Graph Laplacian for image anomaly detection
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ORIGINAL PAPER
Graph Laplacian for image anomaly detection Francesco Verdoja1
· Marco Grangetto2
Received: 9 October 2018 / Revised: 20 December 2019 / Accepted: 14 January 2020 © The Author(s) 2020
Abstract Reed–Xiaoli detector (RXD) is recognized as the benchmark algorithm for image anomaly detection; however, it presents known limitations, namely the dependence over the image following a multivariate Gaussian model, the estimation and inversion of a high-dimensional covariance matrix, and the inability to effectively include spatial awareness in its evaluation. In this work, a novel graph-based solution to the image anomaly detection problem is proposed; leveraging the graph Fourier transform, we are able to overcome some of RXD’s limitations while reducing computational cost at the same time. Tests over both hyperspectral and medical images, using both synthetic and real anomalies, prove the proposed technique is able to obtain significant gains over performance by other algorithms in the state of the art. Keywords Anomaly detection · Graph Fourier transform · Graph-based image processing · Principal component analysis · Hyperspectral images · PET
1 Introduction Anomaly detection is the task of spotting items that do not conform to the expected pattern of the data. In the case of images, it usually refers to the problem of spotting pixels showing a peculiar spectral signature when compared to all other pixels in an image. Image anomaly detection is considered one of the most interesting and crucial tasks for many high-level image- and video-based applications, e.g., surveillance, environmental monitoring, and medical analysis [16]. One of the most used and widely validated techniques for anomaly detection is Reed–Xiaoli detector, often called RX detector for short [56], which is the most known example of covariance-based anomaly detectors. This class of detectors has found wide adoption in many domains, from hyperspectral [49] to medical images [65]; however, methods of this type suffer from crucial drawbacks, most noticeably the need for covariance estimation and inversion. Many situations exist where the drawbacks of these state-of-the-art anomaly
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Francesco Verdoja [email protected] Marco Grangetto [email protected]
1
School of Electrical Engineering, Aalto University, Maarintie 8, Espoo, Finland
2
Department of Computer Science, University of Turin, Via Pessinetto 12, Turin, Italy
detectors lead to poor and unreliable results [67]. Moreover, the operations required by those techniques are is computationally expensive [12]. For all these reasons, the research for a fast and reliable image anomaly detection strategy able to overcome the limitations of covariance-based anomaly detectors deserves further efforts. In this paper, we use graphs to tackle image anomaly detection. Graphs are proved to be natural tools to represent data in many domains, e.g., recommendation systems, social networks, or protein interaction systems [18]. Recently, they have found wide adoption also in compu
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