000 04680nam a22005415i 4500
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020 _a9783031018206
_9978-3-031-01820-6
024 7 _a10.1007/978-3-031-01820-6
_2doi
050 4 _aTA1501-1820
050 4 _aTA1634
072 7 _aUYT
_2bicssc
072 7 _aCOM016000
_2bisacsh
072 7 _aUYT
_2thema
082 0 4 _a006
_223
100 1 _aMinh, Hà Quang.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
_982146
245 1 0 _aCovariances in Computer Vision and Machine Learning
_h[electronic resource] /
_cby Hà Quang Minh, Vittorio Murino.
250 _a1st ed. 2018.
264 1 _aCham :
_bSpringer International Publishing :
_bImprint: Springer,
_c2018.
300 _aXIII, 156 p.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aSynthesis Lectures on Computer Vision,
_x2153-1064
505 0 _aAcknowledgments -- Introduction -- Data Representation by Covariance Matrices -- Geometry of SPD Matrices -- Kernel Methods on Covariance Matrices -- Data Representation by Covariance Operators -- Geometry of Covariance Operators -- Kernel Methods on Covariance Operators -- Conclusion and Future Outlook -- Bibliography -- Authors' Biographies.
520 _aCovariance matrices play important roles in many areas of mathematics, statistics, and machine learning, as well as their applications. In computer vision and image processing, they give rise to a powerful data representation, namely the covariance descriptor, with numerous practical applications. In this book, we begin by presenting an overview of the {\it finite-dimensional covariance matrix} representation approach of images, along with its statistical interpretation. In particular, we discuss the various distances and divergences that arise from the intrinsic geometrical structures of the set of Symmetric Positive Definite (SPD) matrices, namely Riemannian manifold and convex cone structures. Computationally, we focus on kernel methods on covariance matrices, especially using the Log-Euclidean distance. We then show some of the latest developments in the generalization of the finite-dimensional covariance matrix representation to the {\it infinite-dimensional covariance operator} representation via positive definite kernels. We present the generalization of the affine-invariant Riemannian metric and the Log-Hilbert-Schmidt metric, which generalizes the Log-Euclidean distance. Computationally, we focus on kernel methods on covariance operators, especially using the Log-Hilbert-Schmidt distance. Specifically, we present a two-layer kernel machine, using the Log-Hilbert-Schmidt distance and its finite-dimensional approximation, which reduces the computational complexity of the exact formulation while largely preserving its capability. Theoretical analysis shows that, mathematically, the approximate Log-Hilbert-Schmidt distance should be preferred over the approximate Log-Hilbert-Schmidt inner product and, computationally, it should be preferred over the approximate affine-invariant Riemannian distance. Numerical experiments on image classification demonstrate significant improvements of the infinite-dimensional formulation over the finite-dimensional counterpart. Given the numerous applications of covariance matrices in many areas of mathematics, statistics, and machine learning, just to name a few, we expect that the infinite-dimensional covariance operator formulation presented here will have many more applications beyond those in computer vision.
650 0 _aImage processing
_xDigital techniques.
_94145
650 0 _aComputer vision.
_982147
650 0 _aPattern recognition systems.
_93953
650 1 4 _aComputer Imaging, Vision, Pattern Recognition and Graphics.
_931569
650 2 4 _aComputer Vision.
_982148
650 2 4 _aAutomated Pattern Recognition.
_931568
700 1 _aMurino, Vittorio.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
_982149
710 2 _aSpringerLink (Online service)
_982150
773 0 _tSpringer Nature eBook
776 0 8 _iPrinted edition:
_z9783031000775
776 0 8 _iPrinted edition:
_z9783031006920
776 0 8 _iPrinted edition:
_z9783031029486
830 0 _aSynthesis Lectures on Computer Vision,
_x2153-1064
_982151
856 4 0 _uhttps://doi.org/10.1007/978-3-031-01820-6
912 _aZDB-2-SXSC
942 _cEBK
999 _c85304
_d85304