메뉴 건너뛰기




Volumn 3, Issue 1, 2009, Pages 87-122

Discretization-invariant bayesian inversion and besov space priors

Author keywords

Bayesian inversion; Besov space; Discretiza tion invariance; Inverse problem; Reconstruction; Statistical inversion; Wavelet

Indexed keywords


EID: 70450218926     PISSN: 19308337     EISSN: 19308345     Source Type: Journal    
DOI: 10.3934/ipi.2009.3.87     Document Type: Article
Times cited : (176)

References (67)
  • 2
    • 0009727215 scopus 로고    scopus 로고
    • Bayesian approach to wavelet decomposition and shrink-age
    • Lecture Notes in Statist., Springer, New York
    • F. Abramovich and T. B. Sapatinas, Bayesian approach to wavelet decomposition and shrink-age, in “Bayesian Inference in Wavelet-Based Models,” 33–50, Lecture Notes in Statist., 141, Springer, New York, 1999.
    • (1999) Bayesian Inference in Wavelet-Based Models , vol.141 , pp. 33-50
    • Abramovich, F.1    Sapatinas, T. B.2
  • 3
    • 0344172748 scopus 로고    scopus 로고
    • Discretization independent retrieval of atmo-spheric ozone profile
    • B. D’Ambrogi, S. Mäenpää and M. Markkanen, Discretization independent retrieval of atmo-spheric ozone profile, Geophysica, 35 (1999), 87–99.
    • (1999) Geophysica , vol.35 , pp. 87-99
    • D’Ambrogi, B.1    Mäenpää, S.2    Markkanen, M.3
  • 5
    • 0034832060 scopus 로고    scopus 로고
    • A Bayesian approach to crack detection in electrically conducting media
    • K. E. Andersen, S. P. Brooks and M. B. Hansen, A Bayesian approach to crack detection in electrically conducting media, Inverse Problems, 17 (2001), 121–136.
    • (2001) Inverse Problems , vol.17 , pp. 121-136
    • Andersen, K. E.1    Brooks, S. P.2    Hansen, M. B.3
  • 6
    • 57649220862 scopus 로고    scopus 로고
    • 3D Tomographic reconstruction using geometrical models
    • K. M. Hanson, ed., Proc. SPIE
    • X. L. Battle, G. S. Cunningham and K. M. Hanson, 3D Tomographic reconstruction using geometrical models, in “Medical Imaging: Image Processing,” K. M. Hanson, ed., Proc. SPIE, 3034 (1997), 346–357.
    • (1997) Medical Imaging: Image Processing , vol.3034 , pp. 346-357
    • Battle, X. L.1    Cunningham, G. S.2    Hanson, K. M.3
  • 7
    • 28344449679 scopus 로고
    • Bayesian image restoration, with two applications in spatial statistics
    • J. Besag, J. York and A. Mollié, Bayesian image restoration, with two applications in spatial statistics, Ann. Inst. Statist. Math., 43 (1991), 1–59.
    • (1991) Ann. Inst. Statist. Math , vol.43 , pp. 1-59
    • Besag, J.1    York, J.2    Mollié, A.3
  • 8
    • 33645136947 scopus 로고    scopus 로고
    • Bayesian wavelet-based image deconvolution: A GEM algorithm exploiting a class of Heavy-Tailed priors
    • J. M. Bioucas-Dias, Bayesian wavelet-based image deconvolution: A GEM algorithm exploiting a class of Heavy-Tailed priors, IEEE Tr Image Processing, 15 (2006), 937–951.
    • (2006) IEEE Tr Image Processing , vol.15 , pp. 937-951
    • Bioucas-Dias, J. M.1
  • 9
    • 10844236506 scopus 로고    scopus 로고
    • Consistency and rates of convergence of nonlinear Tikhonov regularization with random noise
    • N. Bissantz, T. Hohage and A. Munk, Consistency and rates of convergence of nonlinear Tikhonov regularization with random noise, Inverse Problems, 20 (2004), 1773-1789.
    • (2004) Inverse Problems , vol.20 , pp. 1773-1789
    • Bissantz, N.1    Hohage, T.2    Munk, A.3
  • 10
    • 0004124953 scopus 로고    scopus 로고
    • Gaussian Measures
    • American Mathematical Society, Providence, RI
    • V. I. Bogachev, “Gaussian Measures,” Mathematical Surveys and Monographs, 62, American Mathematical Society, Providence, RI, 1998.
    • (1998) Mathematical Surveys and Monographs , vol.62
    • Bogachev, V. I.1
  • 11
    • 25444526925 scopus 로고    scopus 로고
    • On the continuum limit of a discrete inverse spectral problem on optimal finite difference grids
    • L. Borcea, V. Druskin and L. Knizhnerman, On the continuum limit of a discrete inverse spectral problem on optimal finite difference grids, Communications on Pure and Applied Mathematics, 58 (2005), 1231–1279.
    • (2005) Communications on Pure and Applied Mathematics , vol.58 , pp. 1231-1279
    • Borcea, L.1    Druskin, V.2    Knizhnerman, L.3
  • 12
    • 33947368708 scopus 로고    scopus 로고
    • A Gaussian hypermodel to recover blocky objects
    • D. Calvetti and E. Somersalo, A Gaussian hypermodel to recover blocky objects, Inverse Problems, 23 (2007), 733–754.
    • (2007) Inverse Problems , vol.23 , pp. 733-754
    • Calvetti, D.1    Somersalo, E.2
  • 14
    • 0033346931 scopus 로고    scopus 로고
    • Multiple basis wavelet denoising using Besov projections
    • Kobe, Japan
    • H. Choi and R. Baraniuk, Multiple basis wavelet denoising using Besov projections, Proceedings of ICIP 99, Kobe, Japan, (1999), 595–599.
    • (1999) Proceedings of ICIP , vol.99 , pp. 595-599
    • Choi, H.1    Baraniuk, R.2
  • 16
    • 0003077036 scopus 로고    scopus 로고
    • Ten Lectures on Wavelets
    • I. Daubechies, “Ten Lectures on Wavelets,” SIAM 2006.
    • (2006) SIAM
    • Daubechies, I.1
  • 17
    • 7044231546 scopus 로고    scopus 로고
    • An iterative thresholding algorithm for linear inverse problems with a sparsity constraint
    • I. Daubechies, M. Defrise and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math., 57 (2004), 1413–1457.
    • (2004) Comm. Pure Appl. Math , vol.57 , pp. 1413-1457
    • Daubechies, I.1    Defrise, M.2    De Mol, C.3
  • 18
    • 0003087399 scopus 로고
    • Vector Measures
    • American Mathematical Society
    • J. Diestel and J. J. Uhl, Jr., “Vector Measures,” Mathematical Surveys, 15, American Mathematical Society, 1977.
    • (1977) Mathematical Surveys , vol.15
    • Diestel, J.1    Uhl, J. J.2
  • 20
    • 0030212858 scopus 로고    scopus 로고
    • Recovery of blocky images from noisy and blurred data
    • D. C. Dobson and F. Santosa, Recovery of blocky images from noisy and blurred data, SIAM J. Appl. Math., 56 (1996), 1181–1198.
    • (1996) SIAM J. Appl. Math , vol.56 , pp. 1181-1198
    • Dobson, D. C.1    Santosa, F.2
  • 23
    • 0024580739 scopus 로고
    • Recording and interpretation of cerebral magnetic fields
    • R. Hari and O. Lounasmaa, Recording and interpretation of cerebral magnetic fields, Science, 244 (1989), 432–436.
    • (1989) Science , vol.244 , pp. 432-436
    • Hari, R.1    Lounasmaa, O.2
  • 24
    • 33947240401 scopus 로고    scopus 로고
    • Preconditioning Markov chain Monte Carlo simulations using coarse-scale models
    • Y. Efendiev, T. Hou and W. Luo, Preconditioning Markov chain Monte Carlo simulations using coarse-scale models, SIAM J. Sci. Comput. 28 (2006), 776-803
    • (2006) SIAM J. Sci. Comput , vol.28 , pp. 776-803
    • Efendiev, Y.1    Hou, T.2    Luo, W.3
  • 25
    • 13744252042 scopus 로고    scopus 로고
    • Convergence rates in the Prokhorov metric for assessing uncertainty in ill-posed problems
    • H. Engl, A. Hofinger and S. Kindermann, Convergence rates in the Prokhorov metric for assessing uncertainty in ill-posed problems, Inverse Problems, 21 (2005), 399–412.
    • (2005) Inverse Problems , vol.21 , pp. 399-412
    • Engl, H.1    Hofinger, A.2    Kindermann, S.3
  • 26
    • 13744250840 scopus 로고
    • Generalized inverses of random linear operators in Banach spaces
    • H. W. Engl and M. Z. Nashed, Generalized inverses of random linear operators in Banach spaces, J. Math. Anal. Appl., 83 (1981), 582–610.
    • (1981) J. Math. Anal. Appl , vol.83 , pp. 582-610
    • Engl, H. W.1    Nashed, M. Z.2
  • 27
    • 0036683968 scopus 로고    scopus 로고
    • Inverse problems as statistics
    • S. N. Evans and P. B. Stark, Inverse problems as statistics, Inverse Problems, 18 (2002), R55–R97.
    • (2002) Inverse Problems , vol.18 , pp. R55-R97
    • Evans, S. N.1    Stark, P. B.2
  • 28
    • 0040009224 scopus 로고
    • Bayesian analysis in inverse problems
    • B. G. Fitzpatrick, Bayesian analysis in inverse problems, Inverse Problems, 7 (1991), 675– 702.
    • (1991) Inverse Problems , vol.7 , pp. 675-702
    • Fitzpatrick, B. G.1
  • 31
    • 0034292512 scopus 로고    scopus 로고
    • Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography
    • J. P. Kaipio, V. Kolehmainen, E. Somersalo and M. Vauhkonen, Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography, Inverse Problems, 16 (2000), 1487–1522.
    • (2000) Inverse Problems , vol.16 , pp. 1487-1522
    • Kaipio, J. P.1    Kolehmainen, V.2    Somersalo, E.3    Vauhkonen, M.4
  • 33
    • 84860523889 scopus 로고    scopus 로고
    • Statistical and Computational Inverse Problems
    • Springer-Verlag, New York
    • J. P. Kaipio and E. Somersalo, “Statistical and Computational Inverse Problems,” Applied Mathematical Sciences 160, Springer-Verlag, New York, 2005.
    • (2005) Applied Mathematical Sciences , vol.160
    • Kaipio, J. P.1    Somersalo, E.2
  • 37
    • 3042595166 scopus 로고    scopus 로고
    • Discretizations of Generalized Random Variables with Applications to Inverse Problems
    • Dissertation, University of Oulu
    • S. Lasanen, “Discretizations of Generalized Random Variables with Applications to Inverse Problems,” Dissertation, University of Oulu. Ann. Acad. Sci. Fenn. Math. Diss. No. 130, 2002.
    • (2002) Ann. Acad. Sci. Fenn. Math. Diss , Issue.130
    • Lasanen, S.1
  • 39
    • 6344231795 scopus 로고    scopus 로고
    • Can one use total variation prior for edge preserving Bayesian inversion?
    • M. Lassas and S. Siltanen, Can one use total variation prior for edge preserving Bayesian inversion? Inverse Problems, 20 (2004), 1537–1564.
    • (2004) Inverse Problems , vol.20 , pp. 1537-1564
    • Lassas, M.1    Siltanen, S.2
  • 40
    • 0006684185 scopus 로고
    • Linear inverse problems for generalised random variables
    • M. Lehtinen, L. Päärinta and E. Somersalo, Linear inverse problems for generalised random variables, Inverse Problems, 5 (1989), 599–612.
    • (1989) Inverse Problems , vol.5 , pp. 599-612
    • Lehtinen, M.1    Päärinta, L.2    Somersalo, E.3
  • 41
    • 0034831995 scopus 로고    scopus 로고
    • Bayesian wavelet denoising: Besov priors and non-Gaussian noises
    • D. Leporini and J.-C. Pesquet, Bayesian wavelet denoising: Besov priors and non-Gaussian noises, Signal Processing, 81 (2001), 55–67.
    • (2001) Signal Processing , vol.81 , pp. 55-67
    • Leporini, D.1    Pesquet, J.-C.2
  • 43
    • 34047140155 scopus 로고    scopus 로고
    • Regularization in cardiac source imaging
    • I. Magnin et al. (Eds): Springer-Verlag Berlin Heidel-berg, 2003
    • T. Lunttila, J. Nenonen and E. Somersalo, Regularization in cardiac source imaging, in I. Magnin et al. (Eds.): FIMH 2003, LNCS 2674, 101-111, 2003, Springer-Verlag Berlin Heidel-berg, 2003.
    • (2003) FIMH 2003, LNCS , vol.2674 , pp. 101-111
    • Lunttila, T.1    Nenonen, J.2    Somersalo, E.3
  • 48
    • 0036608343 scopus 로고    scopus 로고
    • Monte Carlo analysis of inverse problems
    • K. Mosegaard and M. Sambridge, Monte Carlo analysis of inverse problems, Inverse Prob-lems, 18 (2002), R29–R54.
    • (2002) Inverse Prob-lems , vol.18 , pp. R29-R54
    • Mosegaard, K.1    Sambridge, M.2
  • 49
    • 0000446730 scopus 로고    scopus 로고
    • Bayesian Inference in Wavelet-Based Models
    • (Eds), Springer, New York
    • P. Mueller and B. Vidakovic (Eds.), “Bayesian Inference in Wavelet-Based Models,” Lecture Notes in Statistics, Springer, New York, 1999.
    • (1999) Lecture Notes in Statistics
    • Mueller, P.1    Vidakovic, B.2
  • 50
    • 0012906846 scopus 로고    scopus 로고
    • Bayesian image analysis with Markov chain Monte Carlo and coloured continuum triangulation models
    • G. Nicholls, Bayesian image analysis with Markov chain Monte Carlo and coloured continuum triangulation models, J. R. Statist. Soc. B, 60 (1998), 643–659.
    • (1998) J. R. Statist. Soc. B , vol.60 , pp. 643-659
    • Nicholls, G.1
  • 51
    • 35848962958 scopus 로고    scopus 로고
    • Bayesian multiresolution method for local tomography in dental x-ray imaging
    • K. Niinimäki, S. Siltanen and V. Kolehmainen, Bayesian multiresolution method for local tomography in dental x-ray imaging, Phys. Med. Biol., 52 (2007), 6663-6678.
    • (2007) Phys. Med. Biol , vol.52 , pp. 6663-6678
    • Niinimäki, K.1    Siltanen, S.2    Kolehmainen, V.3
  • 53
    • 30344445115 scopus 로고    scopus 로고
    • Statistical measurements, experiments and applications
    • P. Piiroinen, Statistical measurements, experiments and applications, Ann. Acad. Sci. Fenn. Math. Diss., (2005).
    • (2005) Ann. Acad. Sci. Fenn. Math. Diss
    • Piiroinen, P.1
  • 54
    • 31844445344 scopus 로고    scopus 로고
    • Stable estimation approach to nonstationary inverse problems: discretiza-tion error and filtering problem
    • H. K. Pikkarainen, Stable estimation approach to nonstationary inverse problems: discretiza-tion error and filtering problem, Inverse Problems, 22 (2006), 365–379.
    • (2006) Inverse Problems , vol.22 , pp. 365-379
    • Pikkarainen, H. K.1
  • 57
    • 44049111982 scopus 로고
    • Nonlinear total variation based noise removal algorithms
    • L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259–268.
    • (1992) Physica D , vol.60 , pp. 259-268
    • Rudin, L. I.1    Osher, S.2    Fatemi, E.3
  • 58
    • 34547233310 scopus 로고    scopus 로고
    • Gaussian free fields for mathematicians
    • S. Sheffield, Gaussian free fields for mathematicians, Probab. Theory Related Fields, 139 (2007), 521–541.
    • (2007) Probab. Theory Related Fields , vol.139 , pp. 521-541
    • Sheffield, S.1
  • 62
    • 0003929280 scopus 로고    scopus 로고
    • MCMC methods for inverse problems
    • Finnish Meteorological Institute
    • J. Tamminen, MCMC methods for inverse problems, Geophysical publications, 48 (1999), Finnish Meteorological Institute.
    • (1999) Geophysical publications , vol.48
    • Tamminen, J.1
  • 63
    • 13144272994 scopus 로고    scopus 로고
    • Inverse Problem Theory and Methods for Model Parameter Estimation
    • A. Tarantola, “Inverse Problem Theory and Methods for Model Parameter Estimation,” SIAM 2005.
    • (2005) SIAM
    • Tarantola, A.1
  • 64
    • 0032096686 scopus 로고    scopus 로고
    • Fast, robust total variation-based reconstruction of noisy, blurred images
    • C. R. Vogel and M. Oman, Fast, robust total variation-based reconstruction of noisy, blurred images, IEEE Trans. Image Process, 7 (1998), 813–824.
    • (1998) IEEE Trans. Image Process , vol.7 , pp. 813-824
    • Vogel, C. R.1    Oman, M.2
  • 66
    • 0021467235 scopus 로고
    • Stochastic inversion of linear first kind equations II. Discrete theory and convergence results
    • C. R. Vogel, Stochastic inversion of linear first kind equations II. Discrete theory and convergence results, Journal of Integral Equations, 7 (1984), 73–92.
    • (1984) Journal of Integral Equations , vol.7 , pp. 73-92
    • Vogel, C. R.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.