Variation Regularization Research Articles

Data-Driven Morozov Regularization of Inverse Problems

The solution of inverse problems is crucial in various fields such as medicine, biology, and engineering, where one seeks to find a solution from noisy observations. These problems often exhibit non-uniqueness and ill-posedness, resulting in instability under noise with standard methods. To address this, regularization techniques have been developed to balance data fitting and prior information. Recently, data-driven variational regularization methods have emerged, mainly analyzed within the framework of Tikhonov regularization, termed Network Tikhonov (NETT). This paper introduces Morozov regularization combined with a learned regularizer, termed DD-Morozov regularization. Our approach employs neural networks to define non-convex regularizers tailored to training data, enabling a convergence analysis in the non-convex context with noise-dependent regularizers. We also propose a refined training strategy that improves adaptation to ill-posed problems compared to NETT’s original strategy, which primarily focuses on addressing non-uniqueness. We present numerical results for attenuation correction in photoacoustic tomography, comparing DD-Morozov regularization with NETT using the same trained regularizer, both with and without an additional total variation regularizer.

Homomorphisms from Functional Equations: The Goldie Equation, II

AbstractThis first of three sequels to Homomorphisms from Functional equations: The Goldie equation (Ostaszewski in Aequationes Math 90:427–448, 2016) by the second author—the second of the resulting quartet—starts from the Goldie functional equation arising in the general regular variation of our joint paper (Bingham et al. in J Math Anal Appl 483:123610, 2020). We extend the work there in two directions. First, we algebraicize the theory, by systematic use of certain groups—the Popa groups arising in earlier work by Popa, and their relatives the Javor groups . Secondly, we extend from the original context on the real line to multi-dimensional (or infinite-dimensional) settings.

Flexible Krylov methods for group sparsity regularization

Abstract This paper introduces new solvers for efficiently computing solutions to large-scale inverse problems with group sparsity regularization, including both non-overlapping and overlapping groups. Group sparsity regularization refers to a type of structured sparsity regularization, where the goal is to impose additional structure in the regularization process by assigning variables to predefined groups that may represent graph or network structures. Special cases of group sparsity regularization include ℓ 1 and isotropic total variation regularization. In this work, we develop hybrid projection methods based on flexible Krylov subspaces, where we first recast the group sparsity regularization term as a sequence of 2-norm penalization terms using adaptive regularization matrices in an iterative reweighted norm fashion. Then we exploit flexible preconditioning techniques to efficiently incorporate the weight updates. The main advantages of these methods are that they are computationally efficient (leveraging the advantages of flexible methods), they are general (and therefore very easily adaptable to new regularization term choices), and they are able to select the regularization parameters automatically and adaptively (exploiting the advantages of hybrid methods). Extensions to multiple regularization terms and solution decomposition frameworks (e.g. for anomaly detection) are described, and a variety of numerical examples demonstrate both the efficiency and accuracy of the proposed approaches compared to existing solvers.

Transitive reasoning: A high-performance computing model for significant pattern discovery in cognitive IoT sensor network

Current research on the Internet of Things (IoT) has given rise to a new field of study called cognitive IoT (CIoT), which aims to incorporate cognition into the designs of IoT systems. Consequently, the CIoT inherits specific attributes and challenges from IoT. The CIoT applications generate vast, diverse, constantly changing, and time-dependent data due to the billions of devices involved. The efficient operation of these CIoT systems requires the extraction of valuable insights from vast data sources in a computationally efficient manner. Therefore, this study proposes transitive reasoning to glean significant concepts and patterns from a 21.25-year environmental dataset. To reduce the effects of missing entries, the proposed methodology includes a grouping of data using probabilistic clustering and applying total variance regularization in the alternate direction method of multipliers (ADMM) to regularize the sensory data. As a result, noisy entries will be less conspicuous. Afterward, it calculates the transitional plausibility value for each cluster using the transited value and then turns it into binary data to create concept lattices. In addition, each concept that is formed is assigned a weight, and the concept with the largest transitive strength value is chosen, followed by calculating the mean value. Therefore, this pattern is seen as significant. Experimental results on 21.25-year environmental data show an accuracy of over 99.5%, outperforming competing methods, as shown by cross-validation using multiple metrics.

Geochemistry and Mineralogy of Phlogopite and Its Implications for Serpentinization of Jian Forsterite Jade in Southern Jilin Province, China

Mica is a kind of important rock-forming mineral in the lithosphere of Earth, which can be a superior tool used to trace the origin and late evolution of rock. The Jian forsterite jade (a kind of geological skarn) is an emerging kind of gemstone in China with a beautiful color and luster, discovered in Ji’an County, Jilin Province, Northeast China. It is mainly composed of rare Mg-rich forsterite (Mg# (Mg/(Mg+Fe2+) up to 99), serpentine and brucite. The source of hydrothermal fluid triggering the late metamorphism (the serpentinization of forsterite) of forsterite jade deposits remains unclear. We report a series of phlogopites with a regular range of mineral compositions in the forsterite jade deposit. Micrographs show that the phlogopites are associated with forsterite and coexist with serpentine in forsterite jade, tourmaline and tremolite in the contact zone, and plagioclase in pegmatite, and the related replacement of phlogopite seems to have not occurred. The phlogopites that occurred as single grains or veinlets in forsterite jade named type I are characterized by high XMg, ranging from ~0.98 to ~0.95, and the phlogopites that occurred in the contact zone of forsterite jade and pegmatite named type II are rich in Fe, with a range of XMg from ~0.82 to ~0.66. Additionally, the type II phlogopites are also rich in Ti, Mn, Cl, Li, Rb, Zn, V, Co, Nb and Ta but poor in Na, Sr and F compared to the type I phlogopite. Petrological and mineralogical characteristics and geochemical compositions suggest that the phlogopites are crystallized from the corresponding fluid component by hydrothermal metasomatism. The abundant Mg of the fluid phase is produced during the serpentinization of forsterite, triggered by pegmatitic hydrothermal fluid, and other main materials like K, Al, Si and H2O are provided by the intrusive pegmatite. With the occurrence of and regular compositional variation in phlogopites in the forsterite jade deposit, we suppose that the hydrothermal fluid triggering the serpentinization of the Jian forsterite jade is produced by the intrusive pegmatite.

Code and Data Repository for On Discrete Subproblems in Integer Optimal Control with Total Variation Regularization in Two Dimensions

The goal of this repository to be able to trace and reproduce the results and plots in the referenced article. For the results in this article, we used the general purpose integer programming solver Gurobi version 10.0.0. In addition, you need to have the python packages pandas, numpy, scipy, and pybind11 installed to be able to build the code for computing the weights of the cutting planes from 5.1.1.

On Discrete Subproblems in Integer Optimal Control with Total Variation Regularization in Two Dimensions

We analyze integer linear programs that we obtain after discretizing two-dimensional subproblems arising from a trust-region algorithm for mixed integer optimal control problems with total variation regularization. We discuss NP-hardness of the discretized problems and the connection to graph-based problems. We show that the underlying polyhedron exhibits structural restrictions in its vertices with regard to which variables can attain fractional values at the same time. Based on this property, we derive cutting planes by employing a relation to shortest-path and minimum bisection problems. We propose a branching rule and a primal heuristic which improves previously found feasible points. We validate the proposed tools with a numerical benchmark in a standard integer programming solver. We observe a significant speedup for medium-sized problems. Our results give hints for scaling toward larger instances in the future. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This work was supported by the Deutsche Forschungsgemeinschaft [Grant MA 10080/2-1]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0680 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0680 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

A modified non-convex Cauchy total variation regularization model for image restoration

A modified non-convex Cauchy total variation regularization model for image restoration

Graph total variation and low-rank regularization for heterogeneous change detection

Heterogeneous change detection (HCD) is challenging because different imaging mechanisms for various sensors make images difficult to compare directly. To address this problem, a graph-based regression algorithm is proposed for HCD, by leveraging the Graph Total Variation regularization and Low-Rank matrices decomposition (GTVLR). Utilizing graph signal processing (GSP) theory, a directed graph (digraph) model is employed to effectively represent the orientation and correlation information of images, thereby enabling direct comparison of heterogeneous data within the same domain after graph filtering. The GTVLR framework facilitates the decomposition of post-event images into regression and changed images. This decomposition ensures that the regression image mirrors the structure similarity of the pre-event image, while the changed image highlights areas of alteration, aiding in change detection. The model characterizes the piecewise smoothness and Low-Rank properties of data through GTV regularization and Low-Rank penalty, respectively. Moreover, by integrating the higher-order neighboring information within the digraph to refine the model. Experiments conducted on three real-world datasets and comparison with several state-of-the-art methods demonstrate the effectiveness of the proposed algorithm.

Denoising sphere‐valued data by relaxed total variation regularization

AbstractCircle‐ and sphere‐valued data play a significant role in inverse problems like magnetic resonance phase imaging and radar interferometry, in the analysis of directional information, and in color restoration tasks. In this paper, we aim to restore ‐sphere‐valued signals exploiting the classical anisotropic total variation on the surrounding ‐dimensional Euclidean space. For this, we propose a novel variational formulation, whose data fidelity is based on inner products instead of the usually employed squared norms. Convexifying the resulting non‐convex problem and using ADMM, we derive an efficient and fast numerical denoiser. In the special case of binary (0‐sphere‐valued) signals, the relaxation is provable tight, that is, the relaxed solution can be used to construct a solution of the original non‐convex problem. Moreover, the tightness can be numerically observed for barcode and QR code denoising as well as in higher dimensional experiments like the color restoration using hue and chromaticity and the recovery of SO(3)‐valued signals.

A Low Transformed Tubal Rank Tensor Model Using a Spatial-Tubal Constraint for Sample Clustering with Cancer Multi-omics Data

Background: Since each dimension of a tensor can store different types of genomics data, compared to matrix methods, utilizing tensor structure can provide a deeper understanding of multi-dimensional data while also facilitating the discovery of more useful information related to cancer. However, in reality, there are issues such as insufficient utilization of prior knowledge in multi-omics data and limitations in the recovery of low-tubal-rank tensors. Therefore, the method proposed in this article was developed. Objective: In this paper, we proposed a low transformed tubal rank tensor model (LTTRT) using a spatial-tubal constraint to accurately partition different types of cancer samples and provide reliable theoretical support for the identification, diagnosis, and treatment of cancer. Method: In the LTTRT method, the transformed tensor nuclear norm based on the transformed tensor singular value decomposition is characterized by the low-rank tensor, which can explore the global low-rank property of the tensor, resolving the challenge of the tensor nuclear norm-based method not achieving the lowest tubal rank. Additionally, the introduction of weighted total variation regularization is conducive to extracting more information from sequencing data in both spatial and tubal dimensions, exploring cross-correlation features of multiple genomic data, and addressing the problem of overlooking prior knowledge from various perspectives. In addition, the L1-norm is used to improve sparsity. A symmetric Gauss‒Seidel-based alternating direction method of multipliers (sGS-ADMM) is used to update the LTTRT model iteratively. Results: The experiments of sample clustering on multiple integrated cancer multi-omics datasets show that the proposed LTTRT method is better than existing methods. Experimental results validate the effectiveness of LTTRT in accurately partitioning different types of cancer samples. result: The experiments of sample clustering on multiple integrated cancer multi-omics datasets show that the proposed LTTRT method is better than existing methods. Experimental results validate the effectiveness of LTTRT in accurately partitioning different types of cancer samples. Conclusion: The LTTRT method achieves precise segmentation of different types of cancer samples.

Impact of Generalized Versus Individualized Load-Velocity Equations on Velocity-Loss Magnitude in Bench-Press Exercise: Mixed-Model and Equivalence Analysis.

This study analyzed the influence of 2 velocity-based training-load prescription strategies (general vs individual load-velocity equations) on the relationship between the magnitude of velocity loss (VL) and the percentage of repetitions completed in the bench-press exercise. Thirty-five subjects completed 6 sessions consisting of performing the maximum number of repetitions to failure against their 40%, 60%, and 80% of 1-repetition maximum (1RM) in the Smith machine bench-press exercise using generalized and individualized equations to adjust the training load. A close relationship and acceptable error were observed between percentage of repetitions completed and the percentage of VL reached for the 3 loading magnitudes and the 2 load-prescription strategies studied (R2 from .83 to .94; standard error of the estimate from 7% to 10%). A simple main effect was observed for load and VL thresholds but not for load-prescription strategies. No significant interaction effects were revealed. The 40% and 60% 1RM showed equivalence on data sets and the most regular variation, whereas the 80% 1-repetition maximum load showed no equivalence and more irregular variation. These results suggest that VL is a useful variable to predict percentage of repetitions completed in the bench-press exercise, regardless of the strategy selected to adjust the relative load. However, caution should be taken when using heavy loads.

Coastline target detection based on UAV hyperspectral remote sensing images

Timely and accurate monitoring of typical coastal targets using remote sensing technology is crucial for maintaining marine ecological stability. Hyperspectral target detection technology proves to be an effective tool in extracting various typical materials along the coastline. Traditional target detection methods using spectral domain information can effectively retain the intrinsic properties of the material. However, it is difficult to effectively recognize targets in homogeneous regions by using only spectral domain information, which may lead to insufficient utilization of spatial information. In this study, a detector based on signal-to-noise ratio fusion constrained energy minimization with low-rank sparse decomposition (SFLRSD) is proposed. This detector improves the separability of background and target by obtaining spatial domain information from hyperspectral images and fusing spectral domain information. First, total variation regularization and fractional Fourier transform are applied to process spatial and spectral domain information, respectively. The constrained energy minimization (CEM) detector is used to improve the separability between the target and background of the processed data. Then, the background and anomalies are represented as low-rank and sparse components, respectively, using low-rank sparse matrix factorization. This transforms the model solution into a covariance matrix problem, which is then solved using marginal distance difference (MDD) to isolate anomalous parts. Subsequently, the anomaly parts are fused with CEM detector results, weighted by their respective signal-to-noise ratios. This detection model leverages unified hyperspectral image features, enhancing spectral discreteness of anomalous targets and backgrounds. Finally, experiments on custom created hyperspectral dataset show that the proposed method outperforms other baseline methods in terms of visualization and quantitative performance. In this paper, we not only propose a new hyperspectral target detection method, but we also collect three typical marine litter of different materials by means of airborne hyperspectral remote sensing and construct four hyperspectral datasets in a real environment. All the simulation experiments in this paper are conducted in these four datasets.

An accelerated alternating direction method of multiplier for MRI with TV regularisation

Compressed Sensing (CS) is important in the field of image processing and signal processing, and CS-Magnetic Resonance Imaging (MRI) is used to reconstruct image from undersampled k-space data. Total Variation (TV) regularisation is a common technique to improve the sparsity of image, and the Alternating Direction Multiplier Method (ADMM) plays a key role in the variational image processing problem. This paper aims to improve the quality of MRI and shorten the reconstruction time. We consider MRI to solve a linear inverse problem, we convert it into a constrained optimization problem based on TV regularisation, then an accelerated ADMM is established. Through a series of theoretical derivations, we verify that the algorithm satisfies the convergence rate of O1/k2 under the condition that one objective function is quadratically convex and the other is strongly convex. We select five undersampled templates for testing in MRI experiment and compare it with other algorithms, experimental results show that our proposed method not only improves the running speed but also gives better reconstruction results.

Better than the Total Variation Regularization

The total variation (TV) regularization is popular in iterative image reconstruction when the piecewise-constant nature of the image is encouraged. As a matter of fact, the TV regularization is not strong enough to enforce the piecewise- constant appearance. This paper suggests a different regularization function that is able to discourage some smooth transitions in the image and to encourage the piecewise-constant behavior. This new regularization function involves a Gaussian function. We use the limited-angle tomography problem to illustrate the effectiveness of this new regularization function. The limited-angle tomography situation considered in this paper uses a scanning angular range of 40°. For two-dimensional parallel-beam imaging, the required angular range is supposed to be 180°.

Segmentation of binary sequence via minimizing least square error with total variation regularization

Segmentation of binary sequence via minimizing least square error with total variation regularization

Adaptive prior image constrained total generalized variation for low-dose dynamic cerebral perfusion CT reconstruction

BACKGROUND: Dynamic cerebral perfusion CT (DCPCT) can provide valuable insight into cerebral hemodynamics by visualizing changes in blood within the brain. However, the associated high radiation dose of the standard DCPCT scanning protocol has been a great concern for the patient and radiation physics. Minimizing the x-ray exposure to patients has been a major effort in the DCPCT examination. A simple and cost-effective approach to achieve low-dose DCPCT imaging is to lower the x-ray tube current in data acquisition. However, the image quality of low-dose DCPCT will be degraded because of the excessive quantum noise. OBJECTIVE: To obtain high-quality DCPCT images, we present a statistical iterative reconstruction (SIR) algorithm based on penalized weighted least squares (PWLS) using adaptive prior image constrained total generalized variation (APICTGV) regularization (PWLS-APICTGV). METHODS: APICTGV regularization uses the precontrast scanned high-quality CT image as an adaptive structural prior for low-dose PWLS reconstruction. Thus, the image quality of low-dose DCPCT is improved while essential features of targe image are well preserved. An alternating optimization algorithm is developed to solve the cost function of the PWLS-APICTGV reconstruction. RESULTS: PWLS-APICTGV algorithm was evaluated using a digital brain perfusion phantom and patient data. Compared to other competing algorithms, the PWLS-APICTGV algorithm shows better noise reduction and structural details preservation. Furthermore, the PWLS-APICTGV algorithm can generate more accurate cerebral blood flow (CBF) map than that of other reconstruction methods. CONCLUSIONS: PWLS-APICTGV algorithm can significantly suppress noise while preserving the important features of the reconstructed DCPCT image, thus achieving a great improvement in low-dose DCPCT imaging.

Smooth robust principal component analysis based on multidimensional transform tensor for dynamic MRI

Dynamic magnetic resonance imaging (DMRI) stands as a sophisticated medical imaging technique pivotal to clinical practice, but the protracted duration of its imaging poses a substantial constraint on its practical application. This paper introduces a smooth robust principal component analysis model based on multidimensional transform tensors for accelerating DMR imaging. Specifically, the proposed method breaks down data into low-rank and sparse parts for reconstruction, respectively. The low-rank part employs a multidimensional adaptive transformation framework to generate transform tensors with favorable low-rank properties along three dimensions of DMR data. As for the sparse part, precise reconstruction can be achieved with the sparsity of the data after sparse transformation. In addition, to enhance the preservation of image details, this paper introduces a novel weighted tensor total variation regularization, imposing varying degrees of constraints based on smoothness in different dimensions. Experimental results demonstrate that the proposed method realizes superior reconstruction effects in comparison to existing advanced methods.

Exact recovery of the support of piecewise constant images via total variation regularization

Exact recovery of the support of piecewise constant images via total variation regularization

Improvement of Anomaly Detection Performance of PaDiM by Fast Fourier Convolution with Total Variation Regularization

Improvement of Anomaly Detection Performance of PaDiM by Fast Fourier Convolution with Total Variation Regularization