Localization error analysis for stereo X-ray image guidance with probability method

The mean value and standard deviation of localization error for the stereo imaging systems are derived based on probability theory. Compared with the maximum error analysis method used in our previous study, the new approach yields more informative and precise results as the guidance for X-ray imaging system design and protocol optimization. The prototype for our current study is a CCD based monoplane digital stereo X-ray imaging system. The imaging model consists of two X-ray sources and one detector plane. With perspective geometry, the least-square solution is derived to reconstruct 3-dimensional object points, such as a biopsy needle tip, from a pair of 2-dimensional digital radiographs. Under the conditions of our specific prototype, the measurement errors of interested points in the radiographs are modeled as random variables with Gaussian distribution. Such variables account for finite image system noise and positioning errors. Then, the 3D localization error, in terms of mean value and standard deviation, is formulated using measurement error, feature point location, and separation between the two X-ray sources and distance from source to detector. Both theoretical analysis and numerical simulation are performed. The mean value and standard deviation of the localization error are first evaluated using numerical simulation under practical imaging conditions. Then, the error estimates are given in simply analytic forms. Simulation and theoretical results are in excellent agreement. The results show that our prototype X-ray stereological imaging system is accurate and reliable to locate feature points in 3D for medical intervention. Imaging protocols can be effectively optimized through the 3D localization error analysis using the approximate formulas proposed in this study.