of different line pairs, (b) is modulation contrast curve. Here, the factory-specified effective focal spot size is 0.4 mm.
Table 4. Improvement rates of modulation contrast.
In summary, the feasibility of the proposed method was demonstrated based on a detailed comparison and analysis. Further, the measured focal spot information exhibited a certain accuracy, and the measurement results improved the resolution of the image.
4. Conclusion
In this study, an effective focal spot measurement method of X-ray source based on the dynamic translation of the light barrier was proposed. In this method, the effective focal spot is logically discretized into multiple subfocal spots, and the size, shape, and intensity distribution of the effective focal spot are obtained by calculating and solving the nonlinear model of the effective focal spot and the measured data. The results of numerical and real experiments indicate that the effective focal spot obtained using this method exhibits higher accuracy.
As indicated by the computational model of the method, the accuracy of measuring the effective focal spot is constrained by the movement accuracy of the electric displacement stage. In real experiments, when measuring the effective focal spots of X-ray sources with their factory-specified sizes are 1.0 and 0.4 mm, the step value was set to 0.01 mm. The electric displacement stage used had a repeatability accuracy of 0.005 mm and absolute positioning accuracy of 0.015 mm, which suffices the measurement requirements. Note that the electric displacement stage used herein is a relatively common equipment, which is simple to operate and easy to obtain. Spatial resolution is a major indicator of CT image quality, and focal spot is an important factor affecting spatial resolution. The characteristics of focal spot under different tube current and tube voltage conditions were found to be different. Thus, the proposed method can be applied to the study of focal spot change rules.
Currently, newer microfocus X-ray source devices with high spatial resolution on the order of microns or even submicrons are being introduced. Measuring the effective focal spot of a microfocus X-ray source is an issue that requires further investigation.
Funding
National Key Research and Development Program of China (2020YFA0712200); National Natural Science Foundation of China (61827809, 61871275, 61971292, 62271330).
Acknowledgments
The authors express their gratitude to Prof. Peng Zhang from Capital Normal University for his valuable suggestions during the initial stage of the study and Eng. Xu Jiang from Beijing Higher Institution Engineering Research Center of Testing and lmaging for his assistance during experiments with pinhole camera radiographic method.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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