Total Variation Minimization Research Articles

A Cubical Persistent Homology-Based Technique for Image Denoising with Topological Feature Preservation

Image denoising is a fundamental challenge in image processing, where the objective is to remove noise while preserving critical image features. Traditional denoising methods, such as Wavelet, Total Variation (TV) minimization, and Non-Local Means (NLM), often struggle to maintain the topological integrity of image features, leading to the loss of essential structures. This study proposes a Cubical Persistent Homology-Based Technique (CPHBT) that leverages persistence barcodes to identify significant topological features and reduce noise. The method selects filtration levels that preserve important features like loops and connected components. Applied to digit images, our method demonstrates superior performance, achieving a Peak Signal-to-Noise Ratio (PSNR) of 46.88 and a Structural Similarity Index Measure (SSIM) of 0.99, outperforming TV (PSNR: 21.52, SSIM: 0.9812) and NLM (PSNR: 22.09, SSIM: 0.9822). These results confirm that cubical persistent homology offers an effective solution for image denoising by balancing noise reduction and preserving critical topological features, thus enhancing overall image quality.

Multiobjective optimization guided by image quality index for limited-angle CT image reconstruction.

Due to the incomplete projection data collected by limited-angle computed tomography (CT), severe artifacts are present in the reconstructed image. Classical regularization methods such as total variation (TV) minimization, ℓ0 minimization, are unable to suppress artifacts at the edges perfectly. Most existing regularization methods are single-objective optimization approaches, stemming from scalarization methods for multiobjective optimization problems (MOP). To further suppress the artifacts and effectively preserve the edge structures of the reconstructed image. This study presents a multiobjective optimization model incorporates both data fidelity term and ℓ0-norm of the image gradient as objective functions. It employs an iterative approach different from traditional scalarization methods, using the maximization of structural similarity (SSIM) values to guide optimization rather than minimizing the objective function.The iterative method involves two steps, firstly, simultaneous algebraic reconstruction technique (SART) optimizes the data fidelity term using SSIM and the Simulated Annealing (SA) algorithm for guidance. The degradation solution is accepted in the form of probability, and guided image filtering (GIF) is introduced to further preserve the image edge when the degradation solution is rejected. Secondly, the result from the first step is integrated into the second objective function as a constraint, we use ℓ0 minimization to optimize ℓ0-norm of the image gradient, and the SSIM, SA algorithm and GIF are introduced to guide optimization process by improving SSIM value like the first step. With visual inspection, the peak signal-to-noise ratio (PSNR), root mean square error (RMSE), and SSIM values indicate that our approach outperforms other traditional methods. The experiments demonstrate the effectiveness of our method and its superiority over other classical methods in artifact suppression and edge detail restoration.

Guaranteed matrix recovery using weighted nuclear norm plus weighted total variation minimization

This work presents a general framework regarding the recovery of matrices equipped with hybrid low-rank and local-smooth properties from just a few measurements consisting of linear combinations of the matrix entries. Concretely, we consider the problem of robust low-rank matrix recovery using Weighted Nuclear Norm plus Weight Total Variation (WNNWTV) minimization. First of all, based on a new restricted isometry property, we prove that the WNNWTV method possesses an error bound consisting of a low-rank approximation term, a total variation approximation term, and an observation error term. It should be noted that although there are many models considering both properties, there are very few recoverable error theories about such models. Specifically, the theoretical error bound provides an automatic mechanism to reducing regularization parameters with no need for cross-validation while keeping almost the same selection result with commonly used cross-validation technique. Subsequently, the proposed method is reformulated into a regularized unconstrained problem, and we study its optimization aspects in detail based on the Alternating Direction Method of Multipliers (ADMM). Extensive experiments on synthetic data and two applications, i.e. hyperspectral image recovery and dynamic magnetic resonance imaging recovery verified our theories and proposed algorithms.

Mitigating adversarial threats in deep CT image diagnosis models via a dual-stage inference-time defense

Artificial intelligence (AI), particularly deep learning (DL) and machine learning (ML) have revolutionized disease diagnosis using complex medical images such as X-rays and CT scans, significantly improving accuracy in identifying various medical conditions. However, concerns arise regarding the trustworthiness of these models due to their vulnerability to adversarial attacks, potentially leading to inaccurate predictions. In response, we propose an innovative pipeline defensive approach that integrates denoising techniques like Total Variation Minimization (TVM) and Non-local (NL) means, followed by a fuzzy logic-based image transformer called Fuzzy Image Transformations (FIT). Deployed during the inferencing phase as a preprocessing module, this strategy aims to fortify DL medical diagnosis models, addressing adversarial vulnerabilities and ensuring their reliability in critical healthcare applications. Our focus is narrowed down to COVID-19 diagnosis, where we initially developed a model for accurately classifying lung CT images, covering both normal and COVID-19 pneumonia cases. Our experimental procedures, employing the Stratified K-Fold cross-validation method with K=5, systematically evaluate the model’s susceptibility to benchmark adversarial attacks such as the Fast Gradient Sign Method (FGSM), revealing a significant drop in aggregate average metrics across all folds under adversarial conditions. Specifically, precision (P), recall (R), accuracy (A), and F1-score (F) metrics decrease from 95.86 %, 96.16 %, 95.85 %, and 96.00–12.54 %, 13.13 %, 12.63 %, and 12.83 %, respectively. Notably, our proposed pipeline defense approach demonstrates substantial efficacy in enhancing diagnostic models, with average P, R, A, and F metrics improving impressively to 92.65 %, 92.39 %, 92.65 %, and 92.51 %, respectively, when applied to the same adversarial images. These promising results underscore the potential of our proposed pipeline defense techniques to enhance the resilience and reliability of AI-based diagnostic systems.

Removing Adversarial Noise in X-ray Images via Total Variation Minimizationand Patch-Based Regularization for Robust DeepLearning-based Diagnosis.

Deep learning has significantly advanced the field of radiology-based disease diagnosis, offering enhanced accuracy and efficiency in detecting various medical conditions through the analysis of complex medical images such as X-rays. This technology's ability to discern subtle patterns and anomalies has proven invaluable for swift and accurate disease identification. The relevance of deep learning in radiology has been particularly highlighted during the COVID-19 pandemic, where rapid and accurate diagnosis is crucial for effective treatment and containment. However, recent research has uncovered vulnerabilities in deep learning models when exposed to adversarial attacks, leading to incorrect predictions. In response to this critical challenge, we introduce a novel approach that leverages total variation minimization to combat adversarial noise within X-ray images effectively. Our focus narrows to COVID-19 diagnosis as a case study, where we initially construct a classification model through transfer learning designed to accurately classify lung X-ray images encompassing no pneumonia, COVID-19 pneumonia, and non-COVID pneumonia cases. Subsequently, we extensively evaluated the model's susceptibility to targeted and un-targetedadversarial attacksby employing the fast gradient sign gradient (FGSM) method. Our findings reveal a substantial reduction in the model's performance, with the average accuracy plummeting from 95.56 to 19.83% under adversarial conditions. However, the experimental results demonstrate the exceptional efficacy of the proposed denoising approach in enhancing the performance of diagnosis models when applied to adversarial examples. Post-denoising, the model exhibits a remarkable accuracy improvement, surging from 19.83 to 88.23% on adversarial images. These promising outcomes underscore the potential of denoising techniques to fortify the resilience and reliability of AI-based COVID-19 diagnostic systems, laying the foundation for their successful deployment in clinical settings.

Cone-beam computed laminography with anisotropic adaptive weighted total variation minimization based on the framework of the Chambolle-Pock algorithm

Cone-beam computed laminography (CL) is still a very challenging problem for the inspection of thin-plate objects. Since CL projections are incomplete, the reconstructed images always suffer from severe aliasing and blurring in the z direction. To mitigate this problem, we propose an anisotropic adaptive weighted total variation (AAwTV) reconstruction model, which takes the edge properties between adjacent voxels into account and introduces different weights in different directions. In addition, we solved the proposed AAwTV using the Chambolle-Pock (CP) framework, since it has good computational efficiency and stable convergence, and is often easy to get a satisfactory reconstruction result. Experiments on simulated PCB phantom and simulated workpiece phantom show that the proposed algorithm can preserve the detailed features of the object well, and can effectively suppress inter-slice aliasing and blurring.

Constrained tensorial total variation problem based on an alternating conditional gradient algorithm

The present paper aims to regularize ill-posed high-dimensional problems using a constrained total variation minimization approach. The analysis of the continuous model via the dual formulation converts this minimization problem into a constrained minimax problem. Three main issues stand before the resolution of this problem: The minimization over a constraint, the high dimensional representation of the model, and the non-linearity and non-differentiability of the cost function. On the one hand, multidimensional data analysis will be performed using the tensor representation. On the other hand, a new alternating approach based on a proposed pseudo-Lagrangian operator will be developed to deal with the non-linearity and the non-differentiability of this constrained optimization problem. The proposed technique aims to decompose the main tensorial problem into feasible subproblems that can be solved using the conditional gradient method. The convergence of the proposed algorithm is proved and some numerical results are given to illustrate the effectiveness of the presented approach in different fields of image and video processing.

Sparse-View Neutron Computed Tomography 3-D Reconstruction via the Fast Gradient Projection Algorithm

Neutron tomography is an efficient nondestructive testing technique. As a complement to X-ray computed tomography, it has been widely used in various fields. Due to the difficulty of obtaining complete neutron projection data in a high-radiation environment and the high noise characteristics of neutron images, it is difficult to reconstruct a high-quality image using the conventional filtered-back projection (FBP) algorithm. Therefore, research on sparse-view reconstruction algorithms in neutron tomography is needed. To improve the quality of neutron three-dimensional reconstructed images, this paper proposes an algorithm that combines the Simultaneous Algebraic Reconstruction Technique (SART) with Fast Gradient Projection (FGP), where the FGP is an algorithm for image denoising and deblurring based on the discrete total variation (TV) minimization model. The algorithm proposed in this paper is compared with other algorithms (FBP, SART, and SART-TV) by simulated experimental data and real neutron experimental data. The experimental results show that the novel algorithm outperforms the other three algorithms in terms of denoising and retaining detailed structural information.

Sparse-view cone-beam computed tomography iterative reconstruction based on new multi-gradient direction total variation.

The accurate reconstruction of cone-beam computed tomography (CBCT) from sparse projections is one of the most important areas for study. The compressed sensing theory has been widely employed in the sparse reconstruction of CBCT. However, the total variation (TV) approach solely uses information from the i-coordinate, j-coordinate, and k-coordinate gradients to reconstruct the CBCT image. It is well recognized that the CBCT image can be reconstructed more accurately with more gradient information from different directions. Thus, this study introduces a novel approach, named the new multi-gradient direction total variation minimization method. The method uses gradient information from the ij-coordinate, ik-coordinate, and jk-coordinate directions to reconstruct CBCT images, which incorporates nine different types of gradient information from nine directions. This study assessed the efficacy of the proposed methodology using under-sampled projections from four different experiments, including two digital phantoms, one patient's head dataset, and one physical phantom dataset. The results indicated that the proposed method achieved the lowest RMSE index and the highest SSIM index. Meanwhile, we compared the voxel intensity curves of the reconstructed images to assess the edge structure preservation. Among the various methods compared, the curves generated by the proposed method exhibited the highest level of consistency with the gold standard image curves. In summary, the proposed method showed significant potential in enhancing the quality and accuracy of CBCT image reconstruction.

Morphing from the TV-Norm to the l 0 -Norm.

For piecewise constant objects, the images can be reconstructed with under-sampled measurements. The gradient image of a piecewise image is sparse. If a sparse solution is a desired solution, an -norm minimization method is effective to solve an under-determined system. However, the -norm is not differentiable, and it is not straightforward to minimize an -norm. This paper suggests a function that is like the -norm function, and we refer to this function as meta -norm. The subdifferential of the meta -norm has a simple explicit expression. Thus, it is straightforward to derive a gradient descent algorithm to enforce the sparseness in the solution. In fact, the proposed meta norm is a transition that varies between the TV-norm and the -norm. As an application, this paper uses the proposed meta -norm for few-view tomography. Computer simulation results indicate that the proposed meta -norm effectively guides the image reconstruction algorithm to a piecewise constant solution. It is not clear whether the TV-norm or the -norm is more effective in producing a sparse solution. Index Terms-Inverse problem, optimization, total-variation minimization, -norm minimization, few-view tomography, iterative algorithm, image reconstruction.

Sparse-view image reconstruction with total-variation minimization applied to sparsely sampled projection data from SiPM-based photon-counting CT

We constructed a sparse-view computed tomography (CT) system that combines a compressed sensing (CS)-based image-reconstruction algorithm and SiPM-based photon-counting (PC) CT. CS-based image-reconstruction algorithms have been extensively studied for X-ray CT image reconstruction using fewer projections because they are expected to reduce CT imaging time and radiation exposure while maintaining CT image quality. In most previous studies, CS-based image-reconstruction algorithms have been applied to data obtained through numerical simulations or conventional dual-energy CT. However, studies on PC-CT have been scarce. Therefore, we applied a CS-based image-reconstruction algorithm to the projection data obtained using our previously established SiPM-based PC-CT system and evaluated its image quality. We prepared static phantoms equivalent to iodine-containing contrast agents and a mouse model injected with iodine-containing contrast agents as subjects. Thereafter, CT scanning was performed. The obtained projection data were downsampled to simulate a sparse-view situation, and a CS-based image-reconstruction algorithm with total-variation minimization was applied. Consequently, sparse-view CT images were successfully reconstructed, and the image quality was maintained even after downsampling the projection data (downsampling ratios of 1/10 and 1/2 for the rod phantom and mouse model, respectively). Thus, the imaging time and exposure dose could be remarkably reduced (by a factor of 10 or 2), indicating that the CS-based image-reconstruction algorithm is effective for PC-CT.

Smooth fusion of multi-spectral images via total variation minimization for traffic scene semantic segmentation

Achieving precise semantic segmentation for traffic scenes relies on adopting multi-spectral image fusion techniques to attain high-quality images. Many existing fusion solutions often aim to enhance the similarity between the input and fusion results at the pixel intensity and texture details stage. However, this can result in smoothness issues that limit semantic segmentation performance. To address these issues, we present a smooth representation learning optimization mechanism (SFLM) that conducts image fusion on two dimensions: inter- and intra-image levels. The former overcomes over- or under-smoothing problems via the mutual information maximization between the fusion result and image samples (i.e., negative and positive). The latter balances under and over-smoothing for fusion results by minimizing the total variation in pixel space and maximizing the total variation in gradient space based on contrast learning. In this way, the proposed method effectively overcomes the fusion quality issues, providing better feature representations for semantic segmentation in autonomous vehicles. Experimental results on four public datasets validate our method’s effectiveness, robustness, and overall superiority.

Reconstruction of Sparse-View X-ray Computed Tomography Based on Adaptive Total Variation Minimization.

Sparse-view reconstruction has garnered significant interest in X-ray computed tomography (CT) imaging owing to its ability to lower radiation doses and enhance detection efficiency. Among current methods for sparse-view CT reconstruction, an algorithm utilizing iterative reconstruction based on full variational regularization demonstrates good performance. The optimized direction and number of computations for the gradient operator of the regularization term play a crucial role in determining not only the reconstructed image quality but also the convergence speed of the iteration process. The conventional TV approach solely accounts for the vertical and horizontal directions of the two-dimensional plane in the gradient direction. When projection data decrease, the edges of the reconstructed image become blurred. Exploring too many gradient directions for TV terms often comes at the expense of more computational costs. To enhance the balance of computational cost and reconstruction quality, this study suggests a novel TV computation model that is founded on a four-direction gradient operator. In addition, selecting appropriate iteration parameters significantly impacts the quality of the reconstructed image. We propose a nonparametric control method utilizing the improved TV approach as a solution to the tedious manual parameter optimization issue. The relaxation parameters of projection onto convex sets (POCS) are determined according to the scanning number and numerical proportion of the projection data; according to the image error before and after iterations, the gradient descent step of the TV item is adaptively adjusted. Compared with several representative iterative reconstruction algorithms, the experimental results show that the algorithm can effectively preserve edges and suppress noise in sparse-view CT reconstruction.

Ultrafast multiple paramagnetic species EPR imaging using a total variation based model

An EPR spectrum or an EPR sinogram for imaging contains information about all the paramagnetic species that are in the analyzed sample. When only one species is present, an image of its spatial repartition can be reconstructed from the sinogram by using the well-known Filtered Back-Projection (FBP). However, in the case of several species, the FBP does not allow the reconstruction of the images of each species from a standard acquisition. One has to use for this spectral-spatial imaging whose acquisition can be very long. A new approach, based on Total Variation minimization, is proposed in order to efficiently extract the spatial repartitions of all the species present in a sample from standard imaging data and therefore drastically reduce the acquisition time. Experiments have been carried out on Tetrathiatriarylmethyl, nitroxide and DPPH.

An edge-preserving total nuclear variation minimization algorithm in EPR image reconstruction

ObjectiveCurrently, slow scanning speed has become a technical bottleneck limiting the development of electron paramagnetic resonance imaging (EPRI), a novel oxygen imaging modality. Image reconstruction from sparse-view projections is an important imaging configuration to achieve fast scanning in EPRI. However, EPR images reconstructed from sparse-view projections using traditional analytic algorithms suffer from sparse artifacts, causing severe image degradation. In this work, we aim to use an optimization-based approach to achieve high-precision sparse reconstruction for EPRI. Methods. Inspired by the success of total nuclear variation (TVN) for denoising multi-channel spectral CT images and adaptive-weighted total variation (awTV) for improving the edge-preserving performance of TV, we proposed an edge-preserving total nuclear variation (EPTVN) minimization algorithm for sparse reconstruction in EPRI. The EPTVN combines the advantage of TVN in modelling structural similarity and low-rank prior between multi-channel images for denoising with the advantage of awTV for edge-preserving, which can better protect the edge structure of images while suppressing sparse artifacts. Results. We designed two numerical phantoms with different properties, a piecewise-constant phantom and a gradient phantom, and a physical phantom, and then performed inverse crime, simulation study and real-data study. Experimental results show that the EPTVN minimization algorithm can reconstruct more accurate EPR images from sparse-view projections for both the numerical and physical phantoms. Significance. The proposed method can effectively suppress sparse artifacts and protect edge structures of reconstructed images, thus achieving high-precision sparse-view reconstruction for EPRI. The insights of this work can also be applied to multi-channel image denoising tasks.

Three-dimensional reconstruction of Y-IrNi rhombic dodecahedron nanoframe by STEM/EDS tomography

The structural analysis of nanocrystals via transmission electron microscopy (TEM) is a valuable technique for the material science field. Recently, two-dimensional images by scanning TEM (STEM) and energy-dispersive X-ray spectroscopy (EDS) have successfully extended to three-dimensional (3D) imaging by tomography. However, despite improving TEM instruments and measurement techniques, detector shadowing, the missing-wedge problem, X-ray absorption effects, etc., significant challenges still remain; therefore, the various required corrections should be considered and applied when performing quantitative tomography. Nonetheless, this 3D reconstruction technique can facilitate active site analysis and the development of nanocatalyst systems, such as water electrolysis and fuel cell. Herein, we present a 3D reconstruction technique to obtain tomograms of IrNi rhombic dodecahedral nanoframes (IrNi-RFs) from STEM and EDS images by applying simultaneous iterative reconstruction technique and total variation minimization algorithms. From characterizing the morphology and spatial chemical composition of the Ir and Ni atoms in the nanoframes, we were able to infer the origin of the physical and catalytic durability of IrNi-RFs. Also, by calculating the surface area and volume of the 3D reconstructed model, we were able to quantify the Ir-to-Ni composition ratio and compare it to the EDS measurement result.

Photoacoustic tomography with a model-based approach involving realistic detector properties

A computational and experimental study is conducted to examine how directivity associated with a finite aperture sensor affects photoacoustic tomography (PAT) image reconstruction. Acoustic signals for the simulation work were computed using a discrete particle approach from three numerical phantoms including a vasculature. The theoretical framework and a Monte Carlo approach for construction of a tissue configuration are discussed in detail. While simulating forward data, the directivity of the sensor was taken into account. The image reconstruction was accomplished using system matrix based methods like l2 norm Tikhonov regularization, l1 norm regularization and total variation (TV) minimization. Accordingly, two different system matrices were constructed- (i) assuming transducer as a point detector (PD) and (ii) retaining properties of a finite detector with directivity (FDWD). Image reconstruction was also performed utilizing experimentally measured PA signals. Both the computational and experimental results demonstrate that blur-free PAT imaging can be achieved with the FDWD method. Additionally, TV minimization provides marginally better image reconstruction compared to the other schemes.

A Uniform Preconditioner for a Newton Algorithm for Total Variation Minimization and Minimum-Surface Problems

A Uniform Preconditioner for a Newton Algorithm for Total Variation Minimization and Minimum-Surface Problems

A high-resolution cone beam computed tomography (HRCBCT) reconstruction framework for CBCT-guided online adaptive therapy.

Kilo-voltage cone-beam computed tomography (CBCT) is a prevalent modality used for adaptive radiotherapy (ART) due to its compatibility with linear accelerators and ability to provide online imaging. However, the widely-used Feldkamp-Davis-Kress (FDK) reconstruction algorithm has several limitations, including potential streak aliasing artifacts and elevated noise levels. Iterative reconstruction (IR) techniques, such as total variation (TV) minimization, dictionary-based methods, and prior information-based methods, have emerged as viable solutions to address these limitations and improve the quality and applicability of CBCT in ART. One of the primary challenges in IR-based techniques is finding the right balance between minimizing image noise and preserving image resolution. To overcome this challenge, we have developed a new reconstruction technique called high-resolution CBCT (HRCBCT) that specifically focuses on improving image resolution while reducing noise levels. The HRCBCT reconstruction technique builds upon the conventional IR approach, incorporating three components: the data fidelity term, the resolution preservation term, and the regularization term. The data fidelity term ensures alignment between reconstructed values and measured projection data, while the resolution preservation term exploits the high resolution of the initial Feldkamp-Davis-Kress (FDK) algorithm. The regularization term mitigates noise during the IR process. To enhance convergence and resolution at each iterative stage, we applied Iterative Filtered Backprojection (IFBP) to the data fidelity minimization process. We evaluated the performance of the proposed HRCBCT algorithm using data from two physical phantoms and one head and neck patient. The HRCBCT algorithm outperformed all four different algorithms; FDK, Iterative Filtered Back Projection (IFBP), Compressed Sensing based Iterative Reconstruction (CSIR), and Prior Image Constrained Compressed Sensing (PICCS) methods in terms of resolution and noise reduction for all data sets. Line profiles across three line pairs of resolution revealed that the HRCBCT algorithm delivered the highest distinguishable line pairs compared to the other algorithms. Similarly, the Modulation Transfer Function (MTF) measurements, obtained from the tungsten wire insert on the CatPhan 600 physical phantom, showed a significant improvement with HRCBCT over traditional algorithms. The proposed HRCBCT algorithm offers a promising solution for enhancing CBCT image quality in adaptive radiotherapy settings. By addressing the challenges inherent in traditional IR methods, the algorithm delivers high-definition CBCT images with improved resolution and reduced noise throughout each iterative step. Implementing the HR CBCT algorithm could significantly impact the accuracy of treatment planning during online adaptive therapy.

Autonomous Electron Tomography Reconstruction with Machine Learning.

Modern electron tomography has progressed to higher resolution at lower doses by leveraging compressed sensing (CS) methods that minimize total variation (TV). However, these sparsity-emphasized reconstruction algorithms introduce tunable parameters that greatly influence the reconstruction quality. Here, Pareto front analysis shows that high-quality tomograms are reproducibly achieved when TV minimization is heavily weighted. However, in excess, CS tomography creates overly smoothed three-dimensional (3D) reconstructions. Adding momentum to the gradient descent during reconstruction reduces the risk of over-smoothing and better ensures that CS is well behaved. For simulated data, the tedious process of tomography parameter selection is efficiently solved using Bayesian optimization with Gaussian processes. In combination, Bayesian optimization with momentum-based CS greatly reduces the required compute time-an 80% reduction was observed for the 3D reconstruction of SrTiO3 nanocubes. Automated parameter selection is necessary for large-scale tomographic simulations that enable the 3D characterization of a wider range of inorganic and biological materials.