A computationally rapid image analysis method, weighted overdetermined regression, is presented for two-dimensional (2D) Gaussian Fitting of particle location with subpixel resolution from a pixelized image of light intensity. Compared to least-squares Gaussian iterative fitting, which is most exact but prohibitively slow for large data sets, the precision of this new method is equivalent when the signal-to-noise ratio is high and approaches it when the signal-to-noise ratio is low, while enjoying a more than 100-fold improvement in computational time. Compared to another widely used approximation method, nine-point regression, we show that precision and speed are both improved. Additionally, weighted regression runs nearly as fast and with greatly improved precision compared to the simplest method, the moment method, which, despite its limited precision, is frequently employed because of its speed. Quantitative comparisons are presented for both circular and elliptical Gaussian intensity distributions. This new image analysis method may be useful when dealing with large data sets such as those frequently met in astronomy or in single-particle and single-molecule tracking using microscopy and may facilitate advances such as real-time quantification of microscopy images.