## Atomic decomposition by basis pursuit (1998)

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Venue: | SIAM JOURNAL ON SCIENTIFIC COMPUTING |

Citations: | 1778 - 52 self |

### BibTeX

@ARTICLE{Chen98atomicdecomposition,

author = {Scott Shaobing Chen and David L. Donoho and Michael A. Saunders},

title = {Atomic decomposition by basis pursuit},

journal = {SIAM JOURNAL ON SCIENTIFIC COMPUTING},

year = {1998},

volume = {20},

pages = {33--61}

}

### Years of Citing Articles

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### Abstract

The time-frequency and time-scale communities have recently developed a large number of overcomplete waveform dictionaries — stationary wavelets, wavelet packets, cosine packets, chirplets, and warplets, to name a few. Decomposition into overcomplete systems is not unique, and several methods for decomposition have been proposed, including the method of frames (MOF), Matching pursuit (MP), and, for special dictionaries, the best orthogonal basis (BOB). Basis Pursuit (BP) is a principle for decomposing a signal into an “optimal ” superposition of dictionary elements, where optimal means having the smallest l 1 norm of coefficients among all such decompositions. We give examples exhibiting several advantages over MOF, MP, and BOB, including better sparsity and superresolution. BP has interesting relations to ideas in areas as diverse as ill-posed problems, in abstract harmonic analysis, total variation denoising, and multiscale edge denoising. BP in highly overcomplete dictionaries leads to large-scale optimization problems. With signals of length 8192 and a wavelet packet dictionary, one gets an equivalent linear program of size 8192 by 212,992. Such problems can be attacked successfully only because of recent advances in linear programming by interior-point methods. We obtain reasonable success with a primal-dual logarithmic barrier method and conjugate-gradient solver.

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Citation Context ...nt shift. A variety of other wavelet bases are possible. The most important variations are smooth wavelet bases, using splines or using wavelets defined recursively from two-scale filtering relations =-=[9]-=-. Although the rules of construction are more complicated (boundary conditions [25], orthogonality versus bi-orthogonality [9], etc.), these have the same indexing structure as the standard Haar dicti... |

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Citation Context ...we set # to the value # p = # # 2 log(p) , where p is the cardinality of the dictionary. This can be motivated as follows. In the case of a dictionary that is an orthonormal basis, a number of papers =-=[13, 16]-=- have carefully studied an approach to denoising by so-called soft thresholding in an orthonormal basis. In detail, suppose that # is an orthogonal matrix, and define empirical #-coe#cients bysy = # T... |

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Citation Context ...entations in overcomplete dictionaries. These range from general approaches, like the MOF [9] and the method of MP [25], to clever schemes derived for specialized dictionaries, like the method of BOB =-=[7]-=-. These methods are described briefly in section 2.3. In our view, these methods have both advantages and shortcomings. The principal emphasis of the proposers of these methods is in achieving su#cien... |

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Citation Context ...ithm could be based on an interior-point method [24]. Since then a wide array of interior-point algorithms have been proposed and considerables152 S. S. CHEN,D. L. DONOHO,AND M. A. SAUNDERS practical =-=[25, 27, 32, 50]-=- and theoretical [49, 35, 40] understanding is now available. In this section we describe our algorithmand our experience with it. 6.1. Duality Theory. We consider the linear programin the standard fo... |

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Citation Context ...omic decomposition problem, BP will typically find exactly that sparse solution. We saw numerous examples of this phenomenon in preparing this paper and in Chen’s thesis [4]. Recently, Donoho and Huo =-=[15]-=- have given a theoretical explanation. They have proven a number of results showing that mathematically exact solution of BP in overcomplete dictionaries can exhibit precisely the phenomenon of ideal ... |

372 |
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Citation Context ...polynomial time algorithm could be based on an interior-point method [18]. Since then a very wide array of interior-point algorithms have been proposed and considerable practical [21] and theoretical =-=[27]-=- understanding is now available. In this section we describe our algorithm and our experience with it. 6.1 Duality Theory We consider the linear program in the standard form min c T x subject to Ax = ... |

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Citation Context ...hen's example. The DeVore--Temlyakov example applies to the original MP algorithm as announced by Mallat and Zhang in 1992. A later refinement of the algorithm (see Pati, Rezaiifar, and Krishnaprasad =-=[34]-=- and Davis, Mallat, and Zhang [11]) involves an extra step of orthogonalization. One takes all m terms that have entered at stage m and solves the least squares problem min (# i ) #s - m # i=1 # i # #... |

260 |
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Citation Context ... that are close to certain heuristic methods. As an important contrast, Meyer has a counterexample to multiscale edge approaches, which shows that the Mallat--Zhong approach may fail in certain cases =-=[29]-=-, but there can be no such counterexamples to BP. 4.2. Dictionary mergers. An important methodological tool is the ability to combine dictionaries to make bigger, more expressive dictionaries. We ment... |

256 | Wavelet shrinkage: Asymptopia
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Citation Context ...we set # to the value # p = # # 2 log(p) , where p is the cardinality of the dictionary. This can be motivated as follows. In the case of a dictionary that is an orthonormal basis, a number of papers =-=[13, 16]-=- have carefully studied an approach to denoising by so-called soft thresholding in an orthonormal basis. In detail, suppose that # is an orthogonal matrix, and define empirical #-coe#cients bysy = # T... |

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Citation Context ... proof that a pseudo-polynomial-time algorithm could be based on an interior-point method [20]. Since then a very wide array of interior-point algorithms have been proposed and considerable practical =-=[21, 23, 27]-=- and theoretical [30] understanding is now available. In this section we describe our algorithm and our experience with it. 6.1. Duality theory. We consider the linear program in the standard form min... |

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145 |
Matching Pursuit in a time-frequency dictionary
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Citation Context ...ding a representation. Several methods have been proposed for obtaining signal representations in overcomplete dictionaries. These range from general approaches, like the MOF [9] and the method of MP =-=[25]-=-, to clever schemes derived for specialized dictionaries, like the method of BOB [7]. These methods are described briefly in section 2.3. In our view, these methods have both advantages and shortcomin... |

128 | Basis Pursuit
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Citation Context ...oising, a method for dealing with problem (1.2). In section 6 we discuss recent advances in large-scale linear programming and resulting algorithms for BP. For reasons of space we refer the reader to =-=[4]-=- for a discussion of related work in statistics and analysis. 2. Overcomplete representations. Let s = (s t : 0 # tsn) be a discretetime signal of length n; this may also be viewed as a vector in R n ... |

126 |
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Citation Context ... been successfully solved using digital computers. A good overview of the recent rapid progress in this field and the current state of the art is a#orded by the article of Lustig, Marsten, and Shanno =-=[23] and -=-the accompanying discussions by Bixby [1], Saunders [36], Todd [38], and Vanderbei [39]. Much of the rapid expansion in the size of linear programs solved is due to the "interior-point revolution... |

104 | Wedgelets: Nearly-minimax estimation of edges
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Citation Context ...possible to consider image processing experiments, at least on a limited scale. Huo’s thesis [23] considered decompositions in an overcomplete dictionary consisting of wavelets and so-called edgelets =-=[14]-=-. This allowed both dictionaries to compete on an equal footing to use exactly the terms that best “explain” the image data. In one experiment, Huo analyzed a digitized image and found that the humanl... |

103 |
Theory and Algorithms for Linear Optimization: An Interior Point Approach
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Citation Context ...r-point method [24]. Since then a wide array of interior-point algorithms have been proposed and considerables152 S. S. CHEN,D. L. DONOHO,AND M. A. SAUNDERS practical [25, 27, 32, 50] and theoretical =-=[49, 35, 40]-=- understanding is now available. In this section we describe our algorithmand our experience with it. 6.1. Duality Theory. We consider the linear programin the standard form (6.1) min c T x subject to... |

102 |
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Citation Context ...Statistics, October 1994, described an example in which the straightforward greedy algorithm is not sparsity preserving. In our adaptation of this example, based on Temlyakov's joint work with DeVore =-=[12]-=-, one constructs a dictionary having n + 1 atoms. The first n are the Dirac basis; the final atom involves a linear combination of the first n with decaying weights. The signal s has an exact decompos... |

101 | Beamlab and reproducible research
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(Show Context)
Citation Context ...akes about 8 hours to perform BP on the signal Gong at the same parameter setting. 6.6 Reproducible Research This paper has been written following the discipline of Reproducible Research described in =-=[3]-=-. As a complement to this article, we are releasing the underlying software environment by placing it on internet for access either by anonymous FTP or WWW browsers. Web Browser: http://playfair.stanf... |

100 | Ideal de-noising in an orthonormal basis chosen from a library of bases
- Donoho, Johnstone
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Citation Context ...e selected atom gets below the threshold # # 2 log(p). Best orthogonal basis denoising (BOBDN) is a thresholding scheme in the best orthogonal basis chosen by the BOB algorithm with a special entropy =-=[14]-=-. 5.3.1. Gong. Figure 5.1 displays denoising results on the signal Gong, at signal to noise ratio 1, using a cosine packet dictionary. Figure 5.1a displays the noiseless signal and Figure 5.1b display... |

90 |
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Citation Context ...(n log(n)) time. 1.2. Finding a representation. Several methods have been proposed for obtaining signal representations in overcomplete dictionaries. These range from general approaches, like the MOF =-=[9]-=- and the method of MP [25], to clever schemes derived for specialized dictionaries, like the method of BOB [7]. These methods are described briefly in section 2.3. In our view, these methods have both... |

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Citation Context ...re fully the effect of BPAccuracy and CGAccuracy using these codes. Comparisons will be reported on the Atomizer web site. 6.6. Alternative Algorithms for BPDN. For certain dictionaries, Sardy et al. =-=[42]-=- showed how to minimize the BPDN function (5.1) using a block coordinate relaxation (BCR) method. They assumed that the columns of Φ are the union of (perhaps many) orthonormal complete matrices Φ (1)... |

85 | Interior methods for constrained optimization
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Citation Context ...r-point method [24]. Since then a wide array of interior-point algorithms have been proposed and considerables152 S. S. CHEN,D. L. DONOHO,AND M. A. SAUNDERS practical [25, 27, 32, 50] and theoretical =-=[49, 35, 40]-=- understanding is now available. In this section we describe our algorithmand our experience with it. 6.1. Duality Theory. We consider the linear programin the standard form (6.1) min c T x subject to... |

77 |
Adaptive time-frequency decompositions
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Citation Context ...ov example applies to the original MP algorithm as announced by Mallat and Zhang in 1992. A later refinement of the algorithm (see Pati, Rezaiifar, and Krishnaprasad [34] and Davis, Mallat, and Zhang =-=[11]-=-) involves an extra step of orthogonalization. One takes all m terms that have entered at stage m and solves the least squares problem min (# i ) #s - m # i=1 # i # # i # 2 for coe#cients (# (m) i ). ... |

77 |
Loan.Matrix Computations, 2nd edition
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Citation Context ...ourse, in general solving systems of equations is not rapid: a general n by n system Bx = c takes order O(n 3 ) time to solve by standard elimination methods or by modern stable factorization schemes =-=[17, 16]-=-. In order for practical algorithms to be based on the interior-point heuristic, it is necessary to be able to solve the systems of equations much more rapidly than one could solve general systems. In... |

76 |
ALGORITHM 583 LSQR: Sparse linear equations and sparse least squares problems, Trans
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Citation Context ...4) iteratively, avoiding the costly step of explicitly forming the matrices (ADA T + # 2 I). Similarly, the algorithms for Au and A T v can be used directly in conjugate-gradient methods such as LSQR =-=[32, 33]-=- for solving the least squares problem (6.5). In our application, we do not really need an exact solution of the optimization problem. Moreover, we have a natural initial solution---from MOF---that wo... |

76 |
The Math Works
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Citation Context ... make it possible, with certain dictionaries, to nearly solve the BP optimization problem in nearly linear time. We have implemented primal-dual log barrier interior-point methods as part of a MATLAB =-=[31]-=- computing environment called Atomizer, which accepts a wide range of dictionaries. Instructions for Internet access to Atomizer are given in section 7.3. Experiments with standard time-frequency dict... |

68 |
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Citation Context ... ; A # (#, -#) ; b # s . Hence the solution of (3.1) can be obtained by solving an equivalent linear program. (The equivalence of minimum # 1 optimizations with LP has been known since the 1950s; see =-=[2]-=-). The connection between BP and LP is useful in several ways. 3.1.1. Solutions as bases. In the LP problem (3.2), suppose A is an n-by-m matrix with m > n, and suppose an optimal solution exists. It ... |

48 | On finding primal- and dual-optimal bases
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Citation Context ... proof that a pseudo-polynomial-time algorithm could be based on an interior-point method [20]. Since then a very wide array of interior-point algorithms have been proposed and considerable practical =-=[21, 23, 27]-=- and theoretical [30] understanding is now available. In this section we describe our algorithm and our experience with it. 6.1. Duality theory. We consider the linear program in the standard form min... |

45 |
Ondelettes sur l'intervalle
- Meyer
(Show Context)
Citation Context ...ions are smooth wavelet bases, using splines or using wavelets defined recursively from two-scale filtering relations [9]. Although the rules of construction are more complicated (boundary conditions =-=[25]-=-, orthogonality versus bi-orthogonality [9], etc.), these have the same indexing structure as the standard Haar dictionary. In this paper, we use Symmlet-8 smooth wavelets, i.e., Daubechies Nearly Sym... |

41 |
Wavelet transform maxima and multiscale edges. In Wavelets and their applications
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Citation Context ...agreement of the BP representation in a stationary wavelet dictionary with ideas about signal representation associated with the "multiscale edges" ideas of Mallat and Hwang [24] and Mallat =-=and Zhong [26]. The mult-=-iscale edge method analyzes the continuous wavelet transform (CWT) at scale 2 -j and identifies the maxima of this transform. Then it selects maxima that are "important" by thresholding base... |

40 |
Numerical Linear Algebra and Optimization
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- 1991
(Show Context)
Citation Context ...h involves considerably more e#ort and sophistication. 3.1. Linear programming (LP). To explain the last comment and BP, we develop a connection with LP. The linear program in so-called standard form =-=[8, 18]-=- is a constrained optimization problem defined in terms of a variable x # R m by min c T x subject to Ax = b, x # 0 , (3.2) 44 S. S. CHEN, D. L. DONOHO, AND M. A. SAUNDERS 0 0.5 1 -4 -2 0 2 4 6 (a) Si... |

38 |
Progress in linear programming
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Citation Context ...crete cosine transforms and sine transforms. 2.1.3. Time-scale dictionaries. There are several types of wavelet dictionaries; to fix ideas, we consider the Haar dictionary with "father wavelet&qu=-=ot; # = 1 [0,1] and "-=-;mother wavelet" = 1 (1/2,1] - 1 [0,1/2] . The dictionary is a collection of translations and dilations of the basic mother wavelet, together with translations of a father wavelet. It is indexed ... |

36 | A computational algorithm for minimizing total variation in image restoration
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- 1996
(Show Context)
Citation Context ...pose the optimization problem min g 1 2 #y - g# 2 2 + # TV (g), (5.4) where TV (g) is a discrete measure of the total variation of g. A solution of this problem is the denoised object. Li and Santosa =-=[22]-=- have developed an alternative algorithm for this problem based on interior-point methods for convex optimization. For the 1-D case (signals rather than images) it is possible to implement what amount... |

30 |
Signal representation using adaptive normalized gaussian functions
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- 1994
(Show Context)
Citation Context ...#R (k-1) , # #k # and R (k) = s - s (k) . After m steps, one has a representation of the form (1.2), with residual R = R (m) . Similar algorithms were proposed by Qian and Chen for Gabor dictionaries =-=[35]-=- and by Villemoes for Walsh dictionaries [40]. A similar algorithm was proposed for Gabor dictionaries by Qian and Chen [35]. For an earlier instance of a related algorithm see the article [5]. An int... |

21 | Solving reduced KKT systems in barrier methods for linear and quadratic programming
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(Show Context)
Citation Context ...y of them are mentioned in [21] and others are covered in the references of that article. Our approach is based on a primal-dual log-barrier algorithm. In order to regularize standard LP, Gill et al. =-=[15]-=- proposed solving the following perturbed LP: min c T x + 1 2 kflxk 2 + 1 2 kpk 2 subject to Ax + ffi p = b; xs0 ; (6.3) where fl and ffi are normally small (e.g. 10 \Gamma4 ) regularization parameter... |

19 |
WaveLab and Reproducible Research, in Wavelets and Statistics
- Buckheit, Donoho
- 1995
(Show Context)
Citation Context ...bout eight hours to perform BP on the signal Gong at the same parameter setting. 6.6. Reproducible research. This paper has been written following the discipline of reproducible research described in =-=[3]-=-. As a complement to this article, we are releasing the underlying software environment by placing it on the Internet for access either by anonymous FTP or WWW browsers. Web Browser: http://www-stat.s... |

15 | An a�ne scaling algorithm for minimizing total variation in image enhancement
- Li, Santosa
- 1994
(Show Context)
Citation Context ...ptimization problem min g 1 2 ky \Gamma gk 2 2 +s\Delta TV (g) (5.4) where TV (g) is a discrete measure of the total variation of g. A solution of this problem is the de-noised object. Li and Santosa =-=[20]-=- have developed an alternative algorithm for this problem based on interior-point methods for convex optimization. For the 1-dimensional case (signals rather than images) it is possible to implement w... |

13 | Best approximation with walsh atoms
- Villemoes
- 1997
(Show Context)
Citation Context ...er m steps, one has a representation of the form (1.2), with residual R = R (m) . Similar algorithms were proposed by Qian and Chen for Gabor dictionaries [35] and by Villemoes for Walsh dictionaries =-=[40]-=-. A similar algorithm was proposed for Gabor dictionaries by Qian and Chen [35]. For an earlier instance of a related algorithm see the article [5]. An intrinsic feature of the algorithm is that when ... |

12 |
Remarques sur l'analyze de Fourier �a Fen^etre
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(Show Context)
Citation Context ...from time-frequency lattices, # k = k## and # # = ### , and # # {0, #/2}; with ## and ## chosen su#ciently fine these are complete. For further discussions see, e.g., [9]. Recently, Coifman and Meyer =-=[6]-=- developed the wavelet packet and cosine packet dictionaries especially to meet the computational demands of discrete-time signal processing. For 1-d discrete-time signals of length n, these dictionar... |

12 |
Commentary: Interior point methods: algorithms and formulations
- Vanderbei
(Show Context)
Citation Context ...ess in this field and the current state of the art is a#orded by the article of Lustig, Marsten, and Shanno [23] and the accompanying discussions by Bixby [1], Saunders [36], Todd [38], and Vanderbei =-=[39]. Much of -=-the rapid expansion in the size of linear programs solved is due to the "interior-point revolution" initiated by Karmarkar's proof that a pseudo-polynomial-time algorithm could be based on a... |

9 |
Ondelettes sur l’intervalle, Rev
- Meyer
- 1992
(Show Context)
Citation Context ...ions are smooth wavelet bases, using splines or using wavelets defined recursively from twoscale filtering relations [10]. Although the rules of construction are more complicated (boundary conditions =-=[28]-=-, orthogonality versus biorthogonality [10], etc.), these have the same indexing structure as the standard Haar dictionary. In this paper, we use symmlet-8 smooth wavelets, i.e., Daubechies nearly sym... |

8 |
Nonlinear total-variation-based noise removal algorithms
- Rudin, Osher, et al.
(Show Context)
Citation Context ...p 1---it selects the average of the two frequencies in the TwinSine signal. BP correctly resolves the nonnegative doublet structure. 5.4. Total variation denoising. Recently, Rudin, Osher, and Fatemi =-=[31]-=- have called attention to the possibility of denoising images using total variation penalized least squares. More specifically, they propose the optimization problem min g 1 2 #y - g# 2 2 + # TV (g), ... |

6 |
Major Cholesky would feel proud
- Saunders
- 1994
(Show Context)
Citation Context ...rview of the recent rapid progress in this field and the current state of the art is a#orded by the article of Lustig, Marsten, and Shanno [23] and the accompanying discussions by Bixby [1], Saunders =-=[36], Todd [38-=-], and Vanderbei [39]. Much of the rapid expansion in the size of linear programs solved is due to the "interior-point revolution" initiated by Karmarkar's proof that a pseudo-polynomial-tim... |

6 |
Empirical atomic decomposition
- Donoho, Johnstone
- 1995
(Show Context)
Citation Context ...e we have suggested here -- to be applied in overcomplete as well as orthogonal settings -- therefore includes soft-thresholding in ortho-bases as a special case. Formal arguments similar to those in =-=[13]-=- can be used to give a proof that mean-squared error properties of the resulting procedure are near-optimal under certain conditions. 20 0 0.5 1 -20 0 20 (d) Recovered: BOB 0 0.5 1 -20 0 20 (e) Recove... |