The underlying piecewise continuous surface of a digital image can be estimated through robust statistical procedures. This thesis contains a systematic Monte Carlo study of M-estimation and LMS estimation for image surface approximation, an examination of the merits of postprocessing and tuning various parameters in the robust estimation procedures, and a new robust variable order facet model paradigm. Several new goodness of fit measures are introduced, and systematically compared. It is shown that the M-estimation tuning parameters are not crucial, postprocessing is cheap and well worth the cost, and the robust variable order facet model algorithm (using M-estimation, new statistical goodness of fit measures, and postprocessing) manages to retain most of the statistical efficiency of Mestimation yet displays good robustness properties, and preserves the main geometric features of an image surface: step edges, roof edges and corners. / Master of Science