## Orthonormal Ridgelets and Linear Singularities (1998)

Citations: | 57 - 16 self |

### BibTeX

@TECHREPORT{Donoho98orthonormalridgelets,

author = {David L. Donoho},

title = {Orthonormal Ridgelets and Linear Singularities},

institution = {},

year = {1998}

}

### Years of Citing Articles

### OpenURL

### Abstract

We construct a new orthonormal basis for L2 (R2), whose elements are angularly integrated ridge functions — orthonormal ridgelets. The new basis functions are in L2 (R2) and so are to be distinguished from the ridge function approximation system called ridgelets by Candès (1997, 1998), as ridge functions are not in L2 (R2). Orthonormal ridgelet expansions have an interesting application in nonlinear approximation: the problem of efficient approximations to objects such as 1 {x1 cos θ+x2 sin θ>a} e−x2 1−x2 2 which are smooth away from a discontinuity along a line. The orthonormal ridgelet coefficients of such an object are sparse: they belong to every ℓp, p>0. This implies that simple thresholding in the ridgelet orthobasis is, in a certain sense, a near-ideal nonlinear approximation scheme. The ridgelet orthobasis is the isometric image of a special wavelet basis for Radon space; as a consequence, ridgelet analysis is equivalent to a special wavelet analysis in the Radon domain. This means that questions of ridgelet analysis of linear singularities can be answered by wavelet analysis of point singularities. At the heart of our nonlinear approximation result is the study of a certain tempered distribution on R2 defined formally by S(u, v) =|v | −1/2σ(u/|v|) with σ a certain smooth bounded function; this is singular at (u, v) =(0,0) and C ∞ elsewhere. The key point is that the analysis of this point singularity by tensor Meyer wavelets yields sparse coefficients at high frequencies; this is reflected in the sparsity of the ridgelet coefficients and the good nonlinear approximation properties of the ridgelet basis.

### Citations

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(Show Context)
Citation Context ...point (u0,v0) fixed. For any desired degree of smoothness, there are orthonormal bases of Daubechies wavelets of compact support where the generating wavelets exhibit the desired degree of smoothness =-=[3]-=-. We now exploit this fact, indexing a sequence of such bases by the desired smoothness. Let then, for each D =1,2,3,...,(ψD jk : j, k ∈ Z) be a one-dimensional wavelet basis made of Daubechies wavele... |

339 |
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(Show Context)
Citation Context ... analogous construction straightforwardly in dimensions d>2, we would need orthonormal wavelets on the sphere S d−1 and an isometry for higher dimensions. The isometry exists in every dimension d ≥ 2 =-=[12]-=-. Unfortunately, orthonormal spherical wavelets are not known for any dimension d>2. The next best thing to an orthonormal system is a tight frame, which obeys a Parseval relation. The article [10] co... |

239 | Wavelet shrinkage: Asymptopia
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(Show Context)
Citation Context ...tly approximated by partial wavelet reconstructions. This fact has significant implications in data compression and in statistical estimation. (Extensive references on these implications are given in =-=[7, 6]-=-). Point singularities are just one possible type of singularity. Consider the Gaussian-windowed halfspace g 0 (x1,x2)=1 {x2>0} e −x21−x22, x ∈R 2 . (1.1) This has a singularity along the line x2 = 0.... |

202 |
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(Show Context)
Citation Context ...ne L (θ,t), expressed using the Dirac mass δ as � (Rf)(t, θ)= f(x)δ(x1cos θ + x2 sinθ − t) dx , (2.1) where we permit θ ∈ [0,2π) andt∈R. For more information about the Radon transform see for example =-=[5, 11]-=-. Observe that the line L (θ,t) is identical to the line L (θ+π,−t). As a result, Rf has the antipodal symmetry (Rf)(−t, θ + π)=(Rf)(t, θ) . (2.2) This is a fundamental fact about the Radon transform ... |

142 | Data compression and harmonic analysis
- Donoho, Vetterli, et al.
(Show Context)
Citation Context ...tly approximated by partial wavelet reconstructions. This fact has significant implications in data compression and in statistical estimation. (Extensive references on these implications are given in =-=[7, 6]-=-). Point singularities are just one possible type of singularity. Consider the Gaussian-windowed halfspace g 0 (x1,x2)=1 {x2>0} e −x21−x22, x ∈R 2 . (1.1) This has a singularity along the line x2 = 0.... |

77 |
Ridgelets: Theory and Applications
- CANDÈS
- 1998
(Show Context)
Citation Context ...ond, putting now ν = ν1, there are constants Cℓ,m so that ∂ (m,n) νβ = ∂ (m,0) (ν · ∂ (0,n) β) = m� Cℓ,mν ℓ=0 (ℓ,0) β (m−ℓ,n) , where ν (ℓ,0) ≡ ∂ (ℓ,0) ν.Nowforℓ>0, ν (ℓ,0) is supported in {ω : |ω1| ∈=-=[1, 2]-=-}. Hence, there is ˜νℓ ∈S(R 2 ) so that We have |∂ (m,n) νβ|≤ ν (ℓ,0) (ω)=˜νℓ(ω)·sgn(ω1) ℓ |ω1| −ℓ . m� ℓ=0 for Fn,m,ℓ ∈S(R). Now as each Fn,m,ℓ ∈S(R), we have |˜νℓ|·|ω1| −m−n−2 |Fn,m,ℓ(ω2/ω1)| Fn,m,l... |

62 | Y.: Ondelettes et bases hilbertiennes - Lemarié, Meyer - 1986 |

61 | Optimal reconstruction of a function from its projections - Logan, Shepp - 1975 |

50 |
A Guide to Distribution Theory and Fourier Transforms, World Scientific
- Strichartz
- 2003
(Show Context)
Citation Context .... So consider now the case ε = 1. To begin, we need a formula for the Fourier Transform of H(u, v), viewed as a tempered distribution in S ′ (R 2 ). The convolution formula for tempered distributions =-=[15]-=- says that if f ∈S ′ (R 2 )andg∈S(R 2 ), then (f · g) ˆ =(2π) −2ˆ f⋆ˆg. Hence ˆH =(2π) −2 ·( �˙ H⋆ ˆ V)=γ1 ˆ V+γ2 ˆ V⋆β, 22swhere ˙ H is as in Lemma 8.1, and where the constants γr, r =1,2 can be obta... |

49 |
Ridgelets: the key to high dimensional intermittency
- Candès, Donoho
- 1999
(Show Context)
Citation Context ...s of objects with discontinuities along curved edges having significantly more rapid decay of coefficients than the traditional wavelet and Fourier methods. For further discussion on applications see =-=[3]-=-. 1.5 On the Ridgelet Concept The orthonormal ridgelets are in L 2 (R 2 ) and so are to be distinguished from the approximation system called ridgelets in the pioneering work by Candès (1997, 1998). I... |

20 |
G.: Generalized Functions. I. Properties and Operations
- Gel’fand, Shilov
- 1964
(Show Context)
Citation Context ...ne L (θ,t), expressed using the Dirac mass δ as � (Rf)(t, θ)= f(x)δ(x1cos θ + x2 sinθ − t) dx , (2.1) where we permit θ ∈ [0,2π) andt∈R. For more information about the Radon transform see for example =-=[5, 11]-=-. Observe that the line L (θ,t) is identical to the line L (θ+π,−t). As a result, Rf has the antipodal symmetry (Rf)(−t, θ + π)=(Rf)(t, θ) . (2.2) This is a fundamental fact about the Radon transform ... |

15 | Tight Frames of k-Plane Ridgelets and the Problem of Representing ddimensional singularities
- Donoho
- 1999
(Show Context)
Citation Context ... 2 [12]. Unfortunately, orthonormal spherical wavelets are not known for any dimension d>2. The next best thing to an orthonormal system is a tight frame, which obeys a Parseval relation. The article =-=[10]-=- constructs tight frames of wavelet-like elements on spheres of all dimensions and so obtains tight frames of ridgelets in all dimensions d>2. It also shows how to construct k-plane ridgelets (tight f... |

14 | Ridge functions and orthonormal ridgelets
- Donoho
- 2001
(Show Context)
Citation Context ...sensible because we can show close connections of orthonormal ridgelets to the original ridge function concept. Theorem 1.2 is an instance of this connection; see Section 4 below. The companion paper =-=[9]-=- explores carefully the connection between orthonormal ridgelets and ridge functions and gives results showing that orthonormal ridgelets are an effective substitute for ridge function approximation. ... |

3 |
Harmonic Analysis of Neural Nets
- Candès
- 1997
(Show Context)
Citation Context ...ond, putting now ν = ν1, there are constants Cℓ,m so that ∂ (m,n) νβ = ∂ (m,0) (ν · ∂ (0,n) β) = m� Cℓ,mν ℓ=0 (ℓ,0) β (m−ℓ,n) , where ν (ℓ,0) ≡ ∂ (ℓ,0) ν.Nowforℓ>0, ν (ℓ,0) is supported in {ω : |ω1| ∈=-=[1, 2]-=-}. Hence, there is ˜νℓ ∈S(R 2 ) so that We have |∂ (m,n) νβ|≤ ν (ℓ,0) (ω)=˜νℓ(ω)·sgn(ω1) ℓ |ω1| −ℓ . m� ℓ=0 for Fn,m,ℓ ∈S(R). Now as each Fn,m,ℓ ∈S(R), we have |˜νℓ|·|ω1| −m−n−2 |Fn,m,ℓ(ω2/ω1)| Fn,m,l... |