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

TL;DR

This paper provides the first rigorous mathematical proof that uncalibrated photometric stereo under general lighting is well-posed under perspective projection, establishing conditions for solution uniqueness and offering a closed-form solution.

Contribution

It proves the well-posedness of uncalibrated photometric stereo with perspective projection and general lighting, extending previous directional lighting results and providing a practical solution formula.

Findings

Orthographic integrability ensures uniqueness up to a global ambiguity.

Perspective integrability makes the problem well-posed.

Closed-form least-squares solution validated by synthetic data.

Abstract

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.