  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 problem
under directional lighting are well-established, there is a lack of
mathematical evidence for the uniqueness of a solution under general lighting.
On the other hand, stable and accurate heuristical solutions of uncalibrated
photometric stereo under such general lighting have recently been proposed. The
quality of the results demonstrated therein tends to indicate that the problem
may actually be well-posed, but this still has to be established. The present
paper addresses this theoretical issue, considering first-order spherical
harmonics approximation of general lighting. Two important theoretical results
are established. First, the orthographic integrability constraint ensures
uniqueness of a solution up to a global concave-convex ambiguity, which had
already been conjectured, yet not proven. Second, the perspective integrability
constraint makes the problem well-posed, which generalizes a previous result
limited to directional lighting. Eventually, a closed-form expression for the
unique least-squares solution of the problem under perspective projection is
provided, allowing numerical simulations on synthetic data to empirically
validate our findings.
