American Institute of Mathematical Sciences

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February  2011, 5(1): 37-57. doi: 10.3934/ipi.2011.5.37

Multimodal inverse problems: Maximum compatibility estimate and shape reconstruction

Mikko Kaasalainen 1,

1. 

Department of Mathematics, Tampere University of Technology, P.O. Box 553, 33101 Tampere, Finland

Received  April 2009 Revised  September 2010 Published  February 2011

We present an optimal strategy for the relative weighting of multiple data modalities in inverse problems, 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 also discuss discontinuities in the solution estimates in multimodal inverse problems, and derive a corresponding self-consistency criterion. As a case study, we consider the problem of reconstructing the shape and the spin state of a body in $\R^3$ from the boundary curves (profiles) and volumes (brightness values) of its generalized projections in $\R^2$. We also show that the generalized profiles uniquely determine a large class of shapes. We present a solution method well suitable for adaptive optics images in particular, and discuss various choices of regularization functions.
Keywords: computational geometry, machine vision, computational methods in astronomy., Inverse problems, three-dimensional polytopes.
Mathematics Subject Classification: 68U05, 68T45, 65D18, 52B10, 49N45, 65J22, 85-0.
Citation: Mikko Kaasalainen. Multimodal inverse problems: Maximum compatibility estimate and shape reconstruction. Inverse Problems & Imaging, 2011, 5 (1) : 37-57. doi: 10.3934/ipi.2011.5.37
References:
[1]

A. Zacharopoulos, S. Arridge, O. Dorn, V. Kolehmainen and J. Sikora, Three-dimensional reconstruction of shape and piecewise constant region values for optical tomography using spherical harmonic parametrization and a boundary element method, Inverse Problems, 22 (2006), 1509-1532. doi: 10.1088/0266-5611/22/5/001.  Google Scholar

[2]

M. Belge, M. Kilmer and E. Miller, Efficient determination of multiple regularization parameters in a generalized L-curve framework, Inverse Problems, 18 (2002), 1161-1183. doi: 10.1088/0266-5611/18/4/314.  Google Scholar

[3]

A. Bottino and A. Laurentini, Introducing a new problem: Shape-from-silhouette when the relative positions of the viewpoints are unknown, IEEE Transact. on Pattern Analysis and Machine Intelligence, 25 (2003), 1484-1493 doi: 10.1109/TPAMI.2003.1240121.  Google Scholar

[4]

B. Carry, C. Dumas, M. Fulchignoni, W. Merline, J. Berthier, D. Hestroffer, T. Fusco and P. Tamblyn, Near-infrared mapping and physical properties of the dwarf-planet Ceres, Astron. Astrophys., 478 (2008), 235-244. doi: 10.1051/0004-6361:20078166.  Google Scholar

[5]

B. Carry, C. Dumas, M. Kaasalainen and 9 colleagues, Physical properties of 2 Pallas, Icarus, 205 (2010), 460-472. doi: 10.1016/j.icarus.2009.08.007.  Google Scholar

[6]

B. Carry, A. Conrad, J. Drummond, M. Kaasalainen, W. Merline, J. Berthier and C. Dumas, The resolved asteroid program: Size and shape of (41) Daphne,, Icarus, ().   Google Scholar

[7]

K. Cheung, S. Baker and T. Kanade, Shape-From-Silhouette across time part I: Theory and algorithms, Int. J. Comp. Vision, 62 (2005), 221-247. doi: 10.1007/s11263-005-4881-5.  Google Scholar

[8]

P. Descamps and 22 colleagues, New insights on the binary asteroid 121 Hermione, Icarus, 203 (2009), 88-101. doi: 10.1016/j.icarus.2009.04.032.  Google Scholar

[9]

A. Dobrovolskis, Inertia of any polyhedron, Icarus, 124 (1996), 698-704. doi: 10.1006/icar.1996.0243.  Google Scholar

[10]

J. Ďurech and M. Kaasalainen., Photometric signatures of highly nonconvex and binary asteroids, Astron. Astrophys., 404 (2003), 709-714. doi: 10.1051/0004-6361:20030505.  Google Scholar

[11]

H. Engl and W. Grever, Using the L-curve for determining optimal regularization parameters, Numer. Math., 69 (1994), 25-31. doi: 10.1007/s002110050078.  Google Scholar

[12]

H. Goldstein, "Classical Mechanics" (second edition), Addison-Wesley, Reading, Mass., 1980. Google Scholar

[13]

M. Hanke, Limitations of the L-curve method in ill-posed problems, BIT, 36 (1996), 287-301. doi: 10.1007/BF01731984.  Google Scholar

[14]

M. Kaasalainen, L. Lamberg, K. Lumme and E. Bowell, Interpretation of lightcurves of atmosphereless bodies. I. General theory and new inversion schemes, Astron. Astrophys., 259 (1992), 318-332. Google Scholar

[15]

M. Kaasalainen and J. Torppa, Optimization methods for asteroid lightcurve inversion. I. Shape determination, Icarus, 153 (2001), 24-36. doi: 10.1006/icar.2001.6673.  Google Scholar

[16]

M. Kaasalainen, J. Torppa and K. Muinonen, Optimization methods for asteroid lightcurve inversion. II. The complete inverse problem, Icarus, 153 (2001), 37-51. doi: 10.1006/icar.2001.6674.  Google Scholar

[17]

M. Kaasalainen, Interpretation of lightcurves of precessing asteroids, Astron. Astrophys., 376 (2001), 302-309. doi: 10.1051/0004-6361:20010935.  Google Scholar

[18]

M. Kaasalainen and L. Lamberg, Inverse problems of generalized projection operators, Inverse Problems, 22 (2006), 749-769. doi: 10.1088/0266-5611/22/3/002.  Google Scholar

[19]

M. Kaasalainen and J. Ďurech, Inverse problems of NEO photometry: Imaging the NEO population, in "Proceedings of IAU: Symposium 236" (Milani, Valsecchi, and Vokrouhlicky, eds.), 2, Cambridge (2007), 151-166. Google Scholar

[20]

M. Kaasalainen, J. Ďurech, B. Warner, Y. Krugly and N. Gaftonyuk, Acceleration of the rotation of asteroid 1862 Apollo by radiation torques, Nature, 446 (2007), 420-422. doi: 10.1038/nature05614.  Google Scholar

[21]

J. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems," Springer, New York, 2005.  Google Scholar

[22]

H. U.Keller and 46 colleagues, E-type asteroid (2867) Steins as imaged by OSIRIS on board Rosetta, Science, 327 (2010), 190-193 doi: 10.1126/science.1179559.  Google Scholar

[23]

D. Levin, The approximation power of moving least squares, Math. Comp., 67 (1998), 1517-1531. doi: 10.1090/S0025-5718-98-00974-0.  Google Scholar

[24]

F. Marchis, M. Kaasalainen, E. Hom, J. Berthier, J. Enriquez, D. Hestroffer, D. Le Mignant and I. de Pater, Shape, size and multiplicity of main-belt asteroids. I. Keck adaptive optics survey, Icarus, 185 (2006), 39-63. doi: 10.1016/j.icarus.2006.06.001.  Google Scholar

[25]

P. Pravec, A. Harris and T. Michalowski, Asteroid rotations, in "Asteroids III" (Bottke, Cellino, Paolicchi and Binzel, eds.), U. Arizona Press, Tucson, (2002), 113-122. Google Scholar

[26]

W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, "Numerical Recipes" (third edition), Cambridge U. Press, New York, 2007.  Google Scholar

[27]

S. Savarese, M. Andretto, H. Rushmeier, F. Bernardini and P. Perona, 3D reconstruction by shadow carving: Theory and practical evaluation, Int. J. Comp. Vision, 71 (2007), 305-336. doi: 10.1007/s11263-006-8323-9.  Google Scholar

show all references

References:
[1]

A. Zacharopoulos, S. Arridge, O. Dorn, V. Kolehmainen and J. Sikora, Three-dimensional reconstruction of shape and piecewise constant region values for optical tomography using spherical harmonic parametrization and a boundary element method, Inverse Problems, 22 (2006), 1509-1532. doi: 10.1088/0266-5611/22/5/001.  Google Scholar

[2]

M. Belge, M. Kilmer and E. Miller, Efficient determination of multiple regularization parameters in a generalized L-curve framework, Inverse Problems, 18 (2002), 1161-1183. doi: 10.1088/0266-5611/18/4/314.  Google Scholar

[3]

A. Bottino and A. Laurentini, Introducing a new problem: Shape-from-silhouette when the relative positions of the viewpoints are unknown, IEEE Transact. on Pattern Analysis and Machine Intelligence, 25 (2003), 1484-1493 doi: 10.1109/TPAMI.2003.1240121.  Google Scholar

[4]

B. Carry, C. Dumas, M. Fulchignoni, W. Merline, J. Berthier, D. Hestroffer, T. Fusco and P. Tamblyn, Near-infrared mapping and physical properties of the dwarf-planet Ceres, Astron. Astrophys., 478 (2008), 235-244. doi: 10.1051/0004-6361:20078166.  Google Scholar

[5]

B. Carry, C. Dumas, M. Kaasalainen and 9 colleagues, Physical properties of 2 Pallas, Icarus, 205 (2010), 460-472. doi: 10.1016/j.icarus.2009.08.007.  Google Scholar

[6]

B. Carry, A. Conrad, J. Drummond, M. Kaasalainen, W. Merline, J. Berthier and C. Dumas, The resolved asteroid program: Size and shape of (41) Daphne,, Icarus, ().   Google Scholar

[7]

K. Cheung, S. Baker and T. Kanade, Shape-From-Silhouette across time part I: Theory and algorithms, Int. J. Comp. Vision, 62 (2005), 221-247. doi: 10.1007/s11263-005-4881-5.  Google Scholar

[8]

P. Descamps and 22 colleagues, New insights on the binary asteroid 121 Hermione, Icarus, 203 (2009), 88-101. doi: 10.1016/j.icarus.2009.04.032.  Google Scholar

[9]

A. Dobrovolskis, Inertia of any polyhedron, Icarus, 124 (1996), 698-704. doi: 10.1006/icar.1996.0243.  Google Scholar

[10]

J. Ďurech and M. Kaasalainen., Photometric signatures of highly nonconvex and binary asteroids, Astron. Astrophys., 404 (2003), 709-714. doi: 10.1051/0004-6361:20030505.  Google Scholar

[11]

H. Engl and W. Grever, Using the L-curve for determining optimal regularization parameters, Numer. Math., 69 (1994), 25-31. doi: 10.1007/s002110050078.  Google Scholar

[12]

H. Goldstein, "Classical Mechanics" (second edition), Addison-Wesley, Reading, Mass., 1980. Google Scholar

[13]

M. Hanke, Limitations of the L-curve method in ill-posed problems, BIT, 36 (1996), 287-301. doi: 10.1007/BF01731984.  Google Scholar

[14]

M. Kaasalainen, L. Lamberg, K. Lumme and E. Bowell, Interpretation of lightcurves of atmosphereless bodies. I. General theory and new inversion schemes, Astron. Astrophys., 259 (1992), 318-332. Google Scholar

[15]

M. Kaasalainen and J. Torppa, Optimization methods for asteroid lightcurve inversion. I. Shape determination, Icarus, 153 (2001), 24-36. doi: 10.1006/icar.2001.6673.  Google Scholar

[16]

M. Kaasalainen, J. Torppa and K. Muinonen, Optimization methods for asteroid lightcurve inversion. II. The complete inverse problem, Icarus, 153 (2001), 37-51. doi: 10.1006/icar.2001.6674.  Google Scholar

[17]

M. Kaasalainen, Interpretation of lightcurves of precessing asteroids, Astron. Astrophys., 376 (2001), 302-309. doi: 10.1051/0004-6361:20010935.  Google Scholar

[18]

M. Kaasalainen and L. Lamberg, Inverse problems of generalized projection operators, Inverse Problems, 22 (2006), 749-769. doi: 10.1088/0266-5611/22/3/002.  Google Scholar

[19]

M. Kaasalainen and J. Ďurech, Inverse problems of NEO photometry: Imaging the NEO population, in "Proceedings of IAU: Symposium 236" (Milani, Valsecchi, and Vokrouhlicky, eds.), 2, Cambridge (2007), 151-166. Google Scholar

[20]

M. Kaasalainen, J. Ďurech, B. Warner, Y. Krugly and N. Gaftonyuk, Acceleration of the rotation of asteroid 1862 Apollo by radiation torques, Nature, 446 (2007), 420-422. doi: 10.1038/nature05614.  Google Scholar

[21]

J. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems," Springer, New York, 2005.  Google Scholar

[22]

H. U.Keller and 46 colleagues, E-type asteroid (2867) Steins as imaged by OSIRIS on board Rosetta, Science, 327 (2010), 190-193 doi: 10.1126/science.1179559.  Google Scholar

[23]

D. Levin, The approximation power of moving least squares, Math. Comp., 67 (1998), 1517-1531. doi: 10.1090/S0025-5718-98-00974-0.  Google Scholar

[24]

F. Marchis, M. Kaasalainen, E. Hom, J. Berthier, J. Enriquez, D. Hestroffer, D. Le Mignant and I. de Pater, Shape, size and multiplicity of main-belt asteroids. I. Keck adaptive optics survey, Icarus, 185 (2006), 39-63. doi: 10.1016/j.icarus.2006.06.001.  Google Scholar

[25]

P. Pravec, A. Harris and T. Michalowski, Asteroid rotations, in "Asteroids III" (Bottke, Cellino, Paolicchi and Binzel, eds.), U. Arizona Press, Tucson, (2002), 113-122. Google Scholar

[26]

W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, "Numerical Recipes" (third edition), Cambridge U. Press, New York, 2007.  Google Scholar

[27]

S. Savarese, M. Andretto, H. Rushmeier, F. Bernardini and P. Perona, 3D reconstruction by shadow carving: Theory and practical evaluation, Int. J. Comp. Vision, 71 (2007), 305-336. doi: 10.1007/s11263-006-8323-9.  Google Scholar

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