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Method for correction of rotation errors in Micro-CT System
Nuclear Instruments and Methods in Physics Research A ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Method for correction of rotation errors in Micro-CT System Jintao Zhao, Xiaodong Hu, Jing Zou, Gengyan Zhao, Hanyu Lv, Linyan Xu, Ying Xu, Xiaotang Hu State Key Laboratory of Precision Measuring Technology and Instrument, Tianjin University, Tianjin 300072, People's Republic of China
art ic l e i nf o
a b s t r a c t
Article history: Received 28 September 2015 Received in revised form 14 December 2015 Accepted 15 January 2016
In Micro-CT (Computed Tomography) system, a series of projection data of sample are collected by the detector as the precision stage rotates step by step. However, the accuracy of projection images is limited by rotation errors during the acquisition process. Therefore, evaluating the performance of precision rotary stage and developing corresponding compensation method are necessary in Micro-CT system. In this paper, a metered system is designed which is composed of four precision capacitive sensors, a precision machined steel cylinder and four flexible hinges. Based on the metered system, a method to calibrate and correct the errors when the precision stage turns is proposed. Firstly, the theoretical analysis is proposed and the imperfect situations are considered. And then, the method has been applied to correct experimental data taken from a microscope type of Micro-CT system. Successful results are shown through evaluating MTF (Modulation Transfer Function) of Micro-CT system. Lastly, a sample of tungsten wire is scanned and the reconstructed images are compared before and after using the calibrated method. & 2016 Published by Elsevier B.V.
Keywords: Micro-CT Rotation errors Precision machined cylinder Resolution Capacitive sensor
1. Introduction Micro-CT can provide 3D images of sample's internal micro structures with spatial resolution down to sub-micron scale. Due to the non-destructive nature, Micro-CT has a wide-spread application include biological bone micro-structure analysis [1], oil and geology core analysis [2], geometric measurement [3] and so on. Spatial resolution is one of its key parameters. A Micro-CT system can satisfy high resolution requirements for sample internal nondestructive testing. In order to achieve this goal, many research works have been done. These mainly include decreasing the size of radiation source's focal spot [4], improving the spatial resolution of X-ray detector [5–9], researching new reconstruction algorithms [10–14], and correcting various kinds of artifacts [15–19]. In the process of developing Micro-CT to Nano-CT, there exists two different trends. One is using zone plate for Nano-CT. Laboratory setups with Fresnel zone plates have been realized [20]. However, it is very expensive and still limited in terms of energy range (8 keV) as well as in spatial resolution (approx. 50 nm in 2D imaging). The other is combing with SEM (Scanning Electron Microscope) [21–24]. By utilizing SEM, the spot size of the x-ray source can be decreased effectively to less than 100 nm [25], and this can increase the spatial resolution less than 100 nm. Generally speaking, Nano-CT puts forward high request to the rotation stage. When rotation stage turns, there exists unavoidable errors, such as wobble, radial and axial run-outs. Therefore, calibrating
rotation errors becomes necessary. These methods can be classified into two categories. One is compensation or correction techniques during reconstruction by image analysis [26]. However, in previous works, all authors assumed the axis of the rotation stage was perfect [36–38], and corrected artifacts caused by geometric misalignment of the system. The other is directly measuring and calibrating rotation errors by using sensors. Compared to the first methods, the latter methods can accurately quantify rotation errors and calibrate these errors in time and precisely. Nicola et al. [27] build an instrument which is based on a capacitive sensor spindle error analyser (Lion Precision), a Digital Signal Processor (Sheldon Instruments) and two piezoelectric actuator assemblies and would compensate in real time the still existing eccentricity and wobble on a commercially available air bearing rotary stage. This method can calibrate the errors to ultimate precision, down to less than 10 nm maximum eccentricity. However, as it uses air bearing rotary stage, when scanning samples which must be scanned in vacuum, this method would be non effective. Kim et al. [28] combines fiber optic interferometry as a sensing element and successfully reduces radial run-out errors down to 40 nm. Wang et al. [29] build a system consisting of a precision cylinder mounted on a low run-out ball bearing rotation stage and three integrated capacitive sensors to measure and calibrate radial and axial run-out errors. Both of the above two methods calibrate the errors to a respectively high level. However, when acquire projections of samples, wobble errors of rotation axis is another necessary factor which must be taken into consideration except
http://dx.doi.org/10.1016/j.nima.2016.01.051 0168-9002/& 2016 Published by Elsevier B.V.
Please cite this article as: J. Zhao, et al., Nuclear Instruments & Methods in Physics Research A (2016), http://dx.doi.org/10.1016/j. nima.2016.01.051i
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radial and axial run-out errors. Xu et al. [30] design a high-stability system with an array of five capacitive sensors, allowing simultaneous measurements of wobble, radial and axial errors by repeating the measurements several times. For rotation stages whose run-out and wobble are highly repeatable, this method works well. Unfortunately, if not, the random errors will be left. So it is necessary to develop a new method to measure and calibrate rotation errors, which can deal system and random errors at the same time. In this paper, we develop a system and corresponding calibration method to calibrate wobble, radial and axial errors, which can deal system and random errors simultaneously in a microscope type of Micro-CT system. Though the resolution of this type of Micro-CT mainly from the detector, the spatial resolution and image quality of Micro-CT system still should be increased through this method. The calibrating system consists of four precision capacitive sensors, four flexible hinges and a high precision machined steel cylinder (HPMSTY). Basing on analysing the influence of all rotation errors on sample projections, the principle of calibrate method is given out. When implement it to actual, the installation eccentricity, machining roundness and so on are calibrated first. In the next step, the readings of all the four sensors are recorded during scanning samples. Then we can use the calibrated data and the readings of sensors to correct every frame image. Experiments show that the method is effective and can increase the spatial resolution and reconstruction image quality. Although the result is not perfect and the residual errors are larger than some other experiments because of focal drifts and variation in temperature, it still should point out that this method has no claim to the rotation stage, and is vacuum compatible.
2. Method and materials
Fig. 1. Micro-CT system used in Tianjin University. (1) X-ray tube; the precision stage consists of (2) rotation table, (3) three dimensional translation platform and (4) sample Table; (5) lens-coupled detector; (6) high precision machined steel cylinder; (7) capacitive sensors.
Table 1 Main parameters of the X-ray source. Parameter
Value
Tube type Target material Voltage (kV) Current (mA) Focal spot size (μm)
Transmission Tungsten 20–160 0.05–1.0 o2
Table 2 Main parameters of the X-ray detector. Parameter
Value
Scintillator material Scintillator thickness (μm) CCD pixel size (μm2) CCD size (pixel)
CsI (Tl) 15 13.5 13.5 2048 2048
2.1. Micro-CT prototype system Fig. 1 shows a photograph of the developed micro-CT system with a micro-focus x-ray source, a high precision stage and a lenscoupled detector. Some important parameters about the x-ray source are shown in Table 1. A lens-coupled detector produced by Tianjin University has been used as the 2-D digital x-ray imager. Some main parameters about the detector are listed in Table 2. In addition, the detector can realize 2 , 40 optical magnification due to the adopted optical lens. The prototype precision stage in our Micro-CT system is shown in Fig. 2. It composes of a highaccuracy rotation table with direct reading encoder and having better than 0.0001° (0.36 arcsec) resolution, a three dimensional translation platform which is convenient for placing the sample at the centre of the rotation table, and a sample table which is for placing the sample. 2.2. Errors analysis of precision stage As we know, a rotation table have 6 degrees of freedom (DOF), see Fig. 3a). These errors (Δx ; Δy ; and Δz ) consist of translation, or linear movement, along any of three perpendicular axes (X, Y and Z), as well as rotation errors (Δα ; Δβ and Δγ )around any of those axes(X, Y and Z). In micro-CT system, we adopt rotation stage with direct reading encoder, whose movement resolution around the axis of rotation (Y) is better than 0.0001° (0.36 arcsec). Therefore, only 5 errors (Δx ; Δy ; Δz ; Δα ; Δγ ) existed, see Fig. 3(b–f). It is worthwhile to carefully examine the effects of these 5 errors. Firstly, main parameters of the rotary sample stage are listed in Table 3. Then, we analyse the effect of 5 errors (Δx ; Δy ; Δz ; Δα ; Δγ ) on the projection images in our Micro-CT system. Fig. 4 shows the scanning geometry; the red line represents the rotation axis; SDD
Fig. 2. The prototype of the precision stage.
is the distance between the source and detector; SOD is the distance between the source and rotation axis. If there is no errors during the process of the rotation table rotates, taking a ball which is placed on the rotation axis as an example, the projections of it will stay in the same place. If not, the location of the projections will change. We will firstly analyse these changes when different errors exist. The effect of the error motions will be discussed under the common scan condition that the SOD is 15 mm; the SDD is 32 mm;
Please cite this article as: J. Zhao, et al., Nuclear Instruments & Methods in Physics Research A (2016), http://dx.doi.org/10.1016/j. nima.2016.01.051i
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Fig. 3. 6 degrees of freedom of a rotation table and the 5 existed DOF. Table 3 Main parameters of the rotary sample stage. Parameter
Value
Bore Diameter of the Bearing (mm) Outside Diameter of the Bearing (mm) Open-loop Radial Error Motion of the Roller Bearing (μm) Open-loop Axial Error Motion of the Roller Bearing (μm) Bearing yaw wobble (rad) Rotation Accuracy (fed back by an encoder) (rad)
190 240 10 10 8.33 10 5 7 3.491 10 4
3) Δz equals to the radial run-out of the rotation table. The height of the sample is represented by R. when the sample have Δz translation along the Z axis, as shown in Fig. 5c), the projection offset on the detector can be expressed by
ΔT ¼
SDD U ΔZ U R UM opt ; SOD U ðSOD þ ΔZ Þ
ð1Þ
Assuming Δz is 10 μm, ΔT ¼ 7:04 μm can be derived, which equals to 0.52 pixel, less than one pixel size of the CCD. 4) Δα is assumed 0.005°, see Fig. 5d, then the translation of the 0 sample along Z axis is ΔZ ¼ h sin ð0:005 3 Þ 13 μm; similar to case 3), using Eq. (1), ΔT ¼ 8:8 μm can be derived. Therefore, the maximum deviation of the edge of the projection will be about 0.65 pixel. 5) Δγ is assumed 0.005°, similar to case 4), the sample will deviate about 13 μm from the ideal condition at X direction. According to case 1), it will generate approximating 40 pixels deviation on the actual projection position from the ideal position.
Rotation axis
Source
Mid-plane Detector Fig. 4. The cone beam geometry.
the magnification of the optical microscope within the detector is 20; the reference point on the sample is 250 μm to the optical axis. 1) Δx equals to the radial run-out of the rotation table whose maximum value can be 10 μm. See Fig. 5a), the actual projection position deviate from the ideal position is Δx SDD/SOD Mopt. Assume Δx is 10 μm, then the distance deviation approximates 427 μm, about 31 pixels offset on the detector; 2) Δy , as well as , equals to the ending run-out of the rotation table. The influence of Δy on the projection is the same to Δx . If Δy is equal to 10 μm, the actual projection position deviates from the ideal position approximating 31 pixels; see Fig. 5b;
When the translation/rotation errors lead to the projection position offset more than 1 pixel, it is worthwhile to carefully examine the effects of the errors and the method of measurement. Therefore, we just need to measure 3 sensitive factors (Δx ; Δy ; Δγ ;) and ignore the others (Δz ; Δα ; Δβ ). 2.3. Theoretical analysis As shown in Fig. 6, we designed a metered system which is composed of four precision capacitive sensors, a precision machined steel cylinder and four flexible hinges. The high precision machined steel cylinder (HPMSTY) is mounted on the rotation table and turns with it. Four capacitive distance sensors are used to measure the distances between the sensors and HPMSTY. S1, S2, S3 and S4 represents sensor1, sensor 2, sensor 3 and sensor 4 in
Please cite this article as: J. Zhao, et al., Nuclear Instruments & Methods in Physics Research A (2016), http://dx.doi.org/10.1016/j. nima.2016.01.051i
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Fig. 5. The influences of different errors on projections. (a) the error translation along X axis; (b) the error translation along Y axis; (c) the error translation along Z axis; (d) tilt about X axis and tip about Z axis is the same.
4 sensor signals, according to the following formulate: 8 0 d 1 ¼ d 1 Δx ; > > > > > < d2 ¼ d02 þ r U sin Δγ þ Δy ; 0 > d3 ¼ d3 cos φ U r U sin Δγ sin φ Ur U sin Δα þ Δy ; > > > > : d ¼ d0 cosψ Ur U sin Δ þ sinψ U r U sin Δ þ Δ ; γ α y 4 4
Fig. 6. A mechanical layout of the measuring system and motion system.
Where d1 through d4 are the sensor-to-cylinder gap spacing, as 0 0 measured by the 4 capacitive sensors. d1 through d4 are the initial gap spacing, determined by the location adjustment of each capacitive sensor. Δx and Δy are the displacements of HPMSTY in the X and Y directions, respectively. φ and ψ are the angles between the straight line passing through S3 and S4 and the centre of the HPMSTY and X axis. And r is 140.0 mm, which represents the lateral distance from the center of the cylinder to the center of the three tip-tilt sensors. See Fig. 7 for a diagram of the physical parameters. To simplify the Eq. (2), we assume a ¼ cos ðφÞ; b ¼ sin ðφÞ; c ¼ cos ðψ Þ; d ¼ sin ðψ Þ;
the following text. S1 is used to monitor the radial run-out, S2, S3 and S4 are used to monitor the axial run-out. Flexible hinges are used to adjust the positions of sensors. When the rotation table turns, HPMSTY will have the same errors as the rotation table, therefore, the distances between the HPMSTY and the sensors will also change. Assuming HPMSTY has a perfect flat top surface (zero axial runout), roundness (zero radial run-out) and verticality, and nomanufacturing errors and installation errors. In this case, the complete position of the HPMSTY could be determined from the
ð2Þ
And Eq. (2) can be written as: 8 0 d ¼ d 1 Δx ; > > > 1 > > < d2 ¼ d02 þ r U sin Δγ þ Δy ; 0 > d3 ¼ d3 a U r U sin Δγ b U r U sin Δα þ Δy ; > > > > : d ¼ d0 c Ur U sin Δ þ d U r U sin Δ þ Δ ; γ α y 4 4
ð3Þ
ð4Þ
According to Eq. (4), error motions of rotating stage can be decomposed into two translational and two rotational motions, which is useful to facilitate the solution of equations, and further
Please cite this article as: J. Zhao, et al., Nuclear Instruments & Methods in Physics Research A (2016), http://dx.doi.org/10.1016/j. nima.2016.01.051i
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reason, if the top sensors are not mounted at the exact same radius, S2, S3, and S4 will also contain additional error signals. 2.4.2. Putting forward a new amendment formula According to 1) and 2) above, each sensor will have an additive factor that varies with rotation angle θ. Because of 3) and 4) above, each sensor will measure distance with a slightly different (but constant) sensitivity and offset. And finally, from 5) above, the additive errors for each sensor are different. Now, add the nonidealities mentioned above to Eq. (4), and Eq. (5) are achieved. 8 V 1 ¼ V 01 ðθÞ μ1 Δx ; > > > > > < V 2 ¼ V 02 ðθÞ þ μ2 r U sin ðΔγ Þ þ Δy ; 0 > > > V 3 ¼ V 3 ðθÞ μ3 a Ur U sin ðΔγ Þ þ b U r U sin ðΔα Þ Δy ; > > : V ¼ V 0 ðθÞ μ c U r U sin ðΔ Þ d U r U sin ðΔ Þ Δ ; γ α y 4 4 4
Fig. 7. The diagram of the physical parameters.
calibration. In Eq. (4), d1 ; d2 ; d3 ; d4 can be acquired from the 4 sensors; Δx ; Δy ; Δγ are parameters we want to know and calibrate; Δα is an intermediate parameter which we don't need to 0 0 0 0 calibrate; d1 ; d2 ; d3 ; d4 are related with the installing of the sensors. 0 0 0 0 To calibrate Δx ; Δy ; Δγ , d1 ; d2 ; d3 ; d4 must be calibrated firstly; After 0 0 0 0 doing this, we can use the values of d1 ; d2 ; d3 ; d4 and d1 ; d2 ; d3 ; d4 to solve Δx ; Δy ; Δγ . 2.4. Practical measurement method It should be noted that Eq. (4) can't be used directly as HPMSTY is a manufactured object which is not “perfect”. Besides, installation errors can't be avoided in practical system. Now, we will investigate the non-idealities of the cylinder and installation errors, and then, put forward a new amendment formula which can be used in practice. 2.4.1. Examining the non-idealities of the cylinder 1) Radial run-out (cylinder roundness). This non-ideality will cause the signals measured by S1 to vary in a deterministic way as the cylinder rotates. The error signal will be additive to the "real" signals for X movement. 2) Axial run-out (top flatness). This will cause the signals from S2, S3, and S4 to vary in a deterministic way as the cylinder rotates. Tcnals for Y and β movement. 3) Axis of rotation tilt. If the entire rotation stage is tilted relative to the capacitive sensors, there will be a fixed offset in S2, S3, and S4 due to the sensors being closer or farther from the top of the cylinder. This tilt will also cause S1 to become less sensitive to displacements, as a fraction of the total motion will be in a direction perpendicular to the capacitive sensors. The sensitivity of these sensors will be multiplied by cos(ξ1), where ξ1 is the angle of the rotation tilt. 4) Sensor tilt. If an individual sensor is tilted relatively to the cylinder (not perfectly parallel to the measurement surface), then its sensitivity will be multiplied by cos(ξ2), where ξ2 is the angle of the sensor tilt. 5) Edge non-perpendicularity. If the edges of the steel cylinder are not perpendicular to the top surface, and further, if S1 is not mounted at the exact same vertical position along the edge, then an additional error signal will exist for this sensor. This is because the sensor will be looking at a different "ring" of the steel cylinder, and due to the non-perpendicularity of the edge, these rings will be different from each other. For the same
ð5Þ
In the above equations, V 1 ; V 2 ; V 3 ; V 4 are the output voltages from the individual sensors, V 01 ðθÞ; V 02 ðθÞ; V 03 ðθÞ; V 04 ðθÞ are the offset voltages of the individual sensors, and μ1 ; μ2 ; μ3 ; μ4 are the individual sensitivities (voltage/distance) of each sensor. Note that the offset voltages V 01 ðθÞ; V 02 ðθÞ; V 03 ðθÞ; V 04 ðθÞ are functions of θ, and θ is the angle of the rotation table turning, as the error terms due to the steel cylinder imperfections have been lumped in. Here, we focus on solving these parameters Δx ; Δy ; Δγ . Looking at Eq. (5), it is apparent that each steel cylinder must be fully characterized, to map out the voltage offsets V 01 ðθÞ; V 02 ðθÞ; V 03 ðθÞ; V 04 ðθÞ; and sensitivities μ1 ; μ2 ; μ3 ; μ4 . Further, because these parameters are highly sensitive to sensor mounting, the assembly must not be disassembled or adjusted after calibration has been performed. From Eq. (5), we can find that it contains more unknowns than equations. Therefore, simplifying approximations must be made. 1) The first reasonable simplification is to assume that all sensor sensitivities are equal to each other. A survey of the calibration certificates for 4 sensors reveals a nominal gain of 10 V/50 μm, and a typical gain error of 0.01%. Therefore, it can safely be assumed that all the values of μi (i¼ 1,2,3,4) are the same. This eliminates the 4 different μi (i¼ 1,2,3,4) unknowns, and replaces them with a single known quantity, μ. This assumption will introduce an error multiplier of cos(ξ1)cos(ξ2), where (ξ1) and (ξ2) are the sensor and axis of rotation tilt angles. Since the tile angles are small (assume a worst case tilt of 1 Mrad), the sensitivity will only change by cos2(0.001) ¼0.0001%, which can be safely ignored. So Eq. (5) can be written as: 8 V ¼ V 01 ðθÞ μΔx ; > > > 1 > > < V 2 ¼ V 02 ðθÞ þ μ r U sin ðΔγ Þ þ Δy ; > V 3 ¼ V 03 ðθÞ μ a Ur U sin ðΔγ Þ þ b Ur U sin ðΔα Þ Δy ; > > > > : V ¼ V 0 ðθÞ μ c U r U sin ðΔ Þ d U r U sin ðΔ Þ Δ ; γ α y 4 4
ð6Þ
2) Another simplification is to assume that the top surface of the steel cylinder is extremely flat, which means that the three sensors on the top surface do not have different V 0x θ functions-they merely measure a phase-shifted version of the same function. I.e. 8 < V 3 0 θ ¼ V 02 ðθ þ 1801 φÞ; : V 4 0 θ ¼ V 02 ðθ þ 1801 þ ψ Þ;
ð7Þ
Please cite this article as: J. Zhao, et al., Nuclear Instruments & Methods in Physics Research A (2016), http://dx.doi.org/10.1016/j. nima.2016.01.051i
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3.1. Experiment for resolution
Merging this assumption into Eq. (6) gives: 8 V ¼ V 01 ðθÞ μΔx ; > > > 1 > > < V 2 ¼ V 02 ðθÞ þ μ r U sin ðΔγ Þ þ Δy ; > V 3 ¼ V 02 ðθ þ 1801 φÞ μ a Ur U sin ðΔγ Þ þb U r U sin ðΔα Þ Δy ; > > > > : V ¼ V 0 ðθ þ 1801 þ ψ Þ μ c U r U sin ðΔ Þ d U r U sin ðΔ Þ Δ ; γ α y 4 2 ð8Þ Using above simplifications, Eq. (8) can now be solved for
Δx ; Δy ; Δα .
To verify the influence of the correction method on spatial resolution of the Micro-CT system, in this part, 360 projections of a ball whose diameter is 500 μm are acquired during 2π angular range and Table 4 shows the basic parameters of the system and experimental conditions. Hough transformation are used to compute the centre of the ball. Fig. 8 shows one projection image of the ball and the extraction of the centre of the projection circle. As the errors of the axis of rotation, the centres of the projection
8 > Δx ¼ V 01 ðθÞ V 1 ; > > h i > < Δy ¼ ðad þ bc1þ b þ dÞ U ðad þ bcÞðV 2 V 02 ðθÞÞ þ dðV 3 V 02 ðθ þ 1801 φÞÞ þ bðV 4 V 02 ðθ þ 1801 þ ψ ÞÞ ; > h i > > 0 0 0 > : sin ðΔα Þ ¼ ðad þ bc1þ b þ dÞr U ðbþ dÞðV 2 V 2 ðθÞÞ dðV 3 V 2 ðθ þ 1801 φÞÞ bðV 4 V 2 ðθ þ 1801 þ ψ ÞÞ ; After we acquire the values of Δx ; Δy ; Δα , it is necessary to build the relationships between Δx ; Δy ; Δα and the shifts of the projection images. According to this, we can get: ( xshif t ¼ Δx þ h sin Δγ ; ð10Þ yshif t ¼ Δy ; Where h is the height (above the steel plate) of the sample; xshift and yshift are the sample image shifts along X and Y directions, respectively. Now it is important to talk about how to correct the X-ray image errors resulting from off-axis motion of the sample stage. We use the software correction method, that is, during the reconstruction phase, utilize the recorded signals from the capacitive position sensors to translate each image to correct for X/Y errors due to the stage. This method does not change the acquisition process; it only requires that signals from the position sensors are recorded. For this correction method, only the sample image shifts-xshift and yshift , can be corrected. Therefore, from Eq. (10), only Δx ; Δy ; Δα are important. Putting Eq. (10) together with Eq. (9) yields:
ð9Þ
circles vary within a certain range. After the correction is done, the centres of the projection circles should be reduced to a smaller area. In this part, we measure the errors through projections and capacitive sensors. To be convenient for the following reconstruction, it is necessary to correct the position of the projections, so the unit of pixel is used in this part, like Tables 5 and 6. In addition, the offset which is in unit of pixel can be converted to the one whose unit is μm by multiplying 13:5 SOD=SDD=M opt . Fig. 9(a) shows horizontal position (X axis) offset before and after calibration. The error range and standard deviation are computed and summarized in Table 5. As can be noticed from Table 5, the variation scope after correction reduces to 5 pixels from 10 pixels, with about 50% improvement. Fig. 9(b) shows the vertical position (Y axis) offset before and after calibration. The error range and standard deviation are computed and summarized in Table 6. As can be noticed from Table 6, the variation scope after correction reduces to 4 pixels from 11 pixels, with about 63.6% improvement. Fig. 9(c) shows the variation of the spatial position distribution of the centre of the ball. The circle points represent the position offset at different angles before correction and the
8 < xshif t ¼ V 01 ðθÞ V 1 þ ðad þ bchþ b þ dÞr U ½ðb þ dÞðV 2 V 02 ðθÞÞ dðV 3 V 02 ðθ þ 1801 φÞÞ bðV 4 V 02 ðθ þ 1801 þ ψ ÞÞ; : yshif t ¼
1 U½ðad þ bcÞðV 2 V 02 ð ðad þ bc þ b þ dÞ
ð11Þ
θÞÞ þ dðV 3 V 02 ðθ þ 1801 φÞÞ þ bðV 4 V 02 ðθ þ 1801þ ψ ÞÞ;
Solving for the two unknowns (V 01 θ ; V 02 θ ):
star points represent the position offset at different angles after
8 < ðad þ bcÞV 02 ðθÞ þ dV 02 ðθ þ 1801 φÞ þ bV 02 ðθ þ 1801 þ ψ Þ ¼ ðad þ bcÞV 2 þ dV 3 þ bV 4 ðad þ bc þ bþ dÞyshif t ;
ð12Þ
: V 01 ðθÞ þ ðad þ bchþ b þ dÞr U ½ ðb þ dÞV 02 ðθÞ þdV 02 ðθ þ 1801 φÞ þ bV 02 ðθ þ 1801 þ ψ Þ ¼ xshif t þ V 1 ðad þ bchþ b þ dÞr U ½ðb þ dÞV 2 dV 3 bV 4 ;
Once the two error vectors are derived, we can correct the projections using Eq. (11).
3. Experiments and results Here, we verify the validity of the method on Micro-CT system of Tianjin University. Fig. 1 shows the system we use. To verify the correction method, two on-line experiments are done. One is for inspecting the influence on spatial resolution, and the other is for verifying the influence on image quality.
Table 4 Primary parameters of the system and experimental conditions. Parameter
Value
Voltage of the X-ray source (kV) Target power of the X-ray source (W) SOD (Source Object Distance) (mm) SDD (Source Detector Distance) (mm) M opt Pixel size in the object space (μm2) Rotation interval/Projection image (degree) Number of projection images Exposure time/Projection image (s) Distance between the ball and cylinder: h (mm)
110 4 15 32 20 0.16*0.16 1 360 10 150
Please cite this article as: J. Zhao, et al., Nuclear Instruments & Methods in Physics Research A (2016), http://dx.doi.org/10.1016/j. nima.2016.01.051i
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correction. From this figure, we can observe that the distribution of the centres are more concentrated after correction. To verify the influence of the correction method on spatial resolution of the Micro-CT system, we reconstruct the small ball by using of the FDK [32] algorithm under two conditions. One is using all the projections before correction and the other is using all the projections after correction. Then the mid-plane of the
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constructed images is extracted to acquire the resolution under each condition. To acquire the resolution, the MTF curve is utilized. The MTF of any system is the frequency response to a delta signal and has been one of the physical characteristics that is commonly used to quantitatively measure the physical performance of a system [31]. Fig. 9(d) shows the MTF curves under the two conditions. The circle line represents the MTF curve before calibration. The value of frequency is 470 lp/mm when the value of MTF equals 0.1, Table 5 Horizontal shift of the images before and after calibration.
Before calibration After calibration Improving effect
Error range (pixels)
Standard deviation
10 5 50%
2.145 1.005 53.1%
Table 6 Vertical shift of the images before and after calibration.
Fig. 8. Projection image of the ball.
Before calibration After calibration Improving effect
Error range (pixels)
Standard deviation
11 4 63.6%
2.556 0.881 65.5%
Fig. 9. The positon offset curves of the projections before and after calibration. (a) the offset curves at X direction; (b) the offset curves at Y direction; (c) the distribution of the projections before and after calibration; (d) the MTF curves before and after calibration.
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Fig. 10. The phantom and its X-ray projection image.
Table 7 Condition of the experiments.
Fig. 11. The direct reconstruction result without correction.
Parameter
Value
Voltage of the X-ray source (kV) Target power of the X-ray source (W) SOD (Source Object Distance) (mm) SDD (Source Detector Distance) (mm) Pixel size in the Object Space (μm2) Rotation interval/Projection image (degree) Number of projection images Exposure time/Projection image (s)
110 4 14 30 0.314 0.314 0.5 720 10
which means the resolution is 2.16 μm under this condition. The star line represents the MTF curve after calibration. The value of frequency is 580 lp/mm at the 10% MTF, which means the resolution is 1.72 μm under this condition. From this, we can deduce the resolution of the system is improved about 20%. However, there still exist some residual errors. Here we mainly consider the influence of temperature and focal drifts, as the scanning time is about several hours and we can't ensure the temperature is constant. Taking the vertical direction as an example, the perpendicular distance between the ball we scanned and the capacitive sensor is about 0.14 m, and the ball is propped up by aluminium, whose linear expansion coefficient is 2.3 10 5 K 1. Then the variation at vertical direction caused by temperature (assuming the variation of temperature is ΔT ¼ 0:4K) is 2:3 10 5 0:14 ΔT SDD=SOD M opt ¼ 5:5 10 5 m, about 4.07 pixels, which is correspond with the residual errors we gotten above. Commonly, the temperature is not controlled in Micro-CT. In the future, how to control the temperature when we use the system for scanning or build the relationship between temperature and the errors is a technical challenge and we will put our focus on it. What's more, in previous work, we measured the focal drifts of the x-ray source by utilizing a cross wire making by 12 μm thick
tungsten wire. After warming up the source for tens of minutes, the focal drifts is about 2–3 μm, and this will bring the locations of images changing in about 5 pixels, which also agree with the results of the experiment. Because of the above analysis, in the future, we will put our focus on how to control the temperature when we use the system for scanning or build the relationship between temperature and the errors, and calibrate focal spot drifts. As both of the two reasons mentioned above may cause residual errors, we will also analyse which is the main factor to cause the residual errors. More importantly, through this, we verified the correction method is effectively. 3.2. Experiment for image quality To test our correction method's effect on image quality, a special phantom has been made using two tungsten filaments, as is shown in Fig. 10. The diameter of each is about 6 μm. They were twisted with each other, so the gap between them is a little wider in the midst, and narrower at each end. Moreover, there is a fracture in its midst. The phantom was scanned in the condition shown in Table 7. Fig. 11 is the reconstruction result of the projection images, in which there are two severe artifacts. If the phantom is a little larger, they will overlap the real reconstruction model, and the result will become chaos totally. This is caused mainly by the mounting error of the rotary sample stage which makes the rotation axis deviate from the optic axis. In the micro-CT reconstruction algorithm FDK, there can be an offset interface where the mounting error of the rotary stage can be compensated [32–35]. This can really correct the mounting
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Fig. 12. The reconstruction result of the projection images processed only by the offset (a1) and that processed by both the offset and our correction method (b1). (a2) and (b2) are the enlarged views of the bottom of (a1) and (b1) respectively. (a3) and (b3) are the enlarged views of the middle of (a1) and (b1) respectively.
error of the rotary stage, but it has no effect on the spindle error motion during the CT scan. In Fig. 12, the reconstruction result only processed by the offset is contrasted with that processed by both the offset and the correction method, which demonstrates the correction method is quite effective in the spindle error motion's correction. The gap between the two filaments in the lower part of the reconstruction result processed by the correction method [Fig. 12 (b1)] is larger than that not [Fig. 12(a1)], and matches what the projection image shows in Fig. 11 correctly. This can be better
observed from their enlarged views [Fig. 12(a2) and (b2)]. Evidently, better horizontal resolution has been achieved by the correction method. Meanwhile, at the fracture the filaments [Fig. 12(b3)] are a little thinner than those not [Fig. 12(a3)]. This is not obvious for the fracture is too small to be distinguished, but the corrected one shows an obvious dent due to the fracture, which is marked by an arrow in Fig. 12(b3). This means a better vertical resolution has been achieved. Fig. 13(a) and (b) show the slices on the same layer in the lower part of each result. In Fig. 13 (a) the two filaments are mixed up and the gap cannot be
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Fig. 13. The slice in the reconstruction result not processed by the correction method (a) and that on the same layer in the reconstruction result processed by the correction method (b). (c) is the gray level diagrams of the pixels on the lines marked in (a) and (b).
distinguished, while in Fig. 13(b) the corrected filaments are separate, and the gap between them is clear. Fig. 13(c) is the gray level diagrams of the voxels on the lines marked in Fig. 13(a) and (b). Firstly, the grey value contrast of the two filaments and their gap in the corrected result is much stronger than that in the uncorrected one, and the grey value of the gap is much lower in the corrected result. Secondly the corrected one has sharper filament edges than the uncorrected one. Therefore, it quantitatively proves that the correction method can help attaining a better image quality. In sum, all these attest that the correction method proposed in this article can get a reconstruction result with better 3D resolution and image quality.
4. Conclusion In this paper, we put our attention on correcting the errors of the rotation axis when it turns, aiming at increase system spatial
resolution and image quality. In order to achieve this goal, we use a precision-manufacturing steel cylinder and 4 capacitive sensors to do the calibration and correction, and analyse possible difficulties when implement it to practice. Finally, we verify the theory on-line. And the results show the theory and method is effective. Also, the resolution of the Micro-CT system is improved about 20% and the image quality is enhanced. In summary, the method is effective in improving the resolution and image quality of the Micro-CT. Moreover, although the result is not perfect and the residual errors are larger than some other experiments because of variation in temperature and focal drifts, it still should point out that this method has no claim to the rotation stage, and is vacuum compatible. When temperature is controlled in Nano-CT, this method can be applied to it in the future, and we will put our focus on how to control the temperature when we use the system for scanning or build the relationship between temperature and the errors in the future, and calibrate focal spot drifts.
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Acknowledgements This work was supported by China's Ministry of Science & Technology (No. 2011YQ030112) and National Natural Science Foundation of China (No. 61372144). The authors would like to acknowledge mechanical engineer Bai Jianguo from Centre of Micro-Nano Manufacturing Technology in Tianjin for helping to design and manufacture the HPMSTY.
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Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Method for correction of rotation errors in Micro-CT System Jintao Zhao, Xiaodong Hu, Jing Zou, Gengyan Zhao, Hanyu Lv, Linyan Xu, Ying Xu, Xiaotang Hu State Key Laboratory of Precision Measuring Technology and Instrument, Tianjin University, Tianjin 300072, People's Republic of China
art ic l e i nf o
a b s t r a c t
Article history: Received 28 September 2015 Received in revised form 14 December 2015 Accepted 15 January 2016
In Micro-CT (Computed Tomography) system, a series of projection data of sample are collected by the detector as the precision stage rotates step by step. However, the accuracy of projection images is limited by rotation errors during the acquisition process. Therefore, evaluating the performance of precision rotary stage and developing corresponding compensation method are necessary in Micro-CT system. In this paper, a metered system is designed which is composed of four precision capacitive sensors, a precision machined steel cylinder and four flexible hinges. Based on the metered system, a method to calibrate and correct the errors when the precision stage turns is proposed. Firstly, the theoretical analysis is proposed and the imperfect situations are considered. And then, the method has been applied to correct experimental data taken from a microscope type of Micro-CT system. Successful results are shown through evaluating MTF (Modulation Transfer Function) of Micro-CT system. Lastly, a sample of tungsten wire is scanned and the reconstructed images are compared before and after using the calibrated method. & 2016 Published by Elsevier B.V.
Keywords: Micro-CT Rotation errors Precision machined cylinder Resolution Capacitive sensor
1. Introduction Micro-CT can provide 3D images of sample's internal micro structures with spatial resolution down to sub-micron scale. Due to the non-destructive nature, Micro-CT has a wide-spread application include biological bone micro-structure analysis [1], oil and geology core analysis [2], geometric measurement [3] and so on. Spatial resolution is one of its key parameters. A Micro-CT system can satisfy high resolution requirements for sample internal nondestructive testing. In order to achieve this goal, many research works have been done. These mainly include decreasing the size of radiation source's focal spot [4], improving the spatial resolution of X-ray detector [5–9], researching new reconstruction algorithms [10–14], and correcting various kinds of artifacts [15–19]. In the process of developing Micro-CT to Nano-CT, there exists two different trends. One is using zone plate for Nano-CT. Laboratory setups with Fresnel zone plates have been realized [20]. However, it is very expensive and still limited in terms of energy range (8 keV) as well as in spatial resolution (approx. 50 nm in 2D imaging). The other is combing with SEM (Scanning Electron Microscope) [21–24]. By utilizing SEM, the spot size of the x-ray source can be decreased effectively to less than 100 nm [25], and this can increase the spatial resolution less than 100 nm. Generally speaking, Nano-CT puts forward high request to the rotation stage. When rotation stage turns, there exists unavoidable errors, such as wobble, radial and axial run-outs. Therefore, calibrating
rotation errors becomes necessary. These methods can be classified into two categories. One is compensation or correction techniques during reconstruction by image analysis [26]. However, in previous works, all authors assumed the axis of the rotation stage was perfect [36–38], and corrected artifacts caused by geometric misalignment of the system. The other is directly measuring and calibrating rotation errors by using sensors. Compared to the first methods, the latter methods can accurately quantify rotation errors and calibrate these errors in time and precisely. Nicola et al. [27] build an instrument which is based on a capacitive sensor spindle error analyser (Lion Precision), a Digital Signal Processor (Sheldon Instruments) and two piezoelectric actuator assemblies and would compensate in real time the still existing eccentricity and wobble on a commercially available air bearing rotary stage. This method can calibrate the errors to ultimate precision, down to less than 10 nm maximum eccentricity. However, as it uses air bearing rotary stage, when scanning samples which must be scanned in vacuum, this method would be non effective. Kim et al. [28] combines fiber optic interferometry as a sensing element and successfully reduces radial run-out errors down to 40 nm. Wang et al. [29] build a system consisting of a precision cylinder mounted on a low run-out ball bearing rotation stage and three integrated capacitive sensors to measure and calibrate radial and axial run-out errors. Both of the above two methods calibrate the errors to a respectively high level. However, when acquire projections of samples, wobble errors of rotation axis is another necessary factor which must be taken into consideration except
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radial and axial run-out errors. Xu et al. [30] design a high-stability system with an array of five capacitive sensors, allowing simultaneous measurements of wobble, radial and axial errors by repeating the measurements several times. For rotation stages whose run-out and wobble are highly repeatable, this method works well. Unfortunately, if not, the random errors will be left. So it is necessary to develop a new method to measure and calibrate rotation errors, which can deal system and random errors at the same time. In this paper, we develop a system and corresponding calibration method to calibrate wobble, radial and axial errors, which can deal system and random errors simultaneously in a microscope type of Micro-CT system. Though the resolution of this type of Micro-CT mainly from the detector, the spatial resolution and image quality of Micro-CT system still should be increased through this method. The calibrating system consists of four precision capacitive sensors, four flexible hinges and a high precision machined steel cylinder (HPMSTY). Basing on analysing the influence of all rotation errors on sample projections, the principle of calibrate method is given out. When implement it to actual, the installation eccentricity, machining roundness and so on are calibrated first. In the next step, the readings of all the four sensors are recorded during scanning samples. Then we can use the calibrated data and the readings of sensors to correct every frame image. Experiments show that the method is effective and can increase the spatial resolution and reconstruction image quality. Although the result is not perfect and the residual errors are larger than some other experiments because of focal drifts and variation in temperature, it still should point out that this method has no claim to the rotation stage, and is vacuum compatible.
2. Method and materials
Fig. 1. Micro-CT system used in Tianjin University. (1) X-ray tube; the precision stage consists of (2) rotation table, (3) three dimensional translation platform and (4) sample Table; (5) lens-coupled detector; (6) high precision machined steel cylinder; (7) capacitive sensors.
Table 1 Main parameters of the X-ray source. Parameter
Value
Tube type Target material Voltage (kV) Current (mA) Focal spot size (μm)
Transmission Tungsten 20–160 0.05–1.0 o2
Table 2 Main parameters of the X-ray detector. Parameter
Value
Scintillator material Scintillator thickness (μm) CCD pixel size (μm2) CCD size (pixel)
CsI (Tl) 15 13.5 13.5 2048 2048
2.1. Micro-CT prototype system Fig. 1 shows a photograph of the developed micro-CT system with a micro-focus x-ray source, a high precision stage and a lenscoupled detector. Some important parameters about the x-ray source are shown in Table 1. A lens-coupled detector produced by Tianjin University has been used as the 2-D digital x-ray imager. Some main parameters about the detector are listed in Table 2. In addition, the detector can realize 2 , 40 optical magnification due to the adopted optical lens. The prototype precision stage in our Micro-CT system is shown in Fig. 2. It composes of a highaccuracy rotation table with direct reading encoder and having better than 0.0001° (0.36 arcsec) resolution, a three dimensional translation platform which is convenient for placing the sample at the centre of the rotation table, and a sample table which is for placing the sample. 2.2. Errors analysis of precision stage As we know, a rotation table have 6 degrees of freedom (DOF), see Fig. 3a). These errors (Δx ; Δy ; and Δz ) consist of translation, or linear movement, along any of three perpendicular axes (X, Y and Z), as well as rotation errors (Δα ; Δβ and Δγ )around any of those axes(X, Y and Z). In micro-CT system, we adopt rotation stage with direct reading encoder, whose movement resolution around the axis of rotation (Y) is better than 0.0001° (0.36 arcsec). Therefore, only 5 errors (Δx ; Δy ; Δz ; Δα ; Δγ ) existed, see Fig. 3(b–f). It is worthwhile to carefully examine the effects of these 5 errors. Firstly, main parameters of the rotary sample stage are listed in Table 3. Then, we analyse the effect of 5 errors (Δx ; Δy ; Δz ; Δα ; Δγ ) on the projection images in our Micro-CT system. Fig. 4 shows the scanning geometry; the red line represents the rotation axis; SDD
Fig. 2. The prototype of the precision stage.
is the distance between the source and detector; SOD is the distance between the source and rotation axis. If there is no errors during the process of the rotation table rotates, taking a ball which is placed on the rotation axis as an example, the projections of it will stay in the same place. If not, the location of the projections will change. We will firstly analyse these changes when different errors exist. The effect of the error motions will be discussed under the common scan condition that the SOD is 15 mm; the SDD is 32 mm;
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Fig. 3. 6 degrees of freedom of a rotation table and the 5 existed DOF. Table 3 Main parameters of the rotary sample stage. Parameter
Value
Bore Diameter of the Bearing (mm) Outside Diameter of the Bearing (mm) Open-loop Radial Error Motion of the Roller Bearing (μm) Open-loop Axial Error Motion of the Roller Bearing (μm) Bearing yaw wobble (rad) Rotation Accuracy (fed back by an encoder) (rad)
190 240 10 10 8.33 10 5 7 3.491 10 4
3) Δz equals to the radial run-out of the rotation table. The height of the sample is represented by R. when the sample have Δz translation along the Z axis, as shown in Fig. 5c), the projection offset on the detector can be expressed by
ΔT ¼
SDD U ΔZ U R UM opt ; SOD U ðSOD þ ΔZ Þ
ð1Þ
Assuming Δz is 10 μm, ΔT ¼ 7:04 μm can be derived, which equals to 0.52 pixel, less than one pixel size of the CCD. 4) Δα is assumed 0.005°, see Fig. 5d, then the translation of the 0 sample along Z axis is ΔZ ¼ h sin ð0:005 3 Þ 13 μm; similar to case 3), using Eq. (1), ΔT ¼ 8:8 μm can be derived. Therefore, the maximum deviation of the edge of the projection will be about 0.65 pixel. 5) Δγ is assumed 0.005°, similar to case 4), the sample will deviate about 13 μm from the ideal condition at X direction. According to case 1), it will generate approximating 40 pixels deviation on the actual projection position from the ideal position.
Rotation axis
Source
Mid-plane Detector Fig. 4. The cone beam geometry.
the magnification of the optical microscope within the detector is 20; the reference point on the sample is 250 μm to the optical axis. 1) Δx equals to the radial run-out of the rotation table whose maximum value can be 10 μm. See Fig. 5a), the actual projection position deviate from the ideal position is Δx SDD/SOD Mopt. Assume Δx is 10 μm, then the distance deviation approximates 427 μm, about 31 pixels offset on the detector; 2) Δy , as well as , equals to the ending run-out of the rotation table. The influence of Δy on the projection is the same to Δx . If Δy is equal to 10 μm, the actual projection position deviates from the ideal position approximating 31 pixels; see Fig. 5b;
When the translation/rotation errors lead to the projection position offset more than 1 pixel, it is worthwhile to carefully examine the effects of the errors and the method of measurement. Therefore, we just need to measure 3 sensitive factors (Δx ; Δy ; Δγ ;) and ignore the others (Δz ; Δα ; Δβ ). 2.3. Theoretical analysis As shown in Fig. 6, we designed a metered system which is composed of four precision capacitive sensors, a precision machined steel cylinder and four flexible hinges. The high precision machined steel cylinder (HPMSTY) is mounted on the rotation table and turns with it. Four capacitive distance sensors are used to measure the distances between the sensors and HPMSTY. S1, S2, S3 and S4 represents sensor1, sensor 2, sensor 3 and sensor 4 in
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Fig. 5. The influences of different errors on projections. (a) the error translation along X axis; (b) the error translation along Y axis; (c) the error translation along Z axis; (d) tilt about X axis and tip about Z axis is the same.
4 sensor signals, according to the following formulate: 8 0 d 1 ¼ d 1 Δx ; > > > > > < d2 ¼ d02 þ r U sin Δγ þ Δy ; 0 > d3 ¼ d3 cos φ U r U sin Δγ sin φ Ur U sin Δα þ Δy ; > > > > : d ¼ d0 cosψ Ur U sin Δ þ sinψ U r U sin Δ þ Δ ; γ α y 4 4
Fig. 6. A mechanical layout of the measuring system and motion system.
Where d1 through d4 are the sensor-to-cylinder gap spacing, as 0 0 measured by the 4 capacitive sensors. d1 through d4 are the initial gap spacing, determined by the location adjustment of each capacitive sensor. Δx and Δy are the displacements of HPMSTY in the X and Y directions, respectively. φ and ψ are the angles between the straight line passing through S3 and S4 and the centre of the HPMSTY and X axis. And r is 140.0 mm, which represents the lateral distance from the center of the cylinder to the center of the three tip-tilt sensors. See Fig. 7 for a diagram of the physical parameters. To simplify the Eq. (2), we assume a ¼ cos ðφÞ; b ¼ sin ðφÞ; c ¼ cos ðψ Þ; d ¼ sin ðψ Þ;
the following text. S1 is used to monitor the radial run-out, S2, S3 and S4 are used to monitor the axial run-out. Flexible hinges are used to adjust the positions of sensors. When the rotation table turns, HPMSTY will have the same errors as the rotation table, therefore, the distances between the HPMSTY and the sensors will also change. Assuming HPMSTY has a perfect flat top surface (zero axial runout), roundness (zero radial run-out) and verticality, and nomanufacturing errors and installation errors. In this case, the complete position of the HPMSTY could be determined from the
ð2Þ
And Eq. (2) can be written as: 8 0 d ¼ d 1 Δx ; > > > 1 > > < d2 ¼ d02 þ r U sin Δγ þ Δy ; 0 > d3 ¼ d3 a U r U sin Δγ b U r U sin Δα þ Δy ; > > > > : d ¼ d0 c Ur U sin Δ þ d U r U sin Δ þ Δ ; γ α y 4 4
ð3Þ
ð4Þ
According to Eq. (4), error motions of rotating stage can be decomposed into two translational and two rotational motions, which is useful to facilitate the solution of equations, and further
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reason, if the top sensors are not mounted at the exact same radius, S2, S3, and S4 will also contain additional error signals. 2.4.2. Putting forward a new amendment formula According to 1) and 2) above, each sensor will have an additive factor that varies with rotation angle θ. Because of 3) and 4) above, each sensor will measure distance with a slightly different (but constant) sensitivity and offset. And finally, from 5) above, the additive errors for each sensor are different. Now, add the nonidealities mentioned above to Eq. (4), and Eq. (5) are achieved. 8 V 1 ¼ V 01 ðθÞ μ1 Δx ; > > > > > < V 2 ¼ V 02 ðθÞ þ μ2 r U sin ðΔγ Þ þ Δy ; 0 > > > V 3 ¼ V 3 ðθÞ μ3 a Ur U sin ðΔγ Þ þ b U r U sin ðΔα Þ Δy ; > > : V ¼ V 0 ðθÞ μ c U r U sin ðΔ Þ d U r U sin ðΔ Þ Δ ; γ α y 4 4 4
Fig. 7. The diagram of the physical parameters.
calibration. In Eq. (4), d1 ; d2 ; d3 ; d4 can be acquired from the 4 sensors; Δx ; Δy ; Δγ are parameters we want to know and calibrate; Δα is an intermediate parameter which we don't need to 0 0 0 0 calibrate; d1 ; d2 ; d3 ; d4 are related with the installing of the sensors. 0 0 0 0 To calibrate Δx ; Δy ; Δγ , d1 ; d2 ; d3 ; d4 must be calibrated firstly; After 0 0 0 0 doing this, we can use the values of d1 ; d2 ; d3 ; d4 and d1 ; d2 ; d3 ; d4 to solve Δx ; Δy ; Δγ . 2.4. Practical measurement method It should be noted that Eq. (4) can't be used directly as HPMSTY is a manufactured object which is not “perfect”. Besides, installation errors can't be avoided in practical system. Now, we will investigate the non-idealities of the cylinder and installation errors, and then, put forward a new amendment formula which can be used in practice. 2.4.1. Examining the non-idealities of the cylinder 1) Radial run-out (cylinder roundness). This non-ideality will cause the signals measured by S1 to vary in a deterministic way as the cylinder rotates. The error signal will be additive to the "real" signals for X movement. 2) Axial run-out (top flatness). This will cause the signals from S2, S3, and S4 to vary in a deterministic way as the cylinder rotates. Tcnals for Y and β movement. 3) Axis of rotation tilt. If the entire rotation stage is tilted relative to the capacitive sensors, there will be a fixed offset in S2, S3, and S4 due to the sensors being closer or farther from the top of the cylinder. This tilt will also cause S1 to become less sensitive to displacements, as a fraction of the total motion will be in a direction perpendicular to the capacitive sensors. The sensitivity of these sensors will be multiplied by cos(ξ1), where ξ1 is the angle of the rotation tilt. 4) Sensor tilt. If an individual sensor is tilted relatively to the cylinder (not perfectly parallel to the measurement surface), then its sensitivity will be multiplied by cos(ξ2), where ξ2 is the angle of the sensor tilt. 5) Edge non-perpendicularity. If the edges of the steel cylinder are not perpendicular to the top surface, and further, if S1 is not mounted at the exact same vertical position along the edge, then an additional error signal will exist for this sensor. This is because the sensor will be looking at a different "ring" of the steel cylinder, and due to the non-perpendicularity of the edge, these rings will be different from each other. For the same
ð5Þ
In the above equations, V 1 ; V 2 ; V 3 ; V 4 are the output voltages from the individual sensors, V 01 ðθÞ; V 02 ðθÞ; V 03 ðθÞ; V 04 ðθÞ are the offset voltages of the individual sensors, and μ1 ; μ2 ; μ3 ; μ4 are the individual sensitivities (voltage/distance) of each sensor. Note that the offset voltages V 01 ðθÞ; V 02 ðθÞ; V 03 ðθÞ; V 04 ðθÞ are functions of θ, and θ is the angle of the rotation table turning, as the error terms due to the steel cylinder imperfections have been lumped in. Here, we focus on solving these parameters Δx ; Δy ; Δγ . Looking at Eq. (5), it is apparent that each steel cylinder must be fully characterized, to map out the voltage offsets V 01 ðθÞ; V 02 ðθÞ; V 03 ðθÞ; V 04 ðθÞ; and sensitivities μ1 ; μ2 ; μ3 ; μ4 . Further, because these parameters are highly sensitive to sensor mounting, the assembly must not be disassembled or adjusted after calibration has been performed. From Eq. (5), we can find that it contains more unknowns than equations. Therefore, simplifying approximations must be made. 1) The first reasonable simplification is to assume that all sensor sensitivities are equal to each other. A survey of the calibration certificates for 4 sensors reveals a nominal gain of 10 V/50 μm, and a typical gain error of 0.01%. Therefore, it can safely be assumed that all the values of μi (i¼ 1,2,3,4) are the same. This eliminates the 4 different μi (i¼ 1,2,3,4) unknowns, and replaces them with a single known quantity, μ. This assumption will introduce an error multiplier of cos(ξ1)cos(ξ2), where (ξ1) and (ξ2) are the sensor and axis of rotation tilt angles. Since the tile angles are small (assume a worst case tilt of 1 Mrad), the sensitivity will only change by cos2(0.001) ¼0.0001%, which can be safely ignored. So Eq. (5) can be written as: 8 V ¼ V 01 ðθÞ μΔx ; > > > 1 > > < V 2 ¼ V 02 ðθÞ þ μ r U sin ðΔγ Þ þ Δy ; > V 3 ¼ V 03 ðθÞ μ a Ur U sin ðΔγ Þ þ b Ur U sin ðΔα Þ Δy ; > > > > : V ¼ V 0 ðθÞ μ c U r U sin ðΔ Þ d U r U sin ðΔ Þ Δ ; γ α y 4 4
ð6Þ
2) Another simplification is to assume that the top surface of the steel cylinder is extremely flat, which means that the three sensors on the top surface do not have different V 0x θ functions-they merely measure a phase-shifted version of the same function. I.e. 8 < V 3 0 θ ¼ V 02 ðθ þ 1801 φÞ; : V 4 0 θ ¼ V 02 ðθ þ 1801 þ ψ Þ;
ð7Þ
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3.1. Experiment for resolution
Merging this assumption into Eq. (6) gives: 8 V ¼ V 01 ðθÞ μΔx ; > > > 1 > > < V 2 ¼ V 02 ðθÞ þ μ r U sin ðΔγ Þ þ Δy ; > V 3 ¼ V 02 ðθ þ 1801 φÞ μ a Ur U sin ðΔγ Þ þb U r U sin ðΔα Þ Δy ; > > > > : V ¼ V 0 ðθ þ 1801 þ ψ Þ μ c U r U sin ðΔ Þ d U r U sin ðΔ Þ Δ ; γ α y 4 2 ð8Þ Using above simplifications, Eq. (8) can now be solved for
Δx ; Δy ; Δα .
To verify the influence of the correction method on spatial resolution of the Micro-CT system, in this part, 360 projections of a ball whose diameter is 500 μm are acquired during 2π angular range and Table 4 shows the basic parameters of the system and experimental conditions. Hough transformation are used to compute the centre of the ball. Fig. 8 shows one projection image of the ball and the extraction of the centre of the projection circle. As the errors of the axis of rotation, the centres of the projection
8 > Δx ¼ V 01 ðθÞ V 1 ; > > h i > < Δy ¼ ðad þ bc1þ b þ dÞ U ðad þ bcÞðV 2 V 02 ðθÞÞ þ dðV 3 V 02 ðθ þ 1801 φÞÞ þ bðV 4 V 02 ðθ þ 1801 þ ψ ÞÞ ; > h i > > 0 0 0 > : sin ðΔα Þ ¼ ðad þ bc1þ b þ dÞr U ðbþ dÞðV 2 V 2 ðθÞÞ dðV 3 V 2 ðθ þ 1801 φÞÞ bðV 4 V 2 ðθ þ 1801 þ ψ ÞÞ ; After we acquire the values of Δx ; Δy ; Δα , it is necessary to build the relationships between Δx ; Δy ; Δα and the shifts of the projection images. According to this, we can get: ( xshif t ¼ Δx þ h sin Δγ ; ð10Þ yshif t ¼ Δy ; Where h is the height (above the steel plate) of the sample; xshift and yshift are the sample image shifts along X and Y directions, respectively. Now it is important to talk about how to correct the X-ray image errors resulting from off-axis motion of the sample stage. We use the software correction method, that is, during the reconstruction phase, utilize the recorded signals from the capacitive position sensors to translate each image to correct for X/Y errors due to the stage. This method does not change the acquisition process; it only requires that signals from the position sensors are recorded. For this correction method, only the sample image shifts-xshift and yshift , can be corrected. Therefore, from Eq. (10), only Δx ; Δy ; Δα are important. Putting Eq. (10) together with Eq. (9) yields:
ð9Þ
circles vary within a certain range. After the correction is done, the centres of the projection circles should be reduced to a smaller area. In this part, we measure the errors through projections and capacitive sensors. To be convenient for the following reconstruction, it is necessary to correct the position of the projections, so the unit of pixel is used in this part, like Tables 5 and 6. In addition, the offset which is in unit of pixel can be converted to the one whose unit is μm by multiplying 13:5 SOD=SDD=M opt . Fig. 9(a) shows horizontal position (X axis) offset before and after calibration. The error range and standard deviation are computed and summarized in Table 5. As can be noticed from Table 5, the variation scope after correction reduces to 5 pixels from 10 pixels, with about 50% improvement. Fig. 9(b) shows the vertical position (Y axis) offset before and after calibration. The error range and standard deviation are computed and summarized in Table 6. As can be noticed from Table 6, the variation scope after correction reduces to 4 pixels from 11 pixels, with about 63.6% improvement. Fig. 9(c) shows the variation of the spatial position distribution of the centre of the ball. The circle points represent the position offset at different angles before correction and the
8 < xshif t ¼ V 01 ðθÞ V 1 þ ðad þ bchþ b þ dÞr U ½ðb þ dÞðV 2 V 02 ðθÞÞ dðV 3 V 02 ðθ þ 1801 φÞÞ bðV 4 V 02 ðθ þ 1801 þ ψ ÞÞ; : yshif t ¼
1 U½ðad þ bcÞðV 2 V 02 ð ðad þ bc þ b þ dÞ
ð11Þ
θÞÞ þ dðV 3 V 02 ðθ þ 1801 φÞÞ þ bðV 4 V 02 ðθ þ 1801þ ψ ÞÞ;
Solving for the two unknowns (V 01 θ ; V 02 θ ):
star points represent the position offset at different angles after
8 < ðad þ bcÞV 02 ðθÞ þ dV 02 ðθ þ 1801 φÞ þ bV 02 ðθ þ 1801 þ ψ Þ ¼ ðad þ bcÞV 2 þ dV 3 þ bV 4 ðad þ bc þ bþ dÞyshif t ;
ð12Þ
: V 01 ðθÞ þ ðad þ bchþ b þ dÞr U ½ ðb þ dÞV 02 ðθÞ þdV 02 ðθ þ 1801 φÞ þ bV 02 ðθ þ 1801 þ ψ Þ ¼ xshif t þ V 1 ðad þ bchþ b þ dÞr U ½ðb þ dÞV 2 dV 3 bV 4 ;
Once the two error vectors are derived, we can correct the projections using Eq. (11).
3. Experiments and results Here, we verify the validity of the method on Micro-CT system of Tianjin University. Fig. 1 shows the system we use. To verify the correction method, two on-line experiments are done. One is for inspecting the influence on spatial resolution, and the other is for verifying the influence on image quality.
Table 4 Primary parameters of the system and experimental conditions. Parameter
Value
Voltage of the X-ray source (kV) Target power of the X-ray source (W) SOD (Source Object Distance) (mm) SDD (Source Detector Distance) (mm) M opt Pixel size in the object space (μm2) Rotation interval/Projection image (degree) Number of projection images Exposure time/Projection image (s) Distance between the ball and cylinder: h (mm)
110 4 15 32 20 0.16*0.16 1 360 10 150
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correction. From this figure, we can observe that the distribution of the centres are more concentrated after correction. To verify the influence of the correction method on spatial resolution of the Micro-CT system, we reconstruct the small ball by using of the FDK [32] algorithm under two conditions. One is using all the projections before correction and the other is using all the projections after correction. Then the mid-plane of the
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constructed images is extracted to acquire the resolution under each condition. To acquire the resolution, the MTF curve is utilized. The MTF of any system is the frequency response to a delta signal and has been one of the physical characteristics that is commonly used to quantitatively measure the physical performance of a system [31]. Fig. 9(d) shows the MTF curves under the two conditions. The circle line represents the MTF curve before calibration. The value of frequency is 470 lp/mm when the value of MTF equals 0.1, Table 5 Horizontal shift of the images before and after calibration.
Before calibration After calibration Improving effect
Error range (pixels)
Standard deviation
10 5 50%
2.145 1.005 53.1%
Table 6 Vertical shift of the images before and after calibration.
Fig. 8. Projection image of the ball.
Before calibration After calibration Improving effect
Error range (pixels)
Standard deviation
11 4 63.6%
2.556 0.881 65.5%
Fig. 9. The positon offset curves of the projections before and after calibration. (a) the offset curves at X direction; (b) the offset curves at Y direction; (c) the distribution of the projections before and after calibration; (d) the MTF curves before and after calibration.
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Fig. 10. The phantom and its X-ray projection image.
Table 7 Condition of the experiments.
Fig. 11. The direct reconstruction result without correction.
Parameter
Value
Voltage of the X-ray source (kV) Target power of the X-ray source (W) SOD (Source Object Distance) (mm) SDD (Source Detector Distance) (mm) Pixel size in the Object Space (μm2) Rotation interval/Projection image (degree) Number of projection images Exposure time/Projection image (s)
110 4 14 30 0.314 0.314 0.5 720 10
which means the resolution is 2.16 μm under this condition. The star line represents the MTF curve after calibration. The value of frequency is 580 lp/mm at the 10% MTF, which means the resolution is 1.72 μm under this condition. From this, we can deduce the resolution of the system is improved about 20%. However, there still exist some residual errors. Here we mainly consider the influence of temperature and focal drifts, as the scanning time is about several hours and we can't ensure the temperature is constant. Taking the vertical direction as an example, the perpendicular distance between the ball we scanned and the capacitive sensor is about 0.14 m, and the ball is propped up by aluminium, whose linear expansion coefficient is 2.3 10 5 K 1. Then the variation at vertical direction caused by temperature (assuming the variation of temperature is ΔT ¼ 0:4K) is 2:3 10 5 0:14 ΔT SDD=SOD M opt ¼ 5:5 10 5 m, about 4.07 pixels, which is correspond with the residual errors we gotten above. Commonly, the temperature is not controlled in Micro-CT. In the future, how to control the temperature when we use the system for scanning or build the relationship between temperature and the errors is a technical challenge and we will put our focus on it. What's more, in previous work, we measured the focal drifts of the x-ray source by utilizing a cross wire making by 12 μm thick
tungsten wire. After warming up the source for tens of minutes, the focal drifts is about 2–3 μm, and this will bring the locations of images changing in about 5 pixels, which also agree with the results of the experiment. Because of the above analysis, in the future, we will put our focus on how to control the temperature when we use the system for scanning or build the relationship between temperature and the errors, and calibrate focal spot drifts. As both of the two reasons mentioned above may cause residual errors, we will also analyse which is the main factor to cause the residual errors. More importantly, through this, we verified the correction method is effectively. 3.2. Experiment for image quality To test our correction method's effect on image quality, a special phantom has been made using two tungsten filaments, as is shown in Fig. 10. The diameter of each is about 6 μm. They were twisted with each other, so the gap between them is a little wider in the midst, and narrower at each end. Moreover, there is a fracture in its midst. The phantom was scanned in the condition shown in Table 7. Fig. 11 is the reconstruction result of the projection images, in which there are two severe artifacts. If the phantom is a little larger, they will overlap the real reconstruction model, and the result will become chaos totally. This is caused mainly by the mounting error of the rotary sample stage which makes the rotation axis deviate from the optic axis. In the micro-CT reconstruction algorithm FDK, there can be an offset interface where the mounting error of the rotary stage can be compensated [32–35]. This can really correct the mounting
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Fig. 12. The reconstruction result of the projection images processed only by the offset (a1) and that processed by both the offset and our correction method (b1). (a2) and (b2) are the enlarged views of the bottom of (a1) and (b1) respectively. (a3) and (b3) are the enlarged views of the middle of (a1) and (b1) respectively.
error of the rotary stage, but it has no effect on the spindle error motion during the CT scan. In Fig. 12, the reconstruction result only processed by the offset is contrasted with that processed by both the offset and the correction method, which demonstrates the correction method is quite effective in the spindle error motion's correction. The gap between the two filaments in the lower part of the reconstruction result processed by the correction method [Fig. 12 (b1)] is larger than that not [Fig. 12(a1)], and matches what the projection image shows in Fig. 11 correctly. This can be better
observed from their enlarged views [Fig. 12(a2) and (b2)]. Evidently, better horizontal resolution has been achieved by the correction method. Meanwhile, at the fracture the filaments [Fig. 12(b3)] are a little thinner than those not [Fig. 12(a3)]. This is not obvious for the fracture is too small to be distinguished, but the corrected one shows an obvious dent due to the fracture, which is marked by an arrow in Fig. 12(b3). This means a better vertical resolution has been achieved. Fig. 13(a) and (b) show the slices on the same layer in the lower part of each result. In Fig. 13 (a) the two filaments are mixed up and the gap cannot be
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Fig. 13. The slice in the reconstruction result not processed by the correction method (a) and that on the same layer in the reconstruction result processed by the correction method (b). (c) is the gray level diagrams of the pixels on the lines marked in (a) and (b).
distinguished, while in Fig. 13(b) the corrected filaments are separate, and the gap between them is clear. Fig. 13(c) is the gray level diagrams of the voxels on the lines marked in Fig. 13(a) and (b). Firstly, the grey value contrast of the two filaments and their gap in the corrected result is much stronger than that in the uncorrected one, and the grey value of the gap is much lower in the corrected result. Secondly the corrected one has sharper filament edges than the uncorrected one. Therefore, it quantitatively proves that the correction method can help attaining a better image quality. In sum, all these attest that the correction method proposed in this article can get a reconstruction result with better 3D resolution and image quality.
4. Conclusion In this paper, we put our attention on correcting the errors of the rotation axis when it turns, aiming at increase system spatial
resolution and image quality. In order to achieve this goal, we use a precision-manufacturing steel cylinder and 4 capacitive sensors to do the calibration and correction, and analyse possible difficulties when implement it to practice. Finally, we verify the theory on-line. And the results show the theory and method is effective. Also, the resolution of the Micro-CT system is improved about 20% and the image quality is enhanced. In summary, the method is effective in improving the resolution and image quality of the Micro-CT. Moreover, although the result is not perfect and the residual errors are larger than some other experiments because of variation in temperature and focal drifts, it still should point out that this method has no claim to the rotation stage, and is vacuum compatible. When temperature is controlled in Nano-CT, this method can be applied to it in the future, and we will put our focus on how to control the temperature when we use the system for scanning or build the relationship between temperature and the errors in the future, and calibrate focal spot drifts.
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Acknowledgements This work was supported by China's Ministry of Science & Technology (No. 2011YQ030112) and National Natural Science Foundation of China (No. 61372144). The authors would like to acknowledge mechanical engineer Bai Jianguo from Centre of Micro-Nano Manufacturing Technology in Tianjin for helping to design and manufacture the HPMSTY.
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