The Annals of Statistics

Wedgelets: nearly minimax estimation of edges

David L. Donoho

Full-text: Open access


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

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)

Minimax estimation edges edgels edgelets fast algorithms complexity penalized estimates recursive partitioning subpixel resolution oracle inequalities


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

Export citation


  • 1 BARRON, A. and COVER, T. 1991. Minimum complexity density estimation. IEEE Trans. Inform Theory 37 1034 1054.
  • 2 BENNETT, N. 1997. Fast algorithms for best anisotropic Walsh bases, and relatives. Ph.D. dissertation, Yale Univ.
  • 3 BIRGE, L. and MASSART, P. 1997. From model selection to adaptive estimation. In Festschrift ´ Z. for Lucien Le Cam D. Pollard, E. Torgersen and G. Yang, eds. 55 89. Springer, New York.
  • 4 BREIMAN, L., FRIEDMAN, J., OLSHEN, R. and STONE, C. J. 1983. Classification and Regression Trees. Wadsworth, Belmont, CA.
  • 5 CANDES, E. and DONOHO, D. 1999. Ridgelets: the key to high-dimensional intermittency? Philos. Trans. Roy. Soc. To appear.
  • 6 CHEN, S., DONOHO, D. L. and SAUNDERS, M. A. 1999. Atomic decomposition by basis pursuit. SIAM J. Sci Comput. 20 33 61.
  • 7 COIFMAN, R. R., MEYER, Y., QUAKE, S. and WICKERHAUSER, M. V. 1994. Wavelet analysis Z and signal processing. In Wavelets and Their Applications J. S. Byrnes, J. L. Byrnes,. K. A. Hargreaves and K. Berry, eds. 363 380. Kluwer, Boston.
  • 8 COIFMAN, R. R. and WICKERHAUSER, M. V. 1992. Entropy-based algorithms for best-basis selection. IEEE Trans. Inform Theory 38 713 718.
  • 9 DAUBECHIES, I. 1992. Ten Lectures on Wavelets. SIAM, Philadelphia.
  • 10 DAVID, G. and SEMMES, S. 1993. Analysis of and on Uniformly Rectifiable Sets. Amer Math Soc., Providence, RI.
  • 11 DEVORE, R. A. and LORENTZ, G. G. 1993. Constructive Approximation. Springer, New York.
  • 12 DONOHO, D. L. 1993. Unconditional bases are optimal bases for data compression and for statistical estimation. Appl. Comput. Harmon. Anal. 1 100 115.
  • 13 DONOHO, D. L. 1995. Abstract statistical estimation and modern harmonic analysis. In Proc. 1994 Internat. Congr. Math. 997 1005. Birkhauser, Basel. ¨
  • 14 DONOHO, D. L. 1996. Unconditional bases and bit-level compression. Appl. Comput. Harmon. Anal. 3 388 392.
  • 15 DONOHO, D. L. 1997. CART and best-ortho-basis: a connection: Ann. Statist. 25 1870 1911.
  • 16 DONOHO, D. L. 1998. Sparse components analysis and optimal atomic decomposition. TechZ nical report, Dept. Statistics, Stanford Univ. http: www-stat.stanford.. edu donoho Reports 1998
  • 17 DONOHO, D. L. and JOHNSTONE, I. M. 1994. Ideal spatial adaptation via wavelet shrinkage. Biometrika 81 425 455.
  • 18 DONOHO, D. L. and JOHNSTONE, I. M. 1994. Ideal de-noising in a basis chosen from a library of orthonormal base. Comp. R. Acad. Sci. Paris A 319 1317 1322.
  • 19 DONOHO, D. L. and JOHNSTONE, I. M. 1994. Empirical atomic decomposition. Unpublished manuscript.
  • 20 DONOHO, D. L. and JOHNSTONE, I. M. 1998. Minimax estimation via wavelet shrinkage. Ann. Statist. 26 879 921.
  • 21 DONOHO, D. L., JOHNSTONE, I. M., KERKYACHARIAN, G. and PICARD, D. 1995. Wavelet shrinkage: asymptopia? J. Roy. Statist. Soc. Ser. B 57 301 369.
  • 22 DONOHO, D. L., JOHNSTONE, I. M., KERKYACHARIAN, G. and PICARD, D. 1996. Density estimation by wavelet thresholding. Ann. Statist. 24 508 539.
  • 23 FOSTER, D. and GEORGE, E. I. 1994. The risk inflation factor in multiple linear regression. Ann. Statist. 22 1947 1975.
  • 24 FRAZIER, M., JAWERTH, B. and WEISS, G. 1991. Littlewood Paley Theory and the Study of Function Spaces. Amer. Math. Soc., Providence, RI.
  • 25 GEMAN, S. and GEMAN, D. 1984. Stochastic relaxation. Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Patt. Anal. Mach. Intel. 6 721 741.
  • 26 JONES, P. W. 1990. Rectifiable sets and the travelling salesman problem. Invent. Math. 102 1 15.
  • 28 KOROSTELEV, A. P. 1987. Minimax estimation of a discontinuous signal. Theory Probab. Appl. 32 796 799.
  • 29 KOROSTELEV, A. P. and TSYBAKOV, A. 1993. Minimax Theory of Image Reconstruction. Lecture Notes in Statist. 82. Springer, New York.
  • 30 MALLAT, S. 1997. A Wavelet Tour of Signal Processing. Academic Press, New York.
  • 31 MALLAT, S. and ZHANG, Z. 1993. Matching pursuit in a time frequency dictionary. IEEE Trans. Signal Processing 41 3397 3415.
  • 32 MARR, D. 1982. Vision. Freeman, New York.
  • 33 MEYER, Y. 1990. Ondelettes et Operateurs I: Ondelettes. Hermann, Paris. English transla. tion: Wavelets and Operators. Cambridge Univ. Press.
  • 34 MEYER, Y. 1993. Wavelets: Algorithms and Applications. SIAM, Philadelphia.
  • 35 MULLER, H. G. and SONG, K. S. 1994. Maximin estimation of multidimensional boundaries. ¨ J. Multivariate Anal. 50 265 281.
  • 36 OLSHAUSEN, B. A. and FIELD, D. J. 1996. Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature 381 607 609.
  • 37 THIELE, C. M. and VILLEMOES, L. F. 1996. A fast algorithm for adapted time frequency tilings. Appl. Comput. Harmon. Anal. 3 91 100.
  • 38 WICKERHAUSER, M. V. 1994. Adapted Wavelet Analysis from Theory to Software. Peters, Boston.