## Wedgelets: nearly-minimax estimation of edges (1999)

Venue: | Ann. Statist |

Citations: | 104 - 8 self |

### BibTeX

@ARTICLE{Donoho99wedgelets:nearly-minimax,

author = {David L. Donoho},

title = {Wedgelets: nearly-minimax estimation of edges},

journal = {Ann. Statist},

year = {1999},

pages = {859--897}

}

### Years of Citing Articles

### OpenURL

### 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 α-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 sub-pixel 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 ≤ α ≤ 2. Wedgelet analysis and de-noising 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.

### Citations

4363 |
Classification and Regression Trees
- Breiman, Friedman, et al.
- 1984
(Show Context)
Citation Context ... dynamic programming/backward induction; it is similar in basic outline to the ‘Best-Ortho-Basis’ algorithm of Coifman and Wickerhauser [7], and to the optimal tree pruning algorithm in the CART book =-=[4]-=-; see also [13]. In the remainder of this section, we first describe a basic decomposability property of the CPRSS, then give an algorithm for minimizing CPRSS. Finally we give the proof of Theorem 6.... |

4020 |
Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images
- Geman, Geman
- 1984
(Show Context)
Citation Context ...the ‘edge’ issue to the fore include Khas’minskii and Lebedev [25], and Korostelev and Tsybakov [27], whose initial efforts were roughly synchronous. There is also interesting work by Geman and Geman =-=[23]-=- and by Müller and Song [33] oriented to a different set of goals. Consider the following very simple “Horizon” model. Suppose there is a function H(x) called the horizon, defined on the interval [0, ... |

2287 |
A wavelet tour of signal processing
- Mallat
- 1999
(Show Context)
Citation Context ...alysis (CHA), a rapidly developing discipline whose recent achievements include the development of wavelets, wavelet packets, cosine packets, brushlets, and other novel schemes of data representation =-=[8, 28, 32, 35]-=-. There is an emerging tradition within CHA, whereby the “way to go about things” is to find the “optimal representation” of the objects underlying a problem and then a “fast algorithm” to compute tha... |

1779 | Atomic decomposition by basis pursuit
- Chen
- 1998
(Show Context)
Citation Context ...to two groups • Practically effective methods, which run efficiently on current computers but can’t guarantee to find representations with near-optimal sparsity – matching pursuit [29], basis pursuit =-=[5]-=-, and best-ortho-basis [7] are examples. • Theoretically effective methods, which do guarantee to find good approximations, but require in principle enumeration of all subsets of a large collection of... |

788 |
Vision
- Marr
- 1982
(Show Context)
Citation Context ...recting our attention to the fact that in real-world image data, the most interesting aspects of the image are the edges. The importance of edges in the vision community goes back to the work of Marr =-=[30]-=- and even earlier; one could say that this issue permeates the field. In the statistical literature pioneers in bringing the ‘edge’ issue to the fore include Khas’minskii and Lebedev [25], and Koroste... |

727 | Constructive Approximation
- DeVore, Lorentz
- 1991
(Show Context)
Citation Context ...d on m edgels. So far in the paper we have controlled L∞ approximation; but the weaker L1 condition is all we really needed. The following lemma is based on well-known properties of Besov spaces, see =-=[10]-=-. Lemma 8.2 Let H ∈ Bα p,q(C), 0 < α ≤ 2, p ≥ 1. interpolant Hm obeys The equispaced knot linear spline ‖H − Hm‖ L 1 [0,1] ≤ K α p,q · C · m −α , m =2,4,8,..., (8.7) where K α p,q depends on (α, p, q)... |

531 | Entropy-based algorithms for best-basis selection
- Coifman, Wickerhauser
- 1992
(Show Context)
Citation Context ...y effective methods, which run efficiently on current computers but can’t guarantee to find representations with near-optimal sparsity – matching pursuit [29], basis pursuit [5], and best-ortho-basis =-=[7]-=- are examples. • Theoretically effective methods, which do guarantee to find good approximations, but require in principle enumeration of all subsets of a large collection of candidate decompositions ... |

265 | Minimax estimation via wavelet shrinkage
- Donoho, Johnstone
- 1998
(Show Context)
Citation Context ... appropriate strategy we can expect recoup the associated log-factors. Indeed, this is a natural analog of the “level-dependent thresholding” idea that recoups log terms in wavelet thresholding – see =-=[15, 20, 3]-=-. Note: we can modify the penalization in this way while still using a fast tree-pruning algorithm. 9.2 Relations to other Work Breiman-Freidman-Olshen-Stone. The bulk of the CART book [4] deals with ... |

260 |
Wavelets, Algorithms and Applications
- Meyer
- 1993
(Show Context)
Citation Context ...alysis (CHA), a rapidly developing discipline whose recent achievements include the development of wavelets, wavelet packets, cosine packets, brushlets, and other novel schemes of data representation =-=[8, 28, 32, 35]-=-. There is an emerging tradition within CHA, whereby the “way to go about things” is to find the “optimal representation” of the objects underlying a problem and then a “fast algorithm” to compute tha... |

212 |
Minimum complexity density estimation
- Barron, Cover
- 1991
(Show Context)
Citation Context ...[23] and by Müller and Song [33] oriented to a different set of goals. Consider the following very simple “Horizon” model. Suppose there is a function H(x) called the horizon, defined on the interval =-=[0, 1]-=- and that the image is of the form f(x1,x2)=1 {x2≥H(x1)} 0≤x1,x2 ≤1. This models a “black and white image” with a horizon, where the image is “white” above the horizon and “black” below. We are intere... |

164 |
Littlewood-Paley theory and the study of function spaces
- Frazier, Jawerth, et al.
- 1991
(Show Context)
Citation Context ...ses Horiz α p,q(C1,Cα) defined by the condition that the horizon function H belong to a Besov ball B α p,q(Cα). Here p, q > 0 are scalars; α is a smoothness index. The scale of Besov spaces, see e.g. =-=[31, 22]-=-, includes various Hölder-type spaces, as well as L 2 Sobolev spaces. B α ∞,∞(C) isverynearlyHölder α (C), and so what we have been calling Horiz α (C1,Cα)isverynearlyHoriz α ∞,∞(C1,Cα). Horizon funct... |

155 | Unconditional bases are optimal bases for data compression and statistical
- Donoho
- 1993
(Show Context)
Citation Context ...with noiseless data and perform data compression. The CHA point of view would say that “optimal representation” is primary, and that statistical and information theoretic applications follow directly =-=[11, 14]-=-. From this very general point of view, a number of asymptotic minimaxity results in mathematical statistics are seen as special cases of a larger picture. Among those results we identify: (a) The fac... |

153 |
Minimax theory of image reconstruction
- Korostelev, Tsybakov
- 1993
(Show Context)
Citation Context ...; one could say that this issue permeates the field. In the statistical literature pioneers in bringing the ‘edge’ issue to the fore include Khas’minskii and Lebedev [25], and Korostelev and Tsybakov =-=[27]-=-, whose initial efforts were roughly synchronous. There is also interesting work by Geman and Geman [23] and by Müller and Song [33] oriented to a different set of goals. Consider the following very s... |

151 | Density estimation by wavelet thresholding
- Donoho, Johnstone, et al.
- 1996
(Show Context)
Citation Context ... appropriate strategy we can expect recoup the associated log-factors. Indeed, this is a natural analog of the “level-dependent thresholding” idea that recoups log terms in wavelet thresholding – see =-=[15, 20, 3]-=-. Note: we can modify the penalization in this way while still using a fast tree-pruning algorithm. 9.2 Relations to other Work Breiman-Freidman-Olshen-Stone. The bulk of the CART book [4] deals with ... |

145 |
Matching Pursuit in a time-frequency dictionary
- Mallat, Zhang
(Show Context)
Citation Context ...lity larger than that of a basis. The vector space of n by n arrays has dimension n 2 ; but #W(n, δ) ≥ 6 · (J +1)·4 K ·n 2 which is larger than n 2 by a logarithmic factor. Following Mallat and Zhang =-=[29]-=-, we call such a collection of elements a dictionary of atoms, with each one possessing a position, scale, and (in some cases) pronounced orientation. This dictionary is complete, since it contains a ... |

83 |
From model selection to adaptive estimation
- Birge, Massart
- 1997
(Show Context)
Citation Context ...are called oracle inequalities, because they compare the risk of valid procedures with the risk achieveable by idealized procedures which depend on oracles. Compare related ideas of Birgé and Massart =-=[3]-=- and of Foster and George [21]. A simple corollary of results in [16, 17, 18, 13] gives an oracle inequality for the estimator ˆ f ∗ of this paper. For each P ∈ ED − RDP (m, δ), consider the fixed par... |

72 | Wavelet analysis and signal processing - Coifman, Meyer, et al. - 1992 |

72 |
Rectifiable sets and the traveling salesman problem
- Jones
- 1990
(Show Context)
Citation Context ...artitions. Jones-David-Semmes-Coifman. The edgelet system we have described here, and the associated dyadic organization of edge data, is closely related to important recent work in harmonic analysis =-=[24, 9]-=-. Peter Jones started off this line of research by showing that one could gather information about approximations to the pieces of a curve defined by intersections with dyadic boxes – recording the er... |

66 | best-ortho-basis: A connection
- CART
- 1997
(Show Context)
Citation Context ...ions [18]. Such “methods” can’t be used on large scale problems. In special cases, there are specific algorithms which run rapidly and which give near-best results for objects in certain classes; see =-=[13]-=- for an example. Our goal in this paper is to develop an algorithm of this form. 3.3 Wedgelet Analysis In a setting where we have a dictionary D = {φ} of atoms, there are two tasks which we can distin... |

64 |
Analysis of and on uniformly rectifiable sets
- David, Semmes
- 1993
(Show Context)
Citation Context ...artitions. Jones-David-Semmes-Coifman. The edgelet system we have described here, and the associated dyadic organization of edge data, is closely related to important recent work in harmonic analysis =-=[24, 9]-=-. Peter Jones started off this line of research by showing that one could gather information about approximations to the pieces of a curve defined by intersections with dyadic boxes – recording the er... |

57 | Wavelet shrinkage - Donoho, Johnstone - 1995 |

23 | Unconditional bases and bit-level compression - Donoho - 1996 |

16 |
Fast algorithm for best anisotropic Walsh bases and relatives
- Bennett
- 2000
(Show Context)
Citation Context ... δn). Fix ξ>8, and set √ λ =(ξ·σ·(1 + 2loge(#W (n, δn)))) 2 . (7.1) Let ˆ f ∗ denote the complexity-penalized estimator produced with this λ. For this estimator where F = Horiz α (C1,Cα) for some α ∈ =-=[1, 2]-=-. sup MSE( F ˆ f ∗ ,f)≤O(log(n)) · M ∗ (n, F), (7.2) This is true, whatever be α ∈ [1, 2], with a simple choice of λ; it is not necessary to adapt λ to the unknown f. Comparing this result with (1.4) ... |

12 | A fast algorithm for adapted time-frequency tilings - Thiele, Villemoes - 1996 |

9 |
Maximin estimation of multidimensional boundaries
- MÜLLER, SONG
- 1994
(Show Context)
Citation Context ... include Khas’minskii and Lebedev [25], and Korostelev and Tsybakov [27], whose initial efforts were roughly synchronous. There is also interesting work by Geman and Geman [23] and by Müller and Song =-=[33]-=- oriented to a different set of goals. Consider the following very simple “Horizon” model. Suppose there is a function H(x) called the horizon, defined on the interval [0, 1] and that the image is of ... |

6 | Minimax estimation of a discontinuous signal - Korostelëv - 1987 |

5 | Abstract Statistical Estimation and Modern Harmonic Analysis - Donoho - 1995 |

5 |
The risk inflation factor in multiple linear regression’, Ann
- Foster, George
- 1994
(Show Context)
Citation Context ...s, because they compare the risk of valid procedures with the risk achieveable by idealized procedures which depend on oracles. Compare related ideas of Birgé and Massart [3] and of Foster and George =-=[21]-=-. A simple corollary of results in [16, 17, 18, 13] gives an oracle inequality for the estimator ˆ f ∗ of this paper. For each P ∈ ED − RDP (m, δ), consider the fixed partition estimator ˆ f(·; P) = Ã... |

5 |
On the properties of parametric estimators for areas of a discontinuous image
- Khas’minskii, Lebedev
- 1990
(Show Context)
Citation Context ...work of Marr [30] and even earlier; one could say that this issue permeates the field. In the statistical literature pioneers in bringing the ‘edge’ issue to the fore include Khas’minskii and Lebedev =-=[25]-=-, and Korostelev and Tsybakov [27], whose initial efforts were roughly synchronous. There is also interesting work by Geman and Geman [23] and by Müller and Song [33] oriented to a different set of go... |

3 |
Ideal De-Noising in a basis chosen from a library of orthonormal bases
- Donoho, Johnstone
- 1994
(Show Context)
Citation Context ...ine a specific principle for processing noisy data (1.1). Our goal is to find a partition with low cardinality which fits the data well. The approach we use is of exactly the same type as employed in =-=[17, 18, 13]-=-. Let y(i1,i2) be an array of pixel-level data. Suppose we are given an ED-RDP P. In the vector space of n-by-n arrays, consider the vector subspace L(P) of all arrays arising from linear combinations... |

3 |
Ondelettes et opérateurs I: Ondelettes. Paris: Hermann, Éditeurs des Sciences et des Arts
- Meyer
- 1990
(Show Context)
Citation Context ...ses Horiz α p,q(C1,Cα) defined by the condition that the horizon function H belong to a Besov ball B α p,q(Cα). Here p, q > 0 are scalars; α is a smoothness index. The scale of Besov spaces, see e.g. =-=[31, 22]-=-, includes various Hölder-type spaces, as well as L 2 Sobolev spaces. B α ∞,∞(C) isverynearlyHölder α (C), and so what we have been calling Horiz α (C1,Cα)isverynearlyHoriz α ∞,∞(C1,Cα). Horizon funct... |