The theoretical foundation is developed for a new technique to determine displacements with subpixel accuracy at each pixel location in a digital image of a deformed object using digital image correlation (DIC). This technique, known as pointwise DIC, is evaluated using ideal sinusoidal images for the cases of rigid body translation, extensional strain, and rigid body rotation. Displacement fields obtained using objective correlation functions with and without intensity gradients are compared. Both bilinear and bicubic interpolation schemes are investigated for reconstructing the subpixel intensity and intensity gradient values in undeformed and deformed images. The effects of transforming the Eulerian description of the intensity gradients in the deformed images to the Lagrangian description in the undeformed images are also investigated. The optimal correlation value and the calculated displacement fields of the two interpolation schemes are compared. Theoretical results demonstrate that pointwise DIC can accurately determine displacement fields. To demonstrate the advantages of pointwise DIC over conventional DIC techniques, an ideal image simulating a twinning deformation is correlated, indicating the pointwise technique is up to two orders of magnitude more accurate at determining discontinuous displacements than the conventional technique. Experiments are also conducted on a polycarbonate dogbone specimen that validate pointwise DIC on real images and determine the inherent accuracy in the digital image acquisition.