The Annals of Statistics

Bayesian detection of image boundaries

Meng Li and Subhashis Ghosal

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Abstract

Detecting boundary of an image based on noisy observations is a fundamental problem of image processing and image segmentation. For a $d$-dimensional image ($d=2,3,\ldots$), the boundary can often be described by a closed smooth $(d-1)$-dimensional manifold. In this paper, we propose a nonparametric Bayesian approach based on priors indexed by $\mathbb{S}^{d-1}$, the unit sphere in $\mathbb{R}^{d}$. We derive optimal posterior contraction rates for Gaussian processes or finite random series priors using basis functions such as trigonometric polynomials for 2-dimensional images and spherical harmonics for 3-dimensional images. For 2-dimensional images, we show a rescaled squared exponential Gaussian process on $\mathbb{S}^{1}$ achieves four goals of guaranteed geometric restriction, (nearly) minimax optimal rate adapting to the smoothness level, convenience for joint inference and computational efficiency. We conduct an extensive study of its reproducing kernel Hilbert space, which may be of interest by its own and can also be used in other contexts. Several new estimates on modified Bessel functions of the first kind are given. Simulations confirm excellent performance and robustness of the proposed method.

Article information

Source
Ann. Statist., Volume 45, Number 5 (2017), 2190-2217.

Dates
Received: September 2015
Revised: May 2016
First available in Project Euclid: 31 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.aos/1509436832

Digital Object Identifier
doi:10.1214/16-AOS1523

Mathematical Reviews number (MathSciNet)
MR3718166

Zentralblatt MATH identifier
06821123

Subjects
Primary: 62G20: Asymptotic properties 62H35: Image analysis
Secondary: 62F15: Bayesian inference 60G15: Gaussian processes

Keywords
Boundary detection Gaussian process on sphere image posterior contraction rate random series squared exponential periodic kernel Bayesian adaptation

Citation

Li, Meng; Ghosal, Subhashis. Bayesian detection of image boundaries. Ann. Statist. 45 (2017), no. 5, 2190--2217. doi:10.1214/16-AOS1523. https://projecteuclid.org/euclid.aos/1509436832


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References

  • [1] Arbel, J., Gayraud, G. and Rousseau, J. (2013). Bayesian optimal adaptive estimation using a sieve prior. Scand. J. Stat. 40 549–570.
  • [2] Banerjee, S. and Gelfand, A. E. (2006). Bayesian wombling: Curvilinear gradient assessment under spatial process models. J. Amer. Statist. Assoc. 101 1487–1501.
  • [3] Basu, M. (2002). Gaussian-based edge-detection methods-a survey. IEEE Trans. Syst. Man Cybern., Part C Appl. Rev. 32 252–260.
  • [4] Bhardwaj, S. and Mittal, A. (2012). A survey on various edge detector techniques. Proc. Technol. 4 220–226.
  • [5] Carlstein, E. and Krishnamoorthy, C. (1992). Boundary estimation. J. Amer. Statist. Assoc. 87 430–438.
  • [6] Castillo, I. (2012). A semiparametric Bernstein–von Mises theorem for Gaussian process priors. Probab. Theory Related Fields 152 53–99.
  • [7] Castillo, I., Kerkyacharian, G. and Picard, D. (2014). Thomas Bayes’ walk on manifolds. Probab. Theory Related Fields 158 665–710.
  • [8] Chen, J. and Gupta, A. K. (2012). Parametric Statistical Change Point Analysis: With Applications to Genetics, Medicine, and Finance, 2nd ed. Birkhäuser/Springer, New York.
  • [9] Dai, F. and Xu, Y. (2013). Approximation Theory and Harmonic Analysis on Spheres and Balls. Springer, New York.
  • [10] Donoho, D. L. (1999). Wedgelets: Nearly minimax estimation of edges. Ann. Statist. 27 859–897.
  • [11] Dudley, R. M. (1974). Metric entropy of some classes of sets with differentiable boundaries. J. Approx. Theory 10 227–236.
  • [12] Ennis, D. (2005). Spherical Harmonics. http://www.mathworks.com/matlabcentral/fileexchange/8638-spherical-harmonics. MATLAB Central File Exchange.
  • [13] Fitzpatrick, M. C., Preisser, E. L., Porter, A., Elkinton, J., Waller, L. A., Carlin, B. P. and Ellison, A. M. (2010). Ecological boundary detection using Bayesian areal wombling. Ecology 91 3448–3455.
  • [14] Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. 6 721–741.
  • [15] Ghosal, S. and van der Vaart, A. (2007). Convergence rates of posterior distributions for non-i.i.d. observations. Ann. Statist. 35 192–223.
  • [16] Gu, K., Pati, D. and Dunson, D. B. (2014). Bayesian multiscale modeling of closed curves in point clouds. J. Amer. Statist. Assoc. 109 1481–1494.
  • [17] Hall, P., Peng, L. and Rau, C. (2001). Local likelihood tracking of fault lines and boundaries. J. R. Stat. Soc. Ser. B. Stat. Methodol. 63 569–582.
  • [18] Hastie, T. and Stuetzle, W. (1989). Principal curves. J. Amer. Statist. Assoc. 84 502–516.
  • [19] Jackson, D. (1994). The Theory of Approximation. American Mathematical Society Colloquium Publications 11. Amer. Math. Soc., Providence, RI. Reprint of the 1930 original.
  • [20] Killick, R. and Eckley, I. A. (2011). Changepoint: An R package for changepoint analysis. R Package Version 0.6, http://CRAN.R-project.Org/package=changepoint.
  • [21] Killick, R., Fearnhead, P. and Eckley, I. A. (2012). Optimal detection of changepoints with a linear computational cost. J. Amer. Statist. Assoc. 107 1590–1598.
  • [22] Korostelëv, A. P. and Tsybakov, A. B. (1993). Minimax Theory of Image Reconstruction. Lecture Notes in Statistics 82. Springer, New York.
  • [23] Kuelbs, J. and Li, W. V. (1993). Metric entropy and the small ball problem for Gaussian measures. J. Funct. Anal. 116 133–157.
  • [24] Li, M. and Ghosal, S. (2017). Supplement to “Bayesian detection of image boundaries.” DOI:10.1214/16-AOS1523SUPP.
  • [25] Li, W. V. and Linde, W. (1999). Approximation, metric entropy and small ball estimates for Gaussian measures. Ann. Probab. 27 1556–1578.
  • [26] Lu, H. and Carlin, B. P. (2005). Bayesian areal wombling for geographical boundary analysis. Geogr. Anal. 37 265–285.
  • [27] MacKay, D. J. (1998). Introduction to Gaussian processes. In NATO ASI Series F Computer and Systems Sciences 168 133–166.
  • [28] Mammen, E. and Tsybakov, A. B. (1995). Asymptotical minimax recovery of sets with smooth boundaries. Ann. Statist. 23 502–524.
  • [29] Müller, H.-G. and Song, K. S. (1994). Maximin estimation of multidimensional boundaries. J. Multivariate Anal. 50 265–281.
  • [30] Neal, R. M. (2003). Slice sampling. Ann. Statist. 31 705–767. With discussions and a rejoinder by the author.
  • [31] Polzehl, J. and Spokoiny, V. (2003). Image denoising: Pointwise adaptive approach. Ann. Statist. 31 30–57.
  • [32] Qiu, P. (2005). Image Processing and Jump Regression Analysis. Wiley-Interscience [John Wiley & Sons], Hoboken, NJ.
  • [33] Qiu, P. (2007). Jump surface estimation, edge detection, and image restoration. J. Amer. Statist. Assoc. 102 745–756.
  • [34] Qiu, P. and Sun, J. (2007). Local smoothing image segmentation for spotted microarray images. J. Amer. Statist. Assoc. 102 1129–1144.
  • [35] Qiu, P. and Sun, J. (2009). Using conventional edge detectors and postsmoothing for segmentation of spotted microarray images. J. Comput. Graph. Statist. 18 147–164.
  • [36] Qiu, P. and Yandell, B. (1997). Jump detection in regression surfaces. J. Comput. Graph. Statist. 6 332–354.
  • [37] Rasmussen, C. E. and Williams, C. K. I. (2006). Gaussian Processes for Machine Learning. MIT Press, Cambridge, MA.
  • [38] Rudemo, M. and Stryhn, H. (1994). Approximating the distribution of maximum likelihood contour estimators in two-region images. Scand. J. Stat. 21 41–55.
  • [39] Shen, W. and Ghosal, S. (2015). Adaptive Bayesian procedures using random series priors. Scand. J. Stat. 42 1194–1213.
  • [40] Terras, A. (2013). Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane, 2nd ed. Springer, New York.
  • [41] van der Vaart, A. and van Zanten, H. (2007). Bayesian inference with rescaled Gaussian process priors. Electron. J. Stat. 1 433–448.
  • [42] van der Vaart, A. W. and van Zanten, J. H. (2008). Rates of contraction of posterior distributions based on Gaussian process priors. Ann. Statist. 36 1435–1463.
  • [43] van der Vaart, A. W. and van Zanten, J. H. (2009). Adaptive Bayesian estimation using a Gaussian random field with inverse gamma bandwidth. Ann. Statist. 37 2655–2675.
  • [44] Waller, L. A. and Gotway, C. A. (2004). Applied Spatial Statistics for Public Health Data. Wiley-Interscience [John Wiley & Sons], Hoboken, NJ.
  • [45] Watson, G. N. (1995). A Treatise on the Theory of Bessel Functions. Cambridge Univ. Press, Cambridge. Reprint of the second (1944) edition.

Supplemental materials

  • Supplement to “Bayesian detection of image boundaries”. The supplementary file contains proofs to all lemmas and propositions in the paper.