Self-calibration of a cone-beam micro-CT system

Abstract

Use of cone-beam computed tomography (CBCT) is becoming more frequent. For proper reconstruction, the geometry of the CBCT systems must be known. While the system can be designed to reduce errors in the geometry, calibration measurements must still be performed and corrections applied. Investigators have proposed techniques using calibration objects for system calibration. In this study, the authors present methods to calibrate a rotary-stage CB micro-CT (CBμCT) system using only the images acquired of the object to be reconstructed, i.e., without the use of calibration objects. Projection images are acquired using a CBμCT system constructed in the authors’ laboratories. Dark- and flat-field corrections are performed. Exposure variations are detected and quantified using analysis of image regions with an unobstructed view of the x-ray source. Translations that occur during the acquisition in the horizontal direction are detected, quantified, and corrected based on sinogram analysis. The axis of rotation is determined using registration of antiposed projection images. These techniques were evaluated using data obtained with calibration objects and phantoms. The physical geometric axis of rotation is determined and aligned with the rotational axis (assumed to be the center of the detector plane) used in the reconstruction process. The parameters describing this axis agree to within 0.1 mm and 0.3 deg with those determined using other techniques. Blurring due to residual calibration errors has a point-spread function in the reconstructed planes with a full-width-at-half-maximum of less than 125 μm in a tangential direction and essentially zero in the radial direction for the rotating object. The authors have used this approach on over 100 acquisitions over the past 2 years and have regularly obtained high-quality reconstructions, i.e., without artifacts and no detectable blurring of the reconstructed objects. This self-calibrating approach not only obviates calibration runs, but it also provides quality control data for each data set.

INTRODUCTION

Computed tomography (CT)^{1, 2} systems provide high quality 3D data of extended structures for visualization and analysis. More recently, cone-beam CT (CBCT) systems have been designed based on image intensifiers^{3, 4, 5} or charge-coupled devices.^{6, 7, 8, 9} CB micro-CT (CBμCT) systems have also been constructed using high-resolution, area-detector systems.^{10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25} In all of these systems, gain and dark signal variations in the detectors and output variations of the x-ray tube must be corrected. In addition, the geometry of the imaging system must be calibrated and taken into account in the reconstruction. After the corrections, 3D tomographic data are reconstructed from these projection data using techniques based on the Feldkamp algorithm.²⁶

Flat-field, dark-field, and exposure corrections are standard components of projection image preprocessing. When considering geometric calibrations, two types of systems must be considered: those in which the object is rotated, such as on a rotary stage, while the source-detector system is stationary, and those in which the object is stationary and the source-detector or gantry system rotates. The geometry of both systems is usually determined using calibration objects.^{27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43} In general, rotations occur about a single axis. While motion about this axis is very stable for the rotating-object systems, the motion for the rotating-gantry systems may not be uniform due to mechanical instabilities in the systems, although this motion has generally been found to be reproducible.⁴

In this study, we propose a method which takes advantage of the stability of the rotating-object system to obviate the extra step of acquiring an additional set of calibration images as well as to provide a routine assessment of the alignments of the system, i.e., each time we perform a reconstruction. With this method, angulation of the rotational axis relative to the vertical (y) axis in a plane parallel to the detector plane and horizontal translations of the rotational axis in that plane are detected during the acquisition process and subsequently corrected. Other parameters that determine the geometry of the total system such as out-of-plane angles, source-to-image distance, source-to-object distance, and center in the vertical direction,²⁷ while not measured and corrected, result in minimal effects on reconstruction. This method determines the geometry of a rotary-stage CBμCT system using only the projection images of the object being scanned, i.e., this is a self-calibrating technique. This method was evaluated qualitatively in terms of reduction of artifacts in the reconstructed data and quantitatively in terms of the accuracy and precision of the calculated alignments. In addition, we propose sinogram-based techniques for determination and correction of translations which may occur during the acquisitions.

METHODS

Images are acquired using the CBμCT system developed in our laboratory.²² Corrections for variations in exposure are made. Flat-field and dark-field corrections are performed, after which a logarithmic transformation is applied. Translations that may have occurred during acquisition are identified, and the data are appropriately corrected. The rotation axis is determined, and the data transformed so that the rotation axis is coincident with the vertical central axis in the images. The corrected data are reconstructed using a Feldkamp algorithm.²⁶

Image acquisition

The CBμCT system (Fig. 1) constructed in our laboratory consists of an x-ray source, a rotary stage, and a CCD based micro-angiographic detector.⁴⁴ X rays are generated by an Oxford Ultra-Bright x-ray tube (Oxford Instruments PLC, Scotts Valley, CA) with a focal spot of 40 μm in this study. The detector consists of a 250 μm thick CsI (Tl) phosphor coupled through a 1.8:1 fiber optic taper to a CCD chip and has a field of view (FOV) of approximately 4.5 cm×4.5 cm. The pixel size is 43 μm, the matrix size is 1024×1024, and the bit depth is 12. The object is placed on a rotary stage which is driven by a stepper motor (Velmex Inc. Bloomfield, NY) with a precision better than 0.1°. The source-to-image plane distance (SID) was carefully measured for the evaluations in this study to be 80±0.1 cm. The source-to-center of rotation distance (SCD) is also carefully measured from the source to the center of the rotary stage, which is assumed to be the center of the object. The rotary stage is usually placed such that the magnification of the object ranges between 1.2 and 2.0. A calibration procedure is performed periodically to ensure the accuracy of the SID and SCD values. As part of this setup procedure, a laser pointer is used to align the centers of the detector, the x-ray beam, and the rotary stage. Then profiles of a wire phantom are acquired. The detector is then adjusted in both x and y directions such that the center of the phantom, in both x and y directions, appears at the center of the detector. These accurate measurements assure us that errors arising from the setup itself are minimal. The entire acquisition system is under computer control.²²

Figure 1.

Schematic of the CBμCT setup consisting of an x-ray source, the object located on the rotary stage, and a detector. The x-ray source and detector are stationary. The central axis of the x-ray beam is assumed to be parallel to the z axis, which is perpendicular to the detector plane, and the rotation axis is assumed to lie in the x-y plane, approximately parallel to the y axis.

Flat-field images are acquired with no objects in the x-ray beam at the same kVp at which the images of the object are obtained. The dark-field images are acquired by measuring the detector output when the x-ray tube is off. The flat-field image, F(x,y), and the dark-field image, D(x,y), used in the correction are obtained by averaging 60 flat- and dark-field images, respectively. The flat- and dark-field corrections are applied to the images, I_orig,θ(x,y), acquired at each angle θ

$$I_{f,\theta }(x,y)=\frac{I_{\mathit{orig},\theta }(x,y)-D(x,y)}{F(x,y)-D(x,y)}\times A,$$ (1)

where A is the average pixel value of the denominator to normalize the flat-field correction and subscript f means flat-field corrected image.

The logarithmic transformed corrected image, I_l,θ(x,y), is given by

$$I_{l,\theta }(x,y)=-\ln \left(I_{f,\theta }\right),$$ (2)

where subscript l means log-transformed image.

Determination of exposure variation

To correct for exposure variations between acquisitions, regions in the projection images with an unobstructed view of the x-ray source are identified. In this study, 50×50 pixel regions from each of the four corners in the image are used. The average pixel value, R_i(θ), is calculated at each projection angle, θ, and region i (i varies from 1 to 4). The average over the four regions at each projection is calculated as

$$PA\left(\theta \right)=\frac{{\sum }_{i=1}^4{R}_i\left(\theta \right)}{4}$$ (3)

and the average of the PA(θ) values over all projections is

$$AR=\frac{{\sum }_{\theta }PA\left(\theta \right)}{N},$$ (4)

where N is the total number of projection images. The images are then corrected for the exposure variations as

$$I_{\theta }(x,y)=I_{l,\theta }(x,y)\left[\frac{AR}{PA\left(\theta \right)}\right].$$ (5)

Translation correction

Motion of the object can occur during acquisition. In general, these motions are three-dimensional in nature. For motions that occur during the acquisition of a particular projection image, the image will be blurred in the direction of motion. For motions that occur between acquisitions, i.e., while the rotary stage is turning, the image will be shifted relative to the previous set of projection images, and this shifting can result in a doubling in the reconstructed features. We identify, measure, and correct for translations of the object in the x direction during the acquisition process by analysis of sinograms.^{45, 46, 47, 48, 49} The object is assumed to move as a rigid body. Translations along the y and the z directions are not dealt with here (see Sec. 4).

To obtain the sinogram at a given y, we extract profiles from the projection data, I_θ(x,y) along the axis (x in our system) which is perpendicular to the rotation axis (y for our system) for each projection angle and combine these one-dimensional profiles as a function of projection angle into a two-dimensional array:

$$S_y(x,\theta )=I_{\theta }(x,y).$$ (6)

The sinogram is so named because an off-axis point in the object will trace a sine curve in the sinogram for parallel-beam geometries. For diverging-beam geometries, the curve deviates from a perfect sine curve, with the deviation depending on the amount of divergence in the field of view. For our analysis, however, we do not use the sinusoidal nature of the sinogram, thus, the deviation from sine curves does not appear to affect our results.

The background level, B, for the sinogram is estimated by calculating the average pixel intensity in the unattenuated region. This background value, B, is then subtracted from all pixels in the sinogram to generate S_B,y(x,θ) so that the signal alone is used in the subsequent calculations:

$$S_{B,y}(x,\theta )=S_y(x,\theta )-B.$$ (7)

The centroid of each of the background-subtracted horizontal profiles is calculated as,

$$C_y\left(\theta \right)=\frac{{\sum }_{x}x{S}_{B,y}(x,\theta )}{{\sum }_x{S}_{B,y}(x,\theta )}.$$ (8)

The change of the centroids, Δ_c, as a function of θ is then calculated as follows:

$${\Delta }_c\left({\theta }_n\right)=C_y\left({\theta }_n\right)-C_y\left({\theta }_{n+1}\right).$$ (9)

The root-mean-square (RMS) value of the Δ_c’s over all angles is calculated, and the projection angles at which the Δ_c is larger than two times the RMS value are identified. We chose twice the RMS to allow detection of outliers and not random translations. The Δ_c at the identified angle is then taken as an abnormal translation. The Δ_c corresponds to the x component of translations of objects during the acquisition process. Note that when a translation occurs, all subsequent images will be translated by that amount. Therefore, all images with identified translations and all the subsequent projection images are shifted by the value of the translation of the identified projections. It should be noted that translations are most likely rigid-body and three-dimensional translations. Here we present a technique to determine the x component of the translation. The y component could be determined using cross correlation techniques which would be more computationally intense. We know of no way to determine the z component without a calibration object.

Geometric calibration

Prior to reconstruction, the imaging geometry is determined. When the axis of rotation (the y axis here) used in the reconstruction is correct, the reconstruction can be represented as

$$M(x,y)=\sum _{\theta }{I}_{\theta }(x,y)\otimes G\left(x\right),$$ (10)

where M(x,y) is the reconstructed 3D data, I_θ(x,y) are the projections of the original object, and G(x) is the filter function, e.g., a Shepp-Logan filter. The summation is over all projection angles, θ, and ⊗ represents convolution. When the axis of rotation used in the reconstruction is displaced in a given slice or incorrectly oriented relative to the true axis, the equation becomes

$$M(x,y)=\sum _{\theta }{I}_{\theta }(x+d\left(y\right),y)\otimes G\left(x\right),$$ (11)

where d is the displacement of the axis of rotation from the y axis and is a function of the plane’s y coordinate. If this displacement is not taken into account, the data from the image will be reconstructed in a locus of points, and point objects will appear as rings having radii equal to d. Note that if the ratio of d and the magnification is larger than a voxel, circular artifacts will be seen, for smaller distances, objects will be effectively blurred. This method does not determine angulation of the rotation axis in the y-z plane, but the effect of this angulation on the reconstruction is small as discussed by Yang et al.²⁷ and would result in slight blurring in the y and z directions (see Sec. 4).

To determine the axis of rotation, we take advantage of the fact that the anti-posed projection images (images separated by 180°) should be identical (except for statistical variations and differences in beam hardening and differences in magnification of the objects in the projection) but reflected on the x axis (horizontal) about the axis of rotation. Two antiposed images and corresponding horizontal lines in those images are selected. Image data along these lines are extracted to generate profiles, I_θ(x,y) and I_θ+180(x,y), for each selected y value. Both profiles are shifted by the distance d, to generate I_θ(x+d,y) and I_θ+180(x+d,y) [see Fig. 2a]. The data in one profile are flipped, pivoting about the x=0 point to generate I_θ+180(−x−d,y). The RMS difference between the two profiles is then calculated as

$$\mathrm{RMS}(d,y,\theta )=\sqrt{\frac{{\sum }_x{(I_{\theta }(x+d,y)-I_{\theta +180}(-x-d,y))}^2}{n}},$$ (12)

where RMS(d,y,θ) represents the RMS difference between the two profiles for the horizontal plane at a position y relative to the central plane, n is the number of points in the summation, and d is the shift of the profiles. The value of d for which the RMS difference is the minimum is taken as the shift between the rotation axis and the x=0 axis for that horizontal-plane. This process is repeated for all antiposed image pairs (i.e., for all θ projections) and for all horizontal planes in those images. For each horizontal plane, the average and standard deviation of the shifts are calculated, and a linear fit to the shift data is performed [see Fig. 2b]. The resulting line is taken as the axis of rotation in the xy plane. The intersection of the calculated axis of rotation with the x axis, T_c, is determined (see Fig. 3). The slope of the line is used to calculate the angle, ϕ, that the line makes with the y axis, where

$$\varphi =-{\tan }^{-1}\left(\frac{1}{\text{slope}}\right).$$ (13)

Figure 2.

(a) Profiles obtained from antiposed projections along lines parallel to the x axis in the image at a particular y coordinate. (b) Calculated shift in the corresponding horizontal planes. The values plotted are the average value of the shift for the entire set of antiposed images, and the error bars are the corresponding standard deviations. The tilt angle is calculated from the slope of the linear regression line, and the x-intercept gives the translation, T_c, at x=0 plane. In this case, the tilt angle according to Eq. 13 is ϕ=0.8012±0.0001 deg and T_c=−4.10±0.04 mm.

Figure 3.

Schematic of the detector plane. The reconstruction process assumes that the axis of rotation is the ideal axis of rotation. However, the true axis of rotation may not coincide with the ideal axis of rotation as shown in the figure, and may be tilted by ϕ degrees and translated by T_c.

All image data are shifted by −T_c to align the rotation axis with the origin of the image in the central plane. A simple correction would be to shift the data in each y plane by the specific d. However, this does not take into account that the data were acquired in a rotated geometry, and artifacts will result. Thus, the image data are then rotated by an amount ϕ so that the axis of rotation in the reconstruction lies along the y axis. The transformation is given by

$$\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right)=\left(\begin{array}{cc}\cos \left(\varphi \right)& \sin \left(\varphi \right)\\ -\sin \left(\varphi \right)& \cos \left(\varphi \right)\end{array}\right)\left(\begin{array}{c}x-T_c\\ y\end{array}\right),$$ (14)

where (x,y) is the position of any point on the image.

Evaluations

To evaluate the results of our self-calibration technique, we compared the results obtained with our self-calibration technique with those obtained using a calibration phantom. The calibration phantom consisted of a Lucite cylindrical phantom (2 cm diameter) containing 11 steel beads (diameter 1 mm) placed in a helical pattern. The calibration phantom was placed on the rotary stage, and projection images were obtained over 360 deg at 1 deg steps. We then used two methods for calibration using the bead phantom.

In Method 1, the rotation axis for the bead phantom is determined using a calibration technique described by Yang et al.²⁷ The mass center for each bead is determined in each of the projections. The trajectory of each of the beads in all the projections is recorded. Using all the anti-posed image pairs (referred as “radial pairs” by Yang et al.²⁷), we determined the average rotation center for each bead. This set of points was fit with a straight line. The intersection of this line with the x axis was taken as the translation of the axis of rotation, T_c, and the slope of the line was taken as the angle, ϕ, that the axis of rotation makes with the y axis.

For Method 2, we used our self-calibration method with the bead phantom. However, the self-calibration method described in Sec. 2D assumes that the object is sufficiently extended along the y axis that the object appears in the given horizontal plane for both antiposed images. If the size of the object is small, e.g., a small bead, the calibration object could move out of the particular y plane being used, and the self-calibration method will not provide accurate results. Therefore, we modified the self-calibration method for small objects such that horizontal profiles were obtained from the antiposed projections at y positions separated by 2^*dy and these were flipped about and shifted along the x direction. dy is the distance of the profiles in a vertical direction from the y plane being used. For each of these pairs of profiles, the RMS difference was calculated as

$$\mathrm{RMS}(dx,dy,y,\theta )=\sqrt{\frac{{\sum }_x{(I_{\theta }(x+dx,y+dy)-I_{\theta +180}(-x-dx,y-dy))}^2}{n}}.$$ (15)

The value of dx and dy for which the RMS difference is the minimum is taken as the shift between the rotation axis and the x=0 axis for the y plane. This process is repeated for the horizontal planes corresponding to each of the beads. For each horizontal plane, the average and standard deviation of the shifts are calculated, and a linear fit to the shift data is performed. The angle, ϕ, the axis of rotation makes with the y axis in the x-y plane is determined from the slope of the resulting line. Then the transformation is applied as shown in Eq 13. The angles and T_c’s calculated using these two methods were compared with those calculated using the self-calibration technique (described above in Sec. 2D), Method 3. These calibration experiments were repeated for various angulations achieved by placing thin spacers under the leveling screws of the rotary stage. In addition, we obtained a projection data set of a vessel phantom where we used each of these angulations, i.e., without repositioning the rotary stage or imaging system. We then reconstructed the vessel phantom using the angles and T_c’s determined using the calibration phantom with Methods 1 and 2 and those determined using the vessel phantom and Method 3. Note that in this last experiment, we are effectively comparing the standard method of calibrating the CBμCT system (i.e., using a calibration phantom and then imaging the object of interest) with our self-calibration technique (imaging only the object of interest).

Spatial resolution study

To study the blurring due to small residual miscalibration errors, the same Lucite phantom was used as described in Sec. 2E and projection images were acquired using a high resolution MA detector described in Sec. 2A.

Projection images of the phantom were acquired at 1 deg intervals for all 360 degrees. The center-of-mass of each bead was computed for each of the projections and this point replaced the bead profile in the projection. The system was calibrated using the above-described technique and the reconstruction was performed. A simulation study was also performed by reconstructing center-of-mass projections of beads with perfect geometry, i.e., free of miscalibration errors, at various y (vertical) locations. Reconstructing only the center-of-mass represented by one pixel in both the real phantom and simulation eliminated the need to consider the blurring effects due to focal spot and detector, as well as magnification differences. Both cases were reconstructed using the same Feldkamp-based reconstruction algorithm and Shepp-Logan filter. Profiles of each bead in both datasets of reconstructed slices were compared in radial and tangential directions at various y (vertical) locations. The difference between the profiles of the simulated center-of-mass and the real center-of-mass reconstructions can be attributed to the blurring due to residual miscalibration errors.

RESULTS

With the self-calibration techniques described above which use only the images themselves, we have been able to substantially reduce the artifacts resulting from inaccurate estimation of the axis of rotation and translations of the imaging-object system during the acquisitions.

In Fig. 4, we present the effects of translations between projection acquisitions for a vessel phantom which had a 3 mm inner diameter and contained a stenosis; the lumen was filled with a mixture of 50% iodinated contrast and a solution of water and glycerin. In Fig. 4a, we present the sinogram of the projection data for the central plane. Apparently, the vessel phantom shifted twice during the acquisition sequence; two discontinuities (indicated by arrows) are visible in the sinogram. In Fig. 4b, we present the reconstruction of the data without translation correction. Double-wall artifacts (indicated by arrows) are seen in the reconstruction. In Fig. 5a, we show the curve representing the centroids of the sinograms, C_y(θ) [Eq. 8] for the sinogram shown in Fig. 4a. Δ_c, calculated according to Eq 9, is shown in Fig. 5b. We see that using two times the RMS value provides detection of significant translations and obviates correcting for smaller random fluctuations. While one could also choose to use a fixed number, we have found that two times the RMS works well. The two relatively large discontinuities, shifts 1 and 2 of approximately −0.55 and 0.4 mm, respectively, can be seen in the data. In Fig. 6a, we present the sinogram of the projection data after translation correction; the sinogram appears continuous, and the artifacts in the reconstructed image [Fig. 6b] are substantially reduced. To date, such a translation has occurred only once; we believe that the object may have slipped during acquisition. However, the translation detection and correction method is run for every study for quality control.

Figure 4.

Illustrations of effects of translations during acquisitions. (a) Sinogram of a vessel phantom with stenosis before translation correction. The arrows indicate the region of the shifts. (b) Reconstructed image before translation correction. A double wall artifact (arrows) is seen.

Figure 5.

(a) Plot of the centroids of the sinogram in Fig. 4a. (b) Plot of Δ_c. Shifts 1 and 2 of approximately 0.55 and −0.4 mm, respectively, indicate projections with discontinuities. The dotted line indicates twice the RMS value.

Figure 6.

Translation correction. (a) Sinogram after translation correction. The sinogram appears more continuous. (b) Reconstructed image after translation correction. The double wall artifacts have been substantially reduced.

In Fig. 7, we show the effectiveness of the self-calibration technique. A vessel phantom was imaged which had a 3 mm inner diameter vessel and contained an aneurysm (approximately 12 mm size) and a Tri-Star stent (Guidant Corporation, Indianapolis, IN) deployed at the neck of the aneurysm; the lumen and aneurysm contained air. In Fig. 7a, we present the projection image before the axis of rotation correction; the axis of the phantom is tilted by about −0.8 deg with respect to the vertical. As a result of the use of the incorrect axis of rotation in the reconstruction, the stent wires [Fig. 7b] appear as circles and there appears to be two vessel-aneurysm walls. For this plane, the center of rotation was shifted from the center of the image by about 4.1 mm. After rotation-axis correction [Fig. 7c], the axis of the vessel aligns closely with the vertical [Fig. 7d], and the stent wires appear as dots and the doubling of the walls of the vessel and aneurysm is substantially reduced.

Figure 7.

Illustrations of the effect of rotation-axis misalignment and correction. When no rotation-axis correction is performed, the projection image (a) of a vessel with an aneurysm and a stent placed across its neck appears tilted and a reconstructed image (b) exhibits a doubling of the vessel and aneurysm boundaries and ringlike structures corresponding to the stent wires. The center of rotation lies about 4.1 mm from the y axis in the plane shown. The tilt angle is about 0.8 deg. After rotation-axis correction, the tilt in the projection image (c) and the artifacts in the reconstructed image (d) are substantially reduced.

To study the effects of residual miscalibration errors, the full-width-at-half-maximum of the profiles of the image of the center-of-mass of each bead in the reconstructed slice of the Lucite phantom are compared with that of simulated beads in the ideal geometry. The difference between the FWHM of the real data and that of the simulated data, which can be attributed to blurring caused by miscalibration errors, is about one voxel (i.e., 125 μm) in a tangential direction for the non-central planes and less than 50 μm for planes near the center of the detector. Little or no blurring due to miscalibration is observed in the radial direction in any of the planes that were studied. This blur is of the same order of magnitude or less than that of the detector for which the FWHM of the point spread function (PSF) at the input plane is approximately 100 microns.⁴⁴ This result indicates that the blurring caused by the small residual miscalibration errors is negligible.

In Table 1, we present the results of the determination of the translation and angle of the rotation axis for the three different translation and angulation setups using the various methods. The translations measured using each of the methods on the bead phantom are comparable (to within 0.1 mm, i.e., one voxel), but differ from those determined using the self-calibration method for setups 1 and 3, which we believe indicates that the system moved by about 0.3 mm between calibration and vessel phantom acquisitions for these cases. For the bead phantom, the angles and translations determined using Methods 1, 2, and 3 are in good agreement (average standard deviation of all three angles and translations across methods is 0.16 deg and 0.09 mm for the measured angles and translations, respectively). The maximum differences between the results of Method 3 and the average of Methods 1 and 2 are 0.35 deg and 0.2 mm for the angles and translations, respectively. The uncertainties for the calculated parameters for all methods (the standard deviation in the slopes and intercepts of the rotational axes determined by regression analysis) were approximately 0.03 deg and 0.05 mm. The differences between the methods are larger than these uncertainties, and we attribute this to systematic differences between the methods themselves, i.e., how they determine the axis of rotation. While it might be interesting to investigate these differences further, they are relatively small (less than the uncertainty in the physically measured angle) and hence beyond the scope of this study. Of greater interest, however, is the comparison between the bead calibrations and the vessel phantom self-calibration. Similar angles were obtained for the three bead calibrations and for our self-calibration method applied to the vessel phantom data acquired with the same imaging geometries [Method 3 (vessel)]. The average angle difference between bead-phantom calibrations and vessel-phantom calibrations is 0.07 deg. However, we see that for setups 1 and 3, the translations determined using the bead phantom and the vessel phantom are substantially different (differences >2 standard deviations). We would conclude from this that some part of the system moved (about 0.3 mm) between the bead-phantom and vessel-phantom acquisitions. Because the angles are comparable and the translations were different, we believe that the rotational stage was stable but that the detector moved. In Fig. 8, we present slices from the vessel-phantom data (Method 3) reconstructed using the calibration parameters determined using Methods 1–3. We see that for the data reconstructed using Methods 1 and 2 the vessel wall is doubled, whereas that reconstructed using Method 3 is sharp. The first two reconstructions are performed with the standard method of calibrating the CBμCT system using a calibration phantom and then imaging the object of interest. The improved results using Method 3 indicate that the self-calibration technique overcomes changes in the system geometry that may occur between calibration and image acquisition.

Table 1.

Comparison of the tilt angle and translation (Trans.) parameters determined for the true rotation axis along with their uncertainties for three cases labeled 1, 2, and 3.

Setup parameter	Angle 1 ϕ (deg)	Angle 2 ϕ (deg)	Angle 3 ϕ (deg)	Trans. 1Tc (mm)	Trans. 2Tc (mm)	Trans. 3Tc (mm)
Measured	0.71±0.5	1.47±0.5	−0.60±0.5	a	a	a
Method 1 (beads) Yang et al. (Ref. 27)	0.57±0.03	1.70±0.05	−0.41±0.05	4.05±0.02	−0.06±0.03	−0.14±0.03
Method 2 (beads) 2D self-calibration	0.43±0.03	1.24±0.03	−0.41±0.03	3.97±0.02	−0.12±0.02	−0.17±0.02
Method 3 (beads) 1D self-calibration	0.54±0.03	1.12±0.04	−0.25±0.01	3.84±0.01	−0.28±0.02	−0.26±0.01
Average±S.D.	0.51±0.07	1.35±0.31	−0.36±0.09	3.95±0.11	−0.15±0.11	−0.19±0.06
Method 3 (vessel) 1D Self calibration	0.420±0.002	1.290±0.002	−0.430±0.002	3.690±0.001	−0.270±0.001	−0.550±0.001

^aThe measured translation is not given here because we could not measure the physical distance of the rotation axis from the center of the detector with the required accuracy.

Figure 8.

Examples of slices from vessel phantom data (angulation and translation 3) reconstructed using calibration values determined using (a) Method 1 (calibration phantom), (b) Method 2 (calibration phantom), and (c) Method 3 (vessel phantom). The edges of the vessel and the phantom wall in the data obtained using calibration values from Method 1 and 2 are doubled (see arrows), whereas they are sharp in the data obtained using our self-calibration (Method 3).

Our CBμCT setup consists of a removable and spatially adjustable detector and rotary stage. Thus, our self-calibration method becomes very useful, eliminating the need for a calibration run every time the setup is modified. In Fig. 9, we present histograms of the angles that the rotation axis made with the y axis and the translations of the central plane determined from 66 of these cases to date. Only 66 cases are reported in Fig. 9 because we did not monitor the parameters initially when use of the self-calibration method began. We see that the alignment of the system is fairly stable, with the standard deviation of the angles and translations being about 0.58 deg and 1.91 mm (indicating the high level of quality of the initial setup). These measurements and their variations along with the actual reconstructions allow us to monitor on a case-by-case basis the quality of the alignments of our system and their variation.

Figure 9.

Histograms of the calculated translations of the central plane (a) and tilt angles that the rotation axis made with the y axis (b) for 66 different CBμCT scans.

DISCUSSION AND CONCLUSION

We have developed automated methods for correcting cone-beam micro-CT data for errors in the geometry of the axis of rotation and translations in the horizontal direction that may occur during the acquisition. These methods use data from the images themselves and do not require calibration devices. Translations of the object itself in the x-direction of greater than 0.2 mm occurring between acquisitions are detected and corrected. As a result of these corrections, artifacts are substantially reduced. Furthermore, we are able to correct for movements that may occur during the scan which is not possible even with a calibration phantom. The parameters describing the axis of rotation agree to within 0.1 mm and 0.3 deg with those determined using other techniques. Thus, these methods are applied automatically for each projection data set obtained in our labs. As a result, we routinely obtain high quality 3D CBμCT data without the use of a calibration object, which is shown by the negligible blurring in the PSF as measured with a bead phantom. In addition, because the calibration data are available for every acquisition, this self-calibrating approach provides a means of quality control (QC).

When applied to the calibration phantom data, each of the methods yields comparable calibration parameters (Table 1). However, when the calibration parameters were used in reconstructing vessel phantom data, average blurring of approximately 4 voxels was observed (see Fig. 8). When reconstructed using calibration parameters determined using the self-calibration technique applied to the vessel phantom, blurring is apparently <125 μm. These results imply (1) that the calculated self-calibration parameters were accurate for the vessel phantom and (2) that the calibrations from the calibration phantom were inaccurate, i.e., that there was apparently a slight change in the geometry (translation) of the system between the acquisition of the calibration and vessel phantom data sets. Perhaps, in the process of changing the phantom, one of the structures (e.g., the detector) moved by 0.3 mm. These results appear to indicate that physical changes can occur after calibration-phantom calibration of our micro-CT system and that the correct calibration can be obtained using our self-calibration technique. These results, thereby, indicate effectiveness and usefulness of our method for calibrating on a scan-by-scan basis. The accuracy of the self-calibration method may not have been fully demonstrated by our comparison of the measured angles and by our evaluation of the blurring effects. More precise measurements would be needed to compare the parameters calculated, and measurements of the MTF of the reconstructed images after correction by each of the three methods might provide a more complete assessment; such evaluations are beyond the scope of this paper.

Similar to CT reconstruction techniques, our technique assumes that the object is rotating as a rigid body. Our technique is capable of correcting the physical translations that might occur during acquisition in only the x-direction as of yet. Because the objects are constrained, large translations in z(>1 mm) cannot occur. Small translations in z alone (<1 mm) cannot be detected by the proposed sinogram. Indeed, x and y translations are more likely because of the setup and constraints on the object. Translation in the y direction could cause blurring or doubling∕splitting in the y direction. While we do not detect or correct for y translations, they could potentially be determined and corrected using image correlation techniques, but these would involve much more analysis of the projection images. To date, we have not found that they are needed. Thus, we believe that the x translation is the dominant source of artifacts and blur and it is detected and corrected with our technique.

The method for determination of the rotation axis assumes that the profiles extracted along the x axis for specific y planes in antiposed images are similar. While the magnification of the objects in these two images is different, the maximum difference in magnification is about 4%–6% for the geometries which we use for objects at the periphery of the object. These maximum differences occur when these objects are near the center of the image. While one could perform the alignment weighting the central region less than the more peripheral regions in the image, we found that a uniform weighting gave us accurate results and good quality images.

In this method, we only measure and correct for rotation and translation of the axis of rotation with respect to the y (vertical) axis. Five other parameters such as out-of-plane angles, SID, SCD, and center in vertical direction that determine the geometry of the total system,²⁷ while not measured and corrected, result in minimal effects on reconstruction. Determination of nutations in the y-z and x-z plane of the rotation axis requires the use of a calibration object. However, even those who use calibration objects report little or no effect on reconstruction because of the out of plane nutations.^{27, 33} Let us consider nutation in y-z plane. This component of the rotation axis results in changes in the magnification (less than 1%) and the projection of the object into the imaging plane. In setting up the system, we visually inspect the rotation of the object during the alignment process. If we take the results of the tilting of the rotation axis in the x-y plane as indicative of the tilting of the rotation axis in the y-z plane, the rotation axis deviates perhaps by approximately 1 deg from the y axis in the y-z plane. Consider a point on a 2 cm diameter phantom, 1 cm from the central plane and in the y-z plane [i.e., at (x,y,z)=(0,1,−1) cm] for a system SID of 80+0.1 cm and a magnification of 1.4. A 1 deg nutation results in a shift of this point by approximately 0.18 mm along the y axis. This shift will change as the object rotates, having its maximum excursions of +0.18 mm at (x,y,z)=(0,1,−1) and (0,1,1) cm. Thus, upon reconstruction, this 1 deg nutation will result in a blurring of the point by approximately a full-width-at-half-maximum of about 0.2 mm. SID and SCD, the other two parameters that determine the geometry, are not computed automatically, but are measured carefully. Also, we assume that the vertical location of the intersection of the x axis and detector plane is the center of the detector plane. In summary, our technique determines and corrects those parameters which are critical for high quality reconstructions, namely the rotation and translation of the rotational axis with the y (vertical) axis, with the effect of the remaining parameters being sufficiently small so as to propagate only into a residual effective widening of the resolution function.

The uncertainties in the angle determination as reported in Table 1 are calculated from the standard deviation of the slope determined from the regression for the rotational axis. The various tilt angles were achieved by placing different size spacers under the leveling screws of the rotary stage. Thus, by measuring the thickness of the spacer and the width of the rotary stage base, the tilt angle was determined. However, there was uncertainty in measurement of approximately 0.5 deg involved in the setup. So, the uncertainties reported in the measured angles are computed from propagation of error arising from the uncertainties in measurements. Although there are differences between the angles calculated by the three methods reported in Table 1 (rows 2–4), the results are within the uncertainty in measurements of 0.5 deg. These differences may be expected because of the diverse nature of the methods themselves.

These techniques have been found to work well for our rotary-stage, micro-CT system. Rotary-stage systems are generally more stable mechanically than rotating gantry systems (e.g., rotational angiography); however, the methods presented here may be a useful basis for rotating gantry calibrations as well. Because these techniques are based on the acquired images, deviation from ideal geometry in the acquisition can be routinely identified and corrected; hence this technique can provide a useful means of QC.