## The Annals of Statistics

### Wedgelets: nearly minimax estimation of edges

David L. Donoho

#### Abstract

We study a simple “horizon model” for the problem of recovering an image from noisy data; in this model the image has an edge with $\alpha$-Hölder regularity. Adopting the viewpoint of computational harmonic analysis, we develop an overcomplete collection of atoms called wedgelets, dyadically organized indicator functions with a variety of locations, scales and orientations. The wedgelet representation provides nearly optimal representations of objects in the horizon model, as measured by minimax description length. We show how to rapidly compute a wedgelet approximation to noisy data by finding a special edgelet-decorated recursive partition which minimizes a complexity-penalized sum of squares. This estimate, using sufficient subpixel resolution, achieves nearly the minimax mean-squared error in the horizon model. In fact, the method is adaptive in the sense that it achieves nearly the minimax risk for any value of the unknown degree of regularity of the horizon, $1 \leq \alpha \leq 2$. Wedgelet analysis and denoising may be used successfully outside the horizon model. We study images modelled as indicators of star-shaped sets with smooth boundaries and show that complexity-penalized wedgelet partitioning achieves nearly the minimax risk in that setting also.

#### Article information

Source
Ann. Statist. Volume 27, Number 3 (1999), 859-897.

Dates
First available in Project Euclid: 5 April 2002

http://projecteuclid.org/euclid.aos/1018031261

Mathematical Reviews number (MathSciNet)
MR1724034

Digital Object Identifier
doi:10.1214/aos/1018031261

Zentralblatt MATH identifier
0957.62029

#### Citation

Donoho, David L. Wedgelets: nearly minimax estimation of edges. Ann. Statist. 27 (1999), no. 3, 859--897. doi:10.1214/aos/1018031261. http://projecteuclid.org/euclid.aos/1018031261.

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• STANFORD, CALIFORNIA 94305 E-MAIL: donoho@stat.stanford.edu