Inverse problems of generalized projection operators

We introduce the concept of generalized projection operators, i.e., projection integrals over a body in 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.

[1]  G. Neukum,et al.  Spectral study of the Eunomia asteroid family-I. Eunomia , 2005 .

[2]  M. Kaasalainen.  Interpretation of lightcurves of precessing asteroids , 2001 .

[3]  M. Weissbluth.  Atoms and Molecules , 1978 .

[4]  Steven J. Ostro,et al.  Doppler radar imaging of spherical planetary surfaces , 1990 .

[5]  Mikko Kaasalainen,et al.  Physical models of large number of asteroids from calibrated photometry sparse in time , 2004 .

[6]  C. C. J. Roothaan,et al.  A Study of Two‐Center Integrals Useful in Calculations on Molecular Structure. I , 1951 .

[7]  Grace Wahba,et al.  Spline Models for Observational Data , 1990 .

[8]  Lance A. M. Benner,et al.  Asteroid Radar Astronomy , 1983 .

[9]  E. Steinborn,et al.  The three‐dimensional convolution of reduced Bessel functions and other functions of physical interest , 1978 .

[10]  Jirí Zára,et al.  Fast approximation to spherical harmonics rotation , 2006, SIGGRAPH '06.

[11]  R. Gardner.  Geometric Tomography: Parallel X-rays of planar convex bodies , 2006 .

[12]  M. Kaasalainen,et al.  Numerical solution of the Minkowski problem , 2001 .

[13]  G. Wahba.  Spline models for observational data , 1990 .

[14]  Klaus Rüdenberg.  A Study of Two‐Center Integrals Useful in Calculations on Molecular Structure. II. The Two‐Center Exchange Integrals , 1951 .

[15]  Karri Muinonen,et al.  Optimization Methods for Asteroid Lightcurve Inversion. II. The Complete Inverse Problem , 2001 .

[16]  Mikko Kaasalainen,et al.  Interpretation of lightcurves of atmosphereless bodies. I : General theory and new inversion schemes , 1992 .

[17]  Herbert H. H. Homeier,et al.  On the Evaluation of Overlap Integrals with Exponential-type Basis Functions , 1992 .

[18]  Klaus Ruedenberg,et al.  Rotation Matrices for Real Spherical Harmonics. Direct Determination by Recursion , 1996 .

[19]  J Berthier,et al.  Shape, size and multiplicity of main-belt asteroids I. Keck Adaptive Optics survey. , 2006, Icarus.

[20]  Robert Connelly,et al.  Convex profiles from asteroid lightcurves , 1983 .

[21]  M. Kaasalainen,et al.  Optimization Methods for Asteroid Lightcurve Inversion: I. Shape Determination , 2001 .

[22]  H. N. Russell.  On the light variations of asteroids and satellites , 1906 .

[23]  Peyman Milanfar,et al.  Reconstruction of Convex Bodies from Brightness Functions , 2003, Discret. Comput. Geom..

[24]  H. Minkowski.  Volumen und Oberfläche , 1903 .

[25]  B. Damtie,et al.  A new incoherent scatter technique in the EISCAT Svalbard Radar , 2002 .

[26]  M. Kaasalainen,et al.  Photometric signatures of highly nonconvex and binary asteroids , 2003 .

[27]  Sanna Kaasalainen,et al.  Ground reference for space remote sensing. Laboratory photometry of an asteroid model , 2005 .

[28]  E. J. Weniger,et al.  The Fourier transforms of some exponential‐type basis functions and their relevance to multicenter problems , 1983 .

[29]  Grotendorst,et al.  Efficient evaluation of infinite-series representations for overlap, two-center nuclear attraction, and Coulomb integrals using nonlinear convergence accelerators. , 1986, Physical review. A, General physics.

[30]  E. Wells,et al.  Asteroid observations with the Hubble Space Telescope and the Space Infrared Telescope Facility , 1989 .

[31]  William H. Press,et al.  Numerical recipes , 1990 .

[32]  E. Somersalo,et al.  Statistical and computational inverse problems , 2004 .

[33]  L. Nirenberg.  The Weyl and Minkowski problems in differential geometry in the large , 1953 .

[34]  H. Minkowski.  Volumen und Oberfläche , 1903 .

[35]  Mark S. Gordon,et al.  Rapid and stable determination of rotation matrices between spherical harmonics by direct recursion , 1999 .

[36]  Grotendorst,et al.  Unified analytical treatment of overlap, two-center nuclear attraction, and Coulomb integrals of B functions via the Fourier-transform method. , 1986, Physical review. A, General physics.

[37]  Mario G. Lattanzi,et al.  Asteroids observations with the Hubble Space Telescope FGS - I. Observing strategy, and data analysis and modeling process , 2002 .

[38]  G. Arfken.  Mathematical Methods for Physicists , 1967 .