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We present a primal-dual method to solve L¹-type nonsmooth optimization problems independently of the grid size. We apply these results to two important problems: the Rudin-Osher-Fatemi image denoising model and the L¹ earth mover's distance from optimal transport. Crucially, we provide analysis that determines the choice of optimal step sizes and we prove that our method converges independently of the grid size. Our approach allows us to solve these problems on grids as large as 4096 × 4096 in a few minutes without parallelization.

The class of L1-regularized optimization problems has received much attention recently because of the introduction of \"compressed sensing,\" which allows images and signals to be reconstructed from small amounts of data. Despite this recent attention, many L1-regularized problems still remain difficult to solve, or require techniques that are very problem-specific. In this paper, we show that Bregman iteration can be used to solve a wide variety of constrained optimization problems. Using this technique, we propose a \"split Bregman\" method, which can solve a very broad class of L1-regularized problems. We apply this technique to the Rudin-Osher-Fatemi functional for image denoising and to a compressed sensing problem that arises in magnetic resonance imaging.

We propose a dynamic programming algorithm for the one-dimensional Fused Lasso Signal Approximator (FLSA). The proposed algorithm has a linear running time in the worst case. A similar approach is developed for the task of least squares segmentation, and simulations indicate substantial performance improvement over existing algorithms. Examples of R and C implementations are provided in the online Supplementary materials, posted on the journal web site.

What are the main findings? * We construct a large-scale, high-resolution optical–SAR registration dataset, pairing 3-m SAR imagery from the HongTu-1 satellite with Google Earth optical imagery at zoom level 17, covering the major geographical regions of China, and we release the standardized pipeline—including full-scene pairing, DEM-based terrain correction, geometric refinement, standardized 512 × 512 slicing and multi-stage quality filtering—that was used to build it. * Our proposed Log-domain reformulation of the Total Variation (Log-TV) filter substantially improves SAR image preprocessing by converting the multiplicative speckle noise model into an additive one, thereby enabling effective suppression of speckle while preserving edge structures and providing a much cleaner foundation for subsequent keypoint detection. * Combining a machine learning-based edge filter (Structured Random Forest, SRF) with the hand-crafted phase congruency filter yields a strong synergistic effect for cross-modal optical–SAR edge filtering, producing more stable and consistent shared structural responses than either component alone. We construct a large-scale, high-resolution optical–SAR registration dataset, pairing 3-m SAR imagery from the HongTu-1 satellite with Google Earth optical imagery at zoom level 17, covering the major geographical regions of China, and we release the standardized pipeline—including full-scene pairing, DEM-based terrain correction, geometric refinement, standardized 512 × 512 slicing and multi-stage quality filtering—that was used to build it. Our proposed Log-domain reformulation of the Total Variation (Log-TV) filter substantially improves SAR image preprocessing by converting the multiplicative speckle noise model into an additive one, thereby enabling effective suppression of speckle while preserving edge structures and providing a much cleaner foundation for subsequent keypoint detection. Combining a machine learning-based edge filter (Structured Random Forest, SRF) with the hand-crafted phase congruency filter yields a strong synergistic effect for cross-modal optical–SAR edge filtering, producing more stable and consistent shared structural responses than either component alone. What are the implications of the main findings? * Large-scale, high-resolution optical–SAR datasets are both essential and scarce for registration and other downstream tasks. Only on larger and more complex benchmarks do the robustness and the true relative performance of competing algorithms become evident, making such datasets a necessary foundation for future research in this area. * Different imaging modalities require different filtering strategies: for heavily speckled data such as SAR imagery, regularisation in the logarithmic domain is more appropriate than directly applying denoisers designed for additive noise, highlighting the importance of modality-aware preprocessing in cross-modal registration. * Hybrid pipelines that integrate learning-based components with hand-crafted filters are a promising direction: beyond edge filtering, similar combinations of deep features and classical hand-crafted operators may also benefit cross-modal feature description and matching stages. Large-scale, high-resolution optical–SAR datasets are both essential and scarce for registration and other downstream tasks. Only on larger and more complex benchmarks do the robustness and the true relative performance of competing algorithms become evident, making such datasets a necessary foundation for future research in this area. Different imaging modalities require different filtering strategies: for heavily speckled data such as SAR imagery, regularisation in the logarithmic domain is more appropriate than directly applying denoisers designed for additive noise, highlighting the importance of modality-aware preprocessing in cross-modal registration. Hybrid pipelines that integrate learning-based components with hand-crafted filters are a promising direction: beyond edge filtering, similar combinations of deep features and classical hand-crafted operators may also benefit cross-modal feature description and matching stages. Optical and Synthetic Aperture Radar (SAR) image registration is a fundamental task in remote sensing information fusion, yet it remains challenging due to significant differences in imaging mechanisms, radiation characteristics, and noise properties between the two modalities. Existing public datasets suffer from limited resolution, small scale, and insufficient scene diversity, and these limitations have hindered algorithm development. This paper constructs a large-scale, high-resolution optical–SAR registration dataset based on the HongTu-1 satellite 3-m SAR imagery and Google Earth optical imagery at zoom level 17, covering diverse scenes across China with a standardized pipeline including terrain correction, geometric alignment, standardized slicing, and quality filtering. Building upon this dataset, a hand-crafted keypoint-based cross-modal registration method is proposed, incorporating multi-level edge filtering and hybrid feature detection. Unlike conventional hand-crafted methods such as RIFT, SRIF, and LNIFT, which mainly refine keypoint detection, description, or matching within a SIFT-style pipeline, the core novelty of this work lies in SAR-specific preprocessing and multi-level hybrid filtering. These components are designed to suppress speckle while extracting more stable and discriminative shared edge responses for cross-modal registration. An improved Log-domain Total Variation (Log-TV) denoising model is introduced for SAR preprocessing. A hybrid edge filtering framework combining phase congruency analysis and Structured Random Forest (SRF) edge detection is constructed within a Gaussian scale space. A dual-branch feature detection scheme integrating blob and corner features is designed with a robust orientation assignment strategy. Feature description uses the Gradient Location–Orientation Histogram (GLOH) descriptor with Principal Component Analysis (PCA) reduction, while geometric estimation employs the Fast Sample Consensus (FSC) algorithm. Experiments on the self-constructed HT dataset and on the public OSdataset and SAR2Opt benchmarks show that the proposed method consistently achieves low RMSE and high success rates. It also maintains competitive efficiency among hand-crafted methods while retaining strong robustness to scale and rotation variations.

Fetal heart rate (FHR) is an important determinant of fetal health. Cardiotocography (CTG) is widely used for measuring the FHR in the clinical field. However, fetal movement and blood flow through the maternal blood vessels can critically influence Doppler ultrasound signals. Moreover, CTG is not suitable for long-term monitoring. Therefore, researchers have been developing algorithms to estimate the FHR using electrocardiograms (ECGs) from the abdomen of pregnant women. However, separating the weak fetal ECG signal from the abdominal ECG signal is a challenging problem. In this paper, we propose a method for estimating the FHR using sequential total variation denoising and compare its performance with that of other single-channel fetal ECG extraction methods via simulation using the Fetal ECG Synthetic Database (FECGSYNDB). Moreover, we used real data from PhysioNet fetal ECG databases for the evaluation of the algorithm performance. The R-peak detection rate is calculated to evaluate the performance of our algorithm. Our approach could not only separate the fetal ECG signals from the abdominal ECG signals but also accurately estimate the FHR.

The reconstruction of biomedical signals from noisy measurements has been an indispensable research topic. A majority of biosignals exhibit typical piecewise characteristics. The recovery of these piecewise biomedical signals embedded in noise through conventional nonlinear filtering schemes fails due to the lack of proper balance between strict sparsity and smoothness-inducing property of regularizers at large noise levels. This work proposes a nonlinear convex optimization-based filtering approach, which incorporates a Moreau envelope-based regularizer using the majorized version of the total variation function. The source signals are restored by exploiting their piecewise characteristics through a majorized cost function. The majorized functions provide some relaxation in solving non-convex functions. The relaxation of the stringent sparsifying penalty provides the balance between the smoothness property and the piecewise features of biosignals. The optimality criterion for the proposed method is analyzed in this work. Furthermore, we evaluate the new method using a standard IoT platform. The recovery performance of this method is found to be superior to various state-of-the-art techniques for piecewise synthetic and real-world physiological signals corrupted by additive noise.

Aero-engine electrostatic monitoring technology (EMT) is a novel and effective condition monitoring technology. With the help of EMT, effective monitoring of early failures can be achieved. Since the electrostatic monitoring of the running engine will be strongly interfered, the sampled electrostatic signal has various noise components and low signal-to-noise ratio (SNR). After analyzing the source of the noise components carried by the electrostatic signal, this paper proposes a method for electrostatic signal denoising in a strong interference environment, which is based on the nonlinear total variation theory. In the experiments, the simulated electrostatic measurement signal and the actual test-run electrostatic measurement signal were used as the analysis objects, and the denoising test was carried out by using the proposed method. Meanwhile, the denoising effect was compared and analyzed with other classical methods. The experimental results show that the proposed denoising method can effectively remove random noise, electromagnetic pulse and periodic noise in electrostatic signal, and is more applicable to the measured electrostatic signal with low SNR than the classical electrostatic signal denoising methods such as wavelet threshold denoising method and empirical mode decomposition method.

We consider the problem of estimating a spatially varying density function, motivated by problems that arise in large-scale radiological survey and anomaly detection. In this context, the density functions to be estimated are the background gamma-ray energy spectra at sites spread across a large geographical area, such as nuclear production and waste-storage sites, military bases, medical facilities, university campuses, or the downtown of a city. Several challenges combine to make this a difficult problem. First, the spectral density at any given spatial location may have both smooth and nonsmooth features. Second, the spatial correlation in these density functions is neither stationary nor locally isotropic. Finally, at some spatial locations, there are very little data. We present a method called multiscale spatial density smoothing that successfully addresses these challenges. The method is based on recursive dyadic partition of the sample space, and therefore shares much in common with other multiscale methods, such as wavelets and Pólya-tree priors. We describe an efficient algorithm for finding a maximum a posteriori (MAP) estimate that leverages recent advances in convex optimization for nonsmooth functions. We apply multiscale spatial density smoothing to real data collected on the background gamma-ray spectra at locations across a large university campus. The method exhibits state-of-the-art performance for spatial smoothing in density estimation, and it leads to substantial improvements in power when used in conjunction with existing methods for detecting the kinds of radiological anomalies that may have important consequences for public health and safety.

The time-frequency and time-scale communities have recently developed a large number of overcomplete waveform dictionaries-stationary wavelets, wavelet packets, cosine packets, chirplets, and warplets, to name a few. Decomposition into overcomplete systems is not unique, and several methods for decomposition have been proposed, including the method of frames (MOF), matching pursuit (MP), and, for special dictionaries, the best orthogonal basis (BOB). Basis pursuit (BP) is a principle for decomposing a signal into an \"optimal\" superposition of dictionary elements, where optimal means having the smallest l1 norm of coefficients among all such decompositions. We give examples exhibiting several advantages over MOF, MP, and BOB, including better sparsity and superresolution. BP has interesting relations to ideas in areas as diverse as ill-posed problems, abstract harmonic analysis, total variation denoising, and multiscale edge denoising. BP in highly overcomplete dictionaries leads to large-scale optimization problems. With signals of length 8192 and a wavelet packet dictionary, one gets an equivalent linear program of size 8192 by 212,992. Such problems can be attacked successfully only because of recent advances in linear and quadratic programming by interior-point methods. We obtain reasonable success with a primal-dual logarithmic barrier method and conjugategradient solver.

A method for obtaining the exact solution for the total variation denoising problem of piecewise constant images in dimension one is presented. The validity of the algorithm relies on some results concerning the behavior of the solution when the parameter λ in front of the fidelity term varies. Albeit some of them are well-known in the community, here they are proved with simple techniques based on qualitative geometrical properties of the solutions.