Robust and rate-optimal Gibbs posterior inference on the boundary of a noisy image. (English) Zbl 1454.62209
The authors develop a robust Gibbsian approach that constructs a posterior distribution for the image boundary directly, without modeling the pixel intensities. With this approach, adaptive to the boundary smoothness, the Gibbs posterior concentrates asymptotically at the minimax optimal rate. Monte Carlo computation of the Gibbs posterior is straightforward, and simulation results show that the corresponding inference is more accurate than that based on existing Bayesian
methodology.
Reviewer: Pavel Stoynov (Sofia)
MSC:
| 62H35 | Image analysis in multivariate analysis |
| 62G20 | Asymptotic properties of nonparametric inference |
| 62G05 | Nonparametric estimation |
Keywords:
adaptation; boundary detection; likelihood-free inference; model misspecification; posterior concentration rateSoftware:
BayesBDReferences:
| [1] | Anam, S., Uchino, E. and Suetake, N. (2013). Image boundary detection using the modified level set method and a diffusion filter. Proc. Comput. Sci. 22 192-200. |
| [2] | Bissiri, P. G., Holmes, C. C. and Walker, S. G. (2016). A general framework for updating belief distributions. J. R. Stat. Soc. Ser. B. Stat. Methodol. 78 1103-1130. · Zbl 1414.62039 · doi:10.1111/rssb.12158 |
| [3] | Catoni, O. (2007). PAC—Bayesian Supervised Classification: The Thermodynamics of Statistical Learning. Institute of Mathematical Statistics Lecture Notes—Monograph Series 56. IMS, Beachwood, OH. · Zbl 1277.62015 |
| [4] | Denison, D. G. T., Mallick, B. K. and Smith, A. F. M. (1998). Automatic Bayesian curve fitting. J. R. Stat. Soc. Ser. B. Stat. Methodol. 60 333-350. · Zbl 0907.62031 · doi:10.1111/1467-9868.00128 |
| [5] | DiMatteo, I., Genovese, C. R. and Kass, R. E. (2001). Bayesian curve-fitting with free-knot splines. Biometrika 88 1055-1071. · Zbl 0986.62026 · doi:10.1093/biomet/88.4.1055 |
| [6] | 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. |
| [7] | Green, P. J. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82 711-732. · Zbl 0861.62023 · doi:10.1093/biomet/82.4.711 |
| [8] | 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. · Zbl 1368.62278 · doi:10.1080/01621459.2014.934825 |
| [9] | Jiang, W. and Tanner, M. A. (2008). Gibbs posterior for variable selection in high-dimensional classification and data mining. Ann. Statist. 36 2207-2231. · Zbl 1274.62227 · doi:10.1214/07-AOS547 |
| [10] | Li, M. and Ghosal, S. (2017). Bayesian detection of image boundaries. Ann. Statist. 45 2190-2217. · Zbl 1486.62194 · doi:10.1214/16-AOS1523 |
| [11] | Li, C., Xu, C., Gui, C. and Fox, M. D. (2010). Distance regularized level set evolution and its application to image segmentation. IEEE Trans. Image Process. 19 3243-3254. · Zbl 1371.94226 · doi:10.1109/TIP.2010.2041414 |
| [12] | Liang, S., Banerjee, S. and Carlin, B. P. (2009). Bayesian wombling for spatial point processes. Biometrics 65 1243-1253. · Zbl 1180.62175 · doi:10.1111/j.1541-0420.2009.01203.x |
| [13] | Lu, H. and Carlin, B. P. (2004). Bayesian areal wombling for geographical boundary analysis. Geogr. Anal. 37 265-285. |
| [14] | Ma, H. and Carlin, B. P. (2007). Bayesian multivariate areal wombling for multiple disease boundary analysis. Bayesian Anal. 2 281-302. · Zbl 1331.62427 · doi:10.1214/07-BA211 |
| [15] | Maini, R. and Aggarwal, H. (2009). Study and comparison of various image edge detection techniques. IJIP 3 1-11. |
| [16] | Mammen, E. and Tsybakov, A. B. (1995). Asymptotical minimax recovery of sets with smooth boundaries. Ann. Statist. 23 502-524. · Zbl 0834.62038 · doi:10.1214/aos/1176324533 |
| [17] | Martin, D. R., Fowlkes, C. C. and Malik, J. (2004). Learning to detect natural image boundaries using local brightness, color, and texture cues. IEEE Trans. Pattern Anal. Mach. Intell. 26 530-549. |
| [18] | Schumaker, L. L. (1981). Spline Functions: Basic Theory. Wiley, New York. · Zbl 0449.41004 |
| [19] | Shen, W. and Ghosal, S. (2015). Adaptive Bayesian procedures using random series priors. Scand. J. Stat. 42 1194-1213. · Zbl 1419.62076 · doi:10.1111/sjos.12159 |
| [20] | Shen, X. and Wasserman, L. (2001). Rates of convergence of posterior distributions. Ann. Statist. 29 687-714. · Zbl 1041.62022 · doi:10.1214/aos/1009210686 |
| [21] | Syring, N. and Li, M. (2017a). BayesBD: Bayesian inference for image boundaries. R package version 1.2. |
| [22] | Syring, N. and Li, M. (2017b). BayesBD: An R package for Bayesian inference on image boundaries. R J. 9 149-162. |
| [23] | Syring, N. and Martin, R. (2017). Gibbs posterior inference on the minimum clinically important difference. J. Statist. Plann. Inference 187 67-77. · Zbl 1391.62251 · doi:10.1016/j.jspi.2017.03.001 |
| [24] | Syring, N. and Martin, R. (2019). Calibrating general posterior credible regions. Biometrika 106 479-486. · Zbl 1454.62105 · doi:10.1093/biomet/asy054 |
| [25] | Yuan, W., Chin, K. S., Hua, M., Dong, G. and Wang, C. (2016). Shape classification of wear particles by image boundary analysis using machine learning algorithms. Mech. Syst. Signal Pr. 72-73 346-358. |
| [26] | Zhang, T. (2006). Information-theoretic upper and lower bounds for statistical estimation. IEEE Trans. Inform. Theory 52 1307-1321. · Zbl 1320.94033 · doi:10.1109/TIT.2005.864439 |
| [27] | Ziou, D. |
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