Positron emission tomography (PET) images are characterized by both poor spatial resolution and high statistical noise. Conventional methods to reduce noise, such as local weighted averaging, produce further deteriorations in spatial resolution, while the use of deconvolution to recover resolution typically amplifies noise to unacceptable levels. We studied the use of two-dimensional Fourier filtering to simultaneously increase quantitative recovery and reduce noise. The filter was based on inversion of the scanner's measured transfer function, coupled with high frequency roll-off. In phantom studies, we found improvements in both "hot" and "cold" sphere quantification. Compared with ramp-only filtering, improvements in hot spot recovery for the highest accuracy filter averaged 13.6% +/- 6.6% for spheres larger than 15 mm; improvements in cold spot recovery averaged 30.7% +/- 4.7%. At the same time, the noise was reduced by a factor of 3 compared with randomly filtering. Fourier-based image restoration filtering is thus capable of improving both accuracy and precision in PET.