TITLE: Practical Improvement of Geometric Parameter Estimation for Cone Beam MicroCT Imaging
AUTHORS: Stacia A. Sawyer and Eric C. Frey
PURPOSE: To reduce the error in our estimated geometric parameters through adjustment of our estimation process, namely sphere center estimation.
METHODS AND MATERIALS: To test whether or not the use of a more accurate fitting function to estimate the projected position of the sphere center would improve the accuracy of these estimates, we implemented a different fitting function based on an analytic expression for the cone-beam projection of a sphere. We hypothesized that this would improve the accuracy of the center estimates since we noticed some asymmetry in the projections of the spheres located far from the central slice. We calculated the true projected position of the sphere centers and compared the results to the results from fitting using a Gaussian and the projection of the sphere as the fitting function.
To test whether or not a smaller sphere size would improve the center accuracy and/or final parameter estimation accuracy, we simulated several phantoms that differed only in the diameter of the spheres: a point source, a 1 mm sphere, a 1.6 mm sphere, and a 2 mm sphere. For the center accuracy comparison, we simulated the phantoms with a higher resolution in a 1024x96 matrix. We then generated projection data for each of these phantoms with a realistic degree of geometric misalignments using an analytic cone-beam projector, a pixel size of 0.05 mm, and a magnification factor of 1.053. We compared the centers calculated for each phantom to the actual projected position for the point phantom.
To investigate the effect of various size spheres on final parameter estimation accuracy, we generated spheres in the same configuration as our physical phantom in a smaller (256x256x256) matrix. We used the same sphere sizes as in the above simulation, i.e. phantoms containing point sources, 1, 1.6, or 2 mm spheres. For each set of phantoms, we generated 3 separate spheres (or points) of the same size which were perpendicular to each other. For each of these sets of spheres, we generated projection data with a realistic degree of geometric misalignments using an analytic cone-beam projector, a pixel size of 0.2 mm, and a magnification factor of 1.068. We then followed our parameter estimation process and estimated the parameters for each phantom. We compared the final parameters estimated to the true parameters.
Finally, to test whether varying the size of the spheres would affect the parameter estimation for real data, we acquired projection data for a physical calibration phantom using three 1.6 mm diameter spheres and for a physical calibration phantom using three 1.0 mm diameter spheres using our microCT system. We followed our parameter estimation method (using the sphere projection function to estimate sphere centers) and estimated parameters for our system with both phantoms. The true system parameters were unknown, so for comparison of the estimated parameter accuracy, we compared the reconstructed images. We reconstructed the projection data for the phantom using the 1.0 mm diameter spheres with compensation for geometric misalignments and using both sets of estimated parameters and compared the resulting images.
RESULTS: We found that the differences between the true and estimated projected center positions were smaller when fitted with the sphere's projection equation. In addition, using smaller spheres resulted in improved estimate accuracy.
CONCLUSIONS: The accuracy of the estimate of the projected sphere center position is a significant limitation in calibration parameter estimation. Although it is important to use a sphere large enough that it does not cause discretization problems in the center position estimation, the use of a smaller sphere results in more accurately estimated parameters. Changing the fitting function from a Gaussian to the projection of a sphere results in more accurate center calculations. Further investigation is needed to determine the optimal pixel size for a given reconstructed voxel size.