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Journal articles on the topic 'Scale decomposition'

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Author: Grafiati

Published: 10 December 2022

Last updated: 28 January 2023

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1

Mendez, M. A., M. Balabane, and J. M. Buchlin. "Multi-scale proper orthogonal decomposition of complex fluid flows." Journal of Fluid Mechanics 870 (May 15, 2019): 988–1036. http://dx.doi.org/10.1017/jfm.2019.212.

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Data-driven decompositions are becoming essential tools in fluid dynamics, allowing for tracking the evolution of coherent patterns in large datasets, and for constructing low-order models of complex phenomena. In this work, we analyse the main limits of two popular decompositions, namely the proper orthogonal decomposition (POD) and the dynamic mode decomposition (DMD), and we propose a novel decomposition which allows for enhanced feature detection capabilities. This novel decomposition is referred to as multi-scale proper orthogonal decomposition (mPOD) and combines multi-resolution analysis (MRA) with a standard POD. Using MRA, the mPOD splits the correlation matrix into the contribution of different scales, retaining non-overlapping portions of the correlation spectra; using the standard POD, the mPOD extracts the optimal basis from each scale. After introducing a matrix factorization framework for data-driven decompositions, the MRA is formulated via one- and two-dimensional filter banks for the dataset and the correlation matrix respectively. The validation of the mPOD, and a comparison with the discrete Fourier transform (DFT), DMD and POD are provided in three test cases. These include a synthetic test case, a numerical simulation of a nonlinear advection–diffusion problem and an experimental dataset obtained by the time-resolved particle image velocimetry (TR-PIV) of an impinging gas jet. For each of these examples, the decompositions are compared in terms of convergence, feature detection capabilities and time–frequency localization.

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2

Schmidt, Marie Foged, Martin Benning, and Carola-Bibiane Schönlieb. "Inverse scale space decomposition." Inverse Problems 34, no. 4 (March 13, 2018): 045008. http://dx.doi.org/10.1088/1361-6420/aab0ae.

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3

Aluie, Hussein. "Scale decomposition in compressible turbulence." Physica D: Nonlinear Phenomena 247, no. 1 (March 2013): 54–65. http://dx.doi.org/10.1016/j.physd.2012.12.009.

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4

Camps, O. I., T. Kanungo, and R. M. Haralick. "Gray-scale structuring element decomposition." IEEE Transactions on Image Processing 5, no. 1 (January 1996): 111–20. http://dx.doi.org/10.1109/83.481675.

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5

Führ, Hartmut, and Azita Mayeli. "Homogeneous Besov Spaces on Stratified Lie Groups and Their Wavelet Characterization." Journal of Function Spaces and Applications 2012 (2012): 1–41. http://dx.doi.org/10.1155/2012/523586.

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We establish wavelet characterizations of homogeneous Besov spaces on stratified Lie groups, both in terms of continuous and discrete wavelet systems. We first introduce a notion of homogeneous Besov spaceB˙p,qsin terms of a Littlewood-Paley-type decomposition, in analogy to the well-known characterization of the Euclidean case. Such decompositions can be defined via the spectral measure of a suitably chosen sub-Laplacian. We prove that the scale of Besov spaces is independent of the precise choice of Littlewood-Paley decomposition. In particular, different sub-Laplacians yield the same Besov spaces. We then turn to wavelet characterizations, first via continuous wavelet transforms (which can be viewed as continuous-scale Littlewood-Paley decompositions), then via discretely indexed systems. We prove the existence of wavelet frames and associated atomic decomposition formulas for all homogeneous Besov spacesB˙p,qswith1≤p,q<∞ands∈ℝ.

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6

Moktadir, Z. "Scale decomposition of molecular beam epitaxy." Journal of Physics: Condensed Matter 20, no. 23 (May 13, 2008): 235240. http://dx.doi.org/10.1088/0953-8984/20/23/235240.

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7

Bertsekas, D. P. "Thevenin decomposition and large-scale optimization." Journal of Optimization Theory and Applications 89, no. 1 (April 1996): 1–15. http://dx.doi.org/10.1007/bf02192638.

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8

Ji, Jingyu, Yuhua Zhang, Yongjiang Hu, Yongke Li, Changlong Wang, Zhilong Lin, Fuyu Huang, and Jiangyi Yao. "Fusion of Infrared and Visible Images Based on Three-Scale Decomposition and ResNet Feature Transfer." Entropy 24, no. 10 (September 24, 2022): 1356. http://dx.doi.org/10.3390/e24101356.

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Image fusion technology can process multiple single image data into more reliable and comprehensive data, which play a key role in accurate target recognition and subsequent image processing. In view of the incomplete image decomposition, redundant extraction of infrared image energy information and incomplete feature extraction of visible images by existing algorithms, a fusion algorithm for infrared and visible image based on three-scale decomposition and ResNet feature transfer is proposed. Compared with the existing image decomposition methods, the three-scale decomposition method is used to finely layer the source image through two decompositions. Then, an optimized WLS method is designed to fuse the energy layer, which fully considers the infrared energy information and visible detail information. In addition, a ResNet-feature transfer method is designed for detail layer fusion, which can extract detailed information such as deeper contour structures. Finally, the structural layers are fused by weighted average strategy. Experimental results show that the proposed algorithm performs well in both visual effects and quantitative evaluation results compared with the five methods.

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9

Li, Ming Jing, Xiao Li Wang, and Yu Bing Dong. "Research and Development of Multi-Scale to Pixel-Level Image Fusion." Applied Mechanics and Materials 448-453 (October 2013): 3625–28. http://dx.doi.org/10.4028/www.scientific.net/amm.448-453.3625.

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Image fusion method based on image multi-scale decomposition is a kind of fusion method of multi-scale, multi-resolution image fusion. Its fusion process realize in different scales and different spatial resolution and different decomposition layer. Fusion effects based on multi-scale decomposition algorithm can obviously improve compared to the simple fusion methods. Among the fusion algorithm based on multi-scale to pixel-level image fusion, Pyramid decomposition and wavelet decomposition are widely used, the original image is decomposed to convert the original image domain to transform domain, and then, fusion process realized in transform domain according to certain rules of image fusion. Basic principle of fusion process was introduced in detail in this paper, and pixel level fusion algorithm at present was summed up. Simulation results on fusion are presented to illustrate the proposed fusion scheme. In practice, fusion algorithm was selected according to imaging characteristics being retained.

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10

Abolfazl Hajisami and Dario Pompili. "MSICA: multi-scale signal decomposition based on independent component analysis with application to denoising and reliable multi-channel." ITU Journal on Future and Evolving Technologies 1, no. 1 (December 11, 2020): 25–35. http://dx.doi.org/10.52953/psmv3163.

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Multi-scale decomposition is a signal description method in which the signal is decomposed into multiple scales, which has been shown to be a valuable method in information preservation. Much focus on multi-scale decomposition has been based on scale-space theory and wavelet transform. In this article, a new powerful method to perform multi-scale decomposition exploiting Independent Component Analysis (ICA), called MSICA, is proposed to translate an original signal into multiple statistically independent scales. It is proven that extracting the independent components of the even and odd samples of a digital signal results in the decomposition of the same into approximation and detail. It is also proven that the whitening procedure in ICA is equivalent to a filter bank structure. Performance results of MSICA in signal denoising are presented; also, the statistical independency of the approximation and detail is exploited to propose a novel signal-denoising strategy for multi-channel noisy transmissions aimed at improving communication reliability by exploiting channel diversity.

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11

Lawler, Joshua J., and Thomas C. Edwards. "A Variance-decomposition Approach to Investigating Multiscale Habitat Associations." Condor 108, no. 1 (February 1, 2006): 47–58. http://dx.doi.org/10.1093/condor/108.1.47.

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Abstract The recognition of the importance of spatial scale in ecology has led many researchers to take multiscale approaches to studying habitat associations. However, few of the studies that investigate habitat associations at multiple spatial scales have considered the potential effects of cross-scale correlations in measured habitat variables. When cross-scale correlations in such studies are strong, conclusions drawn about the relative strength of habitat associations at different spatial scales may be inaccurate. Here we adapt and demonstrate an analytical technique based on variance decomposition for quantifying the influence of cross-scale correlations on multiscale habitat associations. We used the technique to quantify the variation in nest-site locations of Red-naped Sapsuckers (Sphyrapicus nuchalis) and Northern Flickers (Colaptes auratus) associated with habitat descriptors at three spatial scales. We demonstrate how the method can be used to identify components of variation that are associated only with factors at a single spatial scale as well as shared components of variation that represent cross-scale correlations. Despite the fact that no explanatory variables in our models were highly correlated (r < 0.60), we found that shared components of variation reflecting cross-scale correlations accounted for roughly half of the deviance explained by the models. These results highlight the importance of both conducting habitat analyses at multiple spatial scales and of quantifying the effects of cross-scale correlations in such analyses. Given the limits of conventional analytical techniques, we recommend alternative methods, such as the variance-decomposition technique demonstrated here, for analyzing habitat associations at multiple spatial scales.

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12

Lebedinskaya, N. A., and D. M. Lebedinskii. "Multiple-scale decomposition for the Zlamal approximation." Vestnik St. Petersburg University: Mathematics 42, no. 1 (March 2009): 14–18. http://dx.doi.org/10.3103/s1063454109010038.

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13

Gkioulekas, Ioannis, Anat Levin, Frédo Durand, and Todd Zickler. "Micron-scale light transport decomposition using interferometry." ACM Transactions on Graphics 34, no. 4 (July 27, 2015): 1–14. http://dx.doi.org/10.1145/2766928.

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14

Du, Hui, Xiaogang Jin, and Xiaoyang Mao. "Digital Camouflage Images Using Two-scale Decomposition." Computer Graphics Forum 31, no. 7 (September 2012): 2203–12. http://dx.doi.org/10.1111/j.1467-8659.2012.03213.x.

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15

Barman, Siddharth, Xishuo Liu, Stark C. Draper, and Benjamin Recht. "Decomposition Methods for Large Scale LP Decoding." IEEE Transactions on Information Theory 59, no. 12 (December 2013): 7870–86. http://dx.doi.org/10.1109/tit.2013.2281372.

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16

Ammar, Hany H., and Su Deng. "Time warp simulation using time scale decomposition." ACM Transactions on Modeling and Computer Simulation 2, no. 2 (April 1992): 158–77. http://dx.doi.org/10.1145/137926.137959.

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17

Pei, Tao, Jianhuan Gao, Ting Ma, and Chenghu Zhou. "Multi-scale decomposition of point process data." GeoInformatica 16, no. 4 (August 3, 2012): 625–52. http://dx.doi.org/10.1007/s10707-012-0165-8.

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18

Finney, John D., and Bonnie S. Heck. "Matrix scaling for large-scale system decomposition." Automatica 32, no. 8 (August 1996): 1177–81. http://dx.doi.org/10.1016/0005-1098(96)00018-0.

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19

Yeong-Chyang Shih, Frank, and Owen Robert Mitchell. "Decomposition of gray-scale morphological structuring elements." Pattern Recognition 24, no. 3 (January 1991): 195–203. http://dx.doi.org/10.1016/0031-3203(91)90061-9.

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20

Schulze, Christian, Sandile Ngcobo, Michael Duparré, and Andrew Forbes. "Modal decomposition without a priori scale information." Optics Express 20, no. 25 (November 29, 2012): 27866. http://dx.doi.org/10.1364/oe.20.027866.

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21

Evangelatos, D. S. "Large Scale System Decomposition into Reachable Subsystems." IFAC Proceedings Volumes 28, no. 10 (July 1995): 769–74. http://dx.doi.org/10.1016/s1474-6670(17)51613-4.

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22

Hoff, Peter D. "Equivariant and Scale-Free Tucker Decomposition Models." Bayesian Analysis 11, no. 3 (September 2016): 627–48. http://dx.doi.org/10.1214/14-ba934.

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23

Kolda, Tamara G., and David Hong. "Stochastic Gradients for Large-Scale Tensor Decomposition." SIAM Journal on Mathematics of Data Science 2, no. 4 (January 2020): 1066–95. http://dx.doi.org/10.1137/19m1266265.

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24

Vasantharajan, S., and L. T. Biegler. "Large-scale decomposition for successive quadratic programming." Computers & Chemical Engineering 12, no. 11 (November 1988): 1087–101. http://dx.doi.org/10.1016/0098-1354(88)87031-5.

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25

Takriti, Samer, and Paul D. Gader. "Local decomposition of gray-scale morphological templates." Journal of Mathematical Imaging and Vision 2, no. 1 (October 1992): 39–50. http://dx.doi.org/10.1007/bf00123880.

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26

Hu, Shufan, Chen Zhang, Hong Liu, and Fuxin Wang. "Study on vortex shedding mode on the wake of horn/ridge ice contamination under high-Reynolds conditions." Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 233, no. 13 (March 19, 2019): 5045–56. http://dx.doi.org/10.1177/0954410019835971.

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This paper studied the unsteadiness of vortex motion produced by a three-dimensional wing section with horn/ridge ice contamination. Using improved delayed detached eddy simulation method, multi-scale vortex and their associated flow structures were successfully captured. Results have shown a diversity of unsteadiness scales at different time series, including shear layer instability, vortex pairing, co-rotating and breaking up. Proper orthogonal decomposition was then introduced to extract the characteristic vortex shedding modes with scheduling the eigenvalues λi from large to small. The dominate and secondary proper orthogonal decomposition modes under horn ice condition were displayed, which could be illustrated as fluctuations near recirculation zone, and large-scale vortex shedding/reattaching motion, respectively. The proper orthogonal decomposition modal characteristics for ridge ice showed that vortex scales varied from large to small. The trajectory of large-scale vortex reattaching and co-rotating exist simultaneously with the pressure peak and recover, which also verified the association of proper orthogonal decomposition modes with different scales of vortices. Future works would be presented on demonstration of the complex structures and the dynamic features in such flow.

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27

Abdel Kareem, Waleed, Mahmoud Abdel Aty, and Zafer M. Asker. "Fourier Decomposition and Anisotropic Diffusion Filtering of Forced Turbulence." International Journal of Applied Mechanics 09, no. 08 (December 2017): 1750121. http://dx.doi.org/10.1142/s1758825117501216.

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The Fourier decomposition and the anisotropic diffusion filtering model are used to extract various flow field scales and their coherent and incoherent parts. The different flow field scales are identified using the Fourier decomposition. Three cutoff wavenumbers are chosen to extract large, medium and fine scale velocity fields, respectively. Then, the anisotropic diffusion model is applied against the obtained velocity fields for each scale to define the coherent and incoherent parts. The forced turbulent velocities are simulated using the lattice Boltzmann method with resolutions [Formula: see text] and [Formula: see text], respectively. The Fourier decomposition of the velocity fields make the filtering process very difficult, so the anisotropic diffusion parameters should be chosen carefully to overcome the problems arising from the sharp cutoffs process. Although of such difficulties, results show that the anisotropic diffusion model successfully isolate the incoherent parts for each scale. It is shown that the incoherent parts are existed everywhere in the flow fields and they are not limited to the fine scales. The coherent fields that are identified by the anisotropic diffusion filtering method are found similar to the extracted scales by the Fourier decomposition. The incoherent regions are fewer in the large scale fields compared with that found in the intermediate and fine fields. The statistical characteristics of the three flow field scales as well as their coherent and incoherent parts are studied and compared with the universal characteristics of turbulence.

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28

Chen, Yuhao, Alexander Wong, Yuan Fang, Yifan Wu, and Linlin Xu. "Deep Residual Transform for Multi-scale Image Decomposition." Journal of Computational Vision and Imaging Systems 6, no. 1 (January 15, 2021): 1–5. http://dx.doi.org/10.15353/jcvis.v6i1.3537.

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Multi-scale image decomposition (MID) is a fundamental task in computer vision and image processing that involves the transformation of an image into a hierarchical representation comprising of different levels of visual granularity from coarse structures to fine details. A well-engineered MID disentangles the image signal into meaningful components which can be used in a variety of applications such as image denoising, image compression, and object classification. Traditional MID approaches such as wavelet transforms tackle the problem through carefully designed basis functions under rigid decomposition structure assumptions. However, as the information distribution varies from one type of image content to another, rigid decomposition assumptions lead to inefficiently representation, i.e., some scales can contain little to no information. To address this issue, we present Deep Residual Transform (DRT), a data-driven MID strategy where the input signal is transformed into a hierarchy of non-linear representations at different scales, with each representation being independently learned as the representational residual of previous scales at a user-controlled detail level. As such, the proposed DRT progressively disentangles scale information from the original signal by sequentially learning residual representations. The decomposition flexibility of this approach allows for highly tailored representations cater to specific types of image content, and results in greater representational efficiency and compactness. In this study, we realize the proposed transform by leveraging a hierarchy of sequentially trained autoencoders. To explore the efficacy of the proposed DRT, we leverage two datasets comprising of very different types of image content: 1) CelebFaces and 2) Cityscapes. Experimental results show that the proposed DRT achieved highly efficient information decomposition on both datasets amid their very different visual granularity characteristics.

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29

Xi, Yugeng, Luping Chen, and Zhongjun Zhang. "Parallel Multi-Time-Scale Decomposition Algorithm for Large-Scale Linear Systems." IFAC Proceedings Volumes 28, no. 10 (July 1995): 775–80. http://dx.doi.org/10.1016/s1474-6670(17)51614-6.

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30

Ouyang, Kewei, Yi Hou, Shilin Zhou, and Ye Zhang. "Adaptive Multi-Scale Wavelet Neural Network for Time Series Classification." Information 12, no. 6 (June 17, 2021): 252. http://dx.doi.org/10.3390/info12060252.

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Wavelet transform is a well-known multi-resolution tool to analyze the time series in the time-frequency domain. Wavelet basis is diverse but predefined by manual without taking the data into the consideration. Hence, it is a great challenge to select an appropriate wavelet basis to separate the low and high frequency components for the task on the hand. Inspired by the lifting scheme in the second-generation wavelet, the updater and predictor are learned directly from the time series to separate the low and high frequency components of the time series. An adaptive multi-scale wavelet neural network (AMSW-NN) is proposed for time series classification in this paper. First, candidate frequency decompositions are obtained by a multi-scale convolutional neural network in conjunction with a depthwise convolutional neural network. Then, a selector is used to choose the optimal frequency decomposition from the candidates. At last, the optimal frequency decomposition is fed to a classification network to predict the label. A comprehensive experiment is performed on the UCR archive. The results demonstrate that, compared with the classical wavelet transform, AMSW-NN could improve the performance based on different classification networks.

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31

Yang, Fang, Feng Gao, Pingqiao Ruan, and Huijuan Zhao. "Combined domain-decomposition and matrix-decomposition scheme for large-scale diffuse optical tomography." Applied Optics 49, no. 16 (May 28, 2010): 3111. http://dx.doi.org/10.1364/ao.49.003111.

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32

Bespalov and, Dmitriy, Ali Shokoufandeh, William C. Regli, and Wei Sun. "Scale-Space Representation and Classification of 3D Models." Journal of Computing and Information Science in Engineering 3, no. 4 (December 1, 2003): 315–24. http://dx.doi.org/10.1115/1.1633576.

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This paper presents a framework for shape matching and classification through scale-space decomposition of 3D models. The algorithm is based on recent developments in efficient hierarchical decomposition of a point distribution in metric space p,d using its spectral properties. Through spectral decomposition, we reduce the problem of matching to that of computing a mapping and distance measure between vertex-labeled rooted trees. We use a dynamic programming scheme to compute distances between trees corresponding to solid models. Empirical evaluation of the algorithm on an extensive set of 3D matching trials demonstrates both robustness and efficiency of the overall approach. Lastly, a technique for comparing shape matchers and classifiers is introduced and the scale-space method is compared with six other known shape matching algorithms.

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33

MUKADDES, A. M. M., Masao OGINO, Ryuji SHIOYA, and Hiroshi KANAYAMA. "704 A Scalable Balancing Domain Decomposition Based Preconditioner for Large Scale Thermal-Solid Coupling Problems." Proceedings of The Computational Mechanics Conference 2005.18 (2005): 523–24. http://dx.doi.org/10.1299/jsmecmd.2005.18.523.

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34

Yue, H. D., and Y. Sun. "Cooperative Coevolution with Two-Stage Decomposition for Large-Scale Global Optimization Problems." Discrete Dynamics in Nature and Society 2021 (October 29, 2021): 1–16. http://dx.doi.org/10.1155/2021/2653807.

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Cooperative coevolution (CC) is an effective framework for solving large-scale global optimization (LSGO) problems. However, CC with static decomposition method is ineffective for fully nonseparable problems, and CC with dynamic decomposition method to decompose problems is computationally costly. Therefore, a two-stage decomposition (TSD) method is proposed in this paper to decompose LSGO problems using as few computational resources as possible. In the first stage, to decompose problems using low computational resources, a hybrid-pool differential grouping (HPDG) method is proposed, which contains a hybrid-pool-based detection structure (HPDS) and a unit vector-based perturbation (UVP) strategy. In the second stage, to decompose the fully nonseparable problems, a known information-based dynamic decomposition (KIDD) method is proposed. Analytical methods are used to demonstrate that HPDG has lower decomposition complexity compared to state-of-the-art static decomposition methods. Experiments show that CC with TSD is a competitive algorithm for solving LSGO problems.

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35

Kumar, Tej, Saketh Sridhara, Bhagyashree Prabhune, and Krishnan Suresh. "Spectral decomposition for graded multi-scale topology optimization." Computer Methods in Applied Mechanics and Engineering 377 (April 2021): 113670. http://dx.doi.org/10.1016/j.cma.2021.113670.

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36

Ohlberger, Mario, and Barbara Verfürth. "Localized Orthogonal Decomposition for two-scale Helmholtz-typeproblems." AIMS Mathematics 2, no. 3 (2017): 458–78. http://dx.doi.org/10.3934/math.2017.2.458.

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37

Rehák, Branislav. "OPTIMIZATION-BASED DECOMPOSITION OF A LARGE-SCALE SYSTEM." IFAC Proceedings Volumes 39, no. 14 (2006): 47–52. http://dx.doi.org/10.3182/20060830-2-sf-4903.00009.

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38

Yi, Y., S. Deb, and S. Shakkottai. "Time-Scale Decomposition and Equivalent Rate-Based Marking." IEEE/ACM Transactions on Networking 14, no. 5 (October 2006): 938–50. http://dx.doi.org/10.1109/tnet.2006.882862.

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39

Ho, K. C., and Y. T. Chan. "An iterative algorithm for two-scale wavelet decomposition." IEEE Transactions on Signal Processing 49, no. 1 (2001): 254–57. http://dx.doi.org/10.1109/78.890371.

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40

Trusov, P. V., P. S. Volegov, and A. Yu Yanz. "Two-scale models of polycrystals: Macroscale motion decomposition." Physical Mesomechanics 17, no. 2 (April 2014): 116–22. http://dx.doi.org/10.1134/s1029959914020039.

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41

Orthacker, Angelina, Georg Haberfehlner, Johannes Taendl, Maria C. Poletti, Bernhard Sonderegger, and Gerald Kothleitner. "Diffusion-defining atomic-scale spinodal decomposition within nanoprecipitates." Nature Materials 17, no. 12 (November 12, 2018): 1101–7. http://dx.doi.org/10.1038/s41563-018-0209-z.

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42

Luse, D. William, and Joseph A. Ball. "Frequency-Scale Decomposition of$H^\infty $-Disk Problems." SIAM Journal on Control and Optimization 27, no. 4 (July 1989): 814–35. http://dx.doi.org/10.1137/0327043.

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43

Lam, W. K., and C. K. Li. "Scale invariant texture classification by iterative morphological decomposition." Electronics Letters 32, no. 6 (1996): 534. http://dx.doi.org/10.1049/el:19960376.

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44

Contreras, Ivan, Jean-François Cordeau, and Gilbert Laporte. "Benders Decomposition for Large-Scale Uncapacitated Hub Location." Operations Research 59, no. 6 (December 2011): 1477–90. http://dx.doi.org/10.1287/opre.1110.0965.

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45

Fredin, Nathaniel J., Jingtao Zhang, and David M. Lynn. "Nanometer-Scale Decomposition of Ultrathin Multilayered Polyelectrolyte Films." Langmuir 23, no. 5 (February 2007): 2273–76. http://dx.doi.org/10.1021/la0624182.

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46

Barty, Kengy, Pierre Carpentier, and Pierre Girardeau. "Decomposition of large-scale stochastic optimal control problems." RAIRO - Operations Research 44, no. 3 (July 2010): 167–83. http://dx.doi.org/10.1051/ro/2010013.

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47

Tang, Liming, Zhuang Fang, Changcheng Xiang, and Shiqiang Chen. "Image selective restoration using multi-scale variational decomposition." Journal of Visual Communication and Image Representation 40 (October 2016): 638–55. http://dx.doi.org/10.1016/j.jvcir.2016.08.004.

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48

Fischetti, Matteo, Ivana Ljubić, and Markus Sinnl. "Redesigning Benders Decomposition for Large-Scale Facility Location." Management Science 63, no. 7 (July 2017): 2146–62. http://dx.doi.org/10.1287/mnsc.2016.2461.

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49

Shih, Frank Y., and Yi-Ta Wu. "Decomposition of arbitrary gray-scale morphological structuring elements." Pattern Recognition 38, no. 12 (December 2005): 2323–32. http://dx.doi.org/10.1016/j.patcog.2005.04.003.

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50

Jácome, S. S. B., L. R. da Silva, A. A. Moreira, J. S. Andrade, and H. J. Herrmann. "Iterative decomposition of Barabasi–Albert scale-free networks." Physica A: Statistical Mechanics and its Applications 389, no. 17 (September 2010): 3674–77. http://dx.doi.org/10.1016/j.physa.2010.03.052.

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