An Augmented Lagrangian Method for image reconstruction with multiple features

<p style='text-indent:20px;'>Total generalization variation (TGV) is a very powerful and important regularization for various inverse problems and computer vision tasks. In this paper, we propose a semismooth Newton based augmented Lagrangian method for solving this problem. The augmented Lagrangian method (also called as method of multipliers) is widely used for lots of smooth or nonsmooth variational problems. However, its efficiency heavily depends on solving the corresponding coupled and nonlinear system together and simultaneously. With efficient primal-dual semismooth Newton methods for the challenging and highly coupled nonlinear subproblems involving total generalized variation, we develop a highly efficient and competitive augmented Lagrangian method compared with some fast first-order method. With the analysis of the metric subregularities of the corresponding functions, we give both the global convergence and local linear convergence rate for the proposed augmented Lagrangian methods.</p>

Synthetic aperture radar (SAR) image super-resolution (SR) has broad application prospects in military reconnaissance, marine monitoring, crop monitoring, and other fields. However, when the scale factor is large, current SAR image SR has limitations, such as insufficient feature extraction, difficulty of feature representation, and modeling difficulty of SR reconstruction. These problems have severely restricted the SR performance of SAR images. In addition, optical remote sensing images share higher resolution and richer high-frequency information than SAR remote sensing images. Therefore, we adopt high-resolution (HR) optical images as auxiliary images of SAR image SR to improve the performance of SAR image SR. Our research is conducted from two aspects: the extraction efficiency of HR optical image-guided information and the representation ability of low-resolution (LR) SAR image features. We propose an optical guidance residual networks (OGRNs) based on residual learning for SAR image SR. The OGRN uses optical images during network training, which enhances the performance of SAR image SR, even when optical images are missing during network testing. First, we extract the texture feature of SAR images through the designed residual learning module. Second, we extract the weight map of feature information for SAR image SR from HR optical images by the convolutional layer. Finally, the weight map is used to guide the weighted supervised SR reconstruction for SAR images. Extensive experimental results demonstrated that, under the guidance of optical images, OGRN achieves excellent performance in terms of both quantitative assessment metrics and visual quality.

—GeoFEM (Iizuka et al., 1999) is a parallel finite element analysis system intended for multi-physics, multi-scale problems of solid earth field phenomena. Very large linear elastic problems have already been resolved by parallel computation with GeoFEM. The next stage is to examine large-scale nonlinear problems using GeoFEM. The analysis of large-scale contact problems for fault zones is particularly important in the development of models that simulate the occurrence and cycle of earthquakes.¶This paper proposes a parallel FEM using an iterative solver with the augmented Lagrange method for solving large-scale contact problems. Direct solvers are presently applied to contact problem analysis because the matrix for such problems is ill-conditioned. However, direct solvers are not suitable for large-scale matrices. The augmented Lagrange method can improve the matrix conditions. The present study evaluates a parallel FEM using an iterative solver with the augmented Lagrange method. Analysis of a contact problem with the augmented Lagrange method revealed that an optimal penalty parameter exists and that large-scale parallel contact analysis using the iterative solver with localized preconditioning is promising.

This paper presents a new point-to-face contact algorithm for contacts between two polyhedrons with planar boundaries. A new discrete numerical method called three-dimensional discontinuous deformation analysis (3-D DDA) is used and formulations of normal contact submatrices based on the proposed algorithm are derived. The presented algorithm is a simple and efficient method and it can be easily coded into a computer program. This approach does not need to use an iterative algorithm in each time step to obtain the contact plane, unlike the ‘Common-Plane’ method applied in the existing 3-D DDA. In the present 3-D DDA method, block contact constraints are enforced using the penalty method. This approach is quite simple, but may lead to inaccuracies that may be large for small values of the penalty number. The penalty method also creates block contact overlap, which violates the physical constraints of the problem. These limitations are overcome by using the augmented Lagrangian method that is used for normal contacts in this research. This point-to-face contact model has been programmed and some illustrative examples are provided to demonstrate the new contact rule between two blocks. A comparison between results obtained by using the augmented Lagrangian method and the penalty method is presented as well.

Several magnetic resonance parallel imaging techniques require explicit estimates of the receive coil sensitivity profiles. These estimates must be accurate over both the object and its surrounding regions to avoid generating artifacts in the reconstructed images. Regularized estimation methods that involve minimizing a cost function containing both a data-fit term and a regularization term provide robust sensitivity estimates. However, these methods can be computationally expensive when dealing with large problems. In this paper, we propose an iterative algorithm based on variable splitting and the augmented Lagrangian method that estimates the coil sensitivity profile by minimizing a quadratic cost function. Our method, ADMM-Circ, reformulates the finite differencing matrix in the regularization term to enable exact alternating minimization steps. We also present a faster variant of this algorithm using intermediate updating of the associated Lagrange multipliers. Numerical experiments with simulated and real data sets indicate that our proposed method converges approximately twice as fast as the preconditioned conjugate gradient method over the entire field-of-view. These concepts may accelerate other quadratic optimization problems.

In Mathematical Morphology, the reconstruction of images from markers has proven to be useful in morphological filtering and image segmentation. This work investigates the utilization of a criterion in the reconstruction process, whose utilization in the problem of the image reconstruction from an image marker has been partially treated elsewhere. This work further investigates this idea and extends it to the problem of image reconstruction from labelled markers. In the binary case, this allows us to compute the modified influence zones associated to the set of labelled markers. A significant difference with the usual case (i.e., the ”normal” influence zones) is that we generally do not obtain a whole partition of the space, because the criterion added to the reconstruction process causes that some points or pixels are not recovered. In addition, in this paper we consider the gray-level case, and we use the reconstruction with criterion to separate regions from a non-binary input image. This input image is considered as a topographic relief (similarly as in a normal watershed); however, the flooding mechanism is modified by the reconstruction criterion. The benefit is that we can control to some extent how the flooding proceeds and, therefore, how image region shapes are recovered.

The term, "Waveform inversion" (WFI), refers to a collection of techniques that use the information from seismic data to derive high-fidelity earth models for seismic imaging. The attractiveness of WFI lies mainly in its lack of approximations, at least in a theoretical sense, in contrast to other model determination techniques such as semblance or tomography. However, a whole raft of approximations must be made to make the technique viable with today'scomputing technology and restrictions of seismic acquisition. These are collectively referred to as "waveform inversion strategies" and in this paper we mainly discuss regularization and preconditioning strategies. Because the wavefields need to be accurately modeled to represent the kinematics of all the waves during WFI iterations, the effects of anisotropy often help to improve WFI results. In this paper, forward modeling and its adjoint computation are based on acoustic wave equations in vertical transversely isotropic (VTI)media. We introduce a multi-parameter inversion for P-wave velocity and anisotropy parameters. WFI is a highly nonlinear, ill-posed problem. We introduce additional information and turn the unconstrained optimization problem into a constrained optimization problem in order to reduce the ill-posedness. The geophysics of the problem leads to appropriate constraints, such as restriction of model parameters, or information from well logs. In this paper, we use well logs as constraints and solve the problem using the augmented Lagrangian method (ALM), a mathematical method that replaces a constrained optimization problem by a series of unconstrained problems. The ALM with well constraints aims at preserving velocity characteristics from welllogs and providing us with more reliable velocity updates. This paper presents the acoustic anisotropic WFI implementation using ALM with well constraints. It also discusses practical strategies for regularization and preconditioning and their influences on the models that are obtained from WFI. We illustrate these approaches on a 2D synthetic Marmousi example and another application to 3D VSOOBC data from the Green Canyon area of the Gulf of Mexico. From the results, we show that multi-parameter VTI WFI with ALM provides us with more useful and reliable model updates. To further evaluate our WFI results, we also compare offset gathers and RTM images and illustrate their significant improvement susing updated models generated from WFI.

Because balanced constraints can overcome the problems of trivial solutions of data classification via minimum cut method, many techniques with different balanced strategies have been proposed to improve data classification accuracy. However, their performances have not been compared comprehensively so far. In this paper, we investigate seven balanced classification methods under the discrete non-local total variational framework and compare their accuracy performances on graph. The two-class classification problem with equality constraints, inequality constraints and Ratio Cut, Normalized Cut, Cheeger Cut models are investigated. For cases of equality constraint, we firstly compare the Penalty Function Method (PFM) and the Augmented Lagrangian Method (ALM), which can transform the constrained problems into unconstrained ones, to show the advantages of ALM. The other cases are all solved using the ALM also. In order to make the comparison fairly, we solve all models using ALM method and using the same proportion of fidelity points and the same neighborhood size on graph. Experimental results demonstrate ALM with the equality balanced constraint has the best classification accuracy compared with other six constraints. 200 words.

Near-field high-resolution synthetic aperture radar (SAR) imaging is mostly accompanied by a large number of data acquisition processes, which increases the system complexity and device cost. According to extensive reports, reducing the number of sampling points of a radar in space can greatly reduce the amount of data. However, when spatial sparse sampling is carried out, a ghost normally appears in the imaging results due to the high side lobes generated in the azimuth. To address this issue, a technique is introduced in this paper to recover the blank data through amplitude and phase compensation based on the correlation between sparse array sampling through adjacent points. Firstly, the data sampled by the sparse array is compressed in the range direction to obtain the expected data slices in the same range direction. Then, the blank element of the slice is compensated for with amplitude and phase to obtain full aperture data. Finally, the matched filter method is used to aid in the image reconstruction. The simulation results verified that the method proposed in this paper can effectively reconstruct the image under two kinds of sparse sampling conditions. Thus, a simple single-input single-output (SISO) synthetic aperture radar imaging test bench is established. Compared with the results of a 1 mm (1/4 ) sampling interval, the quality of the reconstructed image under the condition of a 4 mm (1 ) sampling interval still stands using our proposed method. Demonstrated by the experiment, the normalized root-mean-square error(NMSE) is 5.75%. Additionally, when the spatial sampling points are sampled randomly with 30% of the full sampling condition, this method can also restore and reconstruct the image with high quality. Due to the decrease of sampling points, the data volume can be reduced, which is beneficial for improving the scanning speed and alleviating the pressure of data transmission for near-field high resolution SAR imaging systems.

In this paper, we propose an image segmentation model where an $L^1$ variant of the Euler's elastica energy is used as boundary regularization. An interesting feature of this model lies in its preference for convex segmentation contours. However, due to the high order and non-differentiability of Euler's elastica energy, it is nontrivial to minimize the associated functional. As in recent work on the ordinary $L^2$-Euler's elastica model in imaging, we propose using an augmented Lagrangian method to tackle the minimization problem. Specifically, we design a novel augmented Lagrangian functional that deals with the mean curvature term differently than in previous works. The new treatment reduces the number of Lagrange multipliers employed, and more importantly, it helps represent the curvature more effectively and faithfully. Numerical experiments validate the efficiency of the proposed augmented Lagrangian method and also demonstrate new features of this particular segmentation model, such as shape driven and data driven properties.

Recent advances in compressed sensing (CS) enable accurate CT image reconstruction from highly undersampled and noisy projection measurements, due to the sparsifiable feature of most CT images using total variation (TV). These novel reconstruction methods have demonstrated advantages in clinical applications where radiation dose reduction is critical, such as onboard cone-beam CT (CBCT) imaging in radiation therapy. The image reconstruction using CS is formulated as either a constrained problem to minimize the TV objective within a small and fixed data fidelity error, or an unconstrained problem to minimize the data fidelity error with TV regularization. However, the conventional solutions to the above two formulations are either computationally inefficient or involved with inconsistent regularization parameter tuning, which significantly limit the clinical use of CS-based iterative reconstruction. In this paper, we propose an optimization algorithm for CS reconstruction which overcomes the above two drawbacks. The data fidelity tolerance of CS reconstruction can be well estimated based on the measured data, as most of the projection errors are from Poisson noise after effective data correction for scatter and beam-hardening effects. We therefore adopt the TV optimization framework with a data fidelity constraint. To accelerate the convergence, we first convert such a constrained optimization using a logarithmic barrier method into a form similar to that of the conventional TV regularization based reconstruction but with an automatically adjusted penalty weight. The problem is then solved efficiently by gradient projection with an adaptive Barzilai-Borwein step-size selection scheme. The proposed algorithm is referred to as accelerated barrier optimization for CS (ABOCS), and evaluated using both digital and physical phantom studies. ABOCS directly estimates the data fidelity tolerance from the raw projection data. Therefore, as demonstrated in both digital Shepp-Logan and physical head phantom studies, consistent reconstruction performances are achieved using the same algorithm parameters on scans with different noise levels and∕or on different objects. On the contrary, the penalty weight in a TV regularization based method needs to be fine-tuned in a large range (up to seven times) to maintain the reconstructed image quality. The improvement of ABOCS on computational efficiency is demonstrated in the comparisons with adaptive-steepest-descent-projection-onto-convex-sets (ASD-POCS), an existing CS reconstruction algorithm also using constrained optimization. ASD-POCS alternatively minimizes the TV objective using adaptive steepest descent (ASD) and the data fidelity error using projection onto convex sets (POCS). For similar image qualities of the Shepp-Logan phantom, ABOCS requires less computation time than ASD-POCS in MATLAB by more than one order of magnitude. We propose ABOCS for CBCT reconstruction. As compared to other published CS-based algorithms, our method has attractive features of fast convergence and consistent parameter settings for different datasets. These advantages have been demonstrated on phantom studies.

The augmented Lagrangian method (ALM) is a classical optimization tool that solves a given “difficult” (constrained) problem via finding solutions of a sequence of “easier” (often unconstrained) subproblems with respect to the original (primal) variable, wherein constraints satisfaction is controlled via the so-called dual variables. ALM is highly flexible with respect to how primal subproblems can be solved, giving rise to a plethora of different primal–dual methods. The powerful ALM mechanism has recently proved to be very successful in various large-scale and distributed applications. In addition, several significant advances have appeared, primarily on precise complexity results with respect to computational and communication costs in the presence of inexact updates and design and analysis of novel optimal methods for distributed consensus optimization. We provide a tutorial-style introduction to ALM and its variants for solving convex optimization problems in large-scale and distributed settings. We describe control-theoretic tools for the algorithms’ analysis and design, survey recent results, and provide novel insights into the context of two emerging applications: federated learning and distributed energy trading.

In general, synthetic aperture radar (SAR) imaging and image processing are two sequential steps in SAR image processing. Due to the large size of SAR images, most image processing algorithms require image segmentation before processing. However, the existence of speckle noise in SAR images, as well as poor contrast and the uneven distribution of gray values in the same target, make SAR images difficult to segment. In order to facilitate the subsequent processing of SAR images, this paper proposes a new method that combines the back-projection algorithm (BPA) and a first-order gradient operator to enhance the edges of SAR images to overcome image segmentation problems. For complex-valued signals, the gradient operator was applied directly to the imaging process. The experimental results of simulated images and real images validate our proposed method. For the simulated scene, the supervised image segmentation evaluation indexes of our method have more than 1.18%, 11.2% and 11.72% improvement on probabilistic Rand index (PRI), variability index (VI), and global consistency error (GCE). The proposed imaging method will make SAR image segmentation and related applications easier.

Synthetic aperture radar (SAR) image classification is an important part in the understanding and interpretation of SAR images. Each patch in SAR images has a scene category, but usually contains multiple land-cover classes or latent properties, which can be represented by topics in the probabilistic topic model (PTM). The representation and selection of discriminative features in PTM have a large impact on the classification results. Most of the existing feature learning methods do not make full use of high-level structure feature and the feature correlation within similar images to mine discriminative features. Therefore, this article proposes a discriminative sketch topic model with structural constraint (C-SSTM) for SAR image classification. In the proposed model, each image patch is characterized by structural and texture features. In particular, the sketch structural feature is based on the sketch map to represent the image local structure pattern. Then, the local image manifold information is preserved in terms of structure and texture. In the structural constraint, the texture and structure of each image patch are combined to learn discriminative latent semantic topics between image patches. Finally, each image patch is quantified by discriminative latent semantic topics instead of low-level representation. The experimental results tested on synthetic and real SAR images demonstrate that the proposed C-SSTM is able to learn effective structural feature representation from SAR images. Compared with other related approaches, C-SSTM produces competitive classification accuracies with high time efficiency.

Reliable and accurate land use/land cover (LULC) map is a crucial data source for the understanding of coupled human–environment systems, monitoring changes, timely low-cost planning, and management of natural resources. Improvements in sensor technologies and machine learning capabilities have shifted the attention of remote sensing community to data complementarity through fusion of multi-sensor data for accurate feature extraction and mapping. Amalgamation of optical and synthetic aperture radar (SAR) images has shown promising advantages in enhancing the accuracy of extracting LULC as such method allows exploitation of information in sensors. This study investigated the potential of using freely available multisource Sentinel images to extract LULC maps in semi-arid environments through multi-scale geographic object-based image analysis (GEOBIA). A multi-scale classification framework that integrates GEOBIA, correlation-based feature selection (CFS), and random forest (RF)-supervised classification was adopted to extract LULC from assimilation of Sentinel multi-sensor products. First, Sentinel-1 and -2 images were pre-processed. Second, optimum multi-scale segmentation levels were selected using F-score segmentation quality measures. Third, 70 features of various spectral indices and derivatives and geometrical features from optical data and multiple ratios and textural features from dual-polarization SAR images were computed, and a CFS based on wrapper approach was used to select the most significant features at multi-scale levels. Finally, a single and multi-scale RF classifier was used to extract LULC classes using the most relevant features extracted from Sentinel SAR and optical images. Results of multi-scale image segmentation optimization showed that scale parameter (SP) values of 40, 60, and 150 were optimal for extraction of LULC classes. Results of feature selection showed that 22, 24, and 27 features were selected at scale SP values of 40, 60, and 150, respectively. Half of the features were common among the three scales. Single RF classification yielded overall accuracy (OA) values of 92.10%, 93%, and 91% and kappa coefficients of 0.901, 0.912, and 0.89 at scale values of 150, 60, and 40, respectively. Multiscale RF classification from scale values of 150 and 60 produced better LULC classification with OA 96.06% and kappa coefficient of 0.95 compared with other scale SP values. The integrated approach demonstrated an effective and promising method for high-quality LULC extraction from coupling optical and SAR images. Overall, multi-sensor Sentinel images along with the adopted approach feature a remarkable potential for improving LULC extraction and can effectively be used to update geographic information system layers for various applications.