Optimal Whitening and Decorrelation

Kessy, Agnan; Lewin, Arie Y.; Strimmer, Korbinian

doi:10.1080/00031305.2016.1277159

Cited by 345 publications

(286 citation statements)

References 14 publications

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“…This is not the case with non-i.i.d noise, where the spectral vectors are correlated, thus the covariance is not a diagonal matrix. In order to convert the non-i.i.d noise to i.i.d noise, a whitening transformation [47,48] is applied to the observation Y prior to the denoising methods in case 2. Let…”

Section: Resultsmentioning

confidence: 99%

A New Low-Rank Representation Based Hyperspectral Image Denoising Method for Mineral Mapping

Gao

Yao

et al. 2017

Remote Sensing

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Abstract:Hyperspectral imaging technology has been used for geological analysis for many years wherein mineral mapping is the dominant application for hyperspectral images (HSIs). The very high spectral resolution of HSIs enables the identification and the diagnosis of different minerals with detection accuracy far beyond that offered by multispectral images. However, HSIs are inevitably corrupted by noise during acquisition and transmission processes. The presence of noise may significantly degrade the quality of the extracted mineral information. In order to improve the accuracy of mineral mapping, denoising is a crucial pre-processing task. By leveraging on low-rank and self-similarity properties of HSIs, this paper proposes a state-of-the-art HSI denoising algorithm that implements two main steps: (1) signal subspace learning via fine-tuned Robust Principle Component Analysis (RPCA); and (2) denoising the images associated with the representation coefficients, with respect to an orthogonal subspace basis, using BM3D, a self-similarity based state-of-the-art denoising algorithm. Accordingly, the proposed algorithm is named Hyperspectral Denoising via Robust principle component analysis and Self-similarity (HyDRoS), which can be considered as a supervised version of FastHyDe. The effectiveness of HyDRoS is evaluated in a series of mineral mapping experiments using noise-reduced AVIRIS and Hyperion HSIs. In these experiments, the proposed denoiser yielded systematically state-of-the-art performance.

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Section: Resultsmentioning

confidence: 99%

A New Low-Rank Representation Based Hyperspectral Image Denoising Method for Mineral Mapping

Gao

Yao

et al. 2017

Remote Sensing

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“…This is choice is usually assumed throughout this paper. In the statistics and signal processing communities, the factoring and row-weighting procedure described above is known as "whitening" or "sphering" and we refer the reader to [50] to explore the benefits of the other possibilities for the change-of-basis matrices W in that context.…”

Section: −1mentioning

confidence: 99%

Discrete least-squares finite element methods

Keith

Petrides

Fuentes

et al. 2017

Computer Methods in Applied Mechanics and Engineering

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ABSTRACT. A finite element methodology for large classes of variational boundary value problems is defined which involves discretizing two linear operators: (1) the differential operator defining the spatial boundary value problem; and (2) a Riesz map on the test space. The resulting linear system is overdetermined. Two different approaches for solving the system are suggested (although others are discussed): (1) solving the associated normal equation with linear solvers for symmetric positive-definite systems (e.g. Cholesky factorization); and (2) solving the overdetermined system with orthogonalization algorithms (e.g. QR factorization). The finite element assembly algorithm for each of these approaches is described in detail. The normal equation approach is usually faster for direct solvers and requires less storage. The second approach reduces the condition number of the system by a power of two and is less sensitive to round-off error. The rectangular stiffness matrix of second approach is demonstrated to have condition number O(h −1 ) for a variety of formulations of Poisson's equation. The stiffness matrix from the normal equation approach is demonstrated to be related to the monolithic stiffness matrices of least-squares finite element methods and it is proved that the two are identical in some cases. An example with Poisson's equation indicates that the solutions of these two different linear systems can be nearly indistinguishable (if round-off error is not an issue) and rapidly converge to each other. The orthogonalization approach is suggested to be beneficial for problems which induce poorly conditioned linear systems. Experiments with Poisson's equation in single-precision arithmetic as well as the linear acoustics problem near resonance in double-precision arithmetic verify this conclusion. The methodology described here was developed as an outgrowth of the discontinuous Petrov-Galerkin (DPG) methodology of Demkowicz and Gopalakrishnan [29]. The strength of DPG is highlighted throughout, however, the majority of theory presented is general. Extensions to constrained minimization principles are also considered throughout but are not analyzed in experiments.

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“…The underlying principle behind the noise whitening procedure is that the noise in the data sample, x, can be effectively represented by the covariance matrix, C, and will be transformed into a random sequence (for example Hom and Johnson, 1985;Belouchrani et al, 1997;Kessy et al, 2015). However, the signals in x we wish to preserve will be invariants of C 1{2 and hence be preserved.…”

Section: Theorymentioning

confidence: 99%

“…This work aims to reduce recorded noise to White, Gaussian Noise (WGN) by removing the covariance of the noise. The process of removing the covariance from a dataset is commonly referred to as noise whitening and is a well established procedure in many aspects of signal processing (Hom and Johnson, 1985;Belouchrani et al, 1997;Kessy et al, 2015). In this paper the noise whitening procedure is tested on both recorded noise, noise free synthetic waveform data and semi-synthetic datasets (datasets where recorded noise has been imposed on top of synthetic waveform data).…”

Section: Introductionmentioning

confidence: 99%

Seismic arrival enhancement through the use of noise whitening

Birnie

Chambers²,

Angus

2017

Physics of the Earth and Planetary Interiors

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A constant feature in seismic data, noise is particularly troublesome for passive seismic monitoring where noise commonly masks microseismic events. We propose a statistics-driven noise suppression technique that whitens the noise through the calculation and removal of the noise's covariance. Noise whitening is shown to reduce the noise energy by a factor of 3.5 resulting in microseismic events being observed and imaged at lower signal to noise ratios than originally possible -whilst having negligible effect on the seismic wavelet. The procedure is shown to be highly resistant to most changes in the noise properties and has the flexibility of being used as a stand-alone technique or as a first step before standard random noise attenuation methods.

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Optimal Whitening and Decorrelation

Cited by 345 publications

References 14 publications

A New Low-Rank Representation Based Hyperspectral Image Denoising Method for Mineral Mapping

A New Low-Rank Representation Based Hyperspectral Image Denoising Method for Mineral Mapping

Discrete least-squares finite element methods

Seismic arrival enhancement through the use of noise whitening

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