To validate the accuracy of the calibration method described in Sec. III, the projection coordinates of BBs on the detector plane were simulated. The calibration method was then applied on the simulated data and the calculated system parameters were compared with their known values. Gaussian noise of different standard deviation was added to the projection coordinates to simulate the effect of noise in the practical measurement environment and to evaluate the consistency of this method in the presence of noise.
In this simulation, the detector matrix size was set to 2048 × 1024 with a pixel size of 48 µm × 48 µm, similar to the mouse CT scanner assembled in our laboratory. The five system parameters were set as: RFD=400.0000 mm, RFI=150.0000 mm, u0=1005.0000 pixel, υ0=480.0000 pixel, η=−1.0000°.
Eight BBs were simulated and the distance between the first and the last BB was l=14.39 mm. Five hundred points were simulated for each projection orbit, corresponding to a 500-view CT scan in 360°. One example of the simulation is shown in .
Simulated projection orbits with Gaussian noise. RFD=400 mm, RFI=150 mm, u0=1005 pixel, v0=480 pixel, η=−1.0°, standard deviation of Gaussian noise=0.4 pixel.
Simulations were performed in three different groups:
- No preset out-of-plane rotation, with different noise levels, results are shown in and ;
R-square values for linear fitting.
Calibration results from computer simulation.
- With different out-of-plane rotation angles, without noise, results are shown in ;
- With specific preset out-of-plane rotations (ϕ=1.5000°,σ=1.2000°), with different noise levels, results are shown in and .
R-square values for linear fitting (ϕ=1.5000°,σ=1.2000°).
Calibration results from computer simulation (ϕ=1.5000°, σ=1.2000°).
Linear regression was used in two steps of the calibration method, and the R-square value or the coefficient of determination, which is the indicator of how well the fitting works, was shown in and . Gaussian noise with standard deviations s of 1%, 20%, 40%, and 100% of the pixel size were independently added into the horizontal and vertical coordinates of every projection position. Most R-square values calculated were close to 1.0, corresponding to an ideal linear fitting. Especially in calculation of RFD and υ0, the R square value was very consistent even with a noise level of s=1.00 pixel.
The system parameters calibrated from computer simulation are listed in , , and . Though the results degraded slightly with increasing noise, the calibrated values were nevertheless accurate in comparison with their true values.
From the results shown in –, when the out-of-plane rotation angles are relatively small (less than 2°), there will be minor effects caused by neglecting these angles compared with the effects caused by the noise in the measurement of BB positions. And normal engineering design and machinery can satisfy this loose requirement by limiting out-of-plane rotations to within 1°. These results demonstrate that good calibration accuracy can be achieved assuming ϕ=0 and σ=0.
Computer simulations were performed with various system parameters including the number of views, number of BBs used, and different combinations of system parameters. All the calibration results under various conditions validated the accuracy of this calibration method.
To compare with previous results, simulations with same rotation angles and noise level as in Smekal’s7
work were performed and calibration results are given in . In , relative errors were calculated by the ratio between uncertainties and mean values. The method presented here has equal or better accuracy as the results reported by Smekal et al.7
Comparison with Smekal’s method (ϕ=1.5000°, σ=1.2000°).