Material-separating regularizer for multi-energy X-ray tomography

Dual-energy X-ray tomography is considered in a context where the target under imaging consists of two distinct materials. The materials are assumed to be possibly intertwined in space, but at any given location there is only one material present. Further, two X-ray energies are chosen so that there is a clear difference in the spectral dependence of the attenuation coefficients of the two materials. A novel regularizer is presented for the inverse problem of reconstructing separate tomographic images for the two materials. A combination of two things, (a) non-negativity constraint, and (b) penalty term containing the inner product between the two material images, promotes the presence of at most one material in a given pixel. A preconditioned interior point method is derived for the minimization of the regularization functional. Numerical tests with digital phantoms suggest that the new algorithm outperforms the baseline method, Joint Total Variation regularization, in terms of correctly material-characterized pixels. While the method is tested only in a two-dimensional setting with two materials and two energies, the approach readily generalizes to three dimensions and more materials. The number of materials just needs to match the number of energies used in imaging.

Unsupervised deep learning with higher-order total-variation regularization for multidimensional seismic data reconstruction

In 3D marine seismic acquisition, the seismic wavefield is not sampled uniformly in the spatial directions. This leads to a seismic wavefield consisting of irregularly and sparsely populated traces with large gaps between consecutive sail-lines especially in the near-offsets. The problem of reconstructing the complete seismic wavefield from a subsampled and incomplete wavefield, is formulated as an underdetermined inverse problem. We investigate unsupervised deep learning based on a convolutional neural network (CNN) for multidimensional wavefield reconstruction of irregularly populated traces defined on a regular grid. The proposed network is based on an encoder-decoder architecture with an overcomplete latent representation, including appropriate regularization penalties to stabilize the solution. We proposed a combination of penalties, which consists of the L2-norm penalty on the network parameters, and a first- and second-order total-variation (TV) penalty on the model. We demonstrate the performance of the proposed method on broad-band synthetic data, and field data represented by constant-offset gathers from a source-over-cable data set from the Barents Sea. In the field data example we compare the results to a full production flow from a contractor company, which is based on a 5D Fourier interpolation approach. In this example, our approach displays improved reconstruction of the wavefield with less noise in the sparse near-offsets compared to the industry approach, which leads to improved structural definition of the near offsets in the migrated sections.

A Comparative Study of Four Total Variational Regularization Reconstruction Algorithms for Sparse-View Photoacoustic Imaging

Photoacoustic imaging (PAI) is a new nonionizing, noninvasive biomedical imaging technology that has been employed to reconstruct the light absorption characteristics of biological tissues. The latest developments in compressed sensing (CS) technology have shown that it is possible to accurately reconstruct PAI images from sparse data, which can greatly reduce scanning time. This study focuses on the comparative analysis of different CS-based total variation regularization reconstruction algorithms, aimed at finding a method suitable for PAI image reconstruction. The performance of four total variation regularization algorithms is evaluated through the reconstruction experiment of sparse numerical simulation signal and agar phantom signal data. The evaluation parameters include the signal-to-noise ratio and normalized mean absolute error of the PAI image and the CPU time. The comparative results demonstrate that the TVAL3 algorithm can well balance the quality and efficiency of the reconstruction. The results of this study can provide some useful guidance for the development of the PAI sparse reconstruction algorithm.

Anderson accelerated augmented Lagrangian for extended waveform inversion

The augmented Lagrangian (AL) method provides a flexible and efficient framework for solving extended-space full-waveform inversion (FWI), a constrained nonlinear optimization problem whereby we seek model parameters and wavefields that minimize the data residuals and satisfy the wave equation constraint. The AL-based wavefield reconstruction inversion, also known as iteratively refined wavefield reconstruction inversion, extends the search space of FWI in the source dimension and decreases sensitivity of the inversion to the initial model accuracy. Furthermore, it benefits from the advantages of the alternating direction method of multipliers (ADMM), such as generality and decomposability for dealing with non-differentiable regularizers, e.g., total variation regularization, and large scale problems, respectively. In practice any extension of the method aiming at improving its convergence and decreasing the number of wave-equation solves would have a great importance. To achieve this goal, we recast the method as a general fixed-point iteration problem, which enables us to apply sophisticated acceleration strategies like Anderson acceleration. The accelerated algorithm stores a predefined number of previous iterates and uses their linear combination together with the current iteration to predict the next iteration. We investigate the performance of the proposed accelerated algorithm on a simple checkerboard model and the benchmark Marmousi II and 2004 BP salt models through numerical examples. These numerical results confirm the effectiveness of the proposed algorithm in terms of convergence rate and the quality of the final estimated model.

Multidirectional Anisotropic Total Variation and Its Application in the Tomography of the Surrounding Rock of Coal Mining Faces

The computed tomography (CT) reconstruction algorithm is one of the crucial components of the CT system. To date, total variation (TV) has been widely used in CT reconstruction algorithms. Although TV utilizes the a priori information of the longitudinal and lateral gradient sparsity of an image, it introduces some staircase artifacts. To overcome the current limitations of TV and improve imaging quality, we propose a multidirectional anisotropic total variation (MATV) that uses multidirectional gradient information. The surrounding rock of coal mining faces uses principles of tomography similar to those of medical X-rays. The velocity distribution for the surrounding rock can be obtained by the first-arrival traveltime tomography of the transmitted waves in the coal mining face. Combined with the geological data, we can interpret the geological hazards in the coal mining face. To perform traveltime tomography, we first established the objective function of the first-arrival traveltime tomography of the transmitted waves based on the MATV regularization and then used the split Bregman method to solve the objective function. The simulated data and real data show that the MATV regularization method proposed in this paper can better maintain the boundaries of geological anomalies and reduce the artifacts compared with the isotropic total variation regularization method and the anisotropic total variation regularization method. Furthermore, this approach describes the distribution of geological anomalies more accurately and effectively and improves imaging accuracy.

Limited-angle computed tomography with deep image and physics priors

Computed tomography is a well-established x-ray imaging technique to reconstruct the three-dimensional structure of objects. It has been used extensively in a variety of fields, from diagnostic imaging to materials and biological sciences. One major challenge in some applications, such as in electron or x-ray tomography systems, is that the projections cannot be gathered over all the angles due to the sample holder setup or shape of the sample. This results in an ill-posed problem called the limited angle reconstruction problem. Typical image reconstruction in this setup leads to distortion and artifacts, thereby hindering a quantitative evaluation of the results. To address this challenge, we use a generative model to effectively constrain the solution of a physics-based approach. Our approach is self-training that can iteratively learn the nonlinear mapping from partial projections to the scanned object. Because our approach combines the data likelihood and image prior terms into a single deep network, it is computationally tractable and improves performance through an end-to-end training. We also complement our approach with total-variation regularization to handle high-frequency noise in reconstructions and implement a solver based on alternating direction method of multipliers. We present numerical results for various degrees of missing angle range and noise levels, which demonstrate the effectiveness of the proposed approach.

Magnetic inversion to recover the subsurface block structures based on L1 norm and total variation regularization

Summary This paper presents a new sparse inversion method based on L1 norm regularization for 3D magnetic data. In isolation, L1 norm regularization yields model elements which are unconstrained by the input data to be exactly zero, leading to a sparse model with compact and focused structure. Here, we complement the L1 norm with a penalty minimizing total variation, the L1 norm of the model gradients; it is expected that the sharp boundaries of the subsurface structure are not compromised by incorporating this penalty. Although this penalty is widely used in the geophysical inversion studies, it is often replaced by an alternative quadratic penalty to ease solution of the penalized inversion problem; in this study, the original definition of the total variation, i.e., form of the L1 norm of the model gradients, is used. To solve the problem with this combined penalty of L1 norm and total variation, this study introduces alternative direction method of multipliers (ADMM), which is a primal-dual optimization algorithm that solves convex penalized problems based on the optimization of an augmented Lagrange function. To improve the computational efficiency of the algorithm to make this method applicable to large-scale magnetic inverse problems, this study applies matrix compression using the wavelet transform and the preconditioned conjugate gradient method. The inversion method is applied to both synthetic tests and real data, the synthetic tests demonstrate that, when subsurface structure is blocky, it can be reproduced almost perfectly.