## The Annals of Statistics

### Outlier robust corner-preserving methods for reconstructing noisy images

#### Abstract

The ability to remove a large amount of noise and the ability to preserve most structure are desirable properties of an image smoother. Unfortunately, they usually seem to be at odds with each other; one can only improve one property at the cost of the other. By combining M-smoothing and least-squares-trimming, the TM-smoother is introduced as a means to unify corner-preserving properties and outlier robustness. To identify edge- and corner-preserving properties, a new theory based on differential geometry is developed. Further, robustness concepts are transferred to image processing. In two examples, the TM-smoother outperforms other corner-preserving smoothers. A software package containing both the TM- and the M-smoother can be downloaded from the Internet.

#### Article information

Source
Ann. Statist., Volume 35, Number 1 (2007), 132-165.

Dates
First available in Project Euclid: 6 June 2007

https://projecteuclid.org/euclid.aos/1181100184

Digital Object Identifier
doi:10.1214/009053606000001109

Mathematical Reviews number (MathSciNet)
MR2332272

Zentralblatt MATH identifier
1114.62050

Subjects
Primary: 62G08: Nonparametric regression
Secondary: 62G20: Asymptotic properties 62G35: Robustness

#### Citation

Hillebrand, Martin; Müller, Christine H. Outlier robust corner-preserving methods for reconstructing noisy images. Ann. Statist. 35 (2007), no. 1, 132--165. doi:10.1214/009053606000001109. https://projecteuclid.org/euclid.aos/1181100184

#### References

• Bednarski, T. and Clarke, B. R. (1993). Trimmed likelihood estimation of location and scale of the normal distribution. Austral. J. Statist. 35 141--153.
• Candès, E. J. and Donoho, D. L. (2000). Ridgelets: A key to higher-dimensional intermittency. In Wavelets: The Key to Intermittent Information? (B. Silvermann and J. Vassilicos, eds.) 111--127. Oxford Univ. Press.
• Chu, C. K., Glad, I. K., Godtliebsen, F. and Marron, J. S. (1998). Edge-preserving smoothers for image processing (with discussion). J. Amer. Statist. Assoc. 93 526--556.
• Donoho, D. L. (1999). Wedgelets: Nearly minimax estimation of edges. Ann. Statist. 27 859--897.
• Donoho, D. L., Johnstone, I. M., Kerkyacharian, G., and Picard, D. (1995). Wavelet shrinkage: Asymptopia? (with discussion). J. Roy. Statist. Soc. Ser. B 57 301--369.
• Hampel, F. R. (1971). A general qualitative definition of robustness. Ann. Math. Statist. 42 1887--1896.
• Hillebrand, M. (2003). On robust corner-preserving smoothing in image processing. Ph.D. dissertation, Univ. Oldenburg, Germany. Available at docserver.bis.uni-oldenburg.de/publikationen/dissertation/2003/hilonr03/hilonr03.html.
• Hillebrand, M. and Müller, Ch. H. (2006). On consistency of redescending $M$-kernel smoothers. Metrika 63 71--90.
• Huber, P. (1981). Robust Statistics. Wiley, New York.
• Koch, I. (1996). On the asymptotic performance of median smoothers in image analysis and nonparametric regression. Ann. Statist. 24 1648--1666.
• Meer, P., Mintz, D. and Rosenfeld, A. (1990). Least median of squares based robust analysis of image structure. In Proc. DARPA Image Understanding Workshop 231--254. Morgan Kaufmann, San Francisco.
• Meer, P., Mintz, D., Rosenfeld, A. and Kim, D.Y. (1991). Robust regression methods for computer vision: A review. Internat. J. Computer Vision 6 59--70.
• Mizera, I. and Müller, Ch. H. (1999). Breakdown points and variation exponents of robust $M$-estimators in linear models. Ann. Statist. 27 1164--1177.
• Müller, Ch. H. (1997). Robust Planning and Analysis of Experiments. Lecture Notes in Statistics 124. Springer, New York.
• Müller, Ch. H. (1999). On the use of high breakdown point estimators in the image analysis. Tatra Mt. Math. Publ. 17 283--293.
• Müller, Ch. H. (2002). Comparison of high-breakdown-point estimators for image denoising. Allg. Stat. Arch. 86 307--321.
• Müller, Ch. H. (2002). Robust estimators for estimating discontinuous functions. Metrika 55 99--109.
• Polzehl, J. and Spokoiny, V. G. (2000). Adaptive weights smoothing with applications to image restoration. J. R. Stat. Soc. Ser. B Stat. Methodol. 62 335--354.
• Polzehl, J. and Spokoiny, V. (2003). Image denoising: Pointwise adaptive approach. Ann. Statist. 31 30--57.
• Riedel, M. (1984). Comparison of break points of estimators. In Robustness of Statistical Methods and Nonparametric Statistics (D. Rasch and M. L. Tiku, eds.) 113--116. Reidel, Dordrecht.
• Rousseeuw, P. J. (1984). Least median of squares regression. J. Amer. Statist. Assoc. 79 871--880.
• Rousseeuw, P. J. and Leroy, A. M. (1987). Robust Regression and Outlier Detection. Wiley, New York.
• Rousseeuw, P. J. and Van Aelst, S. (1999). Positive-breakdown robust methods in computer vision. In Computing Science and Statistics. Models, Predictions and Computing. Proc. 31st Symposium on the Interface (K. Berk and M. Pourahmadi, eds.) 451--460. Interface Foundation of North America, Fairfax Station, VA.
• Shikin, E. V. (1995). Handbook and Atlas of Curves. CRC Press, Boca Raton, FL.
• Smith, S. M. and Brady, J. M. (1995). SUSAN---A new approach to low level image processing. International J. Computer Vision 23 45--78.
• Yohai, V. J. and Maronna, R. A. (1976). Location estimators based on linear combinations of modified order statistics. Comm. Statist. Theory Methods 5 481--486.