Seismic high-resolution processing method based on spectral simulation and total variation regularization constraints

There is little low-and-high frequency information on seismic data in seismic exploration,resulting in narrower bandwidth and lower seismic resolution.It considerably restricts the prediction accuracy of thin reservoirs and thin interbeds.This study proposes a novel method to constrain improving seismic resolution in the time and frequency domain.The expected wavelet spectrum is used in the frequency domain to broaden the seismic spectrum range and increase the octave.In the time domain,the Frobenius vector regularization of the Hessian matrix is used to constrain the horizontal continuity of the seismic data.It eff ectively protects the signal-to-noise ratio of seismic data while the longitudinal seismic resolution is improved.This method is applied to actual post-stack seismic data and pre-stack gathers dividedly.Without abolishing the phase characteristics of the original seismic data,the time resolution is signifi cantly improved,and the structural features are clearer.Compared with the traditional spectral simulation and deconvolution methods,the frequency distribution is more reasonable,and seismic data has higher resolution.

Lipschitz and Total-Variational Regularization for Blind Deconvolution

In[3],Chan and Wong proposed to use total variational regularization for both images and point spread functions in blind deconvolution.Their experimental results show that the detail of the restored images cannot be recovered.In this paper,we consider images in Lipschitz spaces,and propose to use Lipschitz regularization for images and total variational regularization for point spread functions in blind deconvolution.Our experimental results show that such combination of Lipschitz and total variational regularization methods can recover both images and point spread functions quite well.

Total Variation Based Parameter-Free Model for Impulse Noise Removal

We propose a new two-phase method for reconstruction of blurred im-ages corrupted by impulse noise.In the first phase,we use a noise detector to iden-tify the pixels that are contaminated by noise,and then,in the second phase,we reconstruct the noisy pixels by solving an equality constrained total variation mini-mization problem that preserves the exact values of the noise-free pixels.For images that are only corrupted by impulse noise(i.e.,not blurred)we apply the semismooth Newton’s method to a reduced problem,and if the images are also blurred,we solve the equality constrained reconstruction problem using a first-order primal-dual algo-rithm.The proposed model improves the computational efficiency(in the denoising case)and has the advantage of being regularization parameter-free.Our numerical results suggest that the method is competitive in terms of its restoration capabilities with respect to the other two-phase methods.

An Alternating Direction Approximate Newton Algorithm for Ill-Conditioned Inverse Problems with Application to Parallel MRI

Analternating direction approximateNewton(ADAN)method is developed for solving inverse problems of the form min{φ(Bu)+(1/2)||Au−f||^(2)_(2)},whereφis convex and possibly nonsmooth,and A and B arematrices.Problems of this form arise in image reconstruction where A is the matrix describing the imaging device,f is the measured data,φis a regularization term,and B is a derivative operator.The proposed algorithm is designed to handle applications where A is a large dense,ill-conditioned matrix.The algorithm is based on the alternating direction method of multipliers(ADMM)and an approximation to Newton’s method in which a term in Newton’s Hessian is replaced by aBarzilai–Borwein(BB)approximation.It is shown thatADAN converges to a solution of the inverse problem.Numerical results are provided using test problems from parallel magnetic resonance imaging.ADAN was faster than a proximal ADMM scheme that does not employ a BB Hessian approximation,while it was more stable and much simpler than the related Bregman operator splitting algorithm with variable stepsize algorithm which also employs a BB-based Hessian approximation.

Stroke-Based Surface Reconstruction

In this paper,we present a surface reconstruction via 2D strokes and a vector field on the strokes based on a two-step method.In the first step,from sparse strokes drawn by artists and a given vector field on the strokes,we propose a nonlinear vector interpolation combining total variation(TV)and H1 regularization with a curl-free constraint for obtaining a dense vector field.In the second step,a height map is obtained by integrating the dense vector field in the first step.Jump discontinuities in surface and discontinuities of surface gradients can be well reconstructed without any surface distortion.We also provide a fast and efficient algorithm for solving the proposed functionals.Since vectors on the strokes are interpreted as a projection of surface gradients onto the plane,different types of strokes are easily devised to generate geometrically crucial structures such as ridge,valley,jump,bump,and dip on the surface.The stroke types help users to create a surface which they intuitively imagine from 2D strokes.We compare our results with conventional methods via many examples.

ESTIMATION AND UNCERTAINTY QUANTIFICATION FOR PIECEWISE SMOOTH SIGNAL RECOVERY

This paper presents an application of the sparse Bayesian learning(SBL)algorithm to linear inverse problems with a high order total variation(HOTV)sparsity prior.For the problem of sparse signal recovery,SBL often produces more accurate estimates than maximum a posteriori estimates,including those that useℓ1 regularization.Moreover,rather than a single signal estimate,SBL yields a full posterior density estimate which can be used for uncertainty quantification.However,SBL is only immediately applicable to problems having a direct sparsity prior,or to those that can be formed via synthesis.This paper demonstrates how a problem with an HOTV sparsity prior can be formulated via synthesis,and then utilizes SBL.This expands the class of problems available to Bayesian learning to include,e.g.,inverse problems dealing with the recovery of piecewise smooth functions or signals from data.Numerical examples are provided to demonstrate how this new technique is effectively employed.

An Alternating Direction Method of Multipliers for Inverse Lithography Problem

We propose an alternating direction method of multipliers(ADMM)to solve an optimization problem stemming from inverse lithography.The objective functional of the optimization problem includes three terms:the misfit between the imaging on wafer and the target pattern,the penalty term which ensures the mask is binary and the total variation regularization term.By variable splitting,we introduce an augmented Lagrangian for the original objective functional.In the framework of ADMM method,the optimization problem is divided into several subproblems.Each of the subproblems can be solved efficiently.We give the convergence analysis of the proposed method.Specially,instead of solving the subproblem concerning sigmoid,we solve directly the threshold truncation imaging function which can be solved analytically.We also provide many numerical examples to illustrate the effectiveness of the method.