Toward Guaranteed Illumination Models for Non-convex Objects

Illumination variation remains a central challenge in object detection and recognition. Existing analyses of illumination variation typically pertain to convex, Lambertian objects, and guarantee quality of approximation in an average case sense. We show that it is possible to build models for the set of images across illumination variation with worst-case performance guarantees, for nonconvex Lambertian objects. Namely, a natural verification test based on the distance to the model guarantees to accept any image which can be sufficiently well-approximated by an image of the object under some admissible lighting condition, and guarantees to reject any image that does not have a sufficiently good approximation. These models are generated by sampling illumination directions with sufficient density, which follows from a new perturbation bound for directional illuminated images in the Lambertian model. As the number of such images required for guaranteed verification may be large, we introduce a new formulation for cone preserving dimensionality reduction, which leverages tools from sparse and low-rank decomposition to reduce the complexity, while controlling the approximation error with respect to the original model.