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1 A PrimalDual Proximal Algorithm for Sparse TemplateBased Adaptive Filtering: Application to Seismic Multiple Removal
"... Abstract—Unveiling meaningful geophysical information from seismic data requires to deal with both random and structured “noises”. As their amplitude may be greater than signals of interest (primaries), additional prior information is especially important in performing efficient signal separation. W ..."
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Abstract—Unveiling meaningful geophysical information from seismic data requires to deal with both random and structured “noises”. As their amplitude may be greater than signals of interest (primaries), additional prior information is especially important in performing efficient signal separation. We address here the problem of multiple reflections, caused by wavefield bouncing between layers. Since only approximate models of these phenomena are available, we propose a flexible framework for timevarying adaptive filtering of seismic signals, using sparse representations, based on inaccurate templates. We recast the joint estimation of adaptive filters and primaries in a new convex variational formulation. This approach allows us to incorporate plausible knowledge about noise statistics, data sparsity and slow filter variation in parsimonypromoting wavelet frames. The designed primaldual algorithm solves a constrained minimization problem that alleviates standard regularization issues in finding hyperparameters. The approach demonstrates significantly good performance in low signaltonoise ratio conditions, both for simulated and real field seismic data. Index Terms—Convex optimization, Parallel algorithms, Wavelet transforms, Adaptive filters, Geophysical signal processing, Signal restoration, Sparsity, Signal separation.
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"... analysis is the identification of separate components in a multi component signal. WignerVille distribution is the classical tool for representing such signals but suffers from crossterms. There are several methods which aim to remove the cross terms by masking the Ambiguity Function (AF) but they ..."
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analysis is the identification of separate components in a multi component signal. WignerVille distribution is the classical tool for representing such signals but suffers from crossterms. There are several methods which aim to remove the cross terms by masking the Ambiguity Function (AF) but they result in reduced resolution. Most practical timevarying signals are in the form of weighted trajectories on the TF plane and many others are sparse in nature. Therefore the problem is cast as TF distribution reconstruction using a subset of AF domain coefficients and sparsity assumption in recent studies. This corresponds to a constrained
1Denosing Using Wavelets and Projections onto the `1Ball
, 2014
"... Both wavelet denoising and denosing methods using the concept of sparsity are based on softthresholding. In sparsity based denoising methods, it is assumed that the original signal is sparse in some transform domains such as the wavelet domain and the wavelet subsignals of the noisy signal are proj ..."
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Both wavelet denoising and denosing methods using the concept of sparsity are based on softthresholding. In sparsity based denoising methods, it is assumed that the original signal is sparse in some transform domains such as the wavelet domain and the wavelet subsignals of the noisy signal are projected onto `1balls to reduce noise. In this lecture note, it is shown that the size of the `1ball or equivalently the soft threshold value can be determined using linear algebra. The key step is an orthogonal projection onto the epigraph set of the `1 norm cost function. In standard wavelet denoising, a signal corrupted by additive noise is wavelet transformed and resulting wavelet subsignals are soft and/or hard thresholded. After this step the denoised signal is reconstructed from the thresholded wavelet subsignals [1, 2]. Thresholding the wavelet coefficients intuitively makes sense because wavelet subsignals obtained from an orthogonal or biorgthogonal wavelet filterbank exhibit large amplitude coefficients only around edges or change locations of the original signal. Other small amplitude coefficients should be due to noise. Many other related wavelet denoising methods are developed based on Donoho and Johnstone’s idea, see e.g. [1–6]. Most denoising methods take advantage of sparse nature of practical signals in wavelet domain to reduce the noise [7–12]. Consider the following basic denoising framework. Let v[n] be a discretetime signal and x[n] be a
Phase and TV Based Convex Sets for Blind Deconvolution of Microscopic Images
"... Abstract—In this article, two closed and convex sets for blind deconvolution problem are proposed. Most blurring functions in microscopy are symmetric with respect to the origin. Therefore, they do not modify the phase of the Fourier transform (FT) of the original image. As a result blurred image an ..."
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Abstract—In this article, two closed and convex sets for blind deconvolution problem are proposed. Most blurring functions in microscopy are symmetric with respect to the origin. Therefore, they do not modify the phase of the Fourier transform (FT) of the original image. As a result blurred image and the original image have the same FT phase. Therefore, the set of images with a prescribed FT phase can be used as a constraint set in blind deconvolution problems. Another convex set that can be used during the image reconstruction process is the Epigraph Set of Total Variation (ESTV) function. This set does not need a prescribed upper bound on the total variation of the image. The upper bound is automatically adjusted according to the current image of the restoration process. Both the TV of the image and the point spread function are regularized using the ESTV set. Both the phase information set and the ESTV are closed and convex sets. Therefore they can be used as a part of any blind deconvolution algorithm. Simulation examples are presented. Index Terms—Projection onto Convex Sets, Blind Deconvolution, Inverse Problems, Epigraph Sets