American Institute of Mathematical Sciences

Advanced
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.  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.  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.  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.  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.  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.  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.  doi: 10.1016/j.icarus.2009.04.032.  Google Scholar

[9]

A. Dobrovolskis, Inertia of any polyhedron,, Icarus, 124 (1996), 698.  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.  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.  doi: 10.1007/s002110050078.  Google Scholar

[12]

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

[13]

M. Hanke, Limitations of the L-curve method in ill-posed problems,, BIT, 36 (1996), 287.  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.   Google Scholar

[15]

M. Kaasalainen and J. Torppa, Optimization methods for asteroid lightcurve inversion. I. Shape determination,, Icarus, 153 (2001), 24.  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.  doi: 10.1006/icar.2001.6674.  Google Scholar

[17]

M. Kaasalainen, Interpretation of lightcurves of precessing asteroids,, Astron. Astrophys., 376 (2001), 302.  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.  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, 2 (2007), 151.   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.  doi: 10.1038/nature05614.  Google Scholar

[21]

J. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems,", Springer, (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.  doi: 10.1126/science.1179559.  Google Scholar

[23]

D. Levin, The approximation power of moving least squares,, Math. Comp., 67 (1998), 1517.  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.  doi: 10.1016/j.icarus.2006.06.001.  Google Scholar

[25]

P. Pravec, A. Harris and T. Michalowski, Asteroid rotations,, in, (2002), 113.   Google Scholar

[26]

W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, "Numerical Recipes" (third edition),, Cambridge U. Press, (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.  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.  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.  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.  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.  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.  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.  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.  doi: 10.1016/j.icarus.2009.04.032.  Google Scholar

[9]

A. Dobrovolskis, Inertia of any polyhedron,, Icarus, 124 (1996), 698.  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.  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.  doi: 10.1007/s002110050078.  Google Scholar

[12]

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

[13]

M. Hanke, Limitations of the L-curve method in ill-posed problems,, BIT, 36 (1996), 287.  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.   Google Scholar

[15]

M. Kaasalainen and J. Torppa, Optimization methods for asteroid lightcurve inversion. I. Shape determination,, Icarus, 153 (2001), 24.  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.  doi: 10.1006/icar.2001.6674.  Google Scholar

[17]

M. Kaasalainen, Interpretation of lightcurves of precessing asteroids,, Astron. Astrophys., 376 (2001), 302.  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.  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, 2 (2007), 151.   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.  doi: 10.1038/nature05614.  Google Scholar

[21]

J. Kaipio and E. Somersalo, "Statistical and Computational Inverse Problems,", Springer, (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.  doi: 10.1126/science.1179559.  Google Scholar

[23]

D. Levin, The approximation power of moving least squares,, Math. Comp., 67 (1998), 1517.  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.  doi: 10.1016/j.icarus.2006.06.001.  Google Scholar

[25]

P. Pravec, A. Harris and T. Michalowski, Asteroid rotations,, in, (2002), 113.   Google Scholar

[26]

W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, "Numerical Recipes" (third edition),, Cambridge U. Press, (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.  doi: 10.1007/s11263-006-8323-9.  Google Scholar

[1]

Peter Giesl, Sigurdur Hafstein. Computational methods for Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : i-ii. doi: 10.3934/dcdsb.2015.20.8i
[2]

Ian H. Dinwoodie. Computational methods for asynchronous basins. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3391-3405. doi: 10.3934/dcdsb.2016103
[3]

Peter Giesl, Sigurdur Hafstein. Review on computational methods for Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2291-2331. doi: 10.3934/dcdsb.2015.20.2291
[4]

Victor Isakov, Shingyu Leung, Jianliang Qian. A three-dimensional inverse gravimetry problem for ice with snow caps. Inverse Problems & Imaging, 2013, 7 (2) : 523-544. doi: 10.3934/ipi.2013.7.523
[5]

Mohsen Tadi. A computational method for an inverse problem in a parabolic system. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 205-218. doi: 10.3934/dcdsb.2009.12.205
[6]

John T. Betts, Stephen L. Campbell, Claire Digirolamo. Initial guess sensitivity in computational optimal control problems. Numerical Algebra, Control & Optimization, 2020, 10 (1) : 39-41. doi: 10.3934/naco.2019031
[7]

Marek Macák, Róbert Čunderlík, Karol Mikula, Zuzana Minarechová. Computational optimization in solving the geodetic boundary value problems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020381
[8]

Marcin Mazur, Jacek Tabor. Computational hyperbolicity. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1175-1189. doi: 10.3934/dcds.2011.29.1175
[9]

Yang Yu. Introduction: Special issue on computational intelligence methods for big data and information analytics. Big Data & Information Analytics, 2017, 2 (1) : i-ii. doi: 10.3934/bdia.201701i
[10]

Shalela Mohd--Mahali, Song Wang, Xia Lou, Sungging Pintowantoro. Numerical methods for estimating effective diffusion coefficients of three-dimensional drug delivery systems. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 377-393. doi: 10.3934/naco.2012.2.377
[11]

Masaru Ikehata, Mishio Kawashita. An inverse problem for a three-dimensional heat equation in thermal imaging and the enclosure method. Inverse Problems & Imaging, 2014, 8 (4) : 1073-1116. doi: 10.3934/ipi.2014.8.1073
[12]

Wangtao Lu, Shingyu Leung, Jianliang Qian. An improved fast local level set method for three-dimensional inverse gravimetry. Inverse Problems & Imaging, 2015, 9 (2) : 479-509. doi: 10.3934/ipi.2015.9.479
[13]

Michael Herty, Giuseppe Visconti. Kinetic methods for inverse problems. Kinetic & Related Models, 2019, 12 (5) : 1109-1130. doi: 10.3934/krm.2019042
[14]

Mário Bessa, Jorge Rocha. Three-dimensional conservative star flows are Anosov. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 839-846. doi: 10.3934/dcds.2010.26.839
[15]

Bernd Krauskopf, Carlo R. Laing. Novel computational approaches and their applications. Journal of Computational Dynamics, 2020, 7 (2) : ⅰ-ⅰ. doi: 10.3934/jcd.2020007
[16]

Shu Liao, Jin Wang, Jianjun Paul Tian. A computational study of avian influenza. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1499-1509. doi: 10.3934/dcdss.2011.4.1499
[17]

Nur Aidya Hanum Aizam, Louis Caccetta. Computational models for timetabling problem. Numerical Algebra, Control & Optimization, 2014, 4 (3) : 269-285. doi: 10.3934/naco.2014.4.269
[18]

Fiammetta Battaglia and Elisa Prato. Nonrational, nonsimple convex polytopes in symplectic geometry. Electronic Research Announcements, 2002, 8: 29-34.

[19]

Sahani Pathiraja, Sebastian Reich. Discrete gradients for computational Bayesian inference. Journal of Computational Dynamics, 2019, 6 (2) : 385-400. doi: 10.3934/jcd.2019019
[20]

Xue-Li Song, Yan-Ren Hou. Attractors for the three-dimensional incompressible Navier-Stokes equations with damping. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 239-252. doi: 10.3934/dcds.2011.31.239

2019 Impact Factor: 1.373

Tools

Metrics

  • PDF downloads (43)
  • HTML views (0)
  • Cited by (30)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences