Invariant Properties of Straight Homogeneous Generalized Cylinders and Their Contours
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Generalized Cylinders are a flexible, loosely-defined class of parametric shapes capable of modeling many real-world objects. Straight Homogeneous Generalized Cylinders are an important subclass of Generalized Cylinders, whose cross-sections are scaled versions of a reference curve. Although there has been considerable research into recovering the shape of SHGCs from their contour, this work has almost exclusively involved methods that couple contour and heuristic constraints. A rigorous approach to the problem of recovering solid parametric shape from a single intensity view should involve at least two stages: 1) deriving the contour constraints, and 2) determining if additional image constraints, e.g., intensity, can be used to uniquely determine the 3D object shape. In this paper, the authors follow the approach just described. This methodology is also important for the recovery of object classes like tubes, where contour and heuristic constraints are shown to be insufficient for shape recovery. First, we prove that SHGC contours generated under orthography have exactly two degrees of freedom. Next, we show that the remaining free parameters can be resolved using reflectance-based constraints, without knowledge of the number of light sources, their positions, intensities, the amount of ambient light, or the surface albedo. Finally, the reflectance-based recovery algorithm is demonstrated on both synthetic and real SHGC images.