H.C. van Assen, M. Egmont-Petersen, J.H.C. Reiber. "Accurate
object localization in gray level images using the center of gravity measure:
accuracy versus precision," IEEE Transactions on Image
Processing, Vol. 11, No. 12, pp. 1379-1384, 2002.
Computing the accurate localization of objects often requires a two-step
approach. First, the coarse object detection is performed followed by accurate
object localization. A widely used estimate of the object center with sub-pixel
precision is the weighted center of gravity (COG). We derive a
maximum-likelihood estimator for the variance of the (2D and 3D) COG, and an
approximation to the estimate, as a function of the noise in the image. We
assume that the noise in the image is additive, Gaussian distributed and
independent between neighboring pixels.
Experiments using 2,500 generated markers with a cosine profile and superimposed with Gaussian noise with different noise levels were performed. The experiments indicate that misplacing the window that indicates which pixels contribute to the COG computation causes a bias and a larger variance in the estimated COG. This bias can be reduced by applying a threshold on the intensities comprised by the window. The chosen weighing scheme influences the accuracy and precision of the estimated COG, as thresholding results in a higher accuracy, but a worse precision. The desired trade-off between accuracy and precision can be chosen using the derived formula for the variance of the center of gravity. The difference between our estimate and the true variance was always less than 5 % of the true variance and this deviation decreases with increasing signal-to-noise ratio. Our approximation to the estimate performed better than the one presented by Oron et al. by up to a factor ˜10 for a window misplacement of 3.0 pixels.
Electronic reprint, or contact me: michael * egmont-petersen.nl (with * indicating @)