We have developed a method for the linear reconstruction of an image from undersampled, dithered data. The algorithm, known as Variable-Pixel Linear Reconstruction (or informally as "drizzling"), preserves photometry and resolution, can weight input images according to the statistical significance of each pixel, and removes the effects of geometric distortion both on image shape and photometry. In this paper, the method and its implementation are presented and the photometric and astrometric accuracy and image fidelity of the algorithm are discussed. In addition, we describe the use of drizzling to combine dithered images in the presence of cosmic rays.