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Uncalibrated photometric stereo aims at estimating the 3D-shape of a surface,given a set of images captured from the same viewing angle, but under unknown,varying illumination. While the theoretical foundations of this inverse problemunder directional lighting are well-established, there is a lack ofmathematical evidence for the uniqueness of a solution under general lighting.On the other hand, stable and accurate heuristical solutions of uncalibratedphotometric stereo under such general lighting have recently been proposed. Thequality of the results demonstrated therein tends to indicate that the problemmay actually be well-posed, but this still has to be established. The presentpaper addresses this theoretical issue, considering first-order sphericalharmonics approximation of general lighting. Two important theoretical resultsare established. First, the orthographic integrability constraint ensuresuniqueness of a solution up to a global concave-convex ambiguity, which hadalready been conjectured, yet not proven. Second, the perspective integrabilityconstraint makes the problem well-posed, which generalizes a previous resultlimited to directional lighting. Eventually, a closed-form expression for theunique least-squares solution of the problem under perspective projection isprovided, allowing numerical simulations on synthetic data to empiricallyvalidate our findings.

On the well-posedness of uncalibrated photometric stereo under general lighting