Anisotropic Total Variation Regularization Research Articles

Robust Tensor Decomposition Based Background/Foreground Separation in Noisy Videos and Its Applications in Additive Manufacturing

Background/foreground separation is one of the most fundamental tasks in computer vision, especially for video data. Robust PCA (RPCA) and its tensor extension, namely, Robust Tensor PCA (RTPCA), provide an effective framework for background/foreground separation by decomposing the data into low-rank and sparse components, which contain the background and the foreground (moving objects), respectively. However, in real-world applications, the video data is contaminated with noise. For example, in metal additive manufacturing (AM), the processed X-ray video to study melt pool dynamics is very noisy. RPCA and RTPCA are not able to separate the background, foreground, and noise simultaneously. As a result, the noise will contaminate the background or the foreground or both. There is a need to remove the noise from the background and foreground. To achieve the three components decomposition, a smooth sparse Robust Tensor Decomposition (SS-RTD) model is proposed to decompose the data into static background, smooth foreground, and noise, respectively. Specifically, the static background is modeled by the low-rank tucker decomposition, the smooth foreground (moving objects) is modeled by the spatio-temporal continuity, which is enforced by the total variation regularization, and the noise is modeled by the sparsity, which is enforced by the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula> norm. An efficient algorithm based on alternating direction method of multipliers (ADMM) is implemented to solve the proposed model. Extensive experiments on both simulated and real data demonstrate that the proposed method significantly outperforms the state-of-the-art approaches for background/foreground separation in noisy cases. Note to Practitioners—This work is motivated by melt pool detection in metal additive manufacturing where the processed X-ray video from the monitoring system is very noisy. The objective is to recover the background with porosity defects and the foreground with melt pool in the presence of noise. Existing methods fail to separate the noise from the background and foreground since RPCA and RTPCA have only two components, which cannot explain the three components in the data. This paper puts forward a smooth sparse Robust Tensor Decomposition by decomposing the tensor data into low-rank, smooth, and sparse components, respectively. It is a highly effective method for background/foreground separation in noisy case. In the case studies on simulated video and X-ray data, the proposed method can handle non-additive noise, and even the case of high noise-ratio. In the proposed algorithm, there is only one tuning parameter <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lambda $ </tex-math></inline-formula> . Based on the case studies, our method achieves satisfying performance by taking any <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lambda \in [{0.2,1}]$ </tex-math></inline-formula> with anisotropic total variation regularization. With this observation, practitioners can apply the proposed method without extensive parameter tuning work. Furthermore, the proposed method is also applicable to other popular industrial applications. Practitioners can use the proposed SS-RTD for degradation processes monitoring, where the degradation image contains the static background, anomaly, and random disturbance, respectively.

The 3-D Global Prestack Seismic Inversion in the Time–Frequency Mixed Domain

Traditional seismic inversion has the problems of low lateral resolution and poor anti-noise performance. To address these challenges, the advantages of inversion methods in time and frequency domains are combined; the Lp norm that can preserve more sparsity information is introduced into seismic inversion. Meanwhile, multi-trace simultaneous inversion has become the mainstream of inversion methods, which can effectively improve the quality of pre-stack seismic inversion. However, traditional multi-trace simultaneous inversion methods have high computational costs and are difficult to obtain the 3D result. Ensuring spatial continuity of inversion results is a challenging task. To address this issue, a 3D global pre-stack inversion method in the time-frequency mixed domain is proposed. The proposed method adds a frequency domain fidelity constraint the objective function of traditional time domain inversion to improve the resolution of the inversion results. To improve the computational efficiency of multi-trace simultaneous inversion, the anisotropic total variation regularization method under Lp norm constraint is employed. In addition, the 2D Sylvester equation is extended to 3D space to enable 3D global inversion. The method not only improves computational efficiency, but also ensures spatial continuity of the results. The stability and feasibility of the proposed method have been demonstrated through its application on both the 3D overthrust model and field pre-stack seismic data.

A Nonlinear Weighted Anisotropic Total Variation Regularization for Electrical Impedance Tomography

This paper proposes a nonlinear weighted anisotropic total variation (NWATV) regularization technique for electrical impedance tomography (EIT). The key idea is to incorporate the internal inhomogeneity information (e.g., edges of the detected objects) into the EIT reconstruction process, aiming to preserve the conductivity profiles (to be detected). We study the NWATV image reconstruction by employing a novel soft thresholding based reformulation included in the alternating direction method of multipliers (ADMM). To evaluate the proposed approach, numerical simulations and human EIT lung imaging are carried out. It is demonstrated that the properties of the internal inhomogeneity are well preserved and improved with the proposed regularization approach, in comparison to traditional total variation (TV) and recently proposed fidelity embedded regularization approaches. Owing to the simplicity of the proposed method, the computational cost is significantly decreased compared with the well established primal-dual algorithm. Precisely, with the proposed algorithm, we are able to alleviate the staircase effect arising in TV regularization and improve the reconstruction accuracy for FER does. To achieve similar accuracy as TV does, the computational times are reduced from 2.311s to 0.629s and 1.733s to 0.428s in 2D and 3D simulations, respectively, and the computational time is reduced from greater than 2s to less than 0.2s in the human experiment. Meanwhile, it was found that the proposed regularization method is quite robust to the measurement noise, which is one of the main uncertainties in EIT.

A Low-Rank Tensor Decomposition Model With Factors Prior and Total Variation for Impulsive Noise Removal.

Image restoration is a long-standing problem in signal processing and low-level computer vision. Previous studies have shown that imposing a low-rank Tucker decomposition (TKD) constraint could produce impressive performances. However, the TKD-based schemes may lead to the overfitting/underfitting problem because of incorrectly predefined ranks. To address this issue, we prove that the n -rank is upper bounded by the rank of each Tucker factor matrix. Using this relationship, we propose a formulation by imposing the nuclear norm regularization on the latent factors of TKD, which can avoid the burden of rank selection and reduce the computational cost when dealing with large-scale tensors. In this formulation, we adopt the Minimax Concave Penalty to remove the impulsive noise instead of the l1 -norm which may deviate from both the data-acquisition model and the prior model. Moreover, we employ an anisotropic total variation regularization to explore the piecewise smooth structure in both spatial and spectral domains. To solve this problem, we design the symmetric Gauss-Seidel (sGS) based alternating direction method of multipliers (ADMM) algorithm. Compared to the directly extended ADMM, our algorithm can achieve higher accuracy since more structural information is utilized. Finally, we conduct experiments on the three kinds of datasets, numerical results demonstrate the superiority of the proposed method, especially, the average PSNR of the proposed method can improve about 1~5dB for each noise level of color images.

Multidirectional Anisotropic Total Variation and Its Application in the Tomography of the Surrounding Rock of Coal Mining Faces

The computed tomography (CT) reconstruction algorithm is one of the crucial components of the CT system. To date, total variation (TV) has been widely used in CT reconstruction algorithms. Although TV utilizes the a priori information of the longitudinal and lateral gradient sparsity of an image, it introduces some staircase artifacts. To overcome the current limitations of TV and improve imaging quality, we propose a multidirectional anisotropic total variation (MATV) that uses multidirectional gradient information. The surrounding rock of coal mining faces uses principles of tomography similar to those of medical X-rays. The velocity distribution for the surrounding rock can be obtained by the first-arrival traveltime tomography of the transmitted waves in the coal mining face. Combined with the geological data, we can interpret the geological hazards in the coal mining face. To perform traveltime tomography, we first established the objective function of the first-arrival traveltime tomography of the transmitted waves based on the MATV regularization and then used the split Bregman method to solve the objective function. The simulated data and real data show that the MATV regularization method proposed in this paper can better maintain the boundaries of geological anomalies and reduce the artifacts compared with the isotropic total variation regularization method and the anisotropic total variation regularization method. Furthermore, this approach describes the distribution of geological anomalies more accurately and effectively and improves imaging accuracy.

Anisotropic Total Variation Regularization Based NAS-RIF Blind Restoration Method for OCT Image

Based on anisotropic total variation regularization (ATVR), a nonnegativity and support constraints recursive inverse filtering(NAS-RIF) blind restoration method is proposed to enhance the quality of optical coherence tomography (OCT) image. First, ATVR is introduced into the cost function of NAS-RIF to improve the noise robustness and retain the details in the image. Since the split Bregman iterative is used to optimize the ATVR based cost function, the ATVR based NAS-RIF blind restoration method is then constructed. Furthermore, combined with the geometric nonlinear diffusion filter and the Poisson-distribution-based minimum error thresholding, the ATVR based NAS-RIF blind restoration method is used to realize the blind OCT image restoration. The experimental results demonstrate that the ATVR based NAS-RIF blind restoration method can successfully retain the details in the OCT images. In addition, the signal-to-noise ratio of the blind restored OCT images can be improved, along with the noise robustness.

Helical CT Reconstruction from Sparse-view Data through Exploiting the 3D Anatomical Structure Sparsity

Sparse-view scanning has great potential for realizing ultra-low-dose computed tomography (CT) examination. However, noise and artifacts in reconstructed images are big obstacles, which must be handled to maintain the diagnosis accuracy. Existing sparse-view CT reconstruction algorithms were usually designed for circular imaging geometry, whereas the helical imaging geometry is commonly adopted in the clinic. In this paper, we show that the sparse-view helical CT (SHCT) images contain not only noise and artifacts but also severe anatomical distortions. These troubles reduce the applicability of existing sparse-view CT reconstruction algorithms. To deal with this problem, we analyzed the three-dimensional (3D) anatomical structure sparsity in SHCT images. Based on the analyses, we proposed a tensor decomposition and anisotropic total variation regularization model (TDATV) for SHCT reconstruction. Specifically, the tensor decomposition works on nonlocal cube groups to exploit the anatomical structure redundancy; the anisotropic total variation works on the whole volume to exploit the structural piecewise-smooth. Finally, an alternating direction method of multipliers is developed to solve the TDATV model. To our knowledge, the paper presents the first work investigating the reconstruction of sparse-view helical CT. The TDATV model was validated through digital phantom, physical phantom, and clinical patient studies. The results reveal that SHCT could serve as a potential solution for reducing HCT radiation dose to ultra-low level by using the proposed TDATV model.

Anisotropic total variation model for removing oblique stripe noise in remote sensing image

Destriping is one of typical problems in remote sensing image processing which is crucial in subsequent applications. The orientation of the stripe in remote sensing image plays a key role in the destriping methods. However, the direction of the oblique stripe noises is uncertain which makes the destriping methods for vertical/horizontal stripes no longer applicable for the oblique ones. To address the issue, we propose an anisotropic total variation (TV) model for oblique stripe noise removal. The model is constructed by combining L1 norm regularization and anisotropic total variation regularization, where the L1 norm regularizations represent the global characteristics and orientation of stripe noise, anisotropic total variation regularizations describe the different effects of oblique stripe noise on the smoothness of the original clear image along the vertical and horizontal directions, which can preserve the strong edges and geometric features while suppressing stripes. In order to solve this model, we design an effective alternating direction method of multipliers (ADMM) algorithm with guaranteed convergence. The experimental results over simulated and real datasets demonstrate that the proposed approach can not only effectively remove oblique stripe noise, but also for vertical/horizontal ones, and outperform the related state-of-the-art methods.

Step phase reconstruction using an anisotropic total variation regularization method in a diffraction phase microscopy.

This paper presents an approach to estimate the phase distribution with step features or edge characteristics from an interferogram obtained in diffraction phase microscopy. The proposed approach relies on formulation of a cost function for the interferogram and subsequent application of anisotropic total variation regularization based optimization. In our approach, the cost function is minimized by using a proximal gradient method based iterative approach with alternative updates of the amplitude and the phase. The performance of the proposed approach for retrieving phase maps with rapid edge variations is shown via numerical simulations. Further, the experimental validity of the proposed method is demonstrated for surface topography in diffraction phase microscopy.

Nonblind Image Deblurring Based on Bi-Composition Decomposition by Local Smoothness and Nonlocal Self-Similarity Priors

Image deblurring is a classical inverse problem in image processing and computer vision. The vital task is to construct the proper image prior model to obtain the high-quality restored image with salient edges and rich details. A new nonblind image deblurring method by combining local smoothness and nonlocal self-similarity of natural images in the regularization framework is proposed. First, the observed image is decomposed into two components: structure component and detail component by a global gradient extraction scheme. Second, the four-directional anisotropic total variation regularization satisfying the local smoothness property is adopted for the structure component, and a new nonlocal statistical modeling for self-similarity is used for the detail component, respectively. At last, the split Bregman-based iteration algorithm and four-directional fast gradient projection algorithm are introduced to optimize the proposed L 1 -regularized problem. The extensive experiments demonstrate the efficiency and viability of the proposed method for preserving salient edges and texture details while alleviating the artifacts.

Fast multi-trace impedance inversion using anisotropic total p-variation regularization in the frequency domain

In this article, we find that the transformational matrix between the natural logarithm of the acoustic impedance and the reflection coefficient is a Toeplitz matrix. As a result, we can convert the matrix multiplication into convolution operation according to properties of the Toeplitz matrix. In this way, we calculate the convolution of matrices through a dot product operation in the frequency domain based on the convolution theorem. Thus a fast multi-trace impedance inversion method by using two dimensional fast Fourier transform in the frequency domain is proposed, which can greatly improve the inversion speed. Considering the non-convex Lp (0 ≤ p < 1) quasi-norm is more suitable for sparse optimization than the L1 norm, anisotropic total variation regularization based on the Lp quasi-norm is used to improve the inversion result. Synthetic seismic data and field data inversion results show that the proposed method improves the inversion speed more than ten times compared with the multi-trace inversion method with total variation (TV) regularization in the time domain. At the same time, the proposed method has the advantages over the inversion methods with TV regularization in the time domain of less inversion error and better depiction of thin layers.

Isotropic and anisotropic total variation regularization in electrical impedance tomography

This paper focuses on studying the effects of isotropic and anisotropic total variation (TV) regularization in electrical impedance tomography (EIT). A characteristic difference between these two widely used TV regularization methods is that the isotropic TV is rotationally invariant and the anisotropic TV is not. The rotational variance of the anisotropic TV is known to cause geometric distortions by favoring edge orientations that are aligned with co-ordinate axes. In many applications, such as transmission tomography problems, these distortions often play only a minor role in the overall accuracy of reconstructed images, because the measurement data is sensitive to the shapes of the edges in the imaged domain. In EIT and other diffusive image modalities, however, the data is severely less sensitive to the fine details of edges, and it is an open question how large impact the selection of the TV regularization variant has on the reconstructed images. In this work, this effect is investigated based on a set of experiments. The results demonstrate that the choice between isotropic and anisotropic TV regularization indeed has a significant impact on the properties of EIT reconstructions; especially, the tendency of the anisotropic TV to favor edges aligned with co-ordinate axes is shown to yield large geometric distortions in EIT reconstructions.

Computerized triplet beam orientation optimization for MRI-guided Co-60 radiotherapy.

Magnetic resonance imaging (MRI)-guided Co-60 provides daily and intrafractional MRI soft tissue imaging for improved target and critical organ tracking. To increase delivery efficiency, the system uses three Co-60 sources at 120° apart, allowing up to 600 cGy combined dose rate at isocenter. Despite the potential tripling in output, creating a delivery plan that uses all three sources is considerably unintuitive. Here, the authors computerize the triplet orientation optimization using column generation, an approach that was demonstrated effective in integrated beam orientation and fluence optimization for noncoplanar therapies. To achieve a better plan quality without increasing the treatment time, the authors then solve a fluence map optimization (FMO) problem while regularizing the fluence maps to reduce the number of deliverable MLC segments. Three patients-one prostate, one lung, and one head and neck boost plan (H&NBoost)-were evaluated in this study. For each patient, the beamlet doses were calculated using Monte Carlo, under a 0.35 T magnetic field, for 180 equally spaced coplanar beams grouped into 60 triplets. The beamlet size is 1.05 × 0.5 cm determined by the MLC leaf thickness and step size. The triplets were selected using the column generation algorithm. The FMO problem was formulated using an L2-norm dose fidelity term and an L1-norm anisotropic total variation regularization term, which allows controlling the number of MLC segments, and hence the treatment time, with minimal degradation to the dose. The authors' Fluence Regularization and Optimized Selection of Triplets (FROST) plans were compared against the clinical treatment plans (CLNs) produced by an experienced dosimetrist. PTV homogeneity, max dose, mean dose, D95, D98, and D99 were evaluated. OAR max and mean doses, as well as R50, defined as the ratio of the 50% isodose volume over the planning target volume were investigated. The mean PTV D95, D98, and D99 differ by +0.04%, +0.07%, and +0.25% of the prescription dose between planning methods. The mean PTV homogeneity was virtually same with values at 0.8788 (FROST) and 0.8812 (CLN). R50 decreased by 0.67 comparing FROST to CLN. On average, FROST reduced Dmax and Dmean of OARs by 7.30% and 6.08% of the prescription dose, respectively. The manual CLN planning processes required numerous trial and error runs. The FROST plans on the other hand required minimal human intervention. Efficient delivery of MRI-guided Co-60 therapy needs the output of multiple sources yet suffers from unintuitive and laborious manual beam selection processes. Computerized triplet orientation optimization improves both planning efficiency and plan dosimetry. The novel fluence map regularization provides additional controls over the number of MLC segments and treatment time.

Robust total-variation based geophysical inversion using split Bregman and proximity operators

In this paper, we take advantages of split Bregman and proximity operators to formulate geophysical inverse problems in a convex optimization setting. In this type of formulation, we can efficiently consider a general convex data misfit term in order to incorporate more realistic error probability assumptions than the ordinary Gaussian distribution. Furthermore, we can simply impose convex non-quadratic and non-smooth regularization terms as a prior information.Although the proposed formulation can be extended for incorporating any types of convex regularization and data fidelity terms, here we consider misfit term corresponding to the Huber norm and anisotropic total variation regularization as a prior to an unknown model structure. This type of regularization enables us to construct piecewise-constant solutions.Two-dimensional linear and non-linear seismic travel-time tomography are studied to evaluate the efficiency of the proposed formulation for handling a robust measure of the misfit term in recovering blocky solutions. We consider the first arrival travel-times which are contaminated by outliers and also a mixed combination of Gaussian and outliers. Finally, Seinsfeld cross-hole tomography data set is used to investigate the performance of the robust approach on real data sets.

Accurate low-dose iterative CT reconstruction from few projections by Generalized Anisotropic Total Variation minimization for industrial CT.

Reducing the amount of time for data acquisition and reconstruction in industrial CT decreases the operation time of the X-ray machine and therefore increases the sales. This can be achieved by reducing both, the dose and the pulse length of the CT system and the number of projections for the reconstruction, respectively. In this paper, a novel generalized Anisotropic Total Variation regularization for under-sampled, low-dose iterative CT reconstruction is discussed and compared to the standard methods, Total Variation, Adaptive weighted Total Variation and Filtered Backprojection. The novel regularization function uses a priori information about the Gradient Magnitude Distribution of the scanned object for the reconstruction. We provide a general parameterization scheme and evaluate the efficiency of our new algorithm for different noise levels and different number of projection views. When noise is not present, error-free reconstructions are achievable for AwTV and GATV from 40 projections. In cases where noise is simulated, our strategy achieves a Relative Root Mean Square Error that is up to 11 times lower than Total Variation-based and up to 4 times lower than AwTV-based iterative statistical reconstruction (e.g. for a SNR of 223 and 40 projections). To obtain the same reconstruction quality as achieved by Total Variation, the projection number and the pulse length, and the acquisition time and the dose respectively can be reduced by a factor of approximately 3.5, when AwTV is used and a factor of approximately 6.7, when our proposed algorithm is used.

TU‐F‐CAMPUS‐I‐03: Enhancement of 4D CBCT Image Quality Using An Adaptive Prior Image Constrained Compressed Sensing

Purpose:To develop an iterative reconstruction algorithm using a compressed sensing with adaptive prior image constraints to solve 4D CBCT reconstruction problem.Methods:The images reconstructed by the FDK algorithm with a full set of unsorted projections are served as prior images for partial projections in each phase group and are utilized as an initial guess. Additionally, the prior images are clustered into several regions by applying intensity‐based thresholding, which is referred to as the segmented prior images. The segmented prior images are employed to detect any possible mismatched areas compared with the target images generated by partial projection data. With these two prior images, our algorithm alternately performs the simultaneous algebraic reconstruction technique and anisotropic total variation regularization while adjusting a weighted relaxation map during the iterative reconstruction process. The weighted relaxation map depends on binary images created by the voxel‐dependent comparison between the segmented prior and segmented target images. For the segmented target images, the k‐means clustering with a geometric weighting is applied on the reconstruction images generated in each iteration step. The inverse values of the distance map converted from binary images are assigned to be the values of the relaxation map. Evaluations using Catphan504 phantom with a motion platform were carried out.Results:Qualitative and quantitative analyses showed that the method provides high‐quality CBCT reconstruction images when compared with those generated by the FDK, CS, and PICCS algorithms, with higher contrast‐to‐noise ratio and faster convergence caused by minimizing data fidelity. Especially, the proposed method was superior to PICCS in the aspect of updating locally‐mismatched region.Conclusion:The proposed method not only improves the image quality of 4D CBCT by adaptive updates during the reconstruction process, but also leads to a lower imaging dose and faster acquisition time by using a regular 3D CBCT scan.

TH‐EF‐BRD‐05: A New Intensity Modulation Radiation Therapy (IMRT) Optimizer Solution with Robust Fluence Maps for MLC Segmentation

Purpose:Converting theoretical fluence maps to deliverable MLC segments while maintaining plan quality has posed a significant challenge for highly complex IMRT plans involving many fields. Rather than simplifying individual fluence maps, we have formulated the problem as a global dose domain fluence map optimization with anisotropic total variation regularization (FMOATVR) problem and then solve the large scale optimization problem.Methods:The 4π planning platform was used to select and optimize 20 non‐coplanar beams for 2 GBM, 2 head &amp; neck, and 2 lung patients with varying prescription doses, including H&amp;N case #2 that had 3 prescription dose levels of 54 Gy, 59.4 Gy, and 69.96 Gy. The plans were re‐run using FMOATVR and the same 20 4π beams. The L2 norm of dose deviation from the ideal prescription with zero dose to OARs was used as the objective function, which was subjected to anisotropic total variation regularization to encourage piecewise smoothness in the fluence maps. The fluences were stratified into equal step sizes to create 300 MLC segments. The dose distributions were normalized so the prescription dose covered cover 95% of the PTV. H&amp;N case #2 was normalized to cover the 59.4 Gy PTV, which was included in the statistics.Results:Compared to the directly stratified 4π plans (4π DS), the 4π FMOATVR method reduced the average mean and max dose to the OARs by 3.54% and 4.15%, respectively, of the prescription dose. Dose coverage was also improved, evidenced by an average D98 and D99 increase of 1.63% and 2.74% of the prescription dose, respectively. The average PTV homogeneity was increased from 0.833 to 0.905.Conclusion:The global dose domain fluence map optimization formula and its solver provides a straightforward yet potent method to attain high quality while remaining robust to MLC segmentation for complex IMRT.This work is supported in part by Varian Medical Systems, Inc.

TU‐C‐17A‐05: Dose Domain Optimization of MLC Leaf Patterns for Highly Complicated 4Ï€ IMRT Plans

Purpose:Highly conformal non‐coplanar 4π radiotherapy plans typically require more than 20 intensity‐modulated fields to deliver. A novel method to calculate multileaf collimator (MLC) leaf patterns is introduced to maximize delivery efficiency, accuracy and plan quality.Methods:4 GBM patients, with a prescription dose of 59.4 Gy or 60 Gy, were evaluated using the 4π algorithm using 20 beams. The MLC calculation utilized a least square minimization of the dose distribution, with an anisotropic total variation regularization term to encourage piecewise continuity in the fluence maps. Transforming the fluence to the dose domain required multiplying the fluence with a sparse matrix. Exploiting this property made it feasible to solve the problem using CVX, a MATLAB‐based convex modeling framework. The fluence was stratified into even step sizes, and the MLC segments, limited to 300, were calculated. The patients studied were replanned using Eclipse with the same beam angles.Results:Compared to the original 4π plan, the stratified 4π plan increased the maximum/mean dose for, in Gy, by 1.0/0.0 (brainstem), 0.5/0.2 (chiasm), 0.0/0.0 (spinal cord), 1.9/0.3 (L eye), 0.7/0.2 (R eye), 0.4/0.4 (L lens), 0.3/0.3 (R lens), 1.0/0.8 (L Optical Nerve), 0.5/0.3 (R Optical Nerve), 0.3/0.2 (L Cochlea), 0.1/0.1 (R Cochlea), 4.6/0.2 (brain), 2.4/0.1 (brain‐PTV), 5.1/0.9 (PTV). Compared to Eclipse, which generated an average of 607 segments, the stratified plan reduced (−) or increased (+) the maximum/mean dose, in Gy, by −10.2/−4.1 (brainstem), −10.5/−8.9 (chiasm), +0.0/−0.1 (spinal cord), −4.9/−3.4 (L eye), −4.1/−2.5 (R eye), −2.8/−2.7 (L lens), −2.1/−1.9 (R lens), −7.6/−6.5 (L Optical Nerve), −8.9/−6.1 (R Optical Nerve), −1.3/−1.9 (L Cochlea), −1.8/−1.8 (R Cochlea), +1.7/−2.1 (brain), +3.2/−2.6 (brain‐PTV), +1.8/+0.3 Gy (PTV. The stratified plan was also more homogeneous in the PTV.Conclusion:This novel solver can transform complicated fluence maps into significantly fewer deliverable MLC segments than the commercial system while achieving superior dosimetry.Funding support partially contributed by Varian.

Fast algorithm for multiplicative noise removal

In this work, we consider a variational restoration model for multiplicative noise removal problem. By using a maximum a posteriori estimator, we propose a strictly convex objective functional whose minimizer corresponds to the denoised image we want to recover. We incorporate the anisotropic total variation regularization in the objective functional in order to preserve the edges well. A fast alternating minimization algorithm is established to find the minimizer of the objective functional efficiently. We also give the convergence of this minimization algorithm. A broad range of numerical results are given to prove the effectiveness of our proposed model.