The Annals of Statistics
- Ann. Statist.
- Volume 27, Number 3 (1999), 859-897.
Wedgelets: nearly minimax estimation of edges
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.
Ann. Statist. Volume 27, Number 3 (1999), 859-897.
First available in Project Euclid: 5 April 2002
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 62G07: Density estimation 62C20: Minimax procedures
Secondary: 62G20: Asymptotic properties 41A30: Approximation by other special function classes 41A63: Multidimensional problems (should also be assigned at least one other classification number in this section)
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.