Summary: We introduce the concept of generalized projection operators, i.e., projection integrals over a body in \(\mathbb R^3\) that generalize the usual result of projected area in a given direction by taking into account shadowing and scattering effects as well as additional convolution functions in the integral. Such operators arise naturally in connection with various observation instruments and data types.
We review and discuss some properties of these operators and the related inverse problems, particularly in the cases pertaining to photometric and radar data. We also prove an ambiguity theorem for a special observing geometry common in astrophysics, and uniqueness theorems for radar inverse problems of a spherical target. These theorems are obtained by employing the intrinsic rotational properties of the observing geometries and function representations.
We then present examples of the mathematical modelling of the shape and rotation state of a body by simultaneously using complementary data sources corresponding to different generalized projection operators. We show that generalized projection operators unify a number of mathematical considerations and physical observation types under the same concept.