Simultaneous, maximum-likelihood determination of focal length and source position for point-source experiments with pinhole collimation

Experimental measurements with a point source provide a valuable tool for characterizing the point-spread function and sensitivity of pinhole collimation. These results may test new theoretical formulas or provide calibrations for reconstructing data. Extracting results from point-source data is also more tenable than from more complex phantoms. However, one limiting factor with point-source measurements for high-precision tests is the determination of the position of the point source relative to the aperture. A maximum-likelihood fit may be used to determine simultaneously the position of the point source and the collimator´s focal length. In this study, a robotic 1D linear stage was mounted on a box collimator to change the angle of the point source with respect to the aperture´s axis of symmetry; the perpendicular height above the aperture, y, was unchanged by this stage. A second stage controlled y. A total of 66 projections (11 angle; 6 y) were acquired for each of three different focal lengths. The projections´ centroids were fit to equations predicting their value as a function of focal length, point-source position, and electronic read-out. The differences in focal length from a reference were -6.2, 7.6, and 12.8 mm. The focal length to the reference was 176.1 mm. Measured focal lengths in each case agreed well with micrometer measurements. Differences in focal length and y also agreed well with micrometer measurements. Standard deviations of focal length and y were about 0.5 mm. In conclusion, this method can determine point-source position during experiment setup and measure focal length to within a millimeter.