scholarly journals Seismic absolute acoustic impedance inversion with L1 norm reflectivity constraint and combined first- and second-order total variation regularizations

2019 ◽  
Vol 16 (4) ◽  
pp. 773-788 ◽  
Author(s):  
Song Guo ◽  
Huazhong Wang
Keyword(s):  
Total Variation ◽  
Acoustic Impedance ◽  
A Priori ◽  
Intermediate Frequency ◽  
Geological Structure ◽  
Second Order ◽  
L1 Norm ◽  
Split Bregman ◽  
Linear Inversion ◽  
First Order

Abstract Absolute acoustic impedance (AI) is generally divided into background AI and relative AI for linear inversion. In practice, the intermediate frequency components of the AI model are generally poorly reconstructed, so the estimated AI will suffer from an error caused by the frequency gap. To remedy this error, a priori information should be incorporated to narrow down the gap. With the knowledge that underground reflectivity was sparse, we solved an L1 norm constrained problem to extend the bandwidth of the reflectivity section, and an absolute AI model was then estimated with broadband reflectivity section and given background AI. Conventionally, the AI model is regularized with the total variation (TV) norm because of its blocky feature. However, the first-order TV norm that leads to piecewise-constant solutions will cause staircase errors in slanted and smooth regions in the inverted AI model. To better restore the smooth variation while preserving the sharp geological structure of the AI model, we introduced a second-order extension of the first-order TV norm and inverted the absolute AI model with combined first- and second-order TV regularizations. The algorithm used to solve the optimization problem with the combined TV constraints was derived based on split-Bregman iterations. Numerical experiments that were tested on the Marmousi AI model and 2D stacked field data illustrated the effectiveness of the sparse constraint with respect to shrinking the frequency gaps and proved that the proposed combined TV norms had better performances than those with conventional first-order TV norms.

scholarly journals Performance of relaxed iterative methods for image deblurring problems

2019 ◽  
Vol 13 ◽  
pp. 174830261986173 ◽  
Author(s):  
Jae H Yun
Keyword(s):  
Fixed Point ◽  
Iterative Methods ◽  
Total Variation ◽  
Numerical Experiments ◽  
Krylov Subspace ◽  
Image Deblurring ◽  
Total Variation Regularization ◽  
L1 Norm ◽  
Split Bregman ◽  
Relaxation Parameters

In this paper, we consider performance of relaxation iterative methods for four types of image deblurring problems with different regularization terms. We first study how to apply relaxation iterative methods efficiently to the Tikhonov regularization problems, and then we propose how to find good preconditioners and near optimal relaxation parameters which are essential factors for fast convergence rate and computational efficiency of relaxation iterative methods. We next study efficient applications of relaxation iterative methods to Split Bregman method and the fixed point method for solving the L1-norm or total variation regularization problems. Lastly, we provide numerical experiments for four types of image deblurring problems to evaluate the efficiency of relaxation iterative methods by comparing their performances with those of Krylov subspace iterative methods. Numerical experiments show that the proposed techniques for finding preconditioners and near optimal relaxation parameters of relaxation iterative methods work well for image deblurring problems. For the L1-norm and total variation regularization problems, Split Bregman and fixed point methods using relaxation iterative methods perform quite well in terms of both peak signal to noise ratio values and execution time as compared with those using Krylov subspace methods.

scholarly journals Image Deblurring under Impulse Noise via Total Generalized Variation and Non-Convex Shrinkage

Algorithms ◽  
2019 ◽  
Vol 12 (10) ◽  
pp. 221
Author(s):  
Lin ◽  
Chen ◽  
Chen ◽  
Yu
Keyword(s):  
Total Variation ◽  
Impulse Noise ◽  
Signal To Noise Ratio ◽  
Image Deblurring ◽  
L1 Norm ◽  
Signal To Noise ◽  
Total Generalized Variation ◽  
First Order ◽  
Norm Constraint ◽  
Generalized Variation

Image deblurring under the background of impulse noise is a typically ill-posed inverse problem which attracted great attention in the fields of image processing and computer vision. The fast total variation deconvolution (FTVd) algorithm proved to be an effective way to solve this problem. However, it only considers sparsity of the first-order total variation, resulting in staircase artefacts. The L1 norm is adopted in the FTVd model to depict the sparsity of the impulse noise, while the L1 norm has limited capacity of depicting it. To overcome this limitation, we present a new algorithm based on the Lp-pseudo-norm and total generalized variation (TGV) regularization. The TGV regularization puts sparse constraints on both the first-order and second-order gradients of the image, effectively preserving the image edge while relieving undesirable artefacts. The Lp-pseudo-norm constraint is employed to replace the L1 norm constraint to depict the sparsity of the impulse noise more precisely. The alternating direction method of multipliers is adopted to solve the proposed model. In the numerical experiments, the proposed algorithm is compared with some state-of-the-art algorithms in terms of peak signal-to-noise ratio (PSNR), structural similarity (SSIM), signal-to-noise ratio (SNR), operation time, and visual effects to verify its superiority.

Skepticism, A Priori Skepticism, and the Possibility of Error

2013 ◽  
Vol 3 (4) ◽  
pp. 235-252 ◽  
Author(s):  
Hamid Vahid
Keyword(s):  
A Priori ◽  
Second Order ◽  
Main Concern ◽  
A Posteriori ◽  
Standard View ◽  
First Order ◽  
Radical Skepticism ◽  
In Virtue Of

Epistemologists have differed in their assessments of what it is in virtue of which skeptical hypotheses succeed in raising doubts. It is widely thought that skeptical hypotheses must satisfy some sort of possibility constraint and that only putative knowledge of contingent and a posteriori propositions is vulnerable to skeptical challenge. These putative constraints have been disputed by a number of epistemologists advocating what we may call “the non-standard view.” My main concern in this paper is to challenge this view by identifying a general recipe by means of which its proponents generate skeptical scenarios. I will argue that many of the skeptical arguments that are founded on these scenarios undermine at most second-order knowledge and that to that extent the non-standard view’s rejection of the standard constraints on skeptical hypotheses is problematic. It will be argued that, pace the non-standard view, only in their error-inducing capacities can skeptical hypotheses challenge first-order knowledge. I will also dispute the non-standard view’s claim that its skeptical arguments bring to light a neglected form of radical skepticism, namely, “a priori skepticism.” I conclude by contending that the non-standard view’s account of how skeptical hypotheses can raise legitimate doubt actually rides piggyback on the standard ways of challenging the possibility of knowledge.

An Efficient Proximity Point Algorithm for Total-Variation-Based Image Restoration

2014 ◽  
Vol 6 (2) ◽  
pp. 145-164 ◽  
Author(s):  
Wei Zhu ◽  
Shi Shu ◽  
Lizhi Cheng
Keyword(s):  
Image Restoration ◽  
Total Variation ◽  
Projection Algorithm ◽  
Split Bregman ◽  
Fixed Point Algorithm ◽  
First Order ◽  
Special Cases ◽  
Point Operator ◽  
Novel Method ◽  
Split Bregman Algorithm

AbstractIn this paper, we propose a fast proximity point algorithm and apply it to total variation (TV) based image restoration. The novel method is derived from the idea of establishing a general proximity point operator framework based on which new first-order schemes for total variation (TV) based image restoration have been proposed. Many current algorithms for TV-based image restoration, such as Chambolle’s projection algorithm, the split Bregman algorithm, the Bermúdez-Moreno algorithm, the Jia-Zhao denoising algorithm, and the fixed point algorithm, can be viewed as special cases of the new first-order schemes. Moreover, the convergence of the new algorithm has been analyzed at length. Finally, we make comparisons with the split Bregman algorithm which is one of the best algorithms for solving TV-based image restoration at present. Numerical experiments illustrate the efficiency of the proposed algorithms.

scholarly journals Positive solutions of a second-order nonlinear Robin problem involving the first-order derivative

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zhilin Yang
Keyword(s):  
Positive Solutions ◽  
A Priori Estimates ◽  
A Priori ◽  
Index Theory ◽  
Second Order ◽  
Iterative Sequence ◽  
Order Derivative ◽  
Robin Problem ◽  
First Order ◽  
Fixed Point Index Theory

AbstractThis paper is concerned with the second-order nonlinear Robin problem involving the first-order derivative: $$ \textstyle\begin{cases} u''+f(t,u,u^{\prime })=0, \\ u(0)=u'(1)-\alpha u(1)=0,\end{cases} $$ { u ″ + f ( t , u , u ′ ) = 0 , u ( 0 ) = u ′ ( 1 ) − α u ( 1 ) = 0 , where $f\in C([0,1]\times \mathbb{R}^{2}_{+},\mathbb{R}_{+})$ f ∈ C ( [ 0 , 1 ] × R + 2 , R + ) and $\alpha \in ]0,1[$ α ∈ ] 0 , 1 [ . Based on a priori estimates, we use fixed point index theory to establish some results on existence, multiplicity and uniqueness of positive solutions thereof, with the unique positive solution being the limit of of an iterative sequence. The results presented here generalize and extend the corresponding ones for nonlinearities independent of the first-order derivative.

scholarly journals Combined First and Second Order Total Variation Inpainting using Split Bregman

2013 ◽  
Vol 3 ◽  
pp. 112-136 ◽  
Author(s):  
Konstantinos Papafitsoros ◽  
Carola Bibiane Schoenlieb ◽  
Bati Sengul
Keyword(s):  
Total Variation ◽  
Second Order ◽  
Split Bregman

Nonlinear initial value problems with a singular perturbation of hyperbolic type

1981 ◽  
Vol 89 (3-4) ◽  
pp. 333-345 ◽  
Author(s):  
R. Geel
Keyword(s):  
Initial Value Problems ◽  
A Priori ◽  
Second Order ◽  
Initial Value ◽  
Existence Of A Solution ◽  
Nonlinear Operators ◽  
First Order ◽  
Order Part ◽  
The Difference ◽  
Small Factor

SynopsisThis paper deals with initial value problems in ℝ2 which are governed by a hyperbolic differential equation consisting of a nonlinear first order part and a linear second order part. The second order part of the differential operator contains a small factor ε and can therefore be considered as a perturbation of the nonlinear first order part of the operator.The existence of a solution u together with pointwise a priori estimates for this solution are established by applying a fixed point theorem for nonlinear operators in a Banach space.It is shown that the difference between the solution u and the solution w of the unperturbed nonlinear initial value problem (which follows from the original problem by putting ε = 0) is of order ε, uniformly in compact subsets of ℝ2 where w is sufficiently smooth.

Scheler’s Reflections on “What is Good?”

2021 ◽  
Vol 21 ◽  
pp. 349-365
Author(s):  
Wei Zhang ◽  
Keyword(s):  
A Priori ◽  
Moral Values ◽  
Second Order ◽  
Moral Cognition ◽  
Moral Value ◽  
First Order ◽  
A Value ◽  
Relevant Notion

In Max Scheler’s non-formal ethics of value, “good” is a value but by no means a “non-moral value”; rather, it is a second-order “moral value,” always appearing in the realization of first-order non-moral values. According to the relevant notion of the a priori of phenomenology, whilst all the non-moral values are given in “value cognition,” the moral value of good is self-given in “moral cognition”. The reflections and answers offered by Scheler’s non-formal ethics of value on “What is good?” constitute the foundation of a phenomenological “meta-ethics”.

scholarly journals A Unidirectional Total Variation and Second-Order Total Variation Model for Destriping of Remote Sensing Images

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Min Wang ◽  
Ting-Zhu Huang ◽  
Xi-Le Zhao ◽  
Liang-Jian Deng ◽  
Gang Liu
Keyword(s):  
Remote Sensing ◽  
Total Variation ◽  
Second Order ◽  
Total Variation Regularization ◽  
Remote Sensing Images ◽  
Split Bregman ◽  
Directional Characteristic ◽  
Directional Information ◽  
Split Bregman Iteration ◽  
Large Width

Remote sensing images often suffer from stripe noise, which greatly degrades the image quality. Destriping of remote sensing images is to recover a good image from the image containing stripe noise. Since the stripes in remote sensing images have a directional characteristic (horizontal or vertical), the unidirectional total variation has been used to consider the directional information and preserve the edges. The remote sensing image contaminated by heavy stripe noise always has large width stripes and the pixels in the stripes have low correlations with the true pixels. On this occasion, the destriping process can be viewed as inpainting the wide stripe domains. In many works, high-order total variation has been proved to be a powerful tool to inpainting wide domains. Therefore, in this paper, we propose a variational destriping model that combines unidirectional total variation and second-order total variation regularization to employ the directional information and handle the wide stripes. In particular, the split Bregman iteration method is employed to solve the proposed model. Experimental results demonstrate the effectiveness of the proposed method.

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