Digital image correlation techniques are well known for motion extraction from video images. Following a two-stage approach, the pixel-level displacement is first estimated by maximizing the cross-correlation between two images, then the estimation is refined in the vicinity of the cross-correlation peak. Among existing subpixel refinement methods, quadratic surface fitting (QSF) provides good performances in terms of accuracy and computational burden. It estimates subpixel displacement by interpolating cross-correlation values with a quadratic surface. The purpose of this paper is to analytically investigate the QSF method. By means of counterexamples, it is first shown in this paper that, contrary to a widespread intuition, the quadratic surface fitted to the pixel-level cross-correlation values in the neighborhood of the cross-correlation peak does not always have a maximum. The main contribution of this paper then consists in establishing the mathematical conditions ensuring the existence of a maximum of this fitted quadratic surface, based on a rigorous analysis. Algorithm modifications for handling the failure cases of the QSF method are also proposed in this paper, in order to consolidate it for subpixel motion extraction. Experimental results based on two typical types of images are also reported.