IEEE Transactions on Pattern Analysis and Machine Intelligence
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THE FIRST GENERICALLY VALID PROOF OF UNIQUENESS FOR THE SURFACE SOLUTION TO SHAPE FROM SHADING IS PRESENT. ALSO, IT IS PROVEN THAT THE LOCAL SURFACE SOLUTIONS AROUND A SINGULAR POINT HAVE AT MOST A FOUR-FOLD AMBIGUITY. THESE RESULTS APPLY FOR A REFLECTANCE FUNCTION CORRESPONDING TO ILLUMINATION FROM FROM THE VIEWER DIRECTION OF A UNIFORM ALBEDO LAMBERTIAN OBJECT. GENERIC SURFACES ARE STUDIED, AND THEIR PROPERTIES ESTABLISHED. THE PROOF MAY LEAD TO NEW, FASTER ALGORITHMS FOR SHAPE RECOVERY. QUESTIONS OF EXISTENCE ARE ALSO DISCUSSED. IT IS ARGUED THAT MOST IMAGES ARE IMPOSSIBLE, IN THE SENSE THAT THEY CAN NOT BE A DEPICTION OF ANY PHYSICAL OBJECT. THE PROOF IS BASED ON IDEAS OF DYNAMICAL SYSTEMS THEORY AND GLOBAL ANALYSIS.