On the Local Behavior of Spaces of Natural Images

By Gunnar E. Carlsson, Tigran Ishkhanov, Vin de Silva, Afra Zomorodian. 2007

604 citations recorded by the source corpus.

Subfields: Biophysics, Computational Theory and Mathematics, Computer Vision and Pattern Recognition

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Citations in the corpus

A roadmap for the computation of persistent homology (2015, background): “…in images [29, 65], periodicity in time series [102], breast-cancer classiﬁcation [41, 99], viral evolution [25], natural images [23], contagion spread on networks [115], structure of amorphous materials [67], force networks in granular matter [78,79], equities-market networks [84], diverse…”
The Gudhi Library: Simplicial Complexes and Persistent Homology (2014, positive): “This approach has been successfully used in various areas of science and engineering, as for example in sensor networks [10], image analysis [6], and data analysis [8], where one typically needs to deal with big data sets in high dimensions and with general metrics.”
Vector Diffusion Maps and the Connection Laplacian (2011, background): “…and other non-linear dimensionality reduction methods are either directly or indirectly related to the heat kernel for functions over the data, ∗Department of Mathematics and PACM, Princeton University, Fine Hall, Washington Road, Princeton NJ 08544-1000 USA, email: amits@math.princeton.edu…”
Zigzag Persistence (2008, background): “Deep structure is sometimes revealed after thresholding according to a density estimate (see [3] for an example drawn from visual image analysis).”
Homological scaffolds of brain functional networks (2014, methodology): “Persistent homology is a recent technique in computational topology developed for shape recognition and the analysis of high dimensional datasets [36,37].”
Homological illusions of persistence and stability (2008, methodology): “use various input preprocessing heuristics aimed to deal with outlier sensitivity of distance functions [17].”
PixelCNN Models with Auxiliary Variables for Natural Image Modeling (2016, background): “3x3 and 9x9 pixels, and reflected only low-order statistics, such as edge frequency and orientation (Zhu et al., 1998; Roth & Black, 2005; Carlsson et al., 2008; Zoran & Weiss, 2011).”
Topological Data Analysis (2012, methodology): “Due to the size of simplicial complexes, topological analysis has been limited to low dimensional features, such as components [17], tunnels [8], and voids [43].”
Persistent homology analysis of protein structure, flexibility and folding (2014, background): “It has been applied to a variety of domains, including image analysis [48–51], image retrieval [52], chaotic dynamics verification [53, 54], sensor network [55], complex network [56, 57], data analysis [58–62], computer vision [50], shape recognition [63], and computational biology [64–66].”
Spontaneous and driven cortical activity: implications for computation. (2009, positive): “In general, however, the manifold of natural signals is not simple a circle but could be much more complicated (Figure 1d) [36 ,37,38], yet all the same ideas apply.”
Clique topology reveals intrinsic geometric structure in neural correlations (2015, background): “Although persistent homology has only recently emerged a tool for studying features of data, it has already found a broad range of applications [27, 28, 29, 30].”
Weighted persistent homology (2017, background): “[1] develops a framework to use persistent homology to analyze natural images such as digital photographs.”
Two’s company, three (or more) is a simplex (2016, methodology): “These techniques have been applied to uncover structure in the space of natural images [59], to detect subject gender from the shape of brain artery trees [27], and to identify statistically anomolous multi-region activation patterns in fMRI recordings [21].”
The Intrinsic Dimension of Images and Its Impact on Learning (2021, background): “It is widely believed that the combination of natural scenes and sensor properties yields very sparse and concentrated image distributions, as has been supported by several empirical studies on image patches (Lee et al., 2003; Donoho & Grimes, 2005; Carlsson et al., 2008).”