## Multiresolution signal decomposition schemes. Part 1: Linear and morphological pyramids (2000)

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Venue: | IEEE TRANSACTIONS ON IMAGE PROCESSING |

Citations: | 43 - 6 self |

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@ARTICLE{Goutsias00multiresolutionsignal,

author = {John Goutsias and Henk J. A. M. Heijmans},

title = {Multiresolution signal decomposition schemes. Part 1: Linear and morphological pyramids},

journal = {IEEE TRANSACTIONS ON IMAGE PROCESSING},

year = {2000},

volume = {9},

pages = {1897--1913}

}

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### Citations

2359 | A theory for multiresolution signal decomposition: The wavelet representation
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Citation Context ...signal decomposition in term of bandpassfiltered copies of the original signal. Moreover, there is neurophysiological evidence that the human visual system indeed uses a similar kind of decomposition =-=[17]-=-. This tool has been one of the most popular multiresolution schemes used in image processing and computer vision. The previously mentioned scheme leaves a lot to be desired however, due to aliasing a... |

2020 |
A Wavelet Tour of Signal Processing
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Citation Context ...ify signal representation whereas the synthesis operators are designed to undo as much as possible this loss of information. This is a widely accepted approach to multiresolution signal decomposition =-=[3, 33, 16]-=-. Moreover, as discussed in the introduction, the analysis/synthesis operators depend on the application at hand and a sound theory should be able to treat them from a general point of view. Motivated... |

1720 | lectures on wavelets - Daubechies, Ten - 1992 |

1574 | Orthonormal bases of compactly supported wavelets," Communications on pure and applied mathematics - Daubechies - 1988 |

1081 |
Image Analysis and Mathematical Morphology
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Citation Context ...es various existing multiresolution approaches, such as linear (e.g. Laplacian) pyramids [1], morphological pyramids [31, 32, 12, 13, 2, 21, 23, 24, 25], median pyramids [29], morphological skeletons =-=[27, 18, 15]-=-, and granulometries [4, 20, 27, 22, 7]. In the earliest multiresolution approaches to signal and image processing, the most popular way was to obtain a coarse level signal by subsampling a fine resol... |

1014 | The Laplacian Pyramid as a Compact Image Code
- Burt, Adelson
- 1983
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Citation Context ...esis. The operators involved in this scheme can be linear or nonlinear (morphological). Such a scheme encompasses various existing multiresolution approaches, such as linear (e.g. Laplacian) pyramids =-=[1]-=-, morphological pyramids [31, 32, 12, 13, 2, 21, 23, 24, 25], median pyramids [29], morphological skeletons [27, 18, 15], and granulometries [4, 20, 27, 22, 7]. In the earliest multiresolution approac... |

658 | Watersheds in digital spaces: an efficient algorithm based onimmersion simulations - Vincent, Soille - 1991 |

525 |
Human and machine recognition of faces: a survey
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Citation Context ...) are well preserved at lower resolutions. This property may turn out to be particularly useful in wavelet-based pattern recognition approaches as, for example, wavelet-based face recognition schemes =-=[41]-=-. Despite all these attractive properties, a number of open theoretical and practical questions need to be addressed before such tools become useful in signal processing and analysis applications. For... |

473 |
Wavelets and subband coding
- Vetterli, Kovačević
- 1995
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Citation Context ...o signal and image processing, the most popular way was to obtain a coarse level signal by subsampling a fine resolution signal, after linear smoothing, in order to remove high frequencies (e.g., see =-=[33]-=-). A detail pyramid canthenbe derived by subtracting from each level an interpolated version of the next coarser level; the best-known example is the Laplacian pyramid [1]. From a frequency point of v... |

437 | and W.Swwldens, “Factoring wavelet transforms into lifting steps
- Daubechies
- 1998
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Citation Context ...d scaled signal does not satisfy the constraint, we may choose in such a way that , given by (38), does satisfy this constraint. We refer to the work of Sweldens [23]–[25] and Daubechies and Sweldens =-=[38]-=- for more details. The update step in (38) gives rise to the diagrams depicted in Fig. 12. It is clear that the input signal can be reconstructed from and , since Thus, we arrive at the update lifting... |

434 | lifting scheme: A custom- design construction of biorthogonal wavelets - Sweldens - 1996 |

418 |
Random Sets and Integral Geometry
- Matheron
- 1974
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Citation Context ...on approaches, such as linear (e.g. Laplacian) pyramids [1], morphological pyramids [31, 32, 12, 13, 2, 21, 23, 24, 25], median pyramids [29], morphological skeletons [27, 18, 15], and granulometries =-=[4, 20, 27, 22, 7]-=-. In the earliest multiresolution approaches to signal and image processing, the most popular way was to obtain a coarse level signal by subsampling a fine resolution signal, after linear smoothing, i... |

375 | The Lifting Scheme: A Construction Of Second Generation Wavelets - Sweldens |

262 | Wavelet transforms that map integers to integers
- Calderbank, Daubechies, et al.
- 1998
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Citation Context ...inal signal . It furthermore guarantees that, if the original signal is discretevalued, the scaled signals will be discrete-valued as well, a highly desirable property in lossless coding applications =-=[37]-=-. Moreover, the morphological Haar wavelet decomposition scheme may do a better job in preserving edges in , as compared to the linear case. This is expected, since the signal analysis filters in the ... |

218 | The morphological approach to segmentation : the watershed transformation - Beucher, Meyer - 1992 |

200 | Anisotropic diffusion in image processing
- Weickert
- 1998
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Citation Context ...the relationship between the discrete nonlinear approach presented in this paper and another nonlinear multiresolution approach to signal analysis known as nonlinear (morphological) scale spaces [42]–=-=[46]-=-. In fact, due to the popularity of nonlinear scale spaces in signal analysis, it may be attractive to investigate the design of nonlinear filter banks by means of discretizing continuous morphologica... |

172 | An Image Multiresolution Representation for Lossless and Lossy Compression
- Said, Peadman
- 1996
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Citation Context ...iterature for several years, and has been successfully used in medical imaging for lossless compression [33]. During the years, several modifications and generalizations have been proposed, e.g., see =-=[34]-=-. We should point out here that certain continuity issues may arise in the case of an infinite-level wavelet decomposition scheme. However, these issues, which become manifest in the case of infinite ... |

149 | The lifting scheme: A new philosophy in biorthogonal wavelet constructions
- Sweldens
- 1995
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Citation Context .... Although wavelet decomposition is a linear signal analysis tool, it is starting to be recognized that nonlinear extensions are possible [4]–[22]. The lifting scheme, recently introduced by Sweldens =-=[23]-=-–[25] (see also [26] for a predecessor to this scheme, known as a “ladder network”), has provided a useful way to construct nonlinear wavelet decompositions. The enormous flexibility and freedom that ... |

117 | Morphological Image Operators - Heijmans - 1994 |

114 |
Pattern spectrum and multiscale shape representation
- Maragos
- 1989
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Citation Context ...∈ Vj+1, j ≥0 y ′ j =x′ j+1 − x′ j , x = ∞� y ′ j. j=0 In the literature, the decomposition of a signal x into the detail signals {...,y ′ 1,y ′ 0,y0,y1,...} is called the discrete size transform of x =-=[19]-=-. If the space E is finite or countably infinite, then {..., |y ′ 1 |, |y′ 0 |, |y0|, |y1|,...}, where |x| = � n∈E |x(n)|, is called the pattern spectrum of x [19]. 5.2.2. Morphological skeleton decom... |

91 | Nonlinear wavelet transforms for image coding via lifting, submitted to
- Jr, Davis, et al.
(Show Context)
Citation Context ...struction filter banks by considering a triangular form of the polyphase representation of a filter bank. The resulting approach is more or less identical to the lifting scheme. In four recent papers =-=[10]-=-, [12], [13], [22], Claypoole et al. use the lifting scheme to build nonlinear wavelet transforms. In the first paper [10], they propose an adaptive lifting step using a nonlinear selection criterion.... |

69 | Morphological skeleton representation and coding of binary images
- Maragos, Schafer
- 1986
(Show Context)
Citation Context ...es various existing multiresolution approaches, such as linear (e.g. Laplacian) pyramids [1], morphological pyramids [31, 32, 12, 13, 2, 21, 23, 24, 25], median pyramids [29], morphological skeletons =-=[27, 18, 15]-=-, and granulometries [4, 20, 27, 22, 7]. In the earliest multiresolution approaches to signal and image processing, the most popular way was to obtain a coarse level signal by subsampling a fine resol... |

57 | Evolution equations for continuous-scale morphology - Brockett, Maragos - 1992 |

52 | Scale-space properties of the multiscale morphological dilationerosion - Jackway, Deriche - 1996 |

41 |
Heijmans, Morphological Image Operators
- M
- 1994
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Citation Context ...is replaced by an erosion (or dilation), i.e., by taking the minimum (or maximum) over two samples. Readers who are unfamiliar with the basic concepts of mathematical morphology are referred to [35], =-=[36]-=-. Let be the lattice of doubly infinite real-valued sequences. Define the analysis and synthesis operators as (14) (15) (16) (17) Here “ ” denotes minimum and “ ” denotes maximum. In Part I, we have s... |

39 |
Formalization and computational aspects of image analysis
- Alvarez, Morel
- 1999
(Show Context)
Citation Context ...gate the relationship between the discrete nonlinear approach presented in this paper and another nonlinear multiresolution approach to signal analysis known as nonlinear (morphological) scale spaces =-=[42]-=-–[46]. In fact, due to the popularity of nonlinear scale spaces in signal analysis, it may be attractive to investigate the design of nonlinear filter banks by means of discretizing continuous morphol... |

38 |
M-Band Nonlinear Subband Decompositions with Perfect Reconstruction
- Hampson, Pesquet
- 1998
(Show Context)
Citation Context ...onstruction. However, these authors did not have at their disposal the lifting scheme, which was developed during the same period [23]–[25]. The same remark applies to the work of Hampson and Pesquet =-=[8]-=-, [11], [17] who developed nonlinear perfect reconstruction filter banks by considering a triangular form of the polyphase representation of a filter bank. The resulting approach is more or less ident... |

38 |
den Enden, “New networks for perfect inversion and perfect reconstruction
- Bruekers, van
- 1992
(Show Context)
Citation Context ...ecomposition is a linear signal analysis tool, it is starting to be recognized that nonlinear extensions are possible [4]–[22]. The lifting scheme, recently introduced by Sweldens [23]–[25] (see also =-=[26]-=- for a predecessor to this scheme, known as a “ladder network”), has provided a useful way to construct nonlinear wavelet decompositions. The enormous flexibility and freedom that the lifting scheme o... |

36 |
A discrete-time multiresolution theory
- Rioul
- 1993
(Show Context)
Citation Context ...and ψ↓ j are and ψ ↓ τ = τ 2 ψ ↓ . (4.1)s12 A straightforward computation shows that there exist convolution kernels ˜ h,h ∈ ℓ 2 (Z) such that ψ ↑ and ψ ↓ are of the following general form (see Rioul =-=[26]-=-): ψ ↑ (x)(n) = ψ ↓ (x)(n) = ∞� k=−∞ ∞� k=−∞ ˜h(2n − k)x(k) (4.2) h(n − 2k)x(k). (4.3) Note that the analysis operator ψ↑ canberegardedasalinearconvolutionwithkernel˜ hfollowed by a downsampling at ra... |

36 | L.: An overview of morphological filtering
- Serra, Vincent
- 1992
(Show Context)
Citation Context ...s form one of the most practical concepts in mathematical morphology [27, 7]. They fit, in a most natural way, into a pyramid framework. The same appears to be true for alternating sequential filters =-=[28, 7]-=-. • Morphological pyramids: Morphological pyramids have been proposed and applied in [31, 32, 12, 13, 29, 2, 21, 23, 24, 25]. We show how such pyramids fit into our general framework, and present some... |

30 |
High Compression Image Coding Using an Adaptive Morphological Subband Decomposition
- Egger, Kunt
- 1995
(Show Context)
Citation Context ...r approach however does not guarantee perfect reconstruction. In 1994, Egger and Li [4] proposed a nonlinear decomposition scheme with perfect reconstruction based on a median-type operator (see also =-=[6]-=-). Independently, Florêncio and Schafer [5] have presented a similar decomposition; see also [7, Ch. 7]. More recently, Queiroz et al. [21] proposed a nonlinear wavelet decomposition, corresponding to... |

30 | The morphological structure of images: the differential equations of morphological scale–space - Boomgaard, Smeulders - 1994 |

29 | Watersheds in Digital Spaces: An E_cient Al-gorithm Based on Immersion Simulations - Vincent, Soille - 1991 |

28 | The World According to Wavelets: The Story of a - Hubbard - 1996 |

26 | Adaptive wavelet transforms via lifting
- Claypoole, Baraniuk, et al.
- 1998
(Show Context)
Citation Context ...ion filter banks by considering a triangular form of the polyphase representation of a filter bank. The resulting approach is more or less identical to the lifting scheme. In four recent papers [10], =-=[12]-=-, [13], [22], Claypoole et al. use the lifting scheme to build nonlinear wavelet transforms. In the first paper [10], they propose an adaptive lifting step using a nonlinear selection criterion. In th... |

24 |
Morphological representation of discrete and binary images
- Goutsias, Shonfeld
- 1991
(Show Context)
Citation Context ...r larger than the detail signal in the Lantuéjoul formula (5.28). It therefore gives rise to a more efficient compression. This skeleton decomposition has been found earlier by Goutsias and Schonfeld =-=[6]-=-. Figure 9(c) depicts the result of applying this decomposition to the binary image in (a). The resulting image is different than the one depicted in Figure 9(b) in 66 pixels. Since the image depicted... |

23 |
A Morphological Pyramidal Image Decomposition
- Toet
- 1989
(Show Context)
Citation Context ... in this scheme can be linear or nonlinear (morphological). Such a scheme encompasses various existing multiresolution approaches, such as linear (e.g. Laplacian) pyramids [1], morphological pyramids =-=[31, 32, 12, 13, 2, 21, 23, 24, 25]-=-, median pyramids [29], morphological skeletons [27, 18, 15], and granulometries [4, 20, 27, 22, 7]. In the earliest multiresolution approaches to signal and image processing, the most popular way was... |

20 | Anisotropic di usion in image processing - Weickert - 1998 |

19 |
The algebraic basis of mathematical morphology. part i: Dilations and erosions
- Heijmans, Ronse
- 1990
(Show Context)
Citation Context ...n is given in terms of adjunctions (e,d) on the complete lattice T and is closely related to the representation of translation invariant adjunctions for grayscale functions in mathematical morphology =-=[11, 7]-=-. 5.1. Proposition. Let (ψ ↑ ,ψ ↓ ) be an adjunction on Fun(Z d , T ). The translation invariance condition ψ ↑ τ 2 = τψ ↑ implies that ψ ↓ τ = τ 2 ψ ↓ and vice versa. Every adjunction satisfying thes... |

19 |
A binary wavelet decomposition of binary images
- Tewfik, Swanson
- 1996
(Show Context)
Citation Context ... of binary images. This is of particular interest to document image processing, analysis, and compression applications (and other industrial applications) and is important on its own right (e.g., see =-=[40]-=- for a recent work on constructing wavelet decomposition schemes for binary images). 4) Due to the nonlinear nature of the proposed signal analysis operators, important geometric information (e.g., ed... |

12 |
Morphological sampling
- Heijmans, Toet
- 1991
(Show Context)
Citation Context ... in this scheme can be linear or nonlinear (morphological). Such a scheme encompasses various existing multiresolution approaches, such as linear (e.g. Laplacian) pyramids [1], morphological pyramids =-=[31, 32, 12, 13, 2, 21, 23, 24, 25]-=-, median pyramids [29], morphological skeletons [27, 18, 15], and granulometries [4, 20, 27, 22, 7]. In the earliest multiresolution approaches to signal and image processing, the most popular way was... |

10 |
Non-expansive pyramid for image coding using non-linear filter bank
- Queiroz, Florêncio, et al.
- 1998
(Show Context)
Citation Context ... nonlinear wavelet decompositions. The enormous flexibility and freedom that the lifting scheme offers has challenged researchers to develop various nonlinear wavelet transforms [4]–[13], [17], [19], =-=[21]-=-, [22], [27]. The literature on nonlinear wavelet decompositions, or critically decimated nonlinear filter banks as they are sometimes called, is not extensive. In 1991, Pei and Chen [28], [29] were a... |

9 | Reconstructing Non-Linear Filter Banks - Florencio, Schafer - 1996 |

8 | New networks for perfect inversion and perfect reconstruction - Bruekers, Enden - 1992 |

8 | The morphological structure of images: the di erential equations of morphological scale space - Boomgaard, Smeulders - 1994 |

7 |
A Non-Expansive Pyramidal Morphological Image Coder
- Florencio, Schafer
- 1994
(Show Context)
Citation Context ...ct reconstruction. In 1994, Egger and Li [4] proposed a nonlinear decomposition scheme with perfect reconstruction based on a median-type operator (see also [6]). Independently, Florêncio and Schafer =-=[5]-=- have presented a similar decomposition; see also [7, Ch. 7]. More recently, Queiroz et al. [21] proposed a nonlinear wavelet decomposition, corresponding to the quincunx sampling grid, for low-comple... |

7 |
lifting scheme: A construction of second generation wavelets
- “The
- 1998
(Show Context)
Citation Context ...hough wavelet decomposition is a linear signal analysis tool, it is starting to be recognized that nonlinear extensions are possible [4]–[22]. The lifting scheme, recently introduced by Sweldens [23]–=-=[25]-=- (see also [26] for a predecessor to this scheme, known as a “ladder network”), has provided a useful way to construct nonlinear wavelet decompositions. The enormous flexibility and freedom that the l... |

7 |
Blume,“Hierarchical image decomposition and filtering using the S-transform
- Ranganath, H
- 1988
(Show Context)
Citation Context ...he standard addition. The -transform, where “ ” stands for “sequential,” has been known in the literature for several years, and has been successfully used in medical imaging for lossless compression =-=[33]-=-. During the years, several modifications and generalizations have been proposed, e.g., see [34]. We should point out here that certain continuity issues may arise in the case of an infinite-level wav... |

6 | Morphological Image Representation for Coding Applications, D.Sc. Thesis (in English - Kresch - 1995 |

6 | Color image coding using morphological pyramid decomposition
- Overturf, Comer, et al.
- 1995
(Show Context)
Citation Context ... in this scheme can be linear or nonlinear (morphological). Such a scheme encompasses various existing multiresolution approaches, such as linear (e.g. Laplacian) pyramids [1], morphological pyramids =-=[31, 32, 12, 13, 2, 21, 23, 24, 25]-=-, median pyramids [29], morphological skeletons [27, 18, 15], and granulometries [4, 20, 27, 22, 7]. In the earliest multiresolution approaches to signal and image processing, the most popular way was... |