Robust multivariate density estimation under Gaussian noise
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Robust multivariate density estimation under Gaussian noise Jitka Kostková1
· Jan Flusser1,2
Received: 17 June 2019 / Revised: 13 January 2020 / Accepted: 20 January 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Observation of random variables is often corrupted by additive Gaussian noise. Noisereducing data processing is time-consuming and may introduce unwanted artifacts. In this paper, a novel approach to description of random variables insensitive with respect to Gaussian noise is presented. The proposed quantities represent the probability density function of the variable to be observed, while noise estimation, deconvolution or denoising are avoided. Projection operators are constructed, that divide the probability density function into a nonGaussian and a Gaussian part. The Gaussian part is subsequently removed by modifying the characteristic function to ensure the invariance. The descriptors are based on the moments of the probability density function of the noisy random variable. The invariance property and the performance of the proposed method are demonstrated on real image data. Keywords Multivariate density · Gaussian additive noise · Noise-robust estimation · Moments · Invariant characteristics
1 Introduction Observation of random variables in a real-world environment is often corrupted by numerous degradation factors, among which an additive random noise is one of the most frequent ones. The noise may be introduced by measurement device imperfection, by storing and transmitting, and also due to the precision loss when pre-processing the data.
This work has been supported by the Czech Science Foundation (Grant No. GA18-07247S), by the Grant SGS18/188/OHK4/3T/14 provided by the Ministry of Education, Youth, and Sports of the Czech Republic ˇ (MŠMT CR), by Praemium Academiae, and by the Joint Laboratory SALOME2. Thanks to Dr. Cyril Höschl IV for kind providing the test images used in the experiments in Section 8.2.
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Jitka Kostková [email protected] Jan Flusser [email protected]
1
Institute of Information Theory and Automation, Czech Academy of Sciences, Pod vodárenskou vˇeží 4, 182 08 Prague 8, Czech Republic
2
Faculty of Management, University of Economics, Jarosovska 1117/II, 377 01 Jindrichuv Hradec, Czech Republic
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Multidimensional Systems and Signal Processing
Let X be the multivariate random variable to be observed and let N be an additive noise. As a result of the measurement, we actually observe realizations of a random variable Z = X + N instead of X , which is observed only indirectly. If the signal-to-noise ratio is low, the corruption is so heavy that it is very difficult to deduce anything about the observed variable X from the sample data Z . This situation occurs frequently in many application areas such as signal and image processing, econometrics, experimental physics, geoscience, and many others. A large amount of effort has been spent to develop methods that decrease the impact of the noise and allow to estimate either the en
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