Total Variation Term Research Articles

Smooth Compact Tensor Ring Regression

In learning tasks with high order correlations, the low-rank approximation of the regression coefficient tensor has become increasingly important. Tensor ring can capture more correlation information among tensor networks. However, its optimal rank is generally unknown and needs to be tuned from multiple combinations. To address the issue, we propose a novel tensor regression framework with a group sparsity constraint on latent factors for tensor ring rank estimation. Specifically, the proposed group sparsity term constrained matrix factorization problem is first shown to be equivalent to a better approximation of matrix rank, namely Schatten- <inline-formula><tex-math notation="LaTeX">$1/2$</tex-math></inline-formula> quasi-norm. Extending it into tensor, the tensor ring rank can be inferred during the learning process to balance the prediction error and the model complexity. Besides, a total variation term is introduced to enhance the local consistency of the predicted response, which is useful for reducing the adverse effects of random noise. Experiments on the simulation dataset show that the proposed method can exactly obtain the tensor ring rank, and the effectiveness and robustness of the proposed algorithm is further verified on a real dataset for human motion capture tasks.

Non-Coplanar Many-Isocenter Optimization for Radiotherapy on Robotic Arm Platform

4π non-coplanar radiotherapy significantly improves dosimetry, but its clinical implementation on the widely available C-arm gantry system is limited by the restricted non-colliding beam angles and the complex coordination of gantry and couch motion. Alternatively, a robotic arm platform is more favorable for the non-coplanar and non-isocentric treatment, but the conflict between field size and modulation resolution needs to be managed. This study investigates the dosimetry and delivery efficiency by creating many isocenters to achieve simultaneously large field-of-view (FOV) and high modulation resolution for robotic radiotherapy. The many-isocenter optimization framework includes a least-square dose fidelity objective, a total variation term for regularizing the fluence smoothness, and a group sparsity term for simultaneous beam orientation optimization (BOO) and isocenter selection. A minimal number of candidates isocenters were identified before the optimization for efficient target coverage. The proposed algorithm was evaluated on the intensity-modulated radiotherapy (IMRT) treatment plans of 10 Head and Neck (H&N) cancer patients. Colliding beams excluded, high-resolution small-field non-isocentric 4π IMRT plans with 50cm source-to-isocenter distance (SID-50) were compared with low-resolution large-field isocentric plans with 100cm SID (SID-100). The table reports the planning target volume (PTV) and organs at risk (OARs) statistics, and the corresponding Wilcoxon signed-rank test, showing significant difference (p<0.05) in PTV homogeneity, PTV Dmax, R50, Integral dose, and OAR Dmean and Dmax. With better target coverage, SID-50 substantially improved dose conformality and reduced OAR dose. R50 and integral dose were reduced by 5.3% and 9.6% respectively. The average reduction of [Dmean, Dmax] of SID-50 from SID-100 were [2.09 Gy, 1.19 Gy] for OAR overall, [3.05 Gy, 0.04 Gy] for parotid gland, [3.62 Gy, 5.19 Gy] for larynx, and [3.27 Gy, 1.10 Gy] for mandible. The estimated delivery time of 20-beam [SID-50, SID-100] plans were [19, 18] minutes and [14, 9] minutes, assuming 5 fractions and 30 fractions, respectively. With clinically acceptable delivery efficiency, the proposed non-coplanar many-isocenter optimization is dosimetrically favorable for treating large targets with high modulation resolution.Tabled 1Abstract 2832; Table 2832Wilcoxon signed rank test of PTV and OAR statistics of SID-100 (isocentric) and SID-50 (non-isocentric) for all 10 patients.Statisticsp-valueSigned rankNumber of samplesSID-100SID-50MeanSTDMeanSTDPTV Homogeneity0.00385290.9010.0450.9060.045PTV Dmin0.749202290.9740.0190.9750.016PTV Dmax<.001380291.1270.0621.1180.061R500.00653102.3810.2362.2540.256Integral Dose0.0025510139.37738.481126.0236.139OAR Dmean0.002551017.8223.72715.7293.273OAR Dmax0.002551034.5156.31733.3256.173 Open table in a new tab

High-quality multi-wavelength lensfree microscopy based on nonlinear optimization

In multi-wavelength lensfree microscopy (MWLM), a laser source is usually replaced with a light-emitting diode for a low-cost and portable configuration. However, the trade-off between noise suppression and resolution enhancement puts a challenging issue for high-quality imaging. Here we propose a noise-robust multi-wavelength phase retrieval to increase the stability of MWLM. To solve the inverse problem of phase retrieval, we derive an optimization procedure to iteratively minimize the data fidelity and impose a total variation term for noise suppression. The experimental results of resolution target, stained human goblet cell, and label-free microglia cell demonstrate that our method could hold a better noise tolerance without the loss of imaging resolution. We believe that our method will become a powerful tool for a resources-limited scene.

Superiorized method for metal artifact reduction.

Metal artifact reduction (MAR) is a challenging problem in computed tomography (CT) imaging. A popular class of MAR methods replace sinogram measurements that are corrupted by metal with artificial data, typically generated from some combination of interpolation along with other heuristics. While these "projection completion" approaches are successful in eliminating severe artifacts, secondary artifacts may be introduced by the artificial data. In this paper, we propose an approach which uses projection completion to generate a prior image, which is then incorporated into an iterative reconstruction algorithm based on the superiorization framework. The rationale is that the image produced by the iterative algorithm can inherit the desirable properties of the prior image, while also reducing secondary artifacts. The prior image is reconstructed using normalized metal artifact reduction (NMAR), a popular projection completion approach. The iterative algorithm is a modified version of the simultaneous algebraic reconstruction technique (SART), which reduces artifacts by incorporating a polyenergetic forward model, least-squares weighting, and superiorization. The penalty function used for superiorization is a weighted average between a total variation (TV) term and a term promoting similarity with the prior image, similar to penalty functions used in prior image constrained compressive sensing (PICCS). Because the prior is largely free of severe metal artifacts, these artifacts are discouraged from arising during iterative reconstruction; additionally, because the iterative approach uses the original projection data, it is able to recover information that is lost during the NMAR process. We perform numerical experiments modeling a simple geometric object, as well as several more realistic scenarios such as metal pins, bilateral hip implants, and dental fillings placed within an anatomical phantom. The proposed iterative algorithm is largely successful at eliminating severe metal artifacts as well as secondary artifacts introduced by the NMAR process, especially lost edges of bone structures in the neighborhood of the metal regions. In one case modeling severe photon starvation, the NMAR algorithm is found to provide better results. The proposed algorithm is effective in applying the superiorization methodology to the problem of MAR, providing better results than both NMAR and a purely total variation-based superiorization approach in nearly all cases.

Variational model for simultaneously image denoising and contrast enhancement.

The performance of contrast enhancement is degraded when input images are noisy. In this paper, we propose and develop a variational model for simultaneously image denoising and contrast enhancement. The idea is to propose a variational approach containing an energy functional to adjust the pixel values of an input image directly so that the resulting histogram can be redistributed to be uniform and the noise of the image can be removed. In the proposed model, a histogram equalization term is considered for image contrast enhancement, a total variational term is incorporate to remove the noise of the input image, and a fidelity term is added to keep the structure and the texture of the input image. The existence of the minimizer and the convergence of the proposed algorithm are studied and analyzed. Experimental results are presented to show the effectiveness of the proposed model compared with existing methods in terms of several measures: average local contrast, discrete entropy, structural similarity index, measure of enhancement, absolute measure of enhancement, and second derivative like measure of enhancement.

Many-isocenter optimization for robotic radiotherapy

Despite significant dosimetric gains, clinical implementation of the 4π non-coplanar radiotherapy on the widely available C-arm gantry system is hindered by limited clearance, and the need to perform complex coordinated gantry and couch motion. A robotic radiotherapy platform would be conducive to such treatment but a new conflict between field size and MLC modulation resolution needs to be managed for versatile applications. This study investigates the dosimetry and delivery efficiency of purposefully creating many isocenters to achieve simultaneously high MLC modulation resolution and large tumor coverage. An integrated optimization framework was proposed for simultaneous beam orientation optimization (BOO), isocenter selection, and fluence map optimization (FMO). The framework includes a least-square dose fidelity objective, a total variation term for regularizing the fluence smoothness, and a group sparsity term for beam selection. A minimal number of isocenters were identified for efficient target coverage. Colliding beams excluded, high-resolution small-field 4π intensity-modulated radiotherapy (IMRT) treatment plans with 50 cm source-to-isocenter distance (SID-50) on 10 Head and Neck (H&N) cancer patients were compared with low-resolution large-field plans with 100 cm SID (SID-100). With the same or better target coverage, the average reduction of [Dmean, Dmax] of 20-beam SID-50 plans from 20-beam SID-100 plans were [2.09 Gy, 1.19 Gy] for organs at risk (OARs) overall, [3.05 Gy, 0.04 Gy] for parotid gland, [3.62 Gy, 5.19 Gy] for larynx, and [3.27 Gy, 1.10 Gy] for mandible. R50 and integral dose were reduced by 5.3% and 9.6%, respectively. Wilcoxon signed-rank test showed significant difference (p < 0.05) in planning target volume (PTV) homogeneity, PTV Dmax, R50, Integral dose, and OAR Dmean and Dmax. The estimated delivery time of 20-beam [SID-50, SID-100] plans were [19, 18] min and [14, 9] min, assuming 5 fractions and 30 fractions, respectively. With clinically acceptable delivery efficiency, many-isocenter optimization is dosimetrically desirable for treating large targets with high modulation resolution on the robotic platform.

Accelerated alternating minimization algorithm for Poisson noisy image recovery

Restoring images corrupted by Poisson noise have attracted much attention in recent years due to its significant applications in image processing. There are various regularization methods of solving this problem and one of the most famous is the total variation (TV) model. In this paper, we present a new method based on accelerated alternating minimization algorithm (AAMA) which involves minimizing the sum of a Kullback–Leibler divergence term and a TV term for restoring Poisson noise degraded images. Our proposed algorithm is applied in solving the aforementioned problem and its convergence analysis is established under very weak conditions. In addition, the numerical examples reported demonstrate the efficiency and versatility of our method compared to existing methods of restoring images with Poisson noise.

Image restoration from noisy incomplete frequency data by alternative iteration scheme

Consider the image restoration from incomplete noisy frequency data with total variation and sparsity regularizing penalty terms. Firstly, we establish an unconstrained optimization model with different smooth approximations on the regularizing terms. Then, to weaken the amount of computations for cost functional with total variation term, the alternating iterative scheme is developed to obtain the exact solution through shrinkage thresholding in inner loop, while the nonlinear Euler equation is appropriately linearized at each iteration in exterior loop, yielding a linear system with diagonal coefficient matrix in frequency domain. Finally the linearized iteration is proven to be convergent in generalized sense for suitable regularizing parameters, and the error between the linearized iterative solution and the one gotten from the exact nonlinear Euler equation is rigorously estimated, revealing the essence of the proposed alternative iteration scheme. Numerical tests for different configurations show the validity of the proposed scheme, compared with some existing algorithms.

A Variational Approach for Fusion of Panchromatic and Multispectral Images Using a New Spatial–Spectral Consistency Term

In this article, we propose a variational approach for fusion of two coregistered high-resolution panchromatic (HRP) and low-resolution multispectral (LRM) images to reach the high-resolution multispectral (HRM) one, i.e., pan-sharpening. In this fusion technique, there is a tradeoff between structural information of an HRP image and spectral information of LRM one. To reconstruct the HRM image, which benefits from the best characteristics of both images, we consider several fidelity terms. The structural fidelity term is used to transfer structural information of an HRP image to HRM one, and a spectral fidelity term is utilized to preserve spectral consistency between HRM and LRM images throughout the fusion process. To reduce the spectral distortion occurred due to the discrepancy between intensity values of HRP and LRM images, a novel spatial–spectral fidelity term is designed to keep the intensity ratio between multispectral and panchromatic pixels in the high-resolution space as the same as the low-resolution space. Moreover, the total variation (TV) regularization term is employed as a prior to promote the sparseness of gradient in HRM bands. These fidelity terms were formulated in a convex optimization problem. However, the structural and TV terms made this optimization problem nondifferentiable. Therefore, we developed an efficient majorization–minimization algorithm for solving the optimization problem. The proposed method applied to three datasets, acquired by WorldView-3, Deimos-2, and QuickBird satellites. To assess the effectiveness of the proposed method, visual analysis, as well as quantitative comparison to various pan-sharpening methods, was carried out. The experimental results suggested that the proposed method outperformed the competitors visually and quantitatively.

An Efficient Variational-Level-Set Model Based on Adaptive Local Fitted Image for Noisy Image Segmentation

In image processing and computer vision, image segmentation plays a fundamental role since it can make images easier to analyze. However, noise is easily introduced into images and brings great challenges to image segmentation. This paper focuses on the segmentation problem of noisy images and proposes an efficient variational level set model based on adaptive local fitted image to handle it. By utilizing normalized local entropy and local means, an adaptive local fitted image is proposed and introduced into the data term to enhance the robustness of the model against noise. Then a penalty term is proposed to reduce the deviation of the adaptive local fitted image from the original image by punishing the difference between them, so as to guarantee the accuracy of segmentation results. Later, the total variational regularization term is introduced into the model, so the level set function can be smoothed and the effect of noise on the active contour can be further reduced. The energy functional of the whole model is convex rigorously, which can reach the minimum and should have good properties in noisy image segmentation. Numerous experiments on synthetic, natural, synthetic aperture radar and oil spill images demonstrate that the proposed model is strongly robust to different types and levels of noise, which indicates its good performance in noisy image segmentation.

A Retinex-based total variation approach for image segmentation and bias correction

Image segmentation methods usually suffer from intensity inhomogeneity problem caused by many factors such as spatial variations in illumination (or bias fields of imaging devices). In order to address this problem, this paper proposes a Retinex-based variational model for image segmentation and bias correction. According to Retinex theory, the input inhomogeneous image can be decoupled into illumination bias and reflectance parts. The main contribution of this paper is to consider piecewise constant of the reflectance, and thereby introduce the total variation term in the proposed model for correcting and segmenting the input image. This is different from the existing model in which the spatial smoothness of the illumination bias is employed only. The existence of the minimizers to the variational model is established. Furthermore, we develop an efficient algorithm to solve the model numerically by using the alternating minimization method. Our experimental results are reported to demonstrate the effectiveness of the proposed method, and its performance is competitive with that of the other testing methods.

An iterative reconstruction algorithm for pole figure inversion using total variation regularization

This paper describes a new discrete method for inverting X-ray pole figures to estimate the orientation distribution function (ODF). The method employs the equal-volume `cubochoric' representation for uniform discretization of orientation space, SO(3). The forward-projection model is combined with an anisotropic total variation term to iteratively determine the ODF. The efficacy of the new method is evaluated with both model and experimental data and compared with existing discrete and series expansion methods.

Image-domain multimaterial decomposition for dual-energy computed tomography with nonconvex sparsity regularization.

Dual-energy computed tomography (CT) has the potential to decompose tissues into different materials. However, the classic direct inversion (DI) method for multimaterial decomposition (MMD) cannot accurately separate more than two basis materials due to the ill-posed problem and amplified image noise. We propose an integrated MMD method that addresses the piecewise smoothness and intrinsic sparsity property of the decomposition image. The proposed MMD was formulated as an optimization problem including a quadratic data fidelity term, an isotropic total variation term that encourages image smoothness, and a nonconvex penalty function that promotes decomposition image sparseness. The mass and volume conservation rule was formulated as the probability simplex constraint. An accelerated primal-dual splitting approach with line search was applied to solve the optimization problem. The proposed method with different penalty functions was compared against DI on a digital phantom, a Catphan® 600 phantom, a quantitative imaging phantom, and a pelvis patient. The proposed framework distinctly separated the CT image up to 12 basis materials plus air with high decomposition accuracy. The cross talks between two different materials are substantially reduced, as shown by the decreased nondiagonal elements of the normalized cross correlation (NCC) matrix. The mean square error of the measured electron densities was reduced by 72.6%. Across all datasets, the proposed method improved the average volume fraction accuracy from 61.2% to 99.9% and increased the diagonality of the NCC matrix from 0.73 to 0.96. Compared with DI, the proposed MMD framework improved decomposition accuracy and material separation.

Hyperspectral Image Restoration Using Weighted Group Sparsity-Regularized Low-Rank Tensor Decomposition.

Mixed noise (such as Gaussian, impulse, stripe, and deadline noises) contamination is a common phenomenon in hyperspectral imagery (HSI), greatly degrading visual quality and affecting subsequent processing accuracy. By encoding sparse prior to the spatial or spectral difference images, total variation (TV) regularization is an efficient tool for removing the noises. However, the previous TV term cannot maintain the shared group sparsity pattern of the spatial difference images of different spectral bands. To address this issue, this article proposes a group sparsity regularization of the spatial difference images for HSI restoration. Instead of using l1 - or l2 -norm (sparsity) on the difference image itself, we introduce a weighted l2,1 -norm to constrain the spatial difference image cube, efficiently exploring the shared group sparse pattern. Moreover, we employ the well-known low-rank Tucker decomposition to capture the global spatial-spectral correlation from three HSI dimensions. To summarize, a weighted group sparsity-regularized low-rank tensor decomposition (LRTDGS) method is presented for HSI restoration. An efficient augmented Lagrange multiplier algorithm is employed to solve the LRTDGS model. The superiority of this method for HSI restoration is demonstrated by a series of experimental results from both simulated and real data, as compared with the other state-of-the-art TV-regularized low-rank matrix/tensor decomposition methods.

On the existence of weak solutions for a curvature driven elliptic system applied to image inpainting

In this paper, an elliptic system consisting of a weighted total variation (TV) equation and a linear equation is proposed as an approximation of Euler’s elastica image inpainting model. The existence of weak solutions for the system is proved by treating the TV term as the limit p→1+ of the p-Laplacian term. Numerical experiments show the connectivity of the proposed system for image inpainting.

Single-arc VMAT optimization for dual-layer MLC

Dual-layer multi-leaf collimator (DLMLC) has recently attracted renewed interest due to its good balance among resolution, low leakage, and high fabricability. However, existing progressive sampling based volumetric modulated arc therapy (VMAT) algorithm is ineffective for DLMLC, requiring more arcs to achieve dosimetry comparable to VMAT plans with higher resolution single-layer MLC (SLMLC). In this study, we develop a novel single-arc VMAT optimization framework to take advantage of the unique DLMLC characteristics fully. Direct aperture optimization (DAO) for single-arc DLMLC VMAT was formulated as a least square dose fidelity objective, along with an anisotropic total variation term to regulate the fluence smoothness and a single segment term for forming simple apertures. The DAO was solved through alternating optimization approach. The DLMLC deliverability constraint and the MLC leaf speed constraint were formulated as the optimization constraints and solved using a graph optimization algorithm. Feasibility of the proposed framework was tested on a brain, a lung, and a prostate cancer patient. The framework was further adapted for a simultaneous integrated boost (SIB) case. The single-arc DLMLC-10 mm (leaf width) plan was compared against single-arc SLMLC VMAT plans including SLMLC-5mm, SLMLC-10mm, and SLMLC with 10 mm leaf width and 5 mm leaf step size (SLMLC-10mm-5mm). Compared with the SLMLC-10mm plan and the SLMLC-10mm-5mm plan, with the same target coverage, the DLMLC-10 mm plan reduced R50 by 30.7% and 10.0%, the average max OAR dose by 5.79% and 3.7% of the prescription dose, and the average mean OAR dose by 4.18% and 2.1% of the prescription dose, respectively. The plan quality is comparable to that of the SLMLC-5mm plan. The novel single-arc VMAT optimization framework for DLMLC utilizes two MLC layers to improve the effective modulation resolution and afford more sophisticated modulation. Consequently, DLMLC VMAT achieves superior dosimetry to SLMLC VMAT with the same leaf width.

A regional adaptive variational PDE model for computed tomography image reconstruction

Improving CT images by increasing the number of scans, hence increasing the ionizing radiation dose, can increase the probability of inducing cancer in the patient. Using fewer images but improving them by accurate reconstruction is better solution.In this paper, an adaptive variational Partial Differential Equation (PDE) model is proposed for image reconstruction. L2 energy of the image gradient and the Total Variation (TV) are combined to form a new functional, which is introduced to an optimization problem. The dynamic behaviors of the model are formed by a threshold function, and then the L2 term is applied in the lower-density region to increase reconstruction speed, and the TV term is applied in the higher-density region to preserve the most important image features. The threshold function is asymptotically controlled by an evolutionary PDE and is more suitable for complex images. The efficiency and accuracy of the proposed model are demonstrated in numerical experiments.

Hyperspectral Image Restoration Based on Low-Rank Recovery With a Local Neighborhood Weighted Spectral–Spatial Total Variation Model

Hyperspectral image (HSI) is often contaminated by mixed noise, which severely affects the visual quality and subsequent applications of the data. In this paper, HSI restoration based on low-rank recovery with a local neighborhood weighted spectral–spatial total variation (TV) model is proposed, which focuses on the preservation of spatial structure and spectral fidelity. The low-rank matrix model is adopted to exploit the spectral and spatial correlation information, and the $l_{1}$ -norm is used as a prior to remove the sparse noise. Furthermore, a local spatial neighborhood weighted spectral–spatial TV is utilized to jointly model the spectral–spatial prior information; specifically, the spectral and spatial differences are both considered in the TV term, and the weight is computed by considering the local neighborhood information in the spatial domain. Alternating direction method of multipliers optimization procedure is extended to solve the presented model. Experimental results demonstrate that the proposed method can remove the mixed noise, enhance the structural information simultaneously, and offer the best performance compared with several state-of-the-art HSI restoration methods.

An adaptive second-order partial differential equation based on TV equation and p-Laplacian equation for image denoising

This paper introduces an adaptive diffusion partial differential equation (PDE) for noise removal, which combines a total variation (TV) term and a p-Laplacian (1 < p ≤ 2) term. Utilizing the edge indicator, we can adaptively control the diffusion model, which alternates between the TV and the p-Laplacian(1 < p ≤ 2) in accordance with the image feature. The main advantage of the proposed model is able to alleviate the staircase effect in smooth regions and preserve edges while removing the noise. The existence of a weak solution of the proposed model is proved. Experimental results confirm the performance of the proposed method with regard to peak signal-to-noise ratio (PSNR), mean structural similarity (MSSIM) and visual quality.

An Enhanced High-Order Variational Model Based on Speckle Noise Removal With $G^0$ Distribution

Speckle noise removal problem has been researched under the framework of regularization-based approaches. The regularizer is normally defined as total variation (TV) that induces staircase effect. Although higher-order regularizer can conquer the staircase effect to some extent, it often leads to blurred. Considering the upper questions, the combination of first and second-order regularizer will be an effective and prior method to tackle speckle noise removal. So a variational model with hybrid TV and higher-order total curvature (TC) term is proposed in this paper, the data fidelity term is derived based on G 0 distribution. In order to preserve the edge detail better, the boundary detection function is combined with the regularizer. Furthermore, the Mellin transform is used to estimate the parameters of the model. To address the speckle noise removal optimization problem, alternating direction method of multipliers (ADMM) framework is employed to design a convex numerical method for the proposed model. The numerical method can be used to update the variables flexibly as required by the hybrid regularizer. The numerous experiments were performed on both synthetic and real SAR images. Compared with some classical and state-of-theart SAR despeckling methods, experiment results demonstrate the improved performance of the proposed method, including that speckle noise can be removed effectively, and staircase effect can be prevented while preserving image feature.