## On minimum entropy segmentation (1994)

Venue: | In |

Citations: | 17 - 4 self |

### BibTeX

@INPROCEEDINGS{Donoho94onminimum,

author = {David L. Donoho},

title = {On minimum entropy segmentation},

booktitle = {In},

year = {1994},

pages = {233--269},

publisher = {Academic Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

We describe segmented multiresolution analyses of [0; 1]. Such multiresolution analyses lead to segmented wavelet bases which are adapted to discontinuities, cusps, etc., at a given location 2 [0; 1]. Our approach emphasizes the idea of average-interpolation { synthesizing a smooth function on the line having prescribed boxcar averages. This particular approach leads to methods with subpixel resolution and to wavelet transforms with the advantage that, for a signal of length n, all n pixel-level segmented wavelet transforms can be computed simultaneously in a total time and space which are both O(n log(n)). We consider the search for a segmented wavelet basis which, among all such segmented bases, minimizes the \entropy " of the resulting coe cients. Fast access to all segmentations enables fast search for a best segmentation. When the \entropy " is Stein's Unbiased Risk Estimate, one obtains a new method of edge-preserving de-noising. When the \entropy " is the ` 2-energy, one obtains a new multi-resolution edge detector, which works not only for step discontinuities but also for cusp and higherorder discontinuities, and in a near-optimal fashion in the presence of noise. We describe an iterative approach, Segmentation Pursuit, for identifying edges by the fast segmentation algorithm and removing them from the data. 1 Key Words and Phrases. Segmented Multi-Resolution analysis. Edge-Preserving Image processing methods. Edge detection. Sub-pixel resolution.

### Citations

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1047 | Matching pursuits with time-frequency dictionaries
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Citation Context ...is component. r := r \Gamma f ];t i 5. Unless satisfied, set i := i + 1 and go to 2. We call this "segmentation pursuit"; it is in formal analogy with "projection pursuit" [27] and=-= "matching pursuit" [31]-=-. Figure 6.4 gives the result of applying segmentation pursuit to object Bumps; shown are the functions f #;t i extracted in the first ten iterations of the procedure. Several issues deserve comment. ... |

710 |
Image coding using wavelet transform
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Citation Context ...ay from the segmentation points, use the earlier procedure to find the polynomialsj;k of degree D which generates the same averages in the neighborhood (a j;k 0 ; k 0 = k \Gamma D=2; : : : ; k +D=2). =-=[2]-=- At each site k which is at most D=2 sites away from the boundaries 0 and 1 find the polynomialsj;k of degree D which generates the same averages in the neighborhood (a j;k 0 ) k 0 2N(k) , where N(k) ... |

691 |
An htroduction to Wavelets
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Citation Context ...the left half of the sub-interval we get a j+1;2k = Ave j+1;2ksj;k on the right half a j+1;2k+1 = Ave j+1;2k+1sj;k : 13 In case [3b], the steps are the same, only using the piecewise polynomial j;k . =-=[5]-=- After having synthesized all the a j+1;k 's, set j := j + 1 and goto [1] This segmented average-interpolating resolution has a variety of properties; a key one being that ifsis a piecewise polynomial... |

678 | AdaptiriP to Unknown Smoothness via Wavelet Shrinkas - Donoho, Johnstone - 1995 |

550 | Ideal spatial adaption via Wavelet Shrinkage”, Biometrika - Donoho, Johnstone - 1994 |

493 | Entropy-based algorithms for best basis selection - Coifman, Wickerhauser - 1992 |

405 | Projection pursuit regression
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Citation Context ...e segmentation. 4. Remove this component. r := r \Gamma f ];t i 5. Unless satisfied, set i := i + 1 and go to 2. We call this "segmentation pursuit"; it is in formal analogy with "proje=-=ction pursuit" [27] and "-=-;matching pursuit" [31]. Figure 6.4 gives the result of applying segmentation pursuit to object Bumps; shown are the functions f #;t i extracted in the first ten iterations of the procedure. Seve... |

400 |
Fast wavelet transforms and numerical algorithms
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Citation Context ...efine the mock-averages at the next finer scale as averages of that polynomial. On the left half of the sub-interval we get a j+1;2k = Ave j+1;2ksj;k on the right half a j+1;2k+1 = Ave j+1;2k+1sj;k : =-=[3]-=- After having synthesized all the a j+1;k 's, set j := j + 1 and goto [1] This refinement scheme, which is analogous to the interpolating refinement scheme of Deslauriers-Dubuc [13, 26], is discussed ... |

299 | Image compression through wavelet transform coding - DeVore, Jawerth, et al. - 1992 |

239 | Wavelet shrinkage: asymptopia
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Citation Context ...d Wim Sweldens for informing us of the priority of Deng, Jawerth, Peters, and Sweldens in introducing fast pixel-level segmentations. 1 Introduction 1.1 Improved De-Noising Several recent papers (see =-=[25]-=- and references therein) have shown that wavelet methods can be used to de-noise data of various kinds, obtaining a level of theoretical performance not approached by pre-existing methods. In general,... |

164 |
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Citation Context ... Ave j+1;2k+1sj;k : [3] After having synthesized all the a j+1;k 's, set j := j + 1 and goto [1] This refinement scheme, which is analogous to the interpolating refinement scheme of Deslauriers-Dubuc =-=[13, 26]-=-, is discussed at length in Donoho(1993b). The main point is that it defines a sequence of refinements which in some sense converge: the function A j 0 ;D (t) = P k a j 0 ;k 1 [k=2 j 0 ;(k+1)=2 j 0 ] ... |

141 | Unconditional bases are optimal bases for data compression and for statistical estimation - Donoho - 1993 |

106 |
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Citation Context ... Ave j+1;2k+1sj;k : [3] After having synthesized all the a j+1;k 's, set j := j + 1 and goto [1] This refinement scheme, which is analogous to the interpolating refinement scheme of Deslauriers-Dubuc =-=[13, 26]-=-, is discussed at length in Donoho(1993b). The main point is that it defines a sequence of refinements which in some sense converge: the function A j 0 ;D (t) = P k a j 0 ;k 1 [k=2 j 0 ;(k+1)=2 j 0 ] ... |

105 |
Stationary subdivision
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(Show Context)
Citation Context ...d left of the segmentation point, respectively; the constraint is that the piecewise polynomialsj;k which issL on the left ofsandsR on the right ofsshould have an average equal to the average a j;k . =-=[4]-=- In cases [1], [2], and [3a], define the mock-averages at the next finer scale as averages of the polynomial. On the left half of the sub-interval we get a j+1;2k = Ave j+1;2ksj;k on the right half a ... |

97 | P.: Multiresolution analysis, wavelets, and fast algorithms on an interval - Cohen, Daubechies, et al. - 1993 |

69 | Wavelets on a closed subsets of the real line
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Citation Context ...stract We describe segmented multiresolution analyses of [0; 1]. Such multiresolution analyses lead to segmented wavelet bases which are adapted to discontinuities, cusps, etc., at a given locations2 =-=[0; 1]-=-. Our approach emphasizes the idea of average-interpolation -- synthesizing a smooth function on the line having prescribed boxcar averages. This particular approach leads to methods with subpixel res... |

54 | Smooth wavelet decompositions with blocky coefficient kernels
- Donoho
- 1993
(Show Context)
Citation Context ...ystem arose before, without the average-interpolation interpretation, in Cohen, Daubechies, and Feauveau (1990), where it was called a system biorthogonal to spline of degree 0; see the discussion in =-=[19]-=-. Corresponding to these schemes on the line are boundary-corrected multiresolutions on the interval [0; 1]. These are built from a boundary-corrected refinement scheme for the interval. This scheme h... |

42 | Wavelets on a bounded interval, in Numerical Methods of Approximation Theory - Chui, Quak - 1992 |

42 | Fast wavelet techniques for near optimal image processing - DeVore, Lucier - 1992 |

39 | Wavelet transform maxima and multiscale edges,” in Wavelets and their - Mallat, Zhong - 1992 |

13 |
New Minimax Theorems, Thresholding, and Adaptation
- Donoho, Johnstone
- 1992
(Show Context)
Citation Context ...d makes no assumptions whatever, and achieves a near-optimal rate for the given type of singularity without advance knowledge of the type of singularity. We conjecture (based on related experience in =-=[22]-=-) that if one wants to adapt to an unknown type of singularity, the logarithm terms can not be avoided. We carried out a small simulation experiment to assess the performance of the estimator. In the ... |

11 | Minimax estimation by wavelet shrinkage - Donoho, Johnstone - 1998 |

9 | Wavelet Shrinkage and W.V.D. – A Ten-Minute Tour - Donoho - 1993 |

7 | Wavelets and image compression - Lucier - 1992 |

6 | Wavelet Probing for Compression-based segmentation - Deng, Jawerth, et al. - 1993 |

2 | De-Noising via Soft Thresholding. To appear - Donoho - 1992 |

1 | Th`ese - Cohen - 1990 |

1 | Two recent results: Wavelet bases for the interval and biorthogonal wavelets diagonalizing the derivative operator - Daubechies - 1993 |

1 | Second-Generation Compact Coding from Wavelet Edges - Mallat, Froment - 1992 |

1 |
D'etection et localisation d'un obstacle grace `a l'utilisation d'une transform'ee en ondelettes
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(Show Context)
Citation Context ...to obtain for experimental purposes. The author sees two principal issues raised by this work. 39 1. There are many existing edge-detection schemes. A number of these are based on wavelets themselves =-=[32, 33]-=-. The method we have discussed here is different, in that it tries to be true to the internal logic of the waveletswavelet packets paradigm. 2. Fidelity to the internal logic of wavelets puts us in a ... |

1 |
D'etecteurs de contours optimaux bas'es sur la transformation en ondelettes
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- 1993
(Show Context)
Citation Context ...to obtain for experimental purposes. The author sees two principal issues raised by this work. 39 1. There are many existing edge-detection schemes. A number of these are based on wavelets themselves =-=[32, 33]-=-. The method we have discussed here is different, in that it tries to be true to the internal logic of the waveletswavelet packets paradigm. 2. Fidelity to the internal logic of wavelets puts us in a ... |