Maximum compatibility estimates and shape reconstruction with boundary curves and volumes of generalized projections

We show that the boundary curves (profiles) in $\R^2$ of the generalized projections of a body in $\R^3$ uniquely determine a large class of shapes, and that sparse profile data, combined with projection volume (brightness) data, can be used to reconstruct the shape and the spin state of a body. We also present an optimal strategy for the relative weighting of the data modes in the inverse problem, and derive the maximum compatibility estimate (MCE) that corresponds to the maximum likelihood or maximum a posteriori estimates in the case of a single data mode. MCE is not explicitly dependent on the noise levels, scale factors or numbers of data points of the complementary data modes, and can be determined without the mode weight parameters. We present a solution method well suitable for adaptive optics images in particular, and discuss various choices of regularization functions.