Fourier single-pixel imaging (FSI) uses Fourier basis patterns for spatial light modulation to acquire the Fourier spectrum of the object image. The object image can be reconstructed via an inverse Fourier transform. However, the Fourier basis patterns are inherently gray scale, which results in the difficulty that the patterns can hardly be generated at a high speed by using a commonly used spatial light modulator-digital micromirrors device. To tackle this problem, fast FSI, which uses upsampled and dithered Fourier basis patterns to approximate the gray scale patterns, has been reported, but the achievable spatial resolution has to be sacrificed in the pattern upsampling process. Here we propose a method that can achieve not only full-resolution but also full-field-of-view and high-quality FSI. The key to the proposed method is to use a new, to the best of our knowledge, error diffusion dithering algorithm combined with two different scanning strategies to generate two sets of binarized Fourier basis patterns for spatial light modulation. As a result, two images with a sub-pixel shift from each other are reconstructed. It results in the final high-quality reconstruction by synthesizing the two images. We experimentally demonstrate the method can produce a high-quality 1024 × 768-pixel and full resolution image with a digital micromirror device with 1024 × 768 micromirrors.