We provide a mathematical analysis of and a numerical framework for full-field optical coherence elastography, which has unique features including micron-scale resolution, realtime processing, and noninvasive imaging. We develop a novel algorithm for transforming volumetric optical images before and after the mechanical solicitation of a sample with subcellular resolution into quantitative shear modulus distributions. This has the potential to improve sensitivities and specificities in the biological and clinical applications of optical coherence tomography. © by SIAM.