## The Annals of Statistics

### Recovering convex boundaries from blurred and noisy observations

#### Abstract

We consider the problem of estimating convex boundaries from blurred and noisy observations. In our model, the convolution of an intensity function f is observed with additive Gaussian white noise. The function f is assumed to have convex support G whose boundary is to be recovered. Rather than directly estimating the intensity function, we develop a procedure which is based on estimating the support function of the set G. This approach is closely related to the method of geometric hyperplane probing, a well-known technique in computer vision applications. We establish bounds that reveal how the estimation accuracy depends on the ill-posedness of the convolution operator and the behavior of the intensity function near the boundary.

#### Article information

Source
Ann. Statist., Volume 34, Number 3 (2006), 1375-1394.

Dates
First available in Project Euclid: 10 July 2006

https://projecteuclid.org/euclid.aos/1152540752

Digital Object Identifier
doi:10.1214/009053606000000326

Mathematical Reviews number (MathSciNet)
MR2278361

Zentralblatt MATH identifier
1113.62116

Subjects
Primary: 62G05: Estimation 62H35: Image analysis

#### Citation

Goldenshluger, Alexander; Zeevi, Assaf. Recovering convex boundaries from blurred and noisy observations. Ann. Statist. 34 (2006), no. 3, 1375--1394. doi:10.1214/009053606000000326. https://projecteuclid.org/euclid.aos/1152540752

#### References

• Anderssen, R. S. (1980). On the use of linear functionals for Abel-type integral equations in applications. In The Application and Numerical Solution of Integral Equations (R. S. Anderssen, F. R. de Hoog and M. A. Lucas, eds.) 195--221. Nijhoff, The Hague.
• Aubin, J.-P. (1979). Applied Functional Analysis. Wiley, New York.
• Baddeley, A. J. (1992). Errors in binary images and an $L^p$ version of the Hausdorff metric. Nieuw Arch. Wisk. (4) 10 157--183.
• Bertero, M. and Boccacci, P. (1998). Introduction to Inverse Problems in Imaging. Institute of Physics, Bristol and Philadelphia.
• Brandolini, L., Rigoli, M. and Travaglini, G. (1998). Average decay of Fourier transforms and geometry of convex sets. Rev. Mat. Iberoamericana 14 519--560.
• Bruna, J., Nagel, A. and Wainger, S. (1988). Convex hypersurfaces and Fourier transforms. Ann. of Math. (2) 127 333--365.
• Candès, E. J. and Donoho, D. L. (2002). Recovering edges in ill-posed inverse problems: Optimality of curvelet frames. Ann. Statist. 30 784--842.
• Donoho, D. L. (1999). Wedgelets: Nearly minimax estimation of edges. Ann. Statist. 27 859--897.
• Donoho, D. L. and Low, M. G. (1992). Renormalization exponents and optimal pointwise rates of convergence. Ann. Statist. 20 944--970.
• Efromovich, S. (1997). Robust and efficient recovery of a signal passed through a filter and then contaminated by non-Gaussian noise. IEEE Trans. Inform. Theory 43 1184--1191.
• Ermakov, M. (1989). Minimax estimation of the solution of an ill-posed convolution type problem. Problems Inform. Transmission 25 191--200.
• Gihman, I. I. and Skorohod, A. V. (1974). The Theory of Stochastic Processes 1. Springer, Berlin.
• Golberg, M. (1979). A method of adjoints for solving some ill-posed equations of the first kind. Appl. Math. Comput. 5 123--129.
• Goldenshluger, A. (1999). On pointwise adaptive nonparametric deconvolution. Bernoulli 5 907--925.
• Goldenshluger, A. and Spokoiny, V. (2004). On the shape-from-moments problem and recovering edges from noisy Radon data. Probab. Theory Related Fields 128 123--140.
• Goldenshluger, A., Tsybakov, A. B. and Zeevi, A. (2006). Optimal change-point estimation from indirect observations. Ann. Statist. 34 350--372.
• Hall, P. (1990). Optimal convergence rates in signal recovery. Ann. Probab. 18 887--900.
• Hall, P. and Koch, I. (1990). On continuous image models and image analysis in the presence of correlated noise. Adv. in Appl. Probab. 22 332--349.
• Hall, P., Nussbaum, M. and Stern, S. E. (1997). On the estimation of a support curve of indeterminate sharpness. J. Multivariate Anal. 62 204--232.
• Hall, P. and Raimondo, M. (1998). On global performance of approximations to smooth curves using gridded data. Ann. Statist. 26 2206--2217.
• Härdle, W., Park, B. U. and Tsybakov, A. B. (1995). Estimation of non-sharp support boundaries. J. Multivariate Anal. 55 205--218.
• Korostelëv, A. P. and Tsybakov, A. B. (1993). Minimax Theory of Image Reconstruction. Lecture Notes in Statist. 82. Springer, New York.
• Lighthill, M. J. (1958). Introduction to Fourier Analysis and Generalised Functions. Cambridge Univ. Press.
• Lindenbaum, M. and Bruckstein, A. M. (1994). Blind approximation of planar convex sets. IEEE Trans. Robotics and Automation 10 517--529.
• Mammen, E. and Tsybakov, A. (1995). Asymptotical minimax recovery of sets with smooth boundaries. Ann. Statist. 23 502--524.
• Müller, H.-G. and Song, K. S. (1994). Maximin estimation of multidimensional boundaries. J. Multivariate Anal. 50 265--281.
• Neumann, M. H. (1997). Optimal change-point estimation in inverse problems. Scand. J. Statist. 24 503--521.
• Richards, I. and Youn, H. (1990). Theory of Distributions: A Nontechnical Introduction. Cambridge Univ. Press.
• Schneider, R. (1993). Convex Bodies: The Brunn--Minkowski Theory. Cambridge Univ. Press.
• Skiena, S. S. (1992). Interactive reconstruction via geometric probing. Proc. IEEE 80 1364--1383.
• Tsybakov, A. (1994). Multidimensional change-point problems and boundary estimation. In Change-Point Problems (E. Carlstein, H.-G. Müller and D. Siegmund, eds.) 317--329. IMS, Hayward, CA.
• van der Vaart, A. and Wellner, J. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, New York.