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As interest in deep neural networks (DNNs) for image reconstruction tasks grows, their reliability has been called into question [V. Antun, F. Renna, C. Poon, B. Adcock, and A. C. Hansen, Proc. Natl. Acad. Sci. USA, 117 (2020), pp. 30088--30095; N. M. Gottschling, V. Antun, B. Adcock, and A. C. Hansen, The Troublesome Kernel: Why Deep Learning for Inverse Problems Is Typically Unstable, preprint, arXiv:2001.01258, 2020]. However, recent work has shown that, compared to total variation (TV) minimization, when appropriately regularized, DNNs show similar robustness to adversarial noise in terms of \ell 2-reconstruction error [M. Genzel, J. Macdonald, and M. M\"arz, IEEE Trans. Pattern Anal., 45 (2022), pp. 1119--1134]. We consider a different notion of robustness, using the \ell \infty - norm, and argue that localized reconstruction artifacts are a more relevant defect than the \ell 2-error. We create adversarial perturbations to undersampled magnetic resonance imaging measurements (in the frequency domain) which induce severe localized artifacts in the TV-regularized reconstruction. Notably, the same attack method is not as effective against DNN-based reconstruction. Finally, we show that this phenomenon is inherent to reconstruction methods for which exact recovery can be guaranteed, as with compressed sensing reconstructions with \ell 1- or TV-minimization. [ABSTRACT FROM AUTHOR]

We introduce a method for the fast estimation of data-adapted, spatially and temporally dependent regularization parameter-maps for variational image reconstruction, focusing on total variation (TV) minimization. The proposed approach is inspired by recent developments in algorithm unrolling using deep neural networks (NNs) and relies on two distinct subnetworks. The first subnetwork estimates the regularization parameter-map from the input data. The second subnetwork unrolls T iterations of an iterative algorithm which approximately solves the corresponding TV-minimization problem incorporating the previously estimated regularization parameter-map. The overall network is then trained end-to-end in a supervised learning fashion using pairs of clean and corrupted data but crucially without the need for access to labels for the optimal regularization parameter-maps. We first prove consistency of the unrolled scheme by showing that the unrolled minimizing energy functional used for the supervised learning\Gamma-converges, as T tends to infinity, to the corresponding functional that incorporates the exact solution map of the TV-minimization problem. Then, we apply and evaluate the proposed method on a variety of large-scale and dynamic imaging problems with retrospectively simulated measurement data for which the automatic computation of such regularization parameters has been so far challenging using the state-of-the-art methods: a 2D dynamic cardiac magnetic resonance imaging (MRI) reconstruction problem, a quantitative brain MRI reconstruction problem, a low-dose computed tomography problem, and a dynamic image denoising problem. The proposed method consistently improves the TV reconstructions using scalar regularization parameters, and the obtained regularization parameter-maps adapt well to imaging problems and data by leading to the preservation of detailed features. Although the choice of the regularization parameter-maps is data-driven and based on NNs, the subsequent reconstruction algorithm is interpretable since it inherits the properties (e.g., convergence guarantees) of the iterative reconstruction method from which the network is implicitly defined. [ABSTRACT FROM AUTHOR]

In the field of image processing, total variational model is an effective prior model. In order to better eliminate impulse noise, an effective method is to use ℓ 1 - norm total variational model. However, the TV image recovery always produces staircase artifacts, and the ℓ 1 - norm excessively penalizes the signal entries. Therefore, in this paper, a new total variational model is proposed to eliminate the staircase effects and impulse noise. We use ℓ 0 - norm as the data fidelity term to eliminate impulse noise and the hybrid total variation as the regularization term to effectively eliminate staircase artifacts. In order to effectively tackle the proposed ℓ 0 - norm and hybrid total variation, we first express this problem as a Mathematical Program with Equilibrium Constraints(MPEC), and then a Proximal Alternating Direction Method of Multipliers(PADMM) is adopted to solve this problem. In the experimental part, we adopt three indices: SNR 0 , SNR 1 and SNR 2 to measure the quality of image restoration. When the value of the SNR tends to be stable, then each algorithm stops iteration. Numerical simulation results show that the proposed method has better performance in removing impulse noise, suppressing staircase effect and preserving image edge information. [ABSTRACT FROM AUTHOR]

Medikal görüntüler doğası gereği farklı gürültü tipleri ve seviyelerine maruz kalmaktadır. Medikal görüntülerin oluşturulmasında kullanılan rekonstrüksiyon algoritmalarının temel amacı, oluşan bu gürültünün giderilmesi ve çözünürlüğün arttırılması için en verimli yöntemlerin kullanılmasıdır. Bu yöntemler kullanılırken filtreleme, düzenleyiciler ve gürültü giderici operatörler kullanıp gürültünün arındırılması amaçlanmaktadır. Sıkıştırılmış algılamanın medikal görüntülemede aktif olarak kullanılmaya başlanmasından sonra, görüntüyü daha seyrek forma dönüştüren toplam değişinti (TD) en küçüklemesi ile görüntü üzerindeki gürültü azaltılarak ufak detayların ve kenarların daha net biçimde korunması sağlanmıştır. Lokal bir gradyan operatörü olan toplam değişinti algoritması bu çalışmada kısmi gradyan yönlerinde kullanılan komşuluk seviyesi arttırılarak daha güçlü bir gürültü giderici olarak yeniden tasarlanmıştır. Çalışma kapsamında, tasarlanan bu yeni güçlendirilmiş gradyan minimizasyonunun medikal görüntülerde mevcut farklı Gauss, Poisson ve Gauss+Poisson gürültü seviyeleri üzerinde gürültü arındırma başarısı klasik TD ile karşılaştırılmıştır. Sonuçlar pik sinyal-gürültü oranı, yapısal benzerlik ve kontrast-gürültü oranı metrikleri ve görsel analiz kullanılarak karşılaştırılmış ve önerilen yeni güçlendirilmiş gradyan minimizasyonu yönteminin mevcut klasik TD algoritmasından daha iyi gürültü arındırma potansiyeline sahip olduğu gösterilmiştir.

The basis of backscatter X-ray imaging systems is the scattering of backscattered rays that have many applications in industry and inspection. In these images, organic materials with low atomic numbers, such as drugs and explosives, produce many scattered photons and are seen more clearly. These images are blurred due to the multiple scattered X-ray. Image processing methods can help improve contrast and material detection in backscatter X-ray (BX) images. In this research, the total variation method has been used to improve the contrast of the BX images. Firstly, the BX images of the various phantoms have been processed for evaluating the BX system, then many BX images of motorcycles are investigated. The results show that the total variation method increases the contrast of the backscattered X-ray images and can be used as the additional filter in the BX imaging system. Evaluation of experts shows that the contrast of the processed images has been significantly improved and increased between 20 to 40%.

The total variation regularizer is diffusely emerged in statistics, image and signal processing to obtain piecewise constant estimator. The ℓ0 total variation (L0TV) regularized signal denoising model is a nonconvex and discontinuous optimization problem, and it is very difficult to find its global optimal solution. In this paper, we present the global optimality analysis of L0TV signal denoising model, and design an efficient algorithm to pursuit its solution. Firstly, we equivalently rewrite the L0TV denoising model as a partial regularized (PL0R) minimization problem by aid of the structured difference operator. Subsequently, we define a P-stationary point of PL0R, and show that it is a global optimal solution. These theoretical results allow us to find the global optimal solution of the L0TV model. Therefore, an efficient Newton-type algorithm is proposed for the PL0R problem. The algorithm has a considerably low computational complexity in each iteration. Finally, experimental results demonstrate the excellent performance of our approach in comparison with several state-of-the-art methods.

This study proposes a new deep neural network to solve the multi-band image synchronous fusion problem (MBF-Net). Unlike other deep learning-based methods, our network architecture design combines the ideas of model-driven and data-driven methods, so it is more interpretable. First, a new multi-band image synchronous fusion model is proposed. The source image in the data fidelity terms and the prior regularization are implicitly represented by the deep learning network and jointly learned from the training data. The proposed model is then solved using a half quadratic splitting (HQS) algorithm and unfolded into a deep fusion network. In addition, a new saliency loss function is proposed to retain thermal radiation information to enhance the fusion effect. Finally, the experimental results on the TNO dataset demonstrated the effectiveness of the proposed MBF-Net.

Inverse problems describe the task of blending a mathematical model with observational data---a fundamental task in many scientific and engineering disciplines. The solvability of such a task is usually classified through its well-posedness. A problem is well-posed if it has a unique solution that depends continuously on input or data. Inverse problems are usually ill-posed, but can sometimes be approached through a methodology that formulates a possibly well-posed problem. Usual methodologies are the variational and the Bayesian approach to inverse problems. For the Bayesian approach, Stuart [Acta Numer., 19 (2010), pp. 451--559] has given assumptions under which the posterior measure---the Bayesian inverse problem's solution---exists, is unique, and is Lipschitz continuous with respect to the Hellinger distance and, thus, well-posed. In this work, we simplify and generalize this concept: Indeed, we show well-posedness by proving existence, uniqueness, and continuity in Hellinger distance, Wasserstein distance, and total variation distance, and with respect to weak convergence, respectively, under significantly weaker assumptions. An immense class of practically relevant Bayesian inverse problems satisfies those conditions. The conditions can often be verified without analyzing the underlying mathematical model--- the model can be treated as a black box. [ABSTRACT FROM AUTHOR]

We consider a variational approach to the problem of structure + texture decomposition (also known as cartoon + texture decomposition). As usual for many variational problems in image analysis and processing, the energy we minimize consists of two terms: a data-fitting term and a regularization term. The main feature of our approach consists of choosing parameters in the regularization term adaptively. Namely, the regularization term is given by a weighted p (·) -Dirichlet-based energy ∫ a (x) | ∇ u | p (x) , where the weight and exponent functions are determined from an analysis of the spectral content of the image curvature. Our numerical experiments, both qualitative and quantitative, suggest that the proposed approach delivers better results than state-of-the-art methods for extracting the structure from textured and mosaic images, as well as competitive results on image enhancement problems. [ABSTRACT FROM AUTHOR]

Previous studies have shown that Synthetic Aperture Radar (SAR) is able to detect burned areas, serving as a key data source for monitoring active wildfires in situations where optical sensors are hindered by dense smoke or cloud cover. Radar remote sensing provides rich and useful data on historical burned areas, which are critical for large-scale wildfire burned area mapping. This study aims to unveil the potential inherent in Sentinel-1 C-Band SAR data and to investigate the impact of reference masks on wildfire burned area mapping. SAR images frequently exhibit disruptive speckle noise, which manifests as isolated pixels or small regions, thereby hindering the seamless identification of burned areas through SAR data. To mitigate noise and enhance the connectivity of SAR-based burned area delineation, we propose a novel approach: a Total-Variation (TV) regularized U-Net model, tailored to learn from noisy pseudo masks derived from Sentinel-1 C-Band SAR backscattering data. To validate its efficacy, we assembled a dataset encompassing 16 geographically diverse wildfire events, incorporating SAR-based kMap, a SAR-based binary label (designated as SARREF), and an optical-based binary label (referred to as OptREF). The kMap, serving as a relative indicator of SAR backscattering change, is defined as the divergence between post-fire SAR backscattering and the temporal average, normalized by the temporal standard deviation of pre-fire SAR time series. Utilizing SAR-based kMap as input and SARREF or OptREF as labels, we systematically explored the impact of TV regularization on SAR-based burned area mapping. Our experimental findings highlight several key insights: 1) Vanilla U-Net considerably elevates the F1 score from 0.61 to 0.66 when compared to threshold-based approaches; 2) The TV-regularized U-Net exhibits a notable capacity for enhancing both accuracy and the connectivity of SAR-based burned areas. When trained with noisy SARREF, it achieves a substantial F1 score of 0.69 and an IoU score of 0.53. 3) In scenarios involving OptREF, the vanilla U-Net attains peak performance with the highest F1 (0.75) and IoU (0.61) scores, with the expected outcome of TV regularization yielding no additional enhancement. [ABSTRACT FROM AUTHOR]

Compressed sensing (CS) has been used to improve image quality in single-photon emission tomography (SPECT) imaging. However, the effects of CS on image quality parameters in myocardial perfusion imaging (MPI) have not been investigated in detail. This preliminary study aimed to compare the performance of CS-iterative reconstruction (CS-IR) with filtered back-projection (FBP) and maximum likelihood expectation maximization (ML-EM) on their ability to reduce the acquisition time of MPI. A digital phantom that mimicked the left ventricular myocardium was created. Projection images with 120 and 30 directions (360°), and with 60 and 15 directions (180°) were generated. The SPECT images were reconstructed using FBP, ML-EM, and CS-IR. The coefficient of variation (CV) for the uniformity of myocardial accumulation, septal wall thickness, and contrast ratio (Contrast) of the defect/normal lateral wall were calculated for evaluation. The simulation was performed ten times. The CV of CS-IR was lower than that of FBP and ML-EM in both 360° and 180° acquisitions. The septal wall thickness of CS-IR at the 360° acquisition was inferior to that of ML-EM, with a difference of 2.5 mm. Contrast did not differ between ML-EM and CS-IR for the 360° and 180° acquisitions. The CV for the quarter-acquisition time in CS-IR was lower than that for the full-acquisition time in the other reconstruction methods. CS-IR has the potential to reduce the acquisition time of MPI. [ABSTRACT FROM AUTHOR]

Rician noise removal is an important problem in magnetic resonance (MR) imaging. Among the existing approaches, the variational method is an essential mathematical technique for Rician noise reduction. The previous variational methods mainly employ the total variation (TV) regularizer, which is a first-order term. Although the TV regularizer is able to remove noise while preserving object edges, it suffers the staircase effect. Besides, the adaptability has received little research attention. To this end, we propose a spatially variant high-order variational model (SVHOVM) for Rician noise reduction. We introduce a spatially variant TV regularizer, which can adjust the smoothing strength for each pixel depending on its characteristics. Furthermore, SVHOVM utilizes the bounded Hessian (BH) regularizer to diminish the staircase effect generated by the TV term. We develop a split Bregman algorithm to solve the proposed minimization problem. Extensive experiments are performed to demonstrate the superiority of SVHOVM over some existing variational models for Rician noise removal. [ABSTRACT FROM AUTHOR]

Implicit Neural Representations are powerful tools for representing 3D shapes. They encode an implicit field in the parameters of a Neural Network, leveraging the power of auto-differentiation for optimizing the implicit function and avoiding the need for a manually crafted function. So far, Implicit Neural Representations have been mainly designed to extract or render object surfaces and methods primarily focus on improving the implicit function near the surface. In this paper we argue that implicit fields are useful for other shape analysis tasks, in particular skeleton (medial axis) extraction. Indeed, a medial axis is defined through distances to the surface, which can be provided by an implicit neural representation, making it robust to noise and missing data. However this requires the implicit field to be reliable away from the surface, something most representations are not optimized for. To achieve this, inspired by variational image denoising techniques, we propose to add a Total Variation term, to regularize the implicit field. We further design a skeleton sampling method working directly on the GPU, and link the extracted points using a coverage formulation. We show that our resulting neural skeleton is more robust to sample defects such as noise or missing data compared to other medial axis extraction methods. [Display omitted] • A TV regularization term yielding a smoother signed distance function away from the surface. • A skeleton sampling method on the GPU. • A complete pipeline to extract a medial complex from an input point cloud. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel image denoising algorithm, specifically designed to effectively restore both the edges and texture of images. This is achieved through the use of an innovative model known as the overlapping group sparse fractional-order total variation regularization model (OGS-FOTVR). The OGS-FOTVR model ingeniously combines the benefits of the fractional-order (FO) variation domain with an overlapping group sparsity measure, which acts as its regularization component. This is further enhanced by the inclusion of the well-established L2-norm, which serves as the fidelity term. To simplify the model, we employ the alternating direction method of multipliers (ADMM), which breaks down the model into a series of more manageable sub-problems. Each of these sub-problems can then be addressed individually. However, the sub-problem involving the overlapping group sparse FO regularization presents a high level of complexity. To address this, we construct an alternative function for this sub-problem, utilizing the mean inequality principle. Subsequently, we employ the majorize-minimization (MM) algorithm to solve it. Empirical results strongly support the effectiveness of the OGS-FOTVR model, demonstrating its ability to accurately recover texture and edge information in images. Notably, the model performs better than several advanced variational alternatives, as indicated by superior performance metrics across three image datasets, PSNR, and SSIM. [ABSTRACT FROM AUTHOR]

As one of the most significant issues in imaging science, image denoising plays a major role in plenty of image processing applications. Due to the ill-posed nature of image denoising, total variation regularization is widely used in image denoising problems for its capability to suppress noise and preserve image edges. Nevertheless, traditional total variation will inevitably yield undesirable staircase artifacts when applied to recorded images. Inspired by the success of ℓq norm minimization and overlapping group sparsity in image denoising, and the effective staircase artifacts removal by fractional total variation, the hybrid model which combines the fractional order total variation with overlapping group sparsity and higher order total variation with ℓq norm is developed in this paper to restore images corrupted by Gaussian noise. An efficient algorithm based on the parallel linear alternating direction method of multipliers is developed for solving the corresponding model and the numerical experiments demonstrate the effectiveness of the proposed approach against several state-of-the-art methods, in terms of peak signal-to-noise ratio and structure similarity index measure values. [ABSTRACT FROM AUTHOR]

This paper presents a parallel implementation of a fixed-point algorithm for wrapped phase denoising. The model based on total variation efficiently estimates discontinuous phase maps and further incorporates the Pythagorean trigonometric identity between the real and imaginary parts of the phase map, enhancing the quality of the restored phase. In this work, two parallel C/C++ implementations of the sequential version of algorithms were developed. The implementations include execution in a multi-core CPU and a GPU using OpenMP and CUDA, respectively. We show performance comparisons of the parallel implementations with advanced methods using synthetic and experimental data. Results show that our parallel implementations achieve speedups over the serial implementation of 12 × for multi-core CPU and 110 × for GPU. [ABSTRACT FROM AUTHOR]

In recent years, image denoising methods based on total variational regularization have attracted extensive attention. However, the traditional total variational regularization method is an approximate solution based on convex method, and does not consider the particularity of the region with rich details. In this paper, the adaptive total-variation and nonconvex low-rank model for image denoising is proposed, which is a hybrid regularization model. First, the image is decomposed into sparse terms and low rank terms, and then the total variational regularization is used to denoise. At the same time, an adaptive coefficient based on gradient is constructed to adaptively judge the flat area and detail texture area, slow down the denoising intensity of detail area, and then play the role of preserving detail information. Finally, by constructing a nonconvex function, the optimal solution of the function is obtained by using the alternating minimization method. This method not only effectively removes the image noise, but also retains the detailed information of the image. The experimental results show the effectiveness of the proposed model, and SNR and SSIM of the denoised image are improved. [ABSTRACT FROM AUTHOR]

We propose a simple yet efficient sparse unmixing method for hyperspectral images. It exploits the spatial and spectral properties of hyperspectral images by designing a new regularization informed by multiscale analysis. The proposed approach consists of two steps. First, a sparse unmixing is conducted on a coarse hyperspectral image resulting from a spatial smoothing of the original data. The estimated coarse abundance map is subsequently used to design two weighting terms summarizing the spatial and spectral properties of the image. They are combined to define a sparse regularization embedded into a unmixing problem associated with the original hyperspectral image at full resolution. The performance of the proposed method is assessed with numerous experiments conducted on synthetic and real datasets. It is shown to compete favorably with state-of-the-art methods from the literature with lower computational complexity. [ABSTRACT FROM AUTHOR]

Establishing feature correspondences between multimodal remote sensing images is an essential task for realizing diverse applications. Conventional matching methods, which employ gradient or phase congruency (PC) for feature detection and description, produce limited performance when images suffer from strong noises and intensity differences. In this study, we propose a novel modality-invariant structural feature representation (MISFR) method for multimodal remote sensing image matching. First, a maximal/minimal enhanced PC moment (EPCM) representation is designed by incorporating the PC with a multiscale relative total variation (RTV) model for feature detection and description. The EPCM integrates the advantages of these two models to exploit intrinsic structural features while providing robustness against modality variations. Then, to improve the feature stability and repeatability, an aggregation structural feature detector (ASFD) is proposed, in which the corner and edge points are separately extracted on the minimal and maximum EPCMs. Moreover, an adaptive binning-based log-polar descriptor is constructed on the maximal EPCM, named enhanced features of orientated PC (EOPC), to robustly characterize feature points. Various experiments on a series of multimodal remote sensing images demonstrate that our MISFR significantly improves the matching performance compared with four state-of-the-art approaches. [ABSTRACT FROM AUTHOR]

In order to determine the 3D structure of a thick sample, researchers have recently combined ptychography (for high resolution) and tomography (for 3D imaging) in a single experiment. 2-step methods are usually adopted for reconstruction, where the ptychography and tomography problems are often solved independently. In this paper, we provide a novel model and ADMM-based algorithm to jointly solve the ptychography-tomography problem iteratively, also employing total variation regularization. The proposed method permits large scan stepsizes for the ptychography experiment, requiring less measurements and being more robust to noise with respect to other strategies, while achieving higher reconstruction quality results.

This paper presents a variational model for the denoising of wrapped phase images. By enforcing the required Pythagorean trigonometric identity between the real and imaginary components of the signal, this model improves the signal-to-noise ratio of the restored signal. To preserve phase map discontinuities, the model is based on total variation. The existence and uniqueness of the model's solution are demonstrated using standard techniques. In addition, the convergence of a rapid fixed-point method to determine the numerical solution is demonstrated. Experiments on both synthetic and actual patterns validate the model's performance. [ABSTRACT FROM AUTHOR]

Introduction: Accurately detecting cracks is crucial for assessing the health of materials. Manual detection methods are time-consuming, leading to the development of automatic detection techniques based on image processing and machine learning. These methods utilize morphological image processing and material deformation analysis through Digital Image or Volume Correlation techniques (DIC/DVC) to identify cracks. The strain field derived from DIC/DVC tends to be noisy. Traditional denoising methods sacrifice spatial resolution, limiting their effectiveness in capturing abrupt structural deformations such as fractures. Method: In this study, a novel DVC regularization method is proposed to obtain a sharper and less noisy strain field. The method minimizes the total variation of spatial strain field components based on the assumption of approximate strain constancy within material phases. Results: The proposed methodology is validated using simulated data and actual 4D  μ -CT experimental data. Compared to classical denoising methods, the proposed DVC regularization method provides a more reliable crack detection with fewer false positives. Conclusions: These results highlight the possibility of estimating a low-noise strain field without relying on the spatial smoothness assumption, thereby improving accuracy and reliability in crack detection. [ABSTRACT FROM AUTHOR]

Pansharpening refers to the fusion of a panchromatic (PAN) and a multispectral (MS) image aimed at generating a high-quality outcome over the same area. This particular image fusion problem has been widely studied, but until recently, it has been challenging to balance the spatial and spectral fidelity in fused images. The spectral distortion is widespread in the component substitution-based approaches due to the variation in the intensity distribution of spatial components. We lightened the idea using the total variation optimization to improve upon a novel GIHS-TV framework for pansharpening. The framework drew the high spatial fidelity from the GIHS scheme and implemented it with a simpler variational expression. An improved L1-TV constraint to the new spatial–spectral information was introduced to the GIHS-TV framework, along with its fast implementation. The objective function was solved by the Iteratively Reweighted Norm (IRN) method. The experimental results on the "PAirMax" dataset clearly indicated that GIHS-TV could effectively reduce the spectral distortion in the process of component substitution. Our method has achieved excellent results in visual effects and evaluation metrics. [ABSTRACT FROM AUTHOR]

With the spread of aerial laser bathymetry (ALB), seafloor topographies are being measured more frequently. Nevertheless, data deficiencies occur owing to seawater conditions and other factors. Conventional interpolation methods generally need to produce digital elevation models (DEMs) with sufficient accuracy. If the topographic features are considered as a basis, the DEM should be reproducible based on a combination of such features. The purpose of this study is to develop new DEM generation methods based on sparse modeling. Based on a review of the definitions of sparsity, we developed DEM generation methods based on a discrete cosine transform (DCT), DCT with elastic net, K-singular value decomposition (K-SVD), Fourier regularization, wavelet regularization, and total variation (TV) minimization, and conducted a comparative analysis. The developed methods were applied to artificially deficient DEM and ALB data, and their accuracy was evaluated. Thus, as a conclusion, we can confirm that the K-SVD method is appropriate when the percentage of deficiencies is low, and that the TV minimization method is appropriate when the percentage of deficiencies is high. Based on these results, we also developed a method integrating both methods and achieved an RMSE of 0.128 m. [ABSTRACT FROM AUTHOR]

Impulse noise removal is an important problem in the field of image processing. Although many methods exist to remove impulse noise, there is still room for improvement. This paper proposes a new method for removing impulse noise that combines the nuclear norm and the detection ℓ 0 TV model while considering the low-rank structure commonly found in visual images. The nuclear norm maintains this structure, while the detection ℓ 0 TV criterion promotes sparsity in the gradient domain, effectively removing impulse noise while preserving edges and other vital features. To solve the non-convex and non-smooth optimization problem, we use a mathematical process with equilibrium constraints (MPEC) to transform it. Subsequently, the proximal alternating direction multiplication algorithm is used to solve the transformed problem. The convergence of the algorithm is proven under mild conditions. Numerical experiments in denoising and deblurring show that for low-rank images, the proposed method outperforms ℓ 1 TV with detection, ℓ 0 TV and ℓ 0 OGSTV. [ABSTRACT FROM AUTHOR]

Time stepping methods are often required for solving system of ordinary differential equations arising from spatial discretization of partial differential equations. In our prior work, we derived sufficient conditions for a subclass of second derivative general linear methods (SGLMs), so-called second derivative diagonally implicit multistage integration methods, to preserve the strong stability properties of spatial discretizations when coupled with forward Euler and a Taylor series conditions which allows for more flexibility in the choice of spatial discretizations. In this work, we extend this theory to the class of the high number of external stages SGLMs with Runge–Kutta stability. Proposed methods up to order five are examined on some one spatial dimension linear and nonlinear problems in PDEs confirming the verification of the theoretical order and potential of such schemes in maintaining some nonlinear stability properties such as and total variation. [ABSTRACT FROM AUTHOR]

Total variation model is a popular image restoration method due to its capability to preserve edges. However, it is easy to produce staircase artifacts. In this paper, we propose a non-convex and non-separable regularization term combining the overlapping group sparsity (OGS) and the minimax-concave penalty, which is obtained by subtracting the minimax-concave of the OGS term from the OGS term. The OGS term in the new regularization term can remove the staircase artifacts and preserve the image structure. The minimax-concave penalty can promote the sparsity of image in gradient domain and improve the estimation of the high-frequency components. We prove and set the range of the regularization parameter to ensure that the total cost function of the proposed model is convex. In order to solve the proposed model, we develop an adaptive alternating direction method of multipliers algorithm to tune the regularization parameter adaptively. In the experiments, we compare the proposed model with other state-of-the-art models in image denoising and image deblurring. The experimental results show the effectiveness of the proposed model in image restoration. [ABSTRACT FROM AUTHOR]

Purpose: To apply total generalized variation (TGV) and its combination with low‐rank and sparse decomposition (LRSD) (LTGV) to cerebral perfusion studies using low‐dose dynamic contrast‐enhanced (DCE) CT and to quantitatively evaluate their performances through comparisons with those without any regularizers and those of total variation (TV) and its combination with LRSD (LTV) using simulation and clinical data. Methods: The simulation study used a realistic digital brain phantom. Low‐dose DCE‐CT images were reconstructed using the regularizers and primal‐dual algorithm. Subsequently, cerebral perfusion parameter (CPP) images were generated from them. Thereafter, their quality was evaluated based on the peak signal‐to‐noise ratio (PSNR) and structural similarity index measure (SSIM). Further, the accuracy of CPP estimation was evaluated through a linear regression analysis between the CPP values obtained by the above regularizers and those obtained from the noise‐free DCE‐CT images. In addition, the mean and standard deviation of the CPP were calculated (region analysis). In the clinical study, low‐dose DCE‐CT images were generated using normal‐dose images acquired from a patient, and CPP images were generated from them similar to that in the simulation study. Results: When using LTV and LTGV, both PSNR and SSIM were higher than those of the other methods with increasing regularization parameter values. The results of the linear regression and region analyses demonstrated that TGV generally exhibited the best performance, followed by LTGV, and finally that of TV was significantly different from those of the other regularizers. Despite an overall consistency between the simulation and clinical results, certain inconsistencies appeared owing to the difference in generating low‐dose DCE‐CT images. Conclusions: The results implied that TGV and LTGV were useful in improving the accuracy of CPP estimation using low‐dose DCE‐CT. This study provides an improved understanding of the performance of regularizers and is expected to aid in the selection of a suitable regularizer for low‐dose DCE‐CT perfusion studies. [ABSTRACT FROM AUTHOR]

Medical diagnostics rely heavily on ultrasound imaging since it is non‐invasive, inexpensive, and able to provide images in real time. However, the inherent existence of a kind of signal‐dependent noise in these images diminishes the use of ultrasound imaging systems. Speckle noise is inherently present in ultrasound images. Its inherent presence occurs during the image acquisition phase. It decreases the diagnostic usefulness of this imaging modality since it interferes with and reduces the image resolution and contrast. Speckle noise is multiplicative in nature. This multiplicative nature of speckle noise causes adverse effects on ultrasound imaging because it distorts the image quality and causes a loss of the patient's informative content in ultrasound imaging. This causes difficulty in tissue characterization. A lot of research has been done in this field to remove speckle noise while preserving medical information in the image. To handle this issue, research has been grouped into multiple domains. The two main domains include homomorphic and non‐homomorphic filtering. The homomorphic filtering uses a logarithmic transform that converts this multiplicative nature into an additive nature and uses any additive image restoration model to do a despeckling task. Non‐homomorphic filtering encompasses all methods that do not employ a logarithmic transform. Since the presence of speckle noise in ultrasound imaging is an inherent property. Therefore, the despeckling of medical ultrasound images is a mandatory task that cannot be avoided or ignored. This paper proposes a two‐step hybrid and homomorphic despeckling technique, where modified total variation method is applied as step one for speckle reduction purposes. Step two implements method of noise thresholding in the non‐subsampled contourlet transform (NSCT) domain using the bivariate shrinkage rule for edge preservation. The effectiveness and reliability of the proposed method are tested by comparing it with some of the latest methods based on qualitative and quantitative analyses. This paper will help the new researchers understand various problems and their solutions related to ultrasound image despeckling. [ABSTRACT FROM AUTHOR]

The degraded image during the process of image analysis needs more number of iterations to restore it. These iterations take long waiting time and slow scanning, resulting in inefficient image restoration. A few numbers of measurements are enough to recuperate an image with good condition. Due to tree sparsity, a 2D wavelet tree reduces the number of coefficients and iterations to restore the degraded image. All the wavelet coefficients are extracted with overlaps as low and high sub-band space and ordered them such that they are decomposed in the tree ordering structured path. Some articles have addressed the problems with tree sparsity and total variation (TV), but few authors endorsed the benefits of tree sparsity. In this paper, a spatial variation regularization algorithm based on tree order is implemented to change the window size and variation estimators to reduce the loss of image information and to solve the problem of image smoothing operation. The acceptance rate of the tree-structured path relies on local variation estimators to regularize the performance parameters and update them to restore the image. For this, the Localized Total Variation (LTV) method is proposed and implemented on a 2D wavelet tree ordering structured path based on the proposed image smooth adjustment scheme. In the end, a reliable reordering algorithm proposed to reorder the set of pixels and to increase the reliability of the restored image. Simulation results clearly show that the proposed method improved the performance compared to existing methods of image restoration. [ABSTRACT FROM AUTHOR]

Background: With the rapid development of deep learning technology, deep neural networks can effectively enhance the performance of computed tomography (CT) reconstructions. One kind of commonly used method to construct CT reconstruction networks is to unroll the conventional iterative reconstruction (IR) methods to convolutional neural networks (CNNs). However, most unrolling methods primarily unroll the fidelity term of IR methods to CNNs, without unrolling the prior terms. The prior terms are always directly replaced by neural networks. Purpose: In conventional IR methods, the prior terms play a vital role in improving the visual quality of reconstructed images. Unrolling the hand‐crafted prior terms to CNNs may provide a more specialized unrolling approach to further improve the performance of CT reconstruction. In this work, a primal‐dual network (PD‐Net) was proposed by unrolling both the data fidelity term and the total variation (TV) prior term, which effectively preserves the image edges and textures in the reconstructed images. Methods: By further deriving the Chambolle–Pock (CP) algorithm instance for CT reconstruction, we discovered that the TV prior updates the reconstructed images with its divergences in each iteration of the solution process. Based on this discovery, CNNs were applied to yield the divergences of the feature maps for the reconstructed image generated in each iteration. Additionally, a loss function was applied to the predicted divergences of the reconstructed image to guarantee that the CNNs' results were the divergences of the corresponding feature maps in the iteration. In this manner, the proposed CNNs seem to play the same roles in the PD‐Net as the TV prior in the IR methods. Thus, the TV prior in the CP algorithm instance can be directly unrolled to CNNs. Results: The datasets from the Low‐Dose CT Image and Projection Data and the Piglet dataset were employed to assess the effectiveness of our proposed PD‐Net. Compared with conventional CT reconstruction methods, our proposed method effectively preserves the structural and textural information in reference to ground truth. Conclusions: The experimental results show that our proposed PD‐Net framework is feasible for the implementation of CT reconstruction tasks. Owing to the promising results yielded by our proposed neural network, this study is intended to inspire further development of unrolling approaches by enabling the direct unrolling of hand‐crafted prior terms to CNNs. [ABSTRACT FROM AUTHOR]

We consider the question of whether and in what sense, Wardrop equilibria provide a good approximation for Nash equilibria in atomic unsplittable congestion games with a large number of small players. We examine two different definitions of small players. In the first setting, we consider games in which each player's weight is small. We prove that when the number of players goes to infinity and their weights to zero, the random flows in all (mixed) Nash equilibria for the finite games converge in distribution to the set of Wardrop equilibria of the corresponding nonatomic limit game. In the second setting, we consider an increasing number of players with a unit weight that participate in the game with a decreasingly small probability. In this case, the Nash equilibrium flows converge in total variation toward Poisson random variables whose expected values are Wardrop equilibria of a different nonatomic game with suitably defined costs. The latter can be viewed as symmetric equilibria in a Poisson game in the sense of Myerson, establishing a plausible connection between the Wardrop model for routing games and the stochastic fluctuations observed in real traffic. In both settings, we provide explicit approximation bounds, and we study the convergence of the price of anarchy. Beyond the case of congestion games, we prove a general result on the convergence of large games with random players toward Poisson games. Funding: R. Cominetti gratefully acknowledges the support of Proyecto Anillo [Grant ANID/PIA/ACT192094]. M. Scarsini's work was partially supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni, Istituto Nazionale di Alta Matematica [Grant 2020 "Random walks on random games"] and the Progetti di Rilevante Interesse Nazionale 2017 [Grant "Algorithms, Games, and Digital Markets"]. This research also received partial support from the European Cooperation in Science and Technology [Action European Network for Game Theory]. [ABSTRACT FROM AUTHOR]

Various super-resolution (SR) kernels in the degradation model deteriorate the performance of the SR algorithms, showing unpleasant artifacts in the output images. Hence, SR kernel estimation has been studied to improve the SR performance in several ways for more than a decade. In particular, a conventional research named KernelGAN has recently been proposed. To estimate the SR kernel from a single image, KernelGAN introduces generative adversarial networks(GANs) that utilize the recurrence of similar structures across scales. Subsequently, an enhanced version of KernelGAN, named E-KernelGAN, was proposed to consider image sharpness and edge thickness. Although it is stable compared to the earlier method, it still encounters challenges in estimating sizable and anisotropic kernels because the structural information of an input image is not sufficiently considered. In this paper, we propose a kernel estimation algorithm called Total Variation Guided KernelGAN (TVG-KernelGAN), which efficiently enables networks to focus on the structural information of an input image. The experimental results show that the proposed algorithm accurately and stably estimates kernels, particularly sizable and anisotropic kernels, both qualitatively and quantitatively. In addition, we compared the results of the non-blind SR methods, using SR kernel estimation techniques. The results indicate that the performance of the SR algorithms was improved using our proposed method. [ABSTRACT FROM AUTHOR]

In this article, we propose a novel variational model for restoring images in the presence of the mixture of Cauchy and Gaussian noise. The model involves a novel data-fidelity term that features the mixed noise as an infimal convolution of two noise distributions and total variation regularization. This data-fidelity term contributes to suitable separation of Cauchy noise and Gaussian noise components, facilitating simultaneous removal of the mixed noise. Besides, the total variation regularization enables adequate denoising in homogeneous regions while conserving edges. Despite the nonconvexity of the model, the existence of a solution is proven. By employing an alternating minimization approach and the alternating direction method of multipliers, we present an iterative algorithm for solving the proposed model. Experimental results validate the effectiveness of the proposed model compared to other existing models according to both visual quality and some image quality measurements.

Anomaly detection for time series data is one of the important research problems in data analysis.Its main challenge is to detect if there are any anomalies and locate anomalies with low delay according to context.Most of existing anomaly detection methods capture anomalies using the probability density ratio to measure similarity between sequences.These methods need to use the cross-validation method to estimate the parameters of probability density ratio.However,cross-validation can increase the computational complexity,resulting in low computational efficiency and a high time delay.To address these issues,this paper proposes a detection method based on total variation ratio separation distance,in which total variation is adopted to extract sequence fluctuation features.Due to the fact that the total variation ratio is better than probability density ratio,the proposed method achieves higher computational efficiency and lower time delay.To reduce noise interference and further improve the detection accuracy,the proposed method is combined with the relative total variation.Experimental results show that the proposed method performs well in terms of detection accuracy,low delay and computational efficiency.

Image registration is broadly used in various scenarios in which similar scenes in different images are to be aligned. However, image registration becomes challenging when the contrasts and backgrounds in the images are vastly different. This work proposes using the total variation of the difference map between two images (TVDM) as a dissimilarity metric in rigid registration. A method based on TVDM minimization is implemented for image rigid registration. The method is tested with both synthesized and real experimental data that have various noise and background conditions. The performance of the proposed method is compared with the results of other rigid registration methods. It is demonstrated that the proposed method is highly accurate and robust and outperforms other methods in all of the tests. The new algorithm provides a robust option for image registrations that are critical to many nano-scale X-ray imaging and microscopy applications.

Computed tomography (CT) is capable of generating detailed cross-sectional images of the scanned objects non-destructively. So far, CT has become an increasingly vital tool for 3D modelling of cultural relics. Compressed sensing (CS)-based CT reconstruction algorithms, such as the algebraic reconstruction technique (ART) regularized by total variation (TV), enable accurate reconstructions from sparse-view data, which consequently reduces both scanning time and costs. However, the implementation of the ART-TV is considerably slow, particularly in cone-beam reconstruction. In this paper, we propose an efficient and high-quality scheme for cone-beam CT reconstruction based on the traditional ART-TV algorithm. Our scheme employs Joseph's projection method for the computation of the system matrix. By exploiting the geometric symmetry of the cone-beam rays, we are able to compute the weight coefficients of the system matrix for two symmetric rays simultaneously. We then employ multi-threading technology to speed up the reconstruction of ART, and utilize graphics processing units (GPUs) to accelerate the TV minimization. Experimental results demonstrate that, for a typical reconstruction of a 512 × 512 × 512 volume from 60 views of 512 × 512 projection images, our scheme achieves a speedup of 14 × compared to a single-threaded CPU implementation. Furthermore, high-quality reconstructions of ART-TV are obtained by using Joseph's projection compared with that using traditional Siddon's projection. [ABSTRACT FROM AUTHOR]

Low-rank tensor representation (LRTR) has become a significant method for achieving improved multi-view clustering (MVC) performance. Generally, most LRTR methods impose a tensor low-rank constraint (TLRC) on a tensor, which is spliced by the representation matrix of each view, to explore the low-rank prior hidden in these representation matrices. However, two problems remain unsolved. On the one hand, these representation matrices still contain noise since the raw data usually possess noise; on the other hand, and more importantly, is there any other prior information that can be explored among these representation matrices? Since samples located in the same cluster are similar, the coefficients of their representation matrices should exhibit similar. That is, these representation matrices should be local smoothness (LS), which is also verified by our numerical tests. In addition, the use of the LS prior helps to denoise the noise contain in the data. Then, in this paper, we propose a joint LS and LRTR for robust MVC method (LS-LRTR), which can simultaneously exploit the low-rank and LS priors. Specifically, we utilize a TLRC to explore the low-rank prior in the representation matrices. Subsequently, to mine the LS prior and further reduce the influence of noise, we introduce the Total Variation (TV) norm to the constraint representation matrices. Then, we fuse the TLRC and TV norm into a unified framework. Additionally, we apply an Augmented Lagrange Multiplier to solve the optimization problem of LS-LRTR. Experiments conducted on several datasets indicate that LS-LRTR outperforms the state-of-the-art clustering methods. • Our model integrates local smoothness and low-rank tensor representation for clustering. • Our model applies TV norm to explore the local smoothness and denoise. • Our model is the first to leverage both low-rank and local smoothness. • Experimental results outperform other state-of-the-art clustering methods. [ABSTRACT FROM AUTHOR]

This paper investigates a clustering algorithm with Total Variation (TV) constraint for postural regulation from postural coordination signals. The problem addressed aims to automatically segment postural coordination signals into behavioral patterns according to the patients' performances. Starting from the assumption that the strategies of postural regulation by a patient will be almost constant during an experimental session and for the same oscillation frequency of the mechanical horse, we propose to add to the clustering problem a TV regularization which forces two consecutive cycles to belong to the same cluster. The resulting optimization problem is solved by an ADMM approach. An extensive experimental evaluation is presented, where we demonstrate the usefulness of our approach on various tasks. The proposed clustering scheme was tested on several real datasets, including non-temporal data, time-series signals, and action temporal segmentation data. Experimental findings show that the approach significantly increases the clustering accuracy with gains of 3.7% to 51.2% on non-temporal data and compares favorably with other methods on time series clustering. Also, improved clustering precision was reported on action segmentation task. Finally, our TV regularization approach was evaluated on postural data from brain-damaged people. Empirical results illustrate the meaningfulness of the obtained clusters and signal segmentation. • We propose a new clustering algorithm using temporal coherence and TV regularization. • We illustrate the algorithm's effectiveness on time-series data and action segments. • We show that hippotherapy improves postural coordination in brain-damaged patients. [ABSTRACT FROM AUTHOR]

In order to determine the 3D structure of a thick sample, researchers have recently combined ptychography (for high resolution) and tomography (for 3D imaging) in a single experiment. 2-step methods are usually adopted for reconstruction, where the ptychography and tomography problems are often solved independently. In this paper, we provide a novel model and ADMM-based algorithm to jointly solve the ptychography-tomography problem iteratively, also employing total variation regularization. The proposed method permits large scan stepsizes for the ptychography experiment, requiring less measurements and being more robust to noise with respect to other strategies, while achieving higher reconstruction quality results.