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Journal articles on the topic 'Gaussian scale-space'

Author: Grafiati

Published: 4 June 2021

Last updated: 8 February 2022

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1

Lindeberg, Tony. "Generalized Gaussian Scale-Space Axiomatics Comprising Linear Scale-Space, Affine Scale-Space and Spatio-Temporal Scale-Space." Journal of Mathematical Imaging and Vision 40, no. 1 (December 1, 2010): 36–81. http://dx.doi.org/10.1007/s10851-010-0242-2.

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2

Rey Otero, Ives, and Mauricio Delbracio. "Computing an Exact Gaussian Scale-Space." Image Processing On Line 5 (February 2, 2016): 8–26. http://dx.doi.org/10.5201/ipol.2016.117.

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3

Tschirsich, Martin, and Arjan Kuijper. "Notes on Discrete Gaussian Scale Space." Journal of Mathematical Imaging and Vision 51, no. 1 (May 13, 2014): 106–23. http://dx.doi.org/10.1007/s10851-014-0509-0.

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4

Zhang, Meng Meng, Zhihui Yang, Yang Yang, and Yan Sun. "Bifurcation Properties of Image Gaussian Scale-Space Model." Advanced Science Letters 6, no. 1 (March 15, 2012): 430–35. http://dx.doi.org/10.1166/asl.2012.2286.

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5

SAKAI, Tomoya, and Atsushi IMIYA. "Qualitative Descriptions of Image in the Gaussian Scale-Space." Interdisciplinary Information Sciences 11, no. 2 (2005): 157–66. http://dx.doi.org/10.4036/iis.2005.157.

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6

Lionnie, Regina, and Mudrik Alaydrus. "Hierarchical Gaussian Scale-Space on Androgenic Hair Pattern Recognition." TELKOMNIKA (Telecommunication Computing Electronics and Control) 15, no. 1 (March 1, 2016): 522. http://dx.doi.org/10.12928/telkomnika.v15i1.5381.

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7

Babaud, Jean, Andrew P. Witkin, Michel Baudin, and Richard O. Duda. "Uniqueness of the Gaussian Kernel for Scale-Space Filtering." IEEE Transactions on Pattern Analysis and Machine Intelligence PAMI-8, no. 1 (January 1986): 26–33. http://dx.doi.org/10.1109/tpami.1986.4767749.

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8

Kamejima, Kohji. "Laplacian-Gaussian Sub-Correlation Analysis for Scale Space Imaging." Proceedings of the ISCIE International Symposium on Stochastic Systems Theory and its Applications 2005 (May 5, 2005): 277–82. http://dx.doi.org/10.5687/sss.2005.277.

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9

Behrens, T., K. Schmidt, R. A. MacMillan, and R. A. Viscarra Rossel. "Multiscale contextual spatial modelling with the Gaussian scale space." Geoderma 310 (January 2018): 128–37. http://dx.doi.org/10.1016/j.geoderma.2017.09.015.

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10

Mahour, Milad, Valentyn Tolpekin, and Alfred Stein. "Automatic Detection of Individual Trees from VHR Satellite Images Using Scale-Space Methods." Sensors 20, no. 24 (December 15, 2020): 7194. http://dx.doi.org/10.3390/s20247194.

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This research investigates the use of scale-space theory to detect individual trees in orchards from very-high resolution (VHR) satellite images. Trees are characterized by blobs, for example, bell-shaped surfaces. Their modeling requires the identification of local maxima in Gaussian scale space, whereas location of the maxima in the scale direction provides information about the tree size. A two-step procedure relates the detected blobs to tree objects in the field. First, a Gaussian blob model identifies tree crowns in Gaussian scale space. Second, an improved tree crown model modifies this model in the scale direction. The procedures are tested on the following three representative cases: an area with vitellaria trees in Mali, an orchard with walnut trees in Iran, and one case with oil palm trees in Indonesia. The results show that the refined Gaussian blob model improves upon the traditional Gaussian blob model by effectively discriminating between false and correct detections and accurately identifying size and position of trees. A comparison with existing methods shows an improvement of 10–20% in true positive detections. We conclude that the presented two-step modeling procedure of tree crowns using Gaussian scale space is useful to automatically detect individual trees from VHR satellite images for at least three representative cases.

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11

Florack, Luc, Bart Ter Haar Romeny, Max Viergever, and Jan Koenderink. "The Gaussian scale-space paradigm and the multiscale local jet." International Journal of Computer Vision 18, no. 1 (April 1996): 61–75. http://dx.doi.org/10.1007/bf00126140.

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12

Liang, Dong-tai, Wei-yan Deng, Xuan-yin Wang, and Yang Zhang. "Multivariate image analysis in gaussian multi-scale space for defect detection." Journal of Bionic Engineering 6, no. 3 (September 2009): 298–305. http://dx.doi.org/10.1016/s1672-6529(08)60118-3.

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13

Blom, Johan, Bart M. ter Haar Romeny, Arjan Bel, and Jan J. Koenderink. "Spatial Derivatives and the Propagation of Noise in Gaussian Scale Space." Journal of Visual Communication and Image Representation 4, no. 1 (March 1993): 1–13. http://dx.doi.org/10.1006/jvci.1993.1001.

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14

Guerrero-Colon, Jose A., Luis Mancera, and Javier Portilla. "Image Restoration Using Space-Variant Gaussian Scale Mixtures in Overcomplete Pyramids." IEEE Transactions on Image Processing 17, no. 1 (January 2008): 27–41. http://dx.doi.org/10.1109/tip.2007.911473.

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15

Adler, Robert J., Eliran Subag, and Jonathan E. Taylor. "Rotation and scale space random fields and the Gaussian kinematic formula." Annals of Statistics 40, no. 6 (December 2012): 2910–42. http://dx.doi.org/10.1214/12-aos1055.

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16

Long, JianWu, ZeRan Yan, HongFa Chen, and XinLei Song. "Spectrum decomposition in Gaussian scale space for uneven illumination image binarization." PLOS ONE 16, no. 4 (April 30, 2021): e0251014. http://dx.doi.org/10.1371/journal.pone.0251014.

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Although most images in industrial applications have fewer targets and simple image backgrounds, binarization is still a challenging task, and the corresponding results are usually unsatisfactory because of uneven illumination interference. In order to efficiently threshold images with nonuniform illumination, this paper proposes an efficient global binarization algorithm that estimates the inhomogeneous background surface of the original image constructed from the first k leading principal components in the Gaussian scale space (GSS). Then, we use the difference operator to extract the distinct foreground of the original image in which the interference of uneven illumination is effectively eliminated. Finally, the image can be effortlessly binarized by an existing global thresholding algorithm such as the Otsu method. In order to qualitatively and quantitatively verify the segmentation performance of the presented scheme, experiments were performed on a dataset collected from a nonuniform illumination environment. Compared with classical binarization methods, in some metrics, the experimental results demonstrate the effectiveness of the introduced algorithm in providing promising binarization outcomes and low computational costs.

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17

Popa, L. "Power fluctuations in the wavelet space: large-scale CMB non-gaussian statistics." New Astronomy 3, no. 7 (November 1998): 563–70. http://dx.doi.org/10.1016/s1384-1076(98)00029-3.

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18

Guan, Xuewei, Zhenming Peng, Suqi Huang, and Yingpin Chen. "Gaussian Scale-Space Enhanced Local Contrast Measure for Small Infrared Target Detection." IEEE Geoscience and Remote Sensing Letters 17, no. 2 (February 2020): 327–31. http://dx.doi.org/10.1109/lgrs.2019.2917825.

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19

Kuijper, Arjan. "Exploring and exploiting the structure of saddle points in Gaussian scale space." Computer Vision and Image Understanding 112, no. 3 (December 2008): 337–49. http://dx.doi.org/10.1016/j.cviu.2008.06.001.

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20

Al Zaim, Yasser, and Mohammad Reza Faridrohani. "Bayesian random projection-based signal detection for Gaussian scale space random fields." AStA Advances in Statistical Analysis 105, no. 3 (June 28, 2021): 503–32. http://dx.doi.org/10.1007/s10182-021-00408-6.

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21

Doering, Dionísio, and Adalberto Schuck Junior. "A Novel Method for Generating Scale Space Kernels Based on Wavelet Theory." Revista de Informática Teórica e Aplicada 15, no. 2 (December 12, 2008): 121–38. http://dx.doi.org/10.22456/2175-2745.7024.

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The linear scale-space kernel is a Gaussian or Poisson function. These functions were chosen based on several axioms. This representation creates a good base for visualization when there is no information (in advanced) about which scales are more important. These kernels have some deficiencies, as an example, its support region goes from minus to plus infinite. In order to solve these issues several others scale-space kernels have been proposed. In this paper we present a novel method to create scale-space kernels from one-dimensional wavelet functions. In order to do so, we show the scale-space and wavelet fundamental equations and then the relationship between them. We also describe three different methods to generate two-dimensional functions from one-dimensional functions. Then we show results got from scale-space blob detector using the original and two new scale-space bases (Haar and Bi-ortogonal 4.4), and a comparison between the edges detected using the Gaussian kernel and Haar kernel for a noisy image. Finally we show a comparison between the scale space Haar edge detector and the Canny edge detector for an image with one known square in it, for that case we show the Mean Square Error (MSE) of the edges detected with both algorithms.

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22

Cao, Lei, Jun Liu, and Shu Guang Liu. "Remote Sensing Image Fusion of Worldview-2 Satellite Data." Applied Mechanics and Materials 333-335 (July 2013): 1159–63. http://dx.doi.org/10.4028/www.scientific.net/amm.333-335.1159.

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In view of the situation that most image fusion methods make spectral distortion more or less; this paper proposes a spatial projection method by introducing the Gaussian scale space theory. According to mechanism of the human visual system represented by the Gaussian scale space, the spatial details feature information are extracted from the original panchromatic (PAN) and multispectral (MS) images, then the feature differences are projected into the original MS image to obtain the fused image. The experimental results with WorldView-2 images show that the proposed method can improve the spatial resolution of the fused MS image effectively, while it can make little spectral distortion to the fused images so as to maintain the great majority spectral information.

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23

Kuijper, A. "Singularities in Gaussian scale space that are relevant for changes in its hierarchical structure." PAMM 8, no. 1 (December 2008): 10935–36. http://dx.doi.org/10.1002/pamm.200810935.

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24

Zhang, Yue, Chuan Cai Liu, and Jian Zou. "Using Multi-Scale Gaussian Derivatives for Appearance-Based Recognition." Applied Mechanics and Materials 513-517 (February 2014): 1561–64. http://dx.doi.org/10.4028/www.scientific.net/amm.513-517.1561.

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This paper addresses a novel global appearance-based approach to recognize objects in images by using multi-scale Gaussian derivatives (GDs). Because the GDs distributions of filtered images are almost picked, this obstacles obtaining discriminative binned distributions for each image. For this reason, we execute k-means clustering on each scale of pooled Gaussian derivative set of the instances come from all classes to yield k-cluster centroids for partitioning feature space, thus generating normalized binned marginal distributions for all training and testing samples, which are holistically adaptive to underlying distributions. On similarity matching, we identify each image with a point of product multinomial manifold with boundary, and use the direct sum of geodesic distance metric for sets of binned marginal densities. The promising experimental results on Zurich buildings database (ZuBuD) validate the feasibility and effectiveness of our approach.

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25

FANG, SHIAOFEN, and MARWAN ADADA. "MULTI-SCALE ISO-SURFACE EXTRACTION FOR VOLUME VISUALIZATION." International Journal of Image and Graphics 06, no. 02 (April 2006): 173–85. http://dx.doi.org/10.1142/s0219467806002185.

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This paper describes a new multi-scale approach for the extraction of iso-surfaces from volume datasets. The goal is to automatically identify iso-surfaces that best approximate the boundary surfaces at different levels of details. Using histogram analysis, iso-values are extracted from histograms of boundary voxels defined by gradient thresholding or zero-crossing boundaries. Multi-scale smoothing of the histogram using Gaussian filters of various sizes allows the iso-surfaces to be defined hierarchically over a scale space map. It provides an interactive environment and volume navigation tools for the exploration of large volume datasets by visualizing the layers of the volume space in a multi-scale manner.

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26

Owen, Lucy L. W., Tudor A. Muntianu, Andrew C. Heusser, Patrick M. Daly, Katherine W. Scangos, and Jeremy R. Manning. "A Gaussian Process Model of Human Electrocorticographic Data." Cerebral Cortex 30, no. 10 (June 4, 2020): 5333–45. http://dx.doi.org/10.1093/cercor/bhaa115.

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Abstract We present a model-based method for inferring full-brain neural activity at millimeter-scale spatial resolutions and millisecond-scale temporal resolutions using standard human intracranial recordings. Our approach makes the simplifying assumptions that different people’s brains exhibit similar correlational structure, and that activity and correlation patterns vary smoothly over space. One can then ask, for an arbitrary individual’s brain: given recordings from a limited set of locations in that individual’s brain, along with the observed spatial correlations learned from other people’s recordings, how much can be inferred about ongoing activity at other locations throughout that individual’s brain? We show that our approach generalizes across people and tasks, thereby providing a person- and task-general means of inferring high spatiotemporal resolution full-brain neural dynamics from standard low-density intracranial recordings.

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27

KENNEDY, LESA M., and MITRA BASU. "A GAUSSIAN DERIVATIVE OPERATOR FOR AUTHENTIC EDGE DETECTION AND ACCURATE EDGE LOCALIZATION." International Journal of Pattern Recognition and Artificial Intelligence 13, no. 03 (May 1999): 367–80. http://dx.doi.org/10.1142/s0218001499000215.

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One of the nice properties of the Gaussian scale space map is its well behavedness. This rather well-behaved nature is somewhat deceptive, however, as portions of the map may not have any direct relationship to the features in the unfiltered image.4 It has been shown that not all zero-crossing surface patches can be associated with intensity changes in the unfiltered image. Zero-crossings give rise to both authentic and phantom scale map contours. Recently, we proposed an edge enhancement operator, the LWF, which is a weighted combination of the Gaussian and its second derivative.6 In this paper, we prove analytically and demonstrate experimentally that the LWF produces the authentic scale map contours only. We also show that the LWF has excellent edge localization (i.e. the points marked by the operator is very close to center of the true edge). A performance comparison between the Laplacian of Gaussian and LWF operators with respect to the localization property is also presented.

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28

Xiaoping Li and Tongwen Chen. "Optimal ℒ/sub 1/ approximation of the Gaussian kernel with application to scale-space construction." IEEE Transactions on Pattern Analysis and Machine Intelligence 17, no. 10 (1995): 1015–19. http://dx.doi.org/10.1109/34.464565.

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29

Worsley, Keith J. "Testing for signals with unknown location and scale in a χ2 random field, with an application to fMRI." Advances in Applied Probability 33, no. 04 (December 2001): 773–93. http://dx.doi.org/10.1017/s0001867800011198.

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Siegmund and Worsley (1995) considered the problem of testing for signals with unknown location and scale in a Gaussian random field defined on ℝN. The test statistic was the maximum of a Gaussian random field in anN+1 dimensional ‘scale space’,Ndimensions for location and 1 dimension for the scale of a smoothing filter. Scale space is identical to a continuous wavelet transform with a kernel smoother as the wavelet, though the emphasis here is on signal detection rather than image compression or enhancement. Two methods were used to derive an approximate null distribution forN=2 andN=3: one based on the method of volumes of tubes, the other based on the expected Euler characteristic of the excursion set. The purpose of this paper is two-fold: to show how the latter method can be extended to higher dimensions, and to apply this more general result to χ2fields. The result of Siegmund and Worsley (1995) then follows as a special case. In this paper the results are applied to the problem of searching for activation in brain images obtained by functional magnetic resonance imaging (fMRI).

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30

Worsley, Keith J. "Testing for signals with unknown location and scale in a χ2 random field, with an application to fMRI." Advances in Applied Probability 33, no. 4 (December 2001): 773–93. http://dx.doi.org/10.1239/aap/1011994029.

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Siegmund and Worsley (1995) considered the problem of testing for signals with unknown location and scale in a Gaussian random field defined on ℝN. The test statistic was the maximum of a Gaussian random field in an N+1 dimensional ‘scale space’, N dimensions for location and 1 dimension for the scale of a smoothing filter. Scale space is identical to a continuous wavelet transform with a kernel smoother as the wavelet, though the emphasis here is on signal detection rather than image compression or enhancement. Two methods were used to derive an approximate null distribution for N=2 and N=3: one based on the method of volumes of tubes, the other based on the expected Euler characteristic of the excursion set. The purpose of this paper is two-fold: to show how the latter method can be extended to higher dimensions, and to apply this more general result to χ2 fields. The result of Siegmund and Worsley (1995) then follows as a special case. In this paper the results are applied to the problem of searching for activation in brain images obtained by functional magnetic resonance imaging (fMRI).

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SELVAM, A. M. "FRACTAL FLUCTUATIONS AND STATISTICAL NORMAL DISTRIBUTION." Fractals 17, no. 03 (September 2009): 333–49. http://dx.doi.org/10.1142/s0218348x09004272.

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Dynamical systems in nature exhibit self-similar fractal fluctuations and the corresponding power spectra follow inverse power law form signifying long-range space-time correlations identified as self-organized criticality. The physics of self-organized criticality is not yet identified. The Gaussian probability distribution used widely for analysis and description of large data sets underestimates the probabilities of occurrence of extreme events such as stock market crashes, earthquakes, heavy rainfall, etc. The assumptions underlying the normal distribution such as fixed mean and standard deviation, independence of data, are not valid for real world fractal data sets exhibiting a scale-free power law distribution with fat tails. A general systems theory for fractals visualizes the emergence of successively larger scale fluctuations to result from the space-time integration of enclosed smaller scale fluctuations. The model predicts a universal inverse power law incorporating the golden mean for fractal fluctuations and for the corresponding power spectra, i.e., the variance spectrum represents the probabilities, a signature of quantum systems. Fractal fluctuations therefore exhibit quantum-like chaos. The model predicted inverse power law is very close to the Gaussian distribution for small-scale fluctuations, but exhibits a fat long tail for large-scale fluctuations. Extensive data sets of Dow Jones index, human DNA, Takifugu rubripes (Puffer fish) DNA are analyzed to show that the space/time data sets are close to the model predicted power law distribution.

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32

Zhong, Bao Jiang, and Chang Li. "Robust Image Corner Detection Using Curvature Product in Direct Curvature Scale Space." Applied Mechanics and Materials 20-23 (January 2010): 725–30. http://dx.doi.org/10.4028/www.scientific.net/amm.20-23.725.

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In this paper we propose an image corner detector based on the direct curvature scale space (DCSS) technique, referred to as the curvature product DCSS (CP-DCSS) corner detector. After the contours of interested objects are extracted from a real-world image, their curvature functions are respectively convolved with the Gaussian function as its standard deviation gradually increases. By measuring the product of the curvature values computed at several given scales, true corners on the contours can be easily detected since false or insignificant corners have been effectively suppressed. A point is declared as a corner when the absolute value of the curvature product exceeds a given threshold and is a local maximum at the mentioned point. CP-DCSS combines the advantages of two recently proposed corner detectors, namely, the DCSS corner detector and the multi-scale curvature product (MSCP) corner detector. Compared to DCSS, CP-DCSS omits a parsing process of the DCSS map, and hence it has a simpler structure. Compared to MSCP, CP-DCSS works equally well, however, at less computational cost.

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33

Mehdipour, S. Hamid. "Entropic force law in the presence of a noncommutative inspired space–time for a solar system scale." Canadian Journal of Physics 93, no. 10 (October 2015): 1184–89. http://dx.doi.org/10.1139/cjp-2014-0711.

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We first study some aspects of a physically inspired noncommutative spherically symmetric space–time based on the Gaussian-smeared mass distribution for a solar system scale. This leads to the elimination of a singularity apparent in the origin of the space–time. Afterwards, we investigate some features of Verlinde’s scenario in the presence of the mentioned space–time and derive several quantities, such as Unruh–Verlinde temperature, the energy, and the entropic force on three different types of holographic screens, namely, the static, the stretched horizon, and the accelerating surface.

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34

Wang, Jiaolong, Chengxi Zhang, Jin Wu, and Ming Liu. "An Improved Invariant Kalman Filter for Lie Groups Attitude Dynamics with Heavy-Tailed Process Noise." Machines 9, no. 9 (August 27, 2021): 182. http://dx.doi.org/10.3390/machines9090182.

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Attitude estimation is a basic task for most spacecraft missions in aerospace engineering and many Kalman type attitude estimators have been applied to the guidance and navigation of spacecraft systems. By building the attitude dynamics on matrix Lie groups, the invariant Kalman filter (IKF) was developed according to the invariance properties of symmetry groups. However, the Gaussian noise assumption of Kalman theory may be violated when a spacecraft maneuvers severely and the process noise might be heavy-tailed, which is prone to degrade IKF’s performance for attitude estimation. To address the attitude estimation problem with heavy-tailed process noise, this paper proposes a hierarchical Gaussian state-space model for invariant Kalman filtering: The probability density function of state prediction is defined based on student’s t distribution, while the conjugate prior distributions of the scale matrix and degrees of freedom (dofs) parameter are respectively formulated as the inverse Wishart and Gamma distribution. For the constructed hierarchical Gaussian attitude estimation state-space model, the Lie groups rotation matrix of spacecraft attitude is inferred together with the scale matrix and dof parameter using the variational Bayesian iteration. Numerical simulation results illustrate that the proposed approach can significantly improve the filtering robustness of invariant Kalman filter for Lie groups spacecraft attitude estimation problems with heavy-tailed process uncertainty.

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35

Qi, Xianbiao, Guoying Zhao, Jie Chen, and Matti Pietikäinen. "HEp-2 cell classification: The role of Gaussian Scale Space Theory as a pre-processing approach." Pattern Recognition Letters 82 (October 2016): 36–43. http://dx.doi.org/10.1016/j.patrec.2015.12.011.

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36

Lindeberg, Tony. "Provably Scale-Covariant Continuous Hierarchical Networks Based on Scale-Normalized Differential Expressions Coupled in Cascade." Journal of Mathematical Imaging and Vision 62, no. 1 (October 25, 2019): 120–48. http://dx.doi.org/10.1007/s10851-019-00915-x.

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Abstract This article presents a theory for constructing hierarchical networks in such a way that the networks are guaranteed to be provably scale covariant. We first present a general sufficiency argument for obtaining scale covariance, which holds for a wide class of networks defined from linear and nonlinear differential expressions expressed in terms of scale-normalized scale-space derivatives. Then, we present a more detailed development of one example of such a network constructed from a combination of mathematically derived models of receptive fields and biologically inspired computations. Based on a functional model of complex cells in terms of an oriented quasi quadrature combination of first- and second-order directional Gaussian derivatives, we couple such primitive computations in cascade over combinatorial expansions over image orientations. Scale-space properties of the computational primitives are analysed, and we give explicit proofs of how the resulting representation allows for scale and rotation covariance. A prototype application to texture analysis is developed, and it is demonstrated that a simplified mean-reduced representation of the resulting QuasiQuadNet leads to promising experimental results on three texture datasets.

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37

Anderson, Jeffrey L. "A Non-Gaussian Ensemble Filter Update for Data Assimilation." Monthly Weather Review 138, no. 11 (November 1, 2010): 4186–98. http://dx.doi.org/10.1175/2010mwr3253.1.

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Abstract A deterministic square root ensemble Kalman filter and a stochastic perturbed observation ensemble Kalman filter are used for data assimilation in both linear and nonlinear single variable dynamical systems. For the linear system, the deterministic filter is simply a method for computing the Kalman filter and is optimal while the stochastic filter has suboptimal performance due to sampling error. For the nonlinear system, the deterministic filter has increasing error as ensemble size increases because all ensemble members but one become tightly clustered. In this case, the stochastic filter performs better for sufficiently large ensembles. A new method for computing ensemble increments in observation space is proposed that does not suffer from the pathological behavior of the deterministic filter while avoiding much of the sampling error of the stochastic filter. This filter uses the order statistics of the prior observation space ensemble to create an approximate continuous prior probability distribution in a fashion analogous to the use of rank histograms for ensemble forecast evaluation. This rank histogram filter can represent non-Gaussian observation space priors and posteriors and is shown to be competitive with existing filters for problems as large as global numerical weather prediction. The ability to represent non-Gaussian distributions is useful for a variety of applications such as convective-scale assimilation and assimilation of bounded quantities such as relative humidity.

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38

Biro, T., and K. Rönnmark. "Phase-space description of plasma waves. Part 1. Linear theory." Journal of Plasma Physics 47, no. 3 (June 1992): 465–77. http://dx.doi.org/10.1017/s0022377800024351.

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We develop an (r, k) phase-space description of waves in plasmas by introducing Gaussian window functions to separate short-scale oscillations from long-scale modulations of the wave fields and variations in the plasma parameters. To obtain a wave equation that unambiguously separates conservative dynamics from dissipation in an inhomogeneous and time-varying background plasma, we first discuss the proper form of the current response function. In analogy with the particle distribution function f(v, r, t), we introduce a wave density N(k, r, t) on phase space. This function is proved to satisfy a simple continuity equation. Dissipation is also included, and this allows us to describe the damping or growth of wave density along rays. Problems involving geometric optics of continuous media often appear simpler when viewed in phase space, since the flow of N in phase space is incompressible.

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39

Han, Shoudong, Wenbing Tao, and Xianglin Wu. "Texture segmentation using independent-scale component-wise Riemannian-covariance Gaussian mixture model in KL measure based multi-scale nonlinear structure tensor space." Pattern Recognition 44, no. 3 (March 2011): 503–18. http://dx.doi.org/10.1016/j.patcog.2010.09.006.

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40

Philcox, Oliver H. E., and Daniel J. Eisenstein. "Computing the small-scale galaxy power spectrum and bispectrum in configuration space." Monthly Notices of the Royal Astronomical Society 492, no. 1 (November 28, 2019): 1214–42. http://dx.doi.org/10.1093/mnras/stz3335.

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ABSTRACT We present a new class of estimators for computing small-scale power spectra and bispectra in configuration space via weighted pair and triple counts, with no explicit use of Fourier transforms. Particle counts are truncated at $R_0\sim 100\, h^{-1}\, \mathrm{Mpc}$ via a continuous window function, which has negligible effect on the measured power spectrum multipoles at small scales. This gives a power spectrum algorithm with complexity $\mathcal {O}(NnR_0^3)$ (or $\mathcal {O}(Nn^2R_0^6)$ for the bispectrum), measuring N galaxies with number density n. Our estimators are corrected for the survey geometry and have neither self-count contributions nor discretization artefacts, making them ideal for high-k analysis. Unlike conventional Fourier-transform-based approaches, our algorithm becomes more efficient on small scales (since a smaller R0 may be used), thus we may efficiently estimate spectra across k-space by coupling this method with standard techniques. We demonstrate the utility of the publicly available power spectrum algorithm by applying it to BOSS DR12 simulations to compute the high-k power spectrum and its covariance. In addition, we derive a theoretical rescaled-Gaussian covariance matrix, which incorporates the survey geometry and is found to be in good agreement with that from mocks. Computing configuration- and Fourier-space statistics in the same manner allows us to consider joint analyses, which can place stronger bounds on cosmological parameters; to this end we also discuss the cross-covariance between the two-point correlation function and the small-scale power spectrum.

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Prodanov, Dimiter, Tomasz Konopczynski, and Maciej Trojnar. "Selected Applications of Scale Spaces in Microscopic Image Analysis." Cybernetics and Information Technologies 15, no. 7 (December 1, 2015): 5–12. http://dx.doi.org/10.1515/cait-2015-0084.

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Abstract Image segmentation methods can be classified broadly into two classes: intensity-based and geometry-based. Edge detection is the base of many geometry-based segmentation approaches. Scale space theory represents a systematic treatment of the issues of spatially uncorrelated noise with its main application being the detection of edges, using multiple resolution scales, which can be used for subsequent segmentation, classification or encoding. The present paper will give an overview of some recent applications of scale spaces into problems of microscopic image analysis. Particular overviews will be given to Gaussian and alpha-scale spaces. Some applications in the analysis of biomedical images will be presented. The implementation of filters will be demonstrated.

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Pegram, G. G. S., and A. N. Clothier. "Downscaling rainfields in space and time, using the String of Beads model in time series mode." Hydrology and Earth System Sciences 5, no. 2 (June 30, 2001): 175–86. http://dx.doi.org/10.5194/hess-5-175-2001.

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Abstract. The String of Beads model is a space-time model of rainfields measured by weather radar. It is here driven by two auto-regressive time series models, one at the image scale, the other at the pixel scale, to model the temporal correlation structure of the wet-period process. The marginal distribution of the pixel scale intensities on a given radar-rainfall image is described by a log-normal distribution. The spatial dependence structure of each image is defined by a power spectrum approximated by a power law function with a negative exponent. It is demonstrated that this stochastic modelling approach is valid because the images sampled are effectively stationary above a scale of 30 km, which is less than a quarter of the image width. By advecting a simulated sequence of images along the same cumulative advection vector as the observed event and matching the image-scale statistics of each simulated image with those of the corresponding observed image, a simulated sequence of plausible images is generated which mimics (has the same space-time statistics as) the observed event but differs from it in detail. Aggregating the pixel scale intensities in each sequence over a number of time and space intervals and then comparing their spatial and temporal statistics, demonstrates that the model captures the intermediate scale behaviour well, showing satisfactorily its ability to downscale rainfall in space and time. The model thus has potential as an operational space-time model of rainfields. Keywords: Space-time, rainfield modelling, weather radar, multifractals, Gaussian random fields

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Zhu, Ye, and Kai Ming Ting. "Improving the Effectiveness and Efficiency of Stochastic Neighbour Embedding with Isolation Kernel." Journal of Artificial Intelligence Research 71 (August 2, 2021): 667–95. http://dx.doi.org/10.1613/jair.1.12904.

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This paper presents a new insight into improving the performance of Stochastic Neighbour Embedding (t-SNE) by using Isolation kernel instead of Gaussian kernel. Isolation kernel outperforms Gaussian kernel in two aspects. First, the use of Isolation kernel in t-SNE overcomes the drawback of misrepresenting some structures in the data, which often occurs when Gaussian kernel is applied in t-SNE. This is because Gaussian kernel determines each local bandwidth based on one local point only, while Isolation kernel is derived directly from the data based on space partitioning. Second, the use of Isolation kernel yields a more efficient similarity computation because data-dependent Isolation kernel has only one parameter that needs to be tuned. In contrast, the use of data-independent Gaussian kernel increases the computational cost by determining n bandwidths for a dataset of n points. As the root cause of these deficiencies in t-SNE is Gaussian kernel, we show that simply replacing Gaussian kernel with Isolation kernel in t-SNE significantly improves the quality of the final visualisation output (without creating misrepresented structures) and removes one key obstacle that prevents t-SNE from processing large datasets. Moreover, Isolation kernel enables t-SNE to deal with large-scale datasets in less runtime without trading off accuracy, unlike existing methods in speeding up t-SNE.

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Sun, Yiyuan, Qiang Wang, Kevin Tansey, Sana Ullah, Fan Liu, Haimeng Zhao, and Lei Yan. "Multi-Constrained Optimization Method of Line Segment Extraction Based on Multi-Scale Image Space." ISPRS International Journal of Geo-Information 8, no. 4 (April 8, 2019): 183. http://dx.doi.org/10.3390/ijgi8040183.

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Image-based line segment extraction plays an important role in a wide range of applications. Traditional line segment extraction algorithms focus on the accuracy and efficiency, without considering the integrity. Serious line segmentation fracture problems caused by image quality will result in poor subsequent applications. To solve this problem, a multi-constrained line segment extraction method, based on multi-scale image space, is presented. Firstly, using Gaussian down-sampling with a classical line segment detection method, a multi-scale image space is constructed to extract line segments in each image scale and all line segments are projected onto the original image. Then, a new line segment optimization and purification strategy is proposed with the horizontal and vertical distances and angle geometric constraint relationships between line segments to merge fracture line segments and delete redundant line segments. Finally, line segments with adjacent positions are optimized using the grayscale constraint relationship, based on normalized cross-correlation similarity criterion for realizing the second optimization of fracture line segments. Compared with mainstream line segment detector and edge drawing lines methods, experimental results (i.e., indoor, outdoor, and aerial images) indicate the validity and superiority of our proposed methods which can extract longer and more complete line segments.

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Li, Jianhong, Kanoksak Wattanachote, and Yarong Wu. "Maximizing Nonlocal Self-Similarity Prior for Single Image Super-Resolution." Mathematical Problems in Engineering 2019 (February 28, 2019): 1–14. http://dx.doi.org/10.1155/2019/3840285.

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Prior knowledge plays an important role in the process of image super-resolution reconstruction, which can constrain the solution space efficiently. In this paper, we utilized the fact that clear image exhibits stronger self-similarity property than other degradated version to present a new prior called maximizing nonlocal self-similarity for single image super-resolution. For describing the prior with mathematical language, a joint Gaussian mixture model was trained with LR and HR patch pairs extracted from the input LR image and its lower scale, and the prior can be described as a specific Gaussian distribution by derivation. In our algorithm, a large scale of sophisticated training and time-consuming nearest neighbor searching is not necessary, and the cost function of this algorithm shows closed form solution. The experiments conducted on BSD500 and other popular images demonstrate that the proposed method outperforms traditional methods and is competitive with the current state-of-the-art algorithms in terms of both quantitative metrics and visual quality.

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Huang, Mingming, Zhichun Mu, Hui Zeng, and Hongbo Huang. "A Novel Approach for Interest Point Detection via Laplacian-of-Bilateral Filter." Journal of Sensors 2015 (2015): 1–9. http://dx.doi.org/10.1155/2015/685154.

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Scale-invariant feature transform (SIFT) algorithm, one of the most famous and popular interest point detectors, detects extrema by using difference-of-Gaussian (DoG) filter which is an approximation to the Laplacian-of-Gaussian (LoG) for improving speed. However, DoG filter has a strong response along edge, even if the location along the edge is poorly determined and therefore is unstable to small amounts of noise. In this paper, we propose a novel interest point detection algorithm, which detects scale space extrema by using a Laplacian-of-Bilateral (LoB) filter. The LoB filter, which is produced by Bilateral and Laplacian filter, can preserve edge characteristic by fully utilizing the information of intensity variety. Compared with the SIFT algorithm, our algorithm substantially improves the repeatability of detected interest points on a very challenging benchmark dataset, in which images were generated under different imaging conditions. Extensive experimental results show that the proposed approach is more robust to challenging problems such as illumination and viewpoint changes, especially when encountering large illumination change.

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Paillas, Enrique, Yan-Chuan Cai, Nelson Padilla, and Ariel G. Sánchez. "Redshift-space distortions with split densities." Monthly Notices of the Royal Astronomical Society 505, no. 4 (June 12, 2021): 5731–52. http://dx.doi.org/10.1093/mnras/stab1654.

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ABSTRACT Accurate modelling of redshift-space distortions (RSD) is challenging in the non-linear regime for two-point statistics e.g. the two-point correlation function (2PCF). We take a different perspective to split the galaxy density field according to the local density, and cross-correlate those densities with the entire galaxy field. Using mock galaxies, we demonstrate that combining a series of cross-correlation functions (CCFs) offers improvements over the 2PCF as follows: (1) The distribution of peculiar velocities in each split density is nearly Gaussian. This allows the Gaussian streaming model for RSD to perform accurately within the statistical errors of a ($1.5\, h^{-1}$ Gpc)3 volume for almost all scales and all split densities. (2) The probability distribution of the density contrast at small scales is non-Gaussian, but the CCFs of split densities capture the non-Gaussianity, leading to improved cosmological constraints over the 2PCF. We can obtain unbiased constraints on the growth parameter fσ12 at the per cent level, and Alcock–Paczynski (AP) parameters at the sub-per cent level with the minimal scale of $15\, h^{-1}{\rm Mpc}$. This is a ∼30 per cent and ∼6 times improvement over the 2PCF, respectively. The diverse and steep slopes of the CCFs at small scales are likely to be responsible for the improved constraints of AP parameters. (3) Baryon acoustic oscillations (BAO) are contained in all CCFs of split densities. Including BAO scales helps to break the degeneracy between the line-of-sight and transverse AP parameters, allowing independent constraints on them. We discuss and compare models for RSD around spherical densities.

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Zali, Z., and J. Sadeghi. "Large-scale correction and thermal properties of holographic dual background of an adaptive graphene model." International Journal of Geometric Methods in Modern Physics 17, no. 09 (July 20, 2020): 2050135. http://dx.doi.org/10.1142/s0219887820501352.

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In this paper, we consider the particle on curved graphene space-time. In that case, we calculate the geometric form of potential which is known as Gaussian function. Here, we introduce the metric background which completely corresponds to curved graphene space-times. This metric leads us to obtain the geometry potential and we make the Laplace Beltrami equation in the mentioned metric background. We also rearrange such relation in terms of the second-order equation. By using the known polynomial, we solve the particle equation of motion in graphene background. In that case, we arrive the energy spectrum which has three terms. We take advantage from energy spectrum and investigate the thermal properties of system. The additional terms give us an opportunity to obtain the corrected entropy and free energy. So, we show that the additional term comes from geometry potential. This correction is important for the large scale. Hence, we show that correction term is logarithmic as well as small scale corrections.

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Ribli, Dezső, Bálint Ármin Pataki, José Manuel Zorrilla Matilla, Daniel Hsu, Zoltán Haiman, and István Csabai. "Weak lensing cosmology with convolutional neural networks on noisy data." Monthly Notices of the Royal Astronomical Society 490, no. 2 (September 17, 2019): 1843–60. http://dx.doi.org/10.1093/mnras/stz2610.

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ABSTRACT Weak gravitational lensing is one of the most promising cosmological probes of the late universe. Several large ongoing (DES, KiDS, HSC) and planned (LSST, Euclid, WFIRST) astronomical surveys attempt to collect even deeper and larger scale data on weak lensing. Due to gravitational collapse, the distribution of dark matter is non-Gaussian on small scales. However, observations are typically evaluated through the two-point correlation function of galaxy shear, which does not capture non-Gaussian features of the lensing maps. Previous studies attempted to extract non-Gaussian information from weak lensing observations through several higher order statistics such as the three-point correlation function, peak counts, or Minkowski functionals. Deep convolutional neural networks (CNN) emerged in the field of computer vision with tremendous success, and they offer a new and very promising framework to extract information from 2D or 3D astronomical data sets, confirmed by recent studies on weak lensing. We show that a CNN is able to yield significantly stricter constraints of (σ8, Ωm) cosmological parameters than the power spectrum using convergence maps generated by full N-body simulations and ray-tracing, at angular scales and shape noise levels relevant for future observations. In a scenario mimicking LSST or Euclid, the CNN yields 2.4–2.8 times smaller credible contours than the power spectrum, and 3.5–4.2 times smaller at noise levels corresponding to a deep space survey such as WFIRST. We also show that at shape noise levels achievable in future space surveys the CNN yields 1.4–2.1 times smaller contours than peak counts, a higher order statistic capable of extracting non-Gaussian information from weak lensing maps.

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Gossler, F. E., B. R. Oliveira, M. A. Q. Duarte, J. Vieira Filho, F. Villarreal, and R. L. Lamblém. "Gaussian and Golden Wavelets: A Comparative Study and their Applications in Structural Health Monitoring." Trends in Computational and Applied Mathematics 22, no. 1 (March 31, 2021): 139–55. http://dx.doi.org/10.5540/tcam.2021.022.01.00139.

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In this work, a comparative analysis between Gaussian and Golden wavelets is presented. These wavelets are generated by the derivative of specific base functions. In this case, the order of the derivative also indicates the number of vanishing moments of the wavelet. Although these wavelets have a similar waveform, they have several distinct characteristics in time and frequency domains. These distinctions are explored here in the scale space. In order to compare the results provided by these wavelets for a real signal, they are used in the decomposition of a signal inserted in the context of structural health monitoring.

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