Titlebook: Covariances in Computer Vision and Machine Learning; Hà Quang Minh,Vittorio Murino Book 2018 Springer Nature Switzerland AG 2018

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Introduction,eir applications in many disciplines in science and engineering. The practical applications of SPD matrices are numerous, including Diffusion Tensor Imaging (DTI) in brain imaging [5, 29, 66, 95], kernel learning [2, 60] in machine learning, radar signal processing [3, 9, 40], and Brain Computer Interface (BCI) applications [7, 8, 24, 100].

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Data Representation by Covariance Operatorsis chapter, by employing the feature map viewpoint of kernel methods in machine learning, we generalize covariance matrices to infinite-dimensional covariance operators in RKHS. Since they encode . between input features, they can be employed as a powerful form of data representation, which we explore in subsequent chapters.

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Geometry of Covariance Operatorsrators. These distances and divergences can then be directly employed in a practical application, e.g., image classification. We emphasize, however, that the concepts we present below are general and applicable in any application involving the comparison of covariance operators.

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We then present a statistical interpretation of this framework, which shows that assuming that an image can be represented by a covariance matrix is essentially equivalent to assuming that its features are random variables generated by a multivariate Gaussian probability distribution with mean zero

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d images by covariance matrices, this means that we need to have a similarity measure between covariance matrices. Since covariance matrices, properly regularized if necessary, are symmetric, positive definite (SPD matrices), a natural approach to measuring their similarity is via a distance (or dis