Seismic denoising via truncated nuclear norm minimization

Geophysics ◽  
2020 ◽  
pp. 1-60
Author(s):  
Ouyang Shao ◽  
Lingling Wang ◽  
Xiangyun Hu ◽  
Zhidan Long
Keyword(s):  
Objective Function ◽  
Seismic Data ◽  
Nuclear Norm ◽  
Low Rank ◽  
Seismic Structure ◽  
Denoising Method ◽  
Nuclear Norm Minimization ◽  
Norm Minimization ◽  
Rank Of A Matrix ◽  
Truncated Nuclear Norm

Because there are many similar geological structures underground, seismic profiles have an abundance of self-repeating patterns. Thus, we can divide a seismic profile into groups of blocks with similar seismic structure. The matrix formed by stacking together similar blocks in each group should be of low rank. Hence, we can transfer the seismic denoising problem to a serial of low-rank matrix approximation (LRMA) problem. The LRMA-based model commonly adopts the nuclear norm as a convex substitute of the rank of a matrix. However, the nuclear norm minimization (NNM) shrinks the different rank components equally and may cause some biases in practice. Recently introduced truncated nuclear norm (TNN) has been proven to more accurately approximate the rank of a matrix, which is given by the sum of the set of smallest singular values. Based on this, we propose a novel denoising method using truncated nuclear norm minimization (TNNM). The objective function of this method consists of two terms, the F-norm data fidelity and a truncated nuclear norm regularization. We present an efficient two-step iterative algorithm to solve this objective function. Then, we apply the proposed TNNM algorithm to groups of blocks with similar seismic structure, and aggregate all resulting denoised blocks to get the denoised seismic data. We update the denoised results during each iteration to gradually attenuate the heavy noise. Numerical experiments demonstrate that, compared with FX-Decon, the curvelet, and the NNM-based methods, TNNM not only attenuates noise more effectively even when the SNR is as low as -10 dB and seismic data have complex structures, but also accurately preserves the seismic structures without inducing Gibbs artifacts.

scholarly journals Truncated Nuclear Norm Minimization for Image Restoration Based on Iterative Support Detection

2014 ◽  
Vol 2014 ◽  
pp. 1-17 ◽  
Author(s):  
Yilun Wang ◽  
Xinhua Su
Keyword(s):  
Convex Relaxation ◽  
Nuclear Norm ◽  
Low Rank ◽  
Nuclear Norm Minimization ◽  
Matrix Recovery ◽  
Norm Minimization ◽  
Rank Matrix ◽  
Low Rank Matrix Recovery ◽  
Low Rank Matrix ◽  
Truncated Nuclear Norm

Recovering a large matrix from limited measurements is a challenging task arising in many real applications, such as image inpainting, compressive sensing, and medical imaging, and these kinds of problems are mostly formulated as low-rank matrix approximation problems. Due to the rank operator being nonconvex and discontinuous, most of the recent theoretical studies use the nuclear norm as a convex relaxation and the low-rank matrix recovery problem is solved through minimization of the nuclear norm regularized problem. However, a major limitation of nuclear norm minimization is that all the singular values are simultaneously minimized and the rank may not be well approximated (Hu et al., 2013). Correspondingly, in this paper, we propose a new multistage algorithm, which makes use of the concept of Truncated Nuclear Norm Regularization (TNNR) proposed by Hu et al., 2013, and iterative support detection (ISD) proposed by Wang and Yin, 2010, to overcome the above limitation. Besides matrix completion problems considered by Hu et al., 2013, the proposed method can be also extended to the general low-rank matrix recovery problems. Extensive experiments well validate the superiority of our new algorithms over other state-of-the-art methods.

Gradient nuclear norm minimization-based image filter

2019 ◽  
Vol 33 (19) ◽  
pp. 1950214 ◽  
Author(s):  
Saurabh Khare ◽  
Praveen Kaushik
Keyword(s):  
Signal To Noise Ratio ◽  
Similarity Index ◽  
Structural Similarity ◽  
Nuclear Norm ◽  
Low Rank ◽  
Low Rank Approximation ◽  
Edge Preservation ◽  
Nuclear Norm Minimization ◽  
Norm Minimization ◽  
Filtering Technique

Designing an efficient filtering technique is an ill-posed problem especially for image affected from high density of noise. The majority of existing techniques suffer from edge degradation and texture distortion issues. Therefore, in this paper, an efficient weighted nuclear norm minimization (NNM)-based filtering technique to preserve the edges and texture information of filtered images is proposed. The proposed technique significantly improves the quantitative improvements on the low rank approximation of nonlocal self-similarity matrices to deal with the overshrink problem. Extensive experiments reveal that the proposed technique preserves edges and texture details of filtered image with lesser number of visual artifacts on visual quality. The proposed technique outperforms the existing techniques over the competitive filtering techniques in terms of structural similarity index metric (SSIM), peak signal-to-noise ratio (PSNR) and edge preservation index (EPI).

scholarly journals Tensor theta norms and low rank recovery

2020 ◽  
Author(s):  
Holger Rauhut ◽  
Željka Stojanac
Keyword(s):  
Convex Relaxation ◽  
Nuclear Norm ◽  
Low Rank ◽  
Nuclear Norm Minimization ◽  
Computational Algebraic Geometry ◽  
Matrix Recovery ◽  
Norm Minimization ◽  
The Matrix ◽  
Higher Order Tensors ◽  
Tensor Nuclear Norm

AbstractWe study extensions of compressive sensing and low rank matrix recovery to the recovery of tensors of low rank from incomplete linear information. While the reconstruction of low rank matrices via nuclear norm minimization is rather well-understand by now, almost no theory is available so far for the extension to higher order tensors due to various theoretical and computational difficulties arising for tensor decompositions. In fact, nuclear norm minimization for matrix recovery is a tractable convex relaxation approach, but the extension of the nuclear norm to tensors is in general NP-hard to compute. In this article, we introduce convex relaxations of the tensor nuclear norm which are computable in polynomial time via semidefinite programming. Our approach is based on theta bodies, a concept from real computational algebraic geometry which is similar to the one of the better known Lasserre relaxations. We introduce polynomial ideals which are generated by the second-order minors corresponding to different matricizations of the tensor (where the tensor entries are treated as variables) such that the nuclear norm ball is the convex hull of the algebraic variety of the ideal. The theta body of order k for such an ideal generates a new norm which we call the θk-norm. We show that in the matrix case, these norms reduce to the standard nuclear norm. For tensors of order three or higher however, we indeed obtain new norms. The sequence of the corresponding unit-θk-norm balls converges asymptotically to the unit tensor nuclear norm ball. By providing the Gröbner basis for the ideals, we explicitly give semidefinite programs for the computation of the θk-norm and for the minimization of the θk-norm under an affine constraint. Finally, numerical experiments for order-three tensor recovery via θ1-norm minimization suggest that our approach successfully reconstructs tensors of low rank from incomplete linear (random) measurements.

Line survey joint denoising via low-rank minimization

Geophysics ◽  
2019 ◽  
Vol 84 (1) ◽  
pp. V21-V32 ◽  
Author(s):  
Zhao Liu ◽  
Jianwei Ma ◽  
Xueshan Yong
Keyword(s):  
Minimization Problem ◽  
Nuclear Norm ◽  
Low Rank ◽  
Single Shot ◽  
Rank Minimization ◽  
Nuclear Norm Minimization ◽  
Norm Minimization ◽  
Rank Matrix ◽  
Line Survey ◽  
Low Rank Matrix

Prestack seismic data denoising is an important step in seismic processing due to the development of prestack time migration. Reduced-rank filtering is a state-of-the-art method for prestack seismic denoising that uses predictability between neighbor traces for each single frequency. Different from the original way of embedding low-rank matrix based on the Hankel or Toeplitz transform, we have developed a new multishot gathers joint denoising method in a line survey, which used a new way of rearranging data to a matrix with low rank. Inspired by video denoising, each single-shot record in the line survey can be viewed as a frame in the video sequence. Due to high redundancy and similar event structure among the shot gathers, similar patches can be selected from different shot gathers in the line survey to rearrange a low-rank matrix. Then, seismic denoising is formulated into a low-rank minimization problem that can be further relaxed into a nuclear-norm minimization problem. A fast algorithm, called the orthogonal rank-one matrix pursuit, is used to solve the nuclear-norm minimization. Using this method avoids the computation of a full singular value decomposition. Our method is validated using synthetic and field data, in comparison with [Formula: see text] deconvolution and singular spectrum analysis methods.

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