Block Coordinate Relaxation Research Articles

Complex structure reconstruction using segmented random sampling and combined dictionary

The reconstruction of data is a critical preliminary work in the seismic data processing. Compressed sensing (CS) has been well applied in the field of reconstruction. The key point of CS is random sampling, which converts the mutual interference alias caused by regular undersampling into lower-amplitude outside noise. But traditional sampling methods lack constraints on sampling points, emerging too much alias. Segmented random sampling (SRS) effectively controls the distance between sampling points. On the other hand, a single mathematical transformation will lead to incomplete sparse expression and bad restoration effects. Morphological component analysis (MCA) decomposes a signal into several components with outstanding morphological features to approximate the complex internal structure of data. In this paper, we found a new dictionary combination (shearlet + DCT) under the MCA framework and used the block coordinate relaxation algorithm to get the optimal solution to obtain reconstruction results. Tests of 2D data and 3D data have proved that the proposed method can get a better effect when reconstructing the SRS data.

Ground-Roll Separation and Attenuation Using Curvelet-Based Multichannel Variational Mode Decomposition

Ground roll, a source-generated surface wave, is a main type of coherent noise in a land seismic survey. Low-frequency and high-amplitude ground roll often overlays valid reflection events, resulting in obscuring seismic reflections. Ground-roll attenuation is an essential step for seismic data processing, which is based on the accurate separation of the ground roll and reflections without damaging their morphological characteristics. In this study, we effectively separate and suppress ground roll in shot gathers through a proposed workflow. We first identify the major components of the ground roll adopting the multichannel variational mode decomposition (MVMD), which shows significant improvements compared to the conventional single-channel VMD. Ground roll can be identified on the decomposed band-limited intrinsic mode functions (IMFs). Moreover, we propose an adaptive criterion to determine the number of decomposed IMFs. Due to the narrowly concentrated frequency components with the multichannel continuity constraint from MVMD, ground roll is mainly contained in low-frequency IMFs, which benefits the accurate ground-roll suppression. Next, we separate ground roll and reflections on the selected low-frequency IMFs through a curvelet based block-coordinate relaxation method. Afterward, we can obtain a filtered gather by removing the separated ground roll from the original shot gather. Finally, we apply the proposed workflow to synthetic and field gathers to testify its validity and effectiveness for simultaneously attenuating ground roll and preserving valid seismic reflector information.

Seismic data reconstruction method based on morphological component analysis

The real seismic data is usually under-sampled in space domain because of the physical or economic limitations. So the incomplete seismic data need to be reconstructed before the subsequent processing. A method based on morphological component analysis (MCA) is discussed in the paper, which uses the curvelet dictionary and local discrete cosine transform (LDCT) dictionary to reconstruct the smooth components and the singular components respectively. Block coordinate relaxation (BCR) algorithm is adopted to complete the sparse optimisation. The validity of the proposed method was tested by numerical experiments on synthetic and real data demonstrate. As the method based on curvelet combining with POCS is widely used in practice, we compare its reconstructed results with the MCA-based method. The numerical results validate that the proposed method has higher reconstruction performance.

Distributed acoustic sensing coupling noise removal based on sparse optimization

The industry treats the distributed acoustic sensing (DAS) system, which uses an optical fiber cable in vertical seismic profiling (VSP) data acquisition, as a new and fast-growing technology. The high-quality data set acquired from the DAS acquisition system can produce high-precision VSP images and obtain more detailed checkshots. However, in field data, the acquired VSP data set suffers from strong coherent DAS coupling noise. Many factors may cause coherent DAS coupling noise, such as the cable slapping and ringing due to the physical placement, the regular swing of the wireline in the well, and the uncoupling between the fiber cable and the casing. This DAS coupling noise reduces the signal-to-noise ratio and affects the subsequent processing and interpretation. Removing the DAS coupling noise can help to improve the quality of the VSP data set acquired with the DAS system. We have developed a sparse-optimization-based DAS coupling noise removing method. In the DAS-based VSP data set, the effective signal and the coupling noise have distinct morphological characteristics. The effective VSP signal has a wide bandwidth, whereas the DAS coupling noise appears in some narrow frequency bands in the frequency domain. The continuous wavelet transform and the discrete cosine transform can sparsely represent the effective VSP signal and DAS coupling noise, respectively. Therefore, we choose these two transforms as two sparse dictionaries and combine them to form an overcomplete dictionary. The morphological component analysis (MCA) can use the morphological difference between different components and the overcomplete dictionary to sparsely represent all components in the complicated signal. Based on the MCA theory, we use the block coordinate relaxation algorithm to separate the effective VSP signal and DAS coupling noise. Applications on a synthetic data set and two field data sets have validated the effectiveness of our method.

Ground-roll separation of seismic data based on morphological component analysis in two-dimensional domain

Ground roll is an interference wave that severely degrades the signal-to-noise ratio of seismic data and affects its subsequent processing and interpretation. In this study, according to differences in morphological characteristics between ground roll and reflected waves, we use morphological component analysis based on two-dimensional dictionaries to separate ground roll and reflected waves. Because ground roll is characterized by low-frequency, low-velocity, and dispersion, we select two-dimensional undecimated discrete wavelet transform as a sparse representation dictionary of ground roll. Because of a strong local correlation of the reflected wave, we select two-dimensional local discrete cosine transform as the sparse representation dictionary of reflected waves. A sparse representation model of seismic data is constructed based on a two-dimensional joint dictionary then a block coordinate relaxation algorithm is used to solve the model and decompose seismic record into reflected wave part and ground roll part.The good effects for the synthetic seismic data and application of real seismic data indicate that when using the model, strong-energy ground roll is considerably suppressed and the waveform of the reflected wave is effectively protected.

Noise reduction of diffusion tensor images by sparse representation and dictionary learning.

BackgroundThe low quality of diffusion tensor image (DTI) could affect the accuracy of oncology diagnosis.MethodsWe present a novel sparse representation based denoising method for three dimensional DTI by learning adaptive dictionary with the context redundancy between neighbor slices. In this study, the context redundancy among the adjacent slices of the diffusion weighted imaging volumes is utilized to train sparsifying dictionaries. Therefore, higher redundancy could be achieved for better description of image with lower computation complexity. The optimization problem is solved efficiently using an iterative block-coordinate relaxation method.ResultsThe effectiveness of our proposed method has been assessed on both simulated and real experimental DTI datasets. Qualitative and quantitative evaluations demonstrate the performance of the proposed method on the simulated data. The experiments on real datasets with different b-values also show the effectiveness of the proposed method for noise reduction of DTI.ConclusionsThe proposed approach well removes the noise in the DTI, which has high potential to be applied for clinical oncology applications.

Accelerated [formula omitted] regularization based SAR imaging via BCR and reduced Newton skills

Sparse synthetic aperture radar (SAR) imaging has been highlighted in recent studies. As an important sparsity constraint, L1/2 regularizer has been substantiated effectively when applied to SAR imaging. However, L1/2-SAR imaging suffers from a common challenge with other sparse SAR imaging methods: the computational complexity is costly, especially for high dimensional applications. This challenge is mainly due to that L1/2-SAR imaging is a gradient descent based method, of which the convergence is at most linear. Thus, a lot of iterations are often necessary to yield a satisfactory result. In this paper, we propose an accelerated L1/2-SAR imaging method by applying the block coordinate relaxation (BCR) scheme combined with the reduced Newton skill for acceleration. It is numerically shown that the proposed method keeps fast convergence within a very few iterations, and also maintains high reconstruction precision. We provide a series of simulations and two real SAR applications to demonstrate the superiority of the proposed method. Particularly, much faster convergence and higher reconstruction precision in imaging, of the proposed method over the other sparse SAR imaging methods.

Accelerated Block-coordinate Relaxation for Regularized Optimization

We discuss minimization of a smooth function to which is added a separable regularization function that induces structure in the solution. A block-coordinate relaxation approach with proximal linearized subproblems yields convergence to critical points, while identification of the optimal manifold (under a nondegeneracy condition) allows acceleration techniques to be applied on a reduced space. The work is motivated by experience with an algorithm for regularized logistic regression, and computational results for the algorithm on problems of this type are presented.

Density Estimation by Total Variation Penalized Likelihood Driven by the Sparsity ℓ1 Information Criterion

Abstract. We propose a non-linear density estimator, which is locally adaptive, like wavelet estimators, and positive everywhere, without a log- or root-transform. This estimator is based on maximizing a non-parametric log-likelihood function regularized by a total variation penalty. The smoothness is driven by a single penalty parameter, and to avoid cross-validation, we derive an information criterion based on the idea of universal penalty. The penalized log-likelihood maximization is reformulated as an ℓ1-penalized strictly convex programme whose unique solution is the density estimate. A Newton-type method cannot be applied to calculate the estimate because the ℓ1-penalty is non-differentiable. Instead, we use a dual block coordinate relaxation method that exploits the problem structure. By comparing with kernel, spline and taut string estimators on a Monte Carlo simulation, and by investigating the sensitivity to ties on two real data sets, we observe that the new estimator achieves good L1 and L2 risk for densities with sharp features, and behaves well with ties.

Sparsity and persistence: mixed norms provide simple signal models with dependent coefficients

Sparse regression often uses $\ell_p$ norm priors (with p<2). This paper demonstrates that the introduction of mixed-norms in such contexts allows one to go one step beyond in signal models, and promote some different, structured, forms of sparsity. It is shown that the particular case of $\ell_{1,2}$ and $\ell_{2,1}$ norms lead to new group shrinkage operators. Mixed norm priors are shown to be particularly efficient in a generalized basis pursuit denoising approach, and are also used in a context of morphological component analysis. A suitable version of the Block Coordinate Relaxation algorithm is derived for the latter. The group-shrinkage operators are then modified to overcome some limitations of the mixed-norms. The proposed group shrinkage operators are tested on simulated signals in specific situations, to illustrate their different behaviors. Results on real data are also used to illustrate the relevance of the approach.

AMlet, RAMlet, and GAMlet: Automatic Nonlinear Fitting of Additive Models, Robust and Generalized, With Wavelets

A simple and yet powerful method is presented to estimate nonlinearly and nonparametrically the components of additive models using wavelets. The estimator enjoys the good statistical and computational properties of the Waveshrink scatterplot smoother and it can be efficiently computed using the block coordinate relaxation optimization technique. A rule for the automatic selection of the smoothing parameters, suitable for data mining of large datasets, is derived. The wavelet-based method is then extended to estimate generalized additive models. A primal-dual log-barrier interior point algorithm is proposed to solve the corresponding convex programming problem. Based on an asymptotic analysis, a rule for selecting the smoothing parameters is derived, enabling the estimator to be fully automated in practice. We illustrate the finite sample property with a Gaussian and a Poisson simulation.

Block Coordinate Relaxation Methods for Nonparametric Wavelet Denoising

An important class of nonparametric signal processing methods entails forming a set of predictors from an overcomplete set of basis functions associated with a fast transform (e.g., wavelet packets). In these methods, the number of basis functions can far exceed the number of sample values in the signal, leading to an ill-posed prediction problem. The “basis pursuit” denoising method of Chen, Donoho, and Saunders regularizes the prediction problem by adding an l 1 penalty term on the coefficients for the basis functions. Use of an l 1 penalty instead of l 2 has significant benefits, including higher resolution of signals close in time/frequency and a more parsimonious representation. The l 1 penalty, however, poses a challenging optimization problem that was solved by Chen, Donoho and Saunders using a novel application of interior-point algorithms (IP). This article investigates an alternative optimization approach based on block coordinate relaxation (BCR) for sets of basis functions that are the finite union of sets of orthonormal basis functions (e.g., wavelet packets). We show that the BCR algorithm is globally convergent, and empirically, the BCR algorithm is faster than the IP algorithm for a variety of signal denoising problems.