Data-driven Tight Frame Research Articles

Three dictionary learning algorithms and their applications for marine controlled source electromagnetic data denoising

Marine controlled source electromagnetic (MCSEM) is profoundly used for undersea resources exploration. The effective signal is easily contaminated by kinds of noise when the transmitter-receiver offset is large. Suppressing the noise influence is vital to improve data quality and further interpretation accuracy. Denoising becomes a research focus with the widespread application of the MCSEM technique. Many denoising approaches are proposed by different researchers. However, most of them only target a single type of noise, which severely limits the application of these approaches. The fast-developing dictionary learning technique paves a new way for MCSEM data denoising. Currently, typical dictionary learning algorithms include k-means singular value decomposition (K-SVD), data-driven tight frame (DDTF), shift-invariant sparse coding (SISC) and so on. These three algorithms are different in principles and arithmetic processes. Their applications for MCSEM data denoising are explored for the first time in this article. Besides, a comparative analysis of these three noise reduction methods is carried out. The comparison proves the effectiveness and superiority of the K-SVD, followed by the DDTF method. Besides, all these denoising methods are applied to the field data. The results further corroborates the above conclusions.

Fractional-Order Variational Image Fusion and Denoising Based on Data-Driven Tight Frame

Multi-modal image fusion can provide more image information, which improves the image quality for subsequent image processing tasks. Because the images acquired using photon counting devices always suffer from Poisson noise, this paper proposes a new three-step method based on the fractional-order variational method and data-driven tight frame to solve the problem of multi-modal image fusion for images corrupted by Poisson noise. Thus, this article obtains fused high-quality images while removing Poisson noise. The proposed image fusion model can be solved by the split Bregman algorithm which has significant stability and fast convergence. The numerical results on various modal images show the excellent performance of the proposed three-step method in terms of numerical evaluation metrics and visual quality. Extensive experiments demonstrate that our method outperforms state-of-the-art methods on image fusion with Poisson noise.

Random noise attenuation with weak feature preservation via total variation regularization

Dictionary learning (DL) methods have been successfully applied to seismic data denoising. However, when the noise is strong, weak signals cannot be well preserved. Considering the characteristics of seismic data in the spatial-time domain, we propose a dictionary learning method with total variation regularization (DLTV). The DLTV method consists of two steps. The first step is to learn the dictionary; it uses the sparse feature of the data in the dictionary domain and the learned dictionary contains data features. The second step is to use the augmented Lagrange multiplier method to restore the data, and total variation regularization to preserve the weak signal by sampling data information around it. An example with seismic data shows that the proposed method retains the weak features better than the traditional F-X deconvolution (FX-Decon) method and the data-driven tight frame method (DDTF).

DL2: Dictionary learning regularized with deep learning prior for simultaneous denoising and interpolation

The dictionary learning method has been successfully applied to denoise and interpolate seismic data. However, this method cannot be used to adequately interpret weak seismic events and structural features. By combining dictionary learning and a convolutional neural network (CNN) denoiser, we have constructed a new dictionary learning method regularized by a supervised denoiser (DL-SD). In addition to the sparse prior used in previous dictionary learning, the CNN denoiser learns from sizeable amounts of natural images using a deep neural network to help regularize the fine and structural features of data in the DL-SD. We use the plug-and-play alternating directional method of multipliers to solve the net-transform balanced DL-SD model. The results of simultaneous denoising and interpolation indicates that the proposed method is more effective than a deep learning method called the FFDNet and a dictionary learning method known as the data-driven tight frame.

Simple framework for the contrastive learning of visual representations-based data-driven tight frame for seismic denoising and interpolation

Complex seismic data sets benefit from adaptive sparse transformation methods to improve resolution and fidelity. The data-driven tight frame (DDTF) is a fast dictionary learning method that can simplify the sparse representation process and efficiently update the compact frame. However, the DDTF-based method is computationally intensive and relatively ineffective in accurately characterizing high-dimensional data sets with weak features and high noise levels. A new adaptive patch-selection method based on a simple framework for the contrastive learning of visual representations (SimCLR) is designed to further accelerate and improve the dictionary training process in the DDTF. The variance of all of the available patches is first calculated; then, the cosine similarity between the patch with the maximum variance and the other patches is calculated. Finally, the patches with large similarities are used for the filter bank training process. This method can efficiently track seismic events. The efficacy of applying the trained filter bank for seismic data recovery is tested by using the Monte Carlo-DDTF and SimCLR-DDTF methods. The numerical results demonstrate that the SimCLR-DDTF method can intelligently select training patches with more effective information, greatly improve the training efficiency of the DDTF, and obtain high-quality denoising and interpolation results.

Audio magnetotelluric denoising via variational mode decomposition and adaptive dictionary learning

The audio magnetotelluric (AMT) data are inevitably corrupted by cultural noise in the AMT exploration. This noise will distort apparent resistivity and phase, which affects the geophysical interpretation. The current AMT denoising technologies can not perform well when the cultural noise is severe. In order to improve the denoising accuracy, we propose a denoising technology called V-DDTF, which combines the variational mode decomposition (VMD) and the data-driven tight frame (DDTF). In the V-DDTF, The instantaneous frequency of the AMT signal envelope is first used to determine the number of decomposition layers. Then, the AMT signal is decomposed to several intrinsic mode functions (IMFs) and a residual by the VMD, after which the IMFs and a residual are rearranged as the input to learn dictionaries based on a tight frame constraint. To optimize the selection of parameters and improve denoising accuracy, we adopt an adaptive dictionary learning strategy. Finally, the inverse transforms are applied to the coefficients to achieve the AMT denoising. In our experiments, we validate the V-DDTF by synthetic data. We compare the V-DDTF with some other denoising methods and use the signal-to-noise ratio (SNR) and the Nyquist diagram as the assessments. Consequently, the V-DDTF not only creates dependable denoising results but also achieves the best performance among these mentioned methods. Furthermore, the field data experiments show the potential of the V-DDTF in industrial applications.

Data-Driven Tight Frame Construction for Impulsive Noise Removal

Data-Driven Tight Frame Construction for Impulsive Noise Removal

GPR denoising via shearlet transformation and a data‐driven tight frame

ABSTRACTGround‐penetrating radar (GPR) is commonly used to detect buried and near‐surface geophysical structures. GPR denoising is necessary because some level of interference, such as from clutter, random noise and/or the column artefact, are inevitable and can cause false geological interpretations. Existing sparse representation methods, including wavelet transformation, curvelet transformation and dictionary learning, are critical in GPR denoising. However, they perform poorly in some cases because GPR data cannot be represented efficiently under severe interference. Thus, this study proposes an approach that combines shearlet transformation (ST) and a data‐driven tight frame (DDTF) to improve data sparsity. The ST can provide the prior information of GPR data to the DDTF, while the DDTF can self‐adaptively represent GPR data. First, we separate significant reflections and interferences using ST. Second, we apply the DDTF to further suppress the interferences by setting different thresholds in different ST scales and directions. Finally, we adopt inverse transformations to recover the GPR data. In the experiments, ST is used to show the differences between the significant reflections and interferences of the synthetic GPR data. We also sequentially remove each interference of the synthetic GPR data to clearly highlight the performance of the method. To ensure the effectiveness of the ST‐DDTF approach, we test the method using synthetic GPR data from different models, along with some example field GPR data. The ST‐DDTF method, which is aimed at improving data sparsity, shows state‐of‐the‐art results relative to more standard GPR denoising methods. Although our approach is time consuming, it is useful in processing small GPR data and obtaining accurate denoising results.

Seismic Data Regularization on Nonequispaced Grid via a Joint Sparsity-Promotion Method

The seismic data regularization problem is vital to seismic data processing. We propose a joint sparsity-promotion method based on the compressive sensing theory named the curvelet-data-driven-tight-frame-based sparsity-promoting (CDSP) method. The CDSP method regularizes the seismic data directly on the nonequispaced grid along the spatial dimension. The joint sparsity is exploited in the curvelet and data-driven tight frame transform simultaneously on the projected regular spatial grid. The projection from the nonequispaced grid to the equispaced grid is achieved by the nonequispaced discretized Fourier transform. Comparing with the curvelet-sparsity-promotion-based (CSP) regularization method, CDSP combines the advantage of the predefined curvelet transform and adaptive-learned sparse transform into one single optimization model. An alternative directional method of multipliers (ADMM) is applied to solve the optimization problem. One synthetic and two field examples show that the CDSP method performs better on the preservation of the continuity of events and produces less artifacts than the CSP method.

CT Image Reconstruction via Nonlocal Low-Rank Regularization and Data-Driven Tight Frame

X-ray computed tomography (CT) is widely used in medical applications, where many efforts have been made for decades to eliminate artifacts caused by incomplete projection. In this paper, we propose a new CT image reconstruction model based on nonlocal low-rank regularity and data-driven tight frame (NLR-DDTF). Unlike the Spatial-Radon domain data-driven tight frame regularization, the proposed NLR-DDTF model uses an asymmetric treatment for image reconstruction and Radon domain inpainting, which combines the nonlocal low-rank approximation method for spatial domain CT image reconstruction and data-driven tight frame-based regularization for Radon domain image inpainting. An alternative direction minimization algorithm is designed to solve the proposed model. Several numerical experiments and comparisons are provided to illustrate the superior performance of the NLR-DDTF method.

Adaptive Dictionary Learning for Blind Seismic Data Denoising

The data-driven tight frame (DDTF) method is a dictionary learning method which has been used widely in the adaptive sparse representation and the seismic random noise attenuation. In the DDTF method, the thresholding operator setting plays a significant role on balancing the noise removal and preservation of detail information with high frequency. The hard thresholding operator is closely related to the noise variance; however, the noise variance is unknown and unstable which varies spatially. In this letter, we propose a spatially adaptive DDTF (SA-DDTF) method to find an optimal thresholding parameter without knowing the noise variance. The thresholding is determined by the coherence between the dictionary and the reconstruction residual. Furthermore, the thresholding is chosen adaptively for each patch of the seismic data. In case of the synthetic seismic data denoising, we obtain the reconstructed seismic data with a higher signal-to-noise ratio value. Furthermore, compared with the existing random noise reduction methods, the SA-DDTF method performs much better on the amplitude preservation of weak signal when attenuating the blind random noise for field data.

Denoising with weak signal preservation by group-sparsity transform learning

Weak signal preservation is critical in the application of seismic data denoising, especially in deep seismic exploration. It is hard to separate those weak signals in seismic data from random noise because it is less compressible or sparsifiable, although they are usually important for seismic data analysis. Conventional sparse coding models exploit the local sparsity through learning a union of basis, but it does not take into account any prior information about the internal correlation of patches. Motivated by an observation that data patches within a group are expected to share the same sparsity pattern in the transform domain, so-called group sparsity, we have developed a novel transform learning with group sparsity (TLGS) method that jointly exploits local sparsity and internal patch self-similarity. Furthermore, for weak signal preservation, we extended the TLGS method and developed the transform learning with external reference. External clean or denoised patches are applied as the anchored references, which are grouped together with similar corrupted patches. They are jointly modeled under a sparse transform, which is adaptively learned. This is achieved by jointly learning a subset of the transform for each group data. Our method achieves better denoising performance than existing denoising methods, in terms of signal-to-noise ratio values and visual preservation of weak signal. Comparisons of experimental results on one synthetic data and three field data using the [Formula: see text]-[Formula: see text] deconvolution method and the data-driven tight frame method are also provided.

Variational bimodal image fusion with data-driven tight frame

The purpose of multimodal image fusion is to combine information from different modal images of the same investigated object and create an image that is suitable for human vision and subsequent image processing. This paper proposes a three-step method for bimodal image fusion. A tight frame system is first adaptively learned from bimodal images for capturing source images features as much as possible. Further, a fused coefficient set is constructed by integrating the frame coefficients from both modalities. Finally, a variational model is designed to reconstruct a fused image based on the fused coefficients, and the intensity information of those smooth regions. The alternating iteration scheme and alternating direction method of multipliers are used to solve the resulted variational problems. Numerical experiments on multimodal medical image fusion and multifocused natural image fusion indicate that the proposed approach outperforms some existing methods.

Compressed Sensing Magnetic Resonance Imaging Using Data-Driven Tight Frame and Total Generalized Variation

Compressed Sensing Magnetic Resonance Imaging Using Data-Driven Tight Frame and Total Generalized Variation

Seismic data denoising via shearlet transform and data‐driven tight frame

AbstractWe propose a sort of double sparsity dictionary (DSD) to deal with random noise of seismic exploration, which consists of shearlet transform and data‐driven tight frame (DDTF). We train the DDTF dictionary in the domain of shearlet transform to improve the robustness of the dictionary. Furthermore, the function of hard‐thresholding is applied to find dictionary coefficients and cut small shearlet coefficients. Finally, we verified the reliability of the proposed approach by a synthetic data denoising example.

Joint regularization-based image reconstruction by combining data-driven tight frame and total variation for low-dose computed tomography.

Since the excessive radiation dose may induce potential body lesion, the low-dose computed tomography (LDCT) is widely applied for clinical diagnosis and treatment. However, the dose reduction will inevitably cause severe noise and degrade image quality. Most state-of-the-art methods utilize a pre-determined regularizer to account for the prior images, which may be insufficient for the most images acquired in the clinical practice. This study proposed and investigated a joint regularization method combining a data-driven tight frame and total variation (DDTF-TV) to solve this problem. Unlike the existing methods that designed pre-determined sparse transform for image domain, data-driven regularizer introduced a learning strategy to adaptively and iteratively update the framelets of DDTF, which can preferably recover the detailed image structures. The other regularizer, TV term can reconstruct strong edges and suppress noise. The joint term, DDTF-TV, collaboratively affect detail preservation and noise suppression. The proposed new model was efficiently solved by alternating the direction method of the multipliers. Qualitative and quantitative evaluations were carried out in simulation and real data experiments to demonstrate superiority of the proposed DDTF-TV method. Both visual inspection and numerical accuracy analysis show the potential of the proposed method for improving image quality of the LDCT.

Limited angle CT reconstruction by simultaneous spatial and Radon domain regularization based on TV and data-driven tight frame

Limited angle computed tomography (CT) reconstruction is widely performed in medical diagnosis and industrial testing because of the size of objects, engine/armor inspection requirements, and limited scan flexibility. Limited angle reconstruction necessitates usage of optimization-based methods that utilize additional sparse priors. However, most of conventional methods solely exploit sparsity priors of spatial domains. When CT projection suffers from serious data deficiency or various noises, obtaining reconstruction images that meet the requirement of quality becomes difficult and challenging. To solve this problem, this paper developed an adaptive reconstruction method for limited angle CT problem. The proposed method simultaneously uses spatial and Radon domain regularization model based on total variation (TV) and data-driven tight frame. Data-driven tight frame being derived from wavelet transformation aims at exploiting sparsity priors of sinogram in Radon domain. Unlike existing works that utilize pre-constructed sparse transformation, the framelets of the data-driven regularization model can be adaptively learned from the latest projection data in the process of iterative reconstruction to provide optimal sparse approximations for given sinogram. At the same time, an effective alternating direction method is designed to solve the simultaneous spatial and Radon domain regularization model. The experiments for both simulation and real data demonstrate that the proposed algorithm shows better performance in artifacts depression and details preservation than the algorithms solely using regularization model of spatial domain. Quantitative evaluations for the results also indicate that the proposed algorithm applying learning strategy performs better than the dual domains algorithms without learning regularization model

Seismic data interpolation and denoising by learning a tensor tight frame

Seismic data interpolation and denoising plays a key role in seismic data processing. These problems can be understood as sparse inverse problems, where the desired data are assumed to be sparsely representable within a suitable dictionary.In this paper, we present a new method based on a data-driven tight frame (DDTF) of Kronecker type (KronTF) that avoids the vectorization step and considers the multidimensional structure of data in a tensor-product way. It takes advantage of the structure contained in all different modes (dimensions) simultaneously. In order to overcome the limitations of a usual tensor-product approach we also incorporate data-driven directionality. The complete method is formulated as a sparsity-promoting minimization problem. It includes two main steps. In the first step, a hard thresholding algorithm is used to update the frame coefficients of the data in the dictionary; in the second step, an iterative alternating method is used to update the tight frame (dictionary) in each different mode. The dictionary that is learned in this way contains the principal components in each mode. Furthermore, we apply the proposed KronTF to seismic interpolation and denoising. Examples with synthetic and real seismic data show that the proposed method achieves better results than the traditional projection onto convex sets method based on the Fourier transform and the previous vectorized DDTF methods. In particular, the simple structure of the new frame construction makes it essentially more efficient.

What can machine learning do for seismic data processing? An interpolation application

Machine learning (ML) systems can automatically mine data sets for hidden features or relationships. Recently, ML methods have become increasingly used within many scientific fields. We have evaluated common applications of ML, and then we developed a novel method based on the classic ML method of support vector regression (SVR) for reconstructing seismic data from under-sampled or missing traces. First, the SVR method mines a continuous regression hyperplane from training data that indicates the hidden relationship between input data with missing traces and output completed data, and then it interpolates missing seismic traces for other input data by using the learned hyperplane. The key idea of our new ML method is significantly different from that of many previous interpolation methods. Our method depends on the characteristics of the training data, rather than the assumptions of linear events, sparsity, or low rank. Therefore, it can break out the previous assumptions or constraints and show universality to different data sets. In addition, our method dramatically reduces the manual workload; for example, it allows users to avoid selecting the window size parameters, as is required for methods based on the assumption of linear events. The ML method facilitates intelligent interpolation between data sets with similar geomorphological structures, which can significantly reduce costs in engineering applications. Furthermore, we combine a sparse transform called the data-driven tight frame (so-called compressed learning) with the SVR method to improve the training performance, in which the training is implemented in a sparse coefficient domain rather than in the data domain. Numerical experiments show the competitive performance of our method in comparison with the traditional [Formula: see text]-[Formula: see text] interpolation method.

Monte Carlo data-driven tight frame for seismic data recovery

Seismic data denoising and interpolation are essential preprocessing steps in any seismic data processing chain. Sparse transforms with a fixed basis are often used in these two steps. Recently, we have developed an adaptive learning method, the data-driven tight frame (DDTF) method, for seismic data denoising and interpolation. With its adaptability to seismic data, the DDTF method achieves high-quality recovery. For 2D seismic data, the DDTF method is much more efficient than traditional dictionary learning methods. But for 3D or 5D seismic data, the DDTF method results in a high computational expense. The motivation behind this work is to accelerate the filter bank training process in DDTF, while doing less damage to the recovery quality. The most frequently used method involves only a randomly selective subset of the training set. However, this random selection method uses no prior information of the data. We have designed a new patch selection method for DDTF seismic data recovery. We suppose that patches with higher variance contain more information related to complex structures, and should be selected into the training set with higher probability. First, we calculate the variance of all available patches. Then for each patch, a uniformly distributed random number is generated and the patch is preserved if its variance is greater than the random number. Finally, all selected patches are used for filter bank training. We call this procedure the Monte Carlo DDTF method. We have tested the trained filter bank on seismic data denoising and interpolation. Numerical results using this Monte Carlo DDTF method surpass random or regular patch selection DDTF when the sizes of the training sets are the same. We have also used state-of-the-art methods based on the curvelet transform, block matching 4D, and multichannel singular spectrum analysis as comparisons when dealing with field data.