TV Regularization Research Articles

Poisson noise removal based on non-convex hybrid regularizers

The presence of TV regularizer always induces an unsatisfactory staircase effect. To overcome the staircase while better sustaining edge information, this work proposes a novel model for Poisson noise removal. The model is based on non-convex mixed regularizers, which involves introducing a non-convex penalty into a composition of the total variation and the higher-order total variation. The iterative reweighted l1 algorithm was used to convert the non-convex model into a convex one. The classic alternating direction method of multipliers was then employed to obtain approximate solutions of the model. When applying this model to degraded images contaminated by Poisson noise of medium to high intensity, its performance in noise suppression was tested. The regularizer was compared with others in the terms of the visual effect of the picture, time cost, and several commonly accepted quantitative indicators for evaluation, such as peak signal-to-noise ratio, feature similarity index and structural similarity index. Numerical experiments showed that the present model not only eliminates block artifacts but also retains sharp edges.

A Semi-smooth Newton Method for Diffuse Optical Tomography with L1-Norm and Total Variation Regularization

This study introduces a novel multi-measurement Diffuse Optical Tomography reconstruction method that employs both L1-norm and Total Variation regularization, solved by the semi-smooth Newton method. By integrating L1-norm regularization to address sparsity and TV regularization to preserve edges and structural details, our approach effectively combats the ill-posed nature of DOT reconstructions. The results demonstrate that our approach reduces the required iterations while simultaneously maintaining or enhancing reconstruction accuracy and robustness to noise.

Sparse sampling photoacoustic reconstruction with a graph regularization group sparse dictionary

Photoacoustic tomography (PAT) has emerged as a promising biomedical imaging technique. The combination of optical contrast and ultrasound spatial resolution in photoacoustic tomography overcomes the limitations of optical scattering, enabling clear imaging of tissue structures. However, achieving high-resolution photoacoustic images typically requires a large number of sensor detection elements for sufficient angular coverage. This demand for extensive data acquisition and processing raises concerns about efficiency and system complexity. While sparse sampling strategies can improve efficiency, preserving detailed structural information becomes challenging with a minimal number of detectors. To address the challenges of sparse sampling, compressed sensing (CS) techniques have been successfully applied for image reconstructions in 2D and 3D photoacoustic embodiments. In this context, we propose a joint graph regularization group sparse dictionary and total variational regularization (GRGS-TV) algorithm based on our previous work of a group sparse dictionary. It preserves structured information and geometric relationships among dictionary atoms. Moreover, TV regularization effectively preserves edge structures while exhibiting a certain degree of robustness and flexibility. Numerical simulations and in vivo experiments on mice validate the effectiveness of this method in improving photoacoustic image quality and suppressing artifacts. Comparative evaluations against other algorithms show enhanced performance in terms of image reconstruction evaluation indices. This innovative approach holds promise for advancing photoacoustic imaging in biomedical research and clinical diagnostics.

Data-Proximal Complementary ℓ1-TV Reconstruction for Limited Data Computed Tomography

In a number of tomographic applications, data cannot be fully acquired, resulting in severely underdetermined image reconstruction. Conventional methods in such cases lead to reconstructions with significant artifacts. To overcome these artifacts, regularization methods are applied that incorporate additional information. An important example is TV reconstruction, which is known to be efficient in compensating for missing data and reducing reconstruction artifacts. On the other hand, tomographic data are also contaminated by noise, which poses an additional challenge. The use of a single regularizer must therefore account for both the missing data and the noise. A particular regularizer may not be ideal for both tasks. For example, the TV regularizer is a poor choice for noise reduction over multiple scales, in which case ℓ1 curvelet regularization methods are well suited. To address this issue, in this paper, we present a novel variational regularization framework that combines the advantages of different regularizers. The basic idea of our framework is to perform reconstruction in two stages. The first stage is mainly aimed at accurate reconstruction in the presence of noise, and the second stage is aimed at artifact reduction. Both reconstruction stages are connected by a data proximity condition. The proposed method is implemented and tested for limited-view CT using a combined curvelet–TV approach. We define and implement a curvelet transform adapted to the limited-view problem and illustrate the advantages of our approach in numerical experiments.

Sparse-view X-ray CT based on a box-constrained nonlinear weighted anisotropic TV regularization.

Sparse-view computed tomography (CT) is an important way to reduce the negative effect of radiation exposure in medical imaging by skipping some X-ray projections. However, due to violating the Nyquist/Shannon sampling criterion, there are severe streaking artifacts in the reconstructed CT images that could mislead diagnosis. Noting the ill-posedness nature of the corresponding inverse problem in a sparse-view CT, minimizing an energy functional composed by an image fidelity term together with properly chosen regularization terms is widely used to reconstruct a medical meaningful attenuation image. In this paper, we propose a regularization, called the box-constrained nonlinear weighted anisotropic total variation (box-constrained NWATV), and minimize the regularization term accompanying the least square fitting using an alternative direction method of multipliers (ADMM) type method. The proposed method is validated through the Shepp-Logan phantom model, alongisde the actual walnut X-ray projections provided by Finnish Inverse Problems Society and the human lung images. The experimental results show that the reconstruction speed of the proposed method is significantly accelerated compared to the existing $ L_1/L_2 $ regularization method. Precisely, the central processing unit (CPU) time is reduced more than 8 times.

Three-Dimensional Surface Reconstruction from Point Clouds Using Euler’s Elastica Regularization

Euler’s elastica energy regularizer, initially employed in mathematical and physical systems, has recently garnered much attention in image processing and computer vision tasks. Due to the non-convexity, non-smoothness, and high order of its derivative, however, the term has yet to be effectively applied in 3D reconstruction. To this day, the industry is still searching for 3D reconstruction systems that are robust, accurate, efficient, and easy to use. While implicit surface reconstruction methods generally demonstrate superior robustness and flexibility, the traditional methods rely on initialization and can easily become trapped in local minima. Some low-order variational models are able to overcome these issues, but they still struggle with the reconstruction of object details. Euler’s elastica term, on the other hand, has been found to share the advantages of both the TV regularization term and the curvature regularization term. In this paper, we aim to address the problems of missing details and complex computation in implicit 3D reconstruction by efficiently using Euler’s elastica term. The main contributions of this article can be outlined in three aspects. Firstly, Euler’s elastica is introduced as a regularization term in 3D point cloud reconstruction. Secondly, a new dual algorithm is devised for the proposed model, significantly improving solution efficiency compared to the commonly used TV model. Lastly, numerical experiments conducted in 2D and 3D demonstrate the remarkable performance of Euler’s elastica in enhancing features of curved surfaces during point cloud reconstruction. The reconstructed point cloud surface adheres more closely to the initial point cloud surface when compared to the classical TV model. However, it is worth noting that Euler’s elastica exhibits a lesser capability in handling local extrema compared to the TV model.

An Unsupervised Image Denoising Method Using a Nonconvex Low-Rank Model with TV Regularization

In real-world scenarios, images may be affected by additional noise during compression and transmission, which interferes with postprocessing such as image segmentation and feature extraction. Image noise can also be induced by environmental variables and imperfections in the imaging equipment. Robust principal component analysis (RPCA), one of the traditional approaches for denoising images, suffers from a failure to efficiently use the background’s low-rank prior information, which lowers its effectiveness under complex noise backgrounds. In this paper, we propose a robust PCA method based on a nonconvex low-rank approximation and total variational regularization (TV) to model the image denoising problem in order to improve the denoising performance. Firstly, we use a nonconvex γ-norm to address the issue that the traditional nuclear norm penalizes large singular values excessively. The rank approximation is more accurate than the nuclear norm thanks to the elimination of matrix elements with substantial approximation errors to reduce the sparsity error. The method’s robustness is improved by utilizing the low sensitivity of the γ-norm to outliers. Secondly, we use the l1-norm to increase the sparsity of the foreground noise. The TV norm is used to improve the smoothness of the graph structure in accordance with the sparsity of the image in the gradient domain. The denoising effectiveness of the model is increased by employing the alternating direction multiplier strategy to locate the global optimal solution. It is important to note that our method does not require any labeled images, and its unsupervised denoising principle enables the generalization of the method to different scenarios for application. Our method can perform denoising experiments on images with different types of noise. Extensive experiments show that our method can fully preserve the edge structure information of the image, preserve important features of the image, and maintain excellent visual effects in terms of brightness smoothing.

Level Set Segmentation of Images using Block Matching Local SVD Operator based Sparsity and TV Regularization

Level Set Segmentation of Images using Block Matching Local SVD Operator based Sparsity and TV Regularization

Improved TV Image Denoising over Inverse Gradient

Noise in an image can affect one’s extraction of image information, therefore, image denoising is an important image pre-processing process. Many of the existing models have a large number of estimated parameters, which increases the time complexity of the model solution and the achieved denoising effect is less than ideal. As a result, in this paper, an improved image-denoising algorithm is proposed based on the TV model, which effectively solves the above problems. The L1 regularization term can make the solution generated by the model sparser, thus facilitating the recovery of high-quality images. Reducing the number of estimated parameters, while using the inverse gradient to estimate the regularization parameters, enables the parameters to achieve global adaption and improves the denoising effect of the model in combination with the TV regularization term. The split Bregman iteration method is used to decouple the model into several related subproblems, and the solutions of the coordinated subproblems are derived as optimal solutions. It is also shown that the solution of the model converges to a Karush–Kuhn–Tucker point. Experimental results show that the algorithm in this paper is more effective in both preserving image texture structure and suppressing image noise.

Total Variation‐Based Reconstruction and Phase Retrieval for Diffraction Tomography with an Arbitrarily Moving Object

AbstractWe consider the imaging problem of the reconstruction of a three‐dimensional object via optical diffraction tomography under the assumptions of the Born approximation. Our focus lies in the situation that a rigid object performs an irregular, time‐dependent rotation under acoustical or optical forces. In this study, we compare reconstruction algorithm in case i) that two‐dimensional images of the complex‐valued wave are known, or ii) that only the intensity (absolute value) of these images can be measured, which is the case in many practical setups. The latter phase‐retrieval problem can be solved by an all‐at‐once approach based utilizing a hybrid input‐output scheme with TV regularization.

Electromechanical Impedance Temperature Compensation and Bolt Loosening Monitoring Based on Modified Unet and Multitask Learning

Bolts are frequently subjected to loosening due to time varying external loads during service. The electromechanical impedance (EMI) technique based on piezoelectric ceramic wafers (PZT) is sensitive to the initial bolt preload looseness. However, the change in environmental temperature has a great effect on EMI monitoring. Deep convolutional neural network (CNN) is a promising technique for EMI monitoring. Nevertheless, it is difficult to train a deep CNN with limited training data to accurately identify damages under a wide range of temperature variations. To this end, this study proposes a multitask CNN for identifying bolts loosening. The network consists of a temperature compensation subnetwork to compensate for the temperature effect, and a lightweight damage identification subnetwork to identify bolt loosening states. The temperature compensation subnetwork is a modified Unet, and both the impedance and temperature are used as its input. The damage identification subnetwork is connected in series behind the temperature compensation subnetwork. A multiloss function is proposed in which a TV regularizer is used. Experimental results show that the validation accuracy of the multitask network is 97.71% when the network is trained by only about 30 samples from each loosening state. Moreover, the generalization abilities of the proposed multitask model to unexpected temperatures and bolt torques are investigated. The model is interpreted by the integrated gradients method, and is also compared with single-task damage identification CNNs. It is proved that the multitask network trained by limited samples can achieve accurate damage identification in temperature varying environments.

Acoustic Impedance Inversion from Seismic Imaging Profiles Using Self Attention U-Net

Seismic impedance inversion is a vital way of geological interpretation and reservoir investigation from a geophysical perspective. However, it is inevitably an ill-posed problem due to the noise or the band-limited characteristic of seismic data. Artificial neural network have been used to solve nonlinear inverse problems in recent years. This research obtained an acoustic impedance profile by feeding seismic profile and background impedance into a well-trained self-attention U-Net. The U-Net got convergence by appropriate iteration, and the output predicted the impedance profiles in the test. To value the quality of predicted profiles from different perspectives, e.g., correlation, regression, and similarity, we used four kinds of indexes. At the same time, our results were predicted by conventional methods (e.g., deconvolution with recursive inversion, and TV regularization) and a 1D neural network was calculated in contrast. Self-attention U-Net showed to be robust to noise and does not require prior knowledge. Furthermore, spatial continuity is also better than deconvolution, regularization, and 1D deep learning methods in contrast. The U-Net in this paper is a type of full convolutional neural network, so there are no limits to the shape of the input. Based on this, a large impedance profile can be predicted by U-Net, which is trained by a patchy training dataset. In addition, this paper applied the proposed method to the field data obtained by the Ceduna survey without any label. The predictions prove that this well-trained network could be generalized from synthetic data to field data.

Regularized Tensor Representative Coefficient Model for Hyperspectral Target Detection

Target detection based on hyperspectral image (HSI) representations has drawn wide attention given its wide variety of features. The matrix-based approach inevitably loses spatial information and fails to explore the intrinsic multimodal structure of an HSI cube. In this paper, we propose a regularized tensor-based model without altering the data structure. We assume that an observed third-order HSI tensor is decomposed into the sum of a Total Variation regularized Low-rank background tensor and a Sparse (TVLrS) target tensor. The two tensors are represented as the mode-3 product of a third-order tensor, called the Tensor Representation Coefficient (TRC), and a spectra dictionary matrix. Then, the model is coined as TVLrS-TRC. The background TRC has a low-rank property, contributing to the low-rankness characterization in our model. Moreover, as the size of the background TRC term is smaller than the background tensor, characterizing its local smoothness via TV regularization reduces the computational cost compared to that of the background tensor. Extensive experiments on two real hyperspectral datasets demonstrate the advantage of the proposed method compared with the state-of-the-art.

基于加权L1/2范数和TV正则化的图像去噪方法

基于加权L1/2范数和TV正则化的图像去噪方法

Comparisons between Isotropic and Anisotropic TV Regularizations in Inverse Acoustic Scattering

This article compares the isotropic and anisotropic TV regularizations used in inverse acoustic scattering. It is observed that compared with the traditional Tikhonov regularization, isotropic and anisotropic TV regularizations perform better in the sense of edge preserving. While anisotropic TV regularization will cause distortions along axes. To minimize the energy function with isotropic and anisotropic regularization terms, we use split Bregman scheme. We do several 2D numerical experiments to validate the above arguments.

Elaborated-Structure Awareness SAR Imagery Using Hessian-Enhanced TV Regularization

Elaborated-Structure Awareness SAR Imagery Using Hessian-Enhanced TV Regularization

WARPd: A Linearly Convergent First-Order Primal-Dual Algorithm for Inverse Problems with Approximate Sharpness Conditions

Sharpness conditions directly control the recovery performance of restart schemes for first-order optimization methods without the need for restrictive assumptions such as strong convexity. However, they are challenging to apply in the presence of noise or approximate model classes (e.g., approximate sparsity). We provide a first-order method: weighted, accelerated, and restarted primal-dual (WARPd), based on primal-dual iterations and a novel restart-reweight scheme. Under a generic approximate sharpness condition, WARPd achieves stable linear convergence to the desired vector. Many problems of interest fit into this framework. For example, we analyze sparse recovery in compressed sensing, low-rank matrix recovery, matrix completion, TV regularization, minimization of $\|Bx\|_{l^1}$ under constraints ($l^1$-analysis problems for general $B$), and mixed regularization problems. We show how several quantities controlling recovery performance also provide explicit approximate sharpness constants. Numerical experiments show that WARPd compares favorably with specialized state-of-the-art methods and is ideally suited for solving large-scale problems. We also present a noise-blind variant based on a square-root LASSO decoder. Finally, we show how to unroll WARPd as neural networks. This approximation theory result provides lower bounds for stable and accurate neural networks for inverse problems and sheds light on architecture choices. Code and a gallery of examples are available online as a MATLAB package.

Combining low-rank constraint for similar superpixels and total variation sparse unmixing for hyperspectral image

ABSTRACT Mixed pixels are the main reason for the low accuracy of traditional remote sensing applications. Hyperspectral image unmixing can explore the sub-pixel information of mixed pixels, which is an effective measure to solve the problems of low precision. Current unmixing methods only consider the correlation of local or global pixels and do not fully utilize images information. Based on the above problems, we proposed a novel sparse unmixing model named combining low-rank constrain for similar superpixels and total variation sparse unmixing (CLRSS-TV) in this current research paper. The adjacent similar pixels are clustered first into superpixels, then the superpixels with the same semantic information are merged into a collection. The spatial information of the image is fully considered by twice clustering. Weighted low-rank constraint is imposed on each collection to consider the spectral correlation of pixels. Moreover, the TV regularization and l 2,1 norm are utilized to improve the smoothness and row sparsity, respectively. Experimental results reveal that the proposed method is not only competitive with the four state-of-the-art algorithms in simulated and actual datasets, but also has good anti-noise ability and universality. However, this paper still fails to overcome the problem of parameters selection in sparse unmixing, so the future work study intends to adopt the intelligent optimization algorithm to obtain the optimal parameters.

Analysis of the Frank–Wolfe method for convex composite optimization involving a logarithmically-homogeneous barrier

We present and analyze a new generalized Frank–Wolfe method for the composite optimization problem (P): {min }_{xin {mathbb {R}}^n} ; f(mathsf {A} x) + h(x), where f is a theta -logarithmically-homogeneous self-concordant barrier, mathsf {A} is a linear operator and the function h has a bounded domain but is possibly non-smooth. We show that our generalized Frank–Wolfe method requires O((delta _0 + theta + R_h)ln (delta _0) + (theta + R_h)^2/varepsilon ) iterations to produce an varepsilon -approximate solution, where delta _0 denotes the initial optimality gap and R_h is the variation of h on its domain. This result establishes certain intrinsic connections between theta -logarithmically homogeneous barriers and the Frank–Wolfe method. When specialized to the D-optimal design problem, we essentially recover the complexity obtained by Khachiyan (Math Oper Res 21 (2): 307–320, 1996) using the Frank–Wolfe method with exact line-search. We also study the (Fenchel) dual problem of (P), and we show that our new method is equivalent to an adaptive-step-size mirror descent method applied to the dual problem. This enables us to provide iteration complexity bounds for the mirror descent method despite the fact that the dual objective function is non-Lipschitz and has unbounded domain. In addition, we present computational experiments that point to the potential usefulness of our generalized Frank–Wolfe method on Poisson image de-blurring problems with TV regularization, and on simulated PET problem instances.

Correntropy-Based Autoencoder-Like NMF With Total Variation for Hyperspectral Unmixing

In order to unmix the hyperspectral imagery (HSI) with better performance, this letter proposes a correntropy-based autoencoder-like nonnegative matrix factorization (NMF) (CANMF) with total variation (CANMF-TV) method. NMF is extensively applied to unmix the mixed pixels. However, it only reconstructs the original data from the abundances in endmember space. To directly project the original data space into the endmember space, and then achieve the abundance matrix, we first exploit an autoencoder-like NMF for hyperspectral unmixing, which integrates both <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">decoder</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">encoder</i> . Considering that HSI is typically degraded by noise, the correntropy-induced metric (CIM) is introduced to construct a CANMF model. In addition, TV regularizer is imposed into the CANMF model so as to preserve the spatial-contextual information by promoting the piecewise smoothness of abundances. Finally, a series of experiments are conducted on both synthetic and real data sets, demonstrating the effectiveness of the proposed CANMF-TV method over comparison.