Theoretical Limits of Localizing 3-D Landmarks and Features

In this paper, we analyze the accuracy of estimating the location of 3D landmarks and characteristic image structures. Based on nonlinear estimation theory, we study the minimal stochastic errors of the position estimate caused by noisy data. Given analytic models of the image intensities, we derive closed-form expressions of the Cramer-Rao bound for different 3D structures such as 3D edges, 3D ridges, 3D lines, 3D boxes, and 3D blobs. It turns out that the precision of localization depends on the noise level, the size of the region-of-interest, the image contrast, the width of the intensity transitions, as well as on other parameters describing the considered image structure. The derived lower bounds can serve as benchmarks and the performance of existing algorithms can be compared with them. To give an impression of the achievable accuracy, numeric examples are presented. Moreover, by experimental investigations, we demonstrate that the derived lower bounds can be achieved by fitting parametric intensity models directly to the image data.