Additive Operator Splitting Research Articles

A combined first and fractional order regularization method for mixed Poisson-white spike noisy image restoration

In this paper we study the problem of restoring images affected by mixed Poisson-Square Cauchy noise. A multi-convex variational model is derived by integrating infimal convolution likelihood into a process of joint maximum a posteriori estimation, where a modified four-directional fractional-order total variation is united with the first order total variation to characterize the image prior. To solve the proposed model numerically, a block coefficient descent based algorithm is derived, in which variable splitting and alternating direction minimization of multipliers are utilized along with anisotropic diffusion and additive operator splitting to gain efficiency and quality. The obtained numerical results are compared with results obtained from two other total variation regularized models with different fidelities. The numerical discuss confirms the flexibility and validity of the proposed model.

3D PET/CT Tumor Co-Segmentation Based on Background Subtraction Hybrid Active Contour Model

Accurate tumor segmentation in medical images plays an important role in clinical diagnosis and disease analysis. However, medical images usually have great complexity, such as low contrast of computed tomography (CT) or low spatial resolution of positron emission tomography (PET). In the actual radiotherapy plan, multimodal imaging technology, such as PET/CT, is often used. PET images provide basic metabolic information and CT images provide anatomical details. In this paper, we propose a 3D PET/CT tumor co-segmentation framework based on active contour model. First, a new edge stop function (ESF) based on PET image and CT image is defined, which combines the grayscale standard deviation information of the image and is more effective for blurry medical image edges. Second, we propose a background subtraction model to solve the problem of uneven grayscale level in medical images. Apart from that, the calculation format adopts the level set algorithm based on the additive operator splitting (AOS) format. The solution is unconditionally stable and eliminates the dependence on time step size. Experimental results on a dataset of 50 pairs of PET/CT images of non-small cell lung cancer patients show that the proposed method has a good performance for tumor segmentation.

Using the Split Bregman Algorithm to Solve the Self-repelling Snakes Model

Preserving contour topology during image segmentation is useful in many practical scenarios. By keeping the contours isomorphic, it is possible to prevent over-segmentation and under-segmentation, as well as to adhere to given topologies. The Self-repelling Snakes model (SR) is a variational model that preserves contour topology by combining a non-local repulsion term with the geodesic active contour model. The SR is traditionally solved using the additive operator splitting (AOS) scheme. In our paper, we propose an alternative solution to the SR using the Split Bregman method. Our algorithm breaks the problem down into simpler sub-problems to use lower-order evolution equations and a simple projection scheme rather than re-initialization. The sub-problems can be solved via fast Fourier transform or an approximate soft thresholding formula which maintains stability, shortening the convergence time, and reduces the memory requirement. The Split Bregman and AOS algorithms are compared theoretically and experimentally.

A variational method for Abel inversion tomography with mixed Poisson-Laplace-Gaussian noise

<p style='text-indent:20px;'>Abel inversion tomography plays an important role in dynamic experiments, while most known studies are started with a single Gaussian assumption. This paper proposes a mixed Poisson-Laplace-Gaussian distribution to characterize the noise in charge-coupled-device (CCD) sensed radiographic data, and develops a multi-convex optimization model to address the reconstruction problem. The proposed model is derived by incorporating varying amplitude Gaussian approximation and expectation maximization algorithm into an infimal convolution process. To solve it numerically, variable splitting and augmented Lagrangian method are integrated into a block coordinate descent framework, in which anisotropic diffusion and additive operator splitting are employed to gain edge preserving and computation efficiency. Supplementarily, a space of functions of adaptive bounded Hessian is introduced to prove the existence and uniqueness of solution to a higher-order regularized, quadratic subproblem. Moreover, a simplified algorithm with higher order regularizer is derived for Poisson noise removal. To illustrate the performance of the proposed algorithms, numerical tests on synthesized and real digital data are performed.</p>

A new anisotropic fourth-order diffusion equation model based on image features for image denoising

<p style='text-indent:20px;'>Image denoising has always been a challenging task. For performing this task, one of the most effective methods is based on variational PDE. Inspired by the LLT model, we first propose a new adaptive LLT model by adding a weighted function, and then we propose a class of fourth-order diffusion equations based on the new functional. Owing to the adaptive function, the new functional is better than the LLT model and other fourth-order models in terms of edge preservation. While generalizing the Euler-Lagrange equation of the new functional, we discuss a new fourth-order diffusion framework for image denoising. Different from those of other fourth-order diffusion models, the new diffusion coefficients depend on the first-order and second-order derivatives, which can preserve edges and smooth images, respectively. Regarding numerical implementations, we first design an explicit scheme for the proposed model. However, fourth-order diffusion equations require strict stability conditions, and the number of iterations needed is considerable. Consequently, we apply the fast explicit diffusion algorithm (FED) to the explicit scheme to reduce the time consumption of the proposed approach. Furthermore, the additive operator splitting (AOS) scheme is applied for the numerical implementation, and it is the most efficient among all of our algorithms. Finally, compared with other models, the new model exhibits superior effectiveness and efficiency.</p>

Fast implicit active contour model for inverse lithography.

We combine the ideas from level-set methods in computer vision and inverse imaging to derive a generalized active contour model for inverse lithography problems endowed with a locally implemented semi-implicit difference scheme. We introduce a cognitive analogy to move an initial guess of the interesting pattern contour by image-driven forces to the boundaries of the desired layout pattern. We develop an efficient semi-implicit numerical scheme implemented in the vicinity of the zero level-set and apply additive operator splitting (AOS) with respect to coordinate axes to solve consecutive one-dimensional linear systems of equations with the Thomas method. We demonstrate with simulation results that computation and convergence efficiency are jointly improved with reduced optimization dimensionality and a sufficient large step-size.

Enhanced Deep Learning Architectures for Face Liveness Detection for Static and Video Sequences.

Face liveness detection is a critical preprocessing step in face recognition for avoiding face spoofing attacks, where an impostor can impersonate a valid user for authentication. While considerable research has been recently done in improving the accuracy of face liveness detection, the best current approaches use a two-step process of first applying non-linear anisotropic diffusion to the incoming image and then using a deep network for final liveness decision. Such an approach is not viable for real-time face liveness detection. We develop two end-to-end real-time solutions where nonlinear anisotropic diffusion based on an additive operator splitting scheme is first applied to an incoming static image, which enhances the edges and surface texture, and preserves the boundary locations in the real image. The diffused image is then forwarded to a pre-trained Specialized Convolutional Neural Network (SCNN) and the Inception network version 4, which identify the complex and deep features for face liveness classification. We evaluate the performance of our integrated approach using the SCNN and Inception v4 on the Replay-Attack dataset and Replay-Mobile dataset. The entire architecture is created in such a manner that, once trained, the face liveness detection can be accomplished in real-time. We achieve promising results of 96.03% and 96.21% face liveness detection accuracy with the SCNN, and 94.77% and 95.53% accuracy with the Inception v4, on the Replay-Attack, and Replay-Mobile datasets, respectively. We also develop a novel deep architecture for face liveness detection on video frames that uses the diffusion of images followed by a deep Convolutional Neural Network (CNN) and a Long Short-Term Memory (LSTM) to classify the video sequence as real or fake. Even though the use of CNN followed by LSTM is not new, combining it with diffusion (that has proven to be the best approach for single image liveness detection) is novel. Performance evaluation of our architecture on the REPLAY-ATTACK dataset gave 98.71% test accuracy and 2.77% Half Total Error Rate (HTER), and on the REPLAY-MOBILE dataset gave 95.41% accuracy and 5.28% HTER.

On local active contour model for automatic detection of tumor in MRI and mammogram images

Tumor segmentation in medical images is an important step to determine and predict the stage, size and progression of tumors in realistic geometries and may be used for diagnosis and treatment follow-up in MRI and mammogram images. Automating this challenging task helps radiologists to reduce the high manual workload of brain or breast cancer analysis. Detecting tumor in medical images means to quantify the structure contents of the tumor. In this paper, we propose a model for quantifying the structure contents of the tumor in MRI and mammogram images. The proposed model is based on optimized Laplacian of Gaussian, which is useful in smoothing the homogeneous region and highlight the boundaries of the tumor structure. The proposed model uses local intensity information in later stage of segmentation as image data fitting. For this purpose, local Gaussian distribution is used for fitting image data and is combined with level set function. Due to usage of local intensities and level set, the proposed model is able to deal with intensity inhomogeneity and capture different topologies. To stop the leakage of contour at the boundary of tumor, we take some geometrical points near the tumor's boundary and introduce a distance constraint in model. The gradient flow equation and other optimal values are obtained through minimization of the energy functional. The gradient flow equation is solved by using additive operator splitting method. The experimental results of the proposed model are validated by comparing it with existing state of the art models both qualitatively and quantitatively. The proposed model achieves average values of 99.96%, 99.88%, 99.91%, 99.91%, 99.95% for Jaccard similarity, dice coefficient, accuracy, sensitivity and specificity respectively, which are better than the existing models in comparison.

3D Model Visualization for Brain Tumour using MRI Images

The inaccurate prediction of tumour is because of the 2D images do not present the complete natural tumour representation as in 3D visualization. In order to construct the 3D model, the contour of 2D MRI images must be merged together. In this work, the edge detection of brain tumour on 2D images is done using the Geodesic Active Contour (GAC) model based on Additive Operator Splitting (AOS), then 3D model is visualized using Image Manifold (IM) and Volume Estimation (VE) methods. The comparison of performance for both methods is calculated. In conclusion, the execution time for VE method is higher than IM method since more images are used in VE method to calculate the volume of the tumour accurately. In terms of iteration number, Volume Estimation has a higher number of iterations, but for error and root mean square error, both methods have roughly the same value.

Optimizing Deep CNN Architectures for Face Liveness Detection.

Face recognition is a popular and efficient form of biometric authentication used in many software applications. One drawback of this technique is that it is prone to face spoofing attacks, where an impostor can gain access to the system by presenting a photograph of a valid user to the sensor. Thus, face liveness detection is a necessary step before granting authentication to the user. In this paper, we have developed deep architectures for face liveness detection that use a combination of texture analysis and a convolutional neural network (CNN) to classify the captured image as real or fake. Our development greatly improved upon a recent approach that applies nonlinear diffusion based on an additive operator splitting scheme and a tridiagonal matrix block-solver algorithm to the image, which enhances the edges and surface texture in the real image. We then fed the diffused image to a deep CNN to identify the complex and deep features for classification. We obtained 100% accuracy on the NUAA Photograph Impostor dataset for face liveness detection using one of our enhanced architectures. Further, we gained insight into the enhancement of the face liveness detection architecture by evaluating three different deep architectures, which included deep CNN, residual network, and the inception network version 4. We evaluated the performance of each of these architectures on the NUAA dataset and present here the experimental results showing under what conditions an architecture would be better suited for face liveness detection. While the residual network gave us competitive results, the inception network version 4 produced the optimal accuracy of 100% in liveness detection (with nonlinear anisotropic diffused images with a smoothness parameter of 15). Our approach outperformed all current state-of-the-art methods.

Efficient optical proximity correction based on semi-implicit additive operator splitting

Inverse lithography techniques (ILT) have been extensively used by the semiconductor industry to compensate for the inherent image distortions in optical lithography. However, the iterative ILT optimization procedure requires rather prohibitive time steps leading to poor efficiency with explicit time discretization. In this paper, a semi-implicit time discretization scheme is applied, enabling stable computation of mask synthesis with large time steps. Additive operator splittering (AOS) is implemented with respect to coordinate axes, reducing mask synthesis to consecutive one-dimensional updates represented by tridiagonal linear equations, which is solved efficiently by the Thomas algorithm. Simulation results merit the superiority of the proposed semi-implicit approach with improved convergence performance.

Multiplicative Noise Removal Based on the Smooth Diffusion Equation

The multiplicative noise removal problem is of momentous significance in various image processing applications. In this paper, a nonlinear diffusion equation with smooth solution is proposed to remove multiplicative Gamma noise. The diffusion coefficient takes full advantage of two features of multiplicative noise image, namely, gradient information and gray level information, which makes the model has the ability to remove high level noise effectively and protect the edges. The existence of the solution has been analyzed by Schauder’s fixed-point theorem. Some other theoretical properties such as the maximum principle are also presented in the paper. In the numerical aspect, the explicit finite difference method, fast explicit diffusion method, additive operator splitting method and Krylov subspace spectral method are employed to implement the proposed model. Experimental results show that the fast explicit diffusion method achieves a better trade-off between computational time and denoising performance, and the Krylov subspace spectral method gets better restored results in the visual aspect. In addition, the capability of the proposed model for denoising is illustrated by comparison with other denoising models.

An alternating direction explicit method for time evolution equations with applications to fractional differential equations

We derive and analyze the alternating direction explicit (ADE) method for time evolution equations with the time-dependent Dirichlet boundary condition and with the zero Neumann boundary condition. The original ADE method is an additive operator splitting (AOS) method, which has been developed for treating a wide range of linear and nonlinear time evolution equations with the zero Dirichlet boundary condition. For linear equations, it has been shown to achieve the second order accuracy in time yet is unconditionally stable for an arbitrary time step size. For the boundary conditions considered in this work, we carefully construct the updating formula at grid points near the boundary of the computational domain and show that these formulas maintain the desired accuracy and the property of unconditional stability. We also construct numerical methods based on the ADE scheme for two classes of fractional differential equations. We will give numerical examples to demonstrate the simplicity and the computational efficiency of the method.

Underwater image segmentation based on fast level set method

Image segmentation is a fundamental process in image processing that has found application in many fields, such as neural image analysis, and underwater image analysis. In this paper, we propose a novel fast level set method (FLSM)-based underwater image segmentation method to improve the traditional level set methods by avoiding the calculation of signed distance function (SDF). The proposed method can speed up the computational complexity without re-initialisation. We also provide a fast semi-implicit additive operator splitting (AOS) algorithm to improve the computational complex. The experiments show that the proposed FLSM performs well in selecting local or global segmentation regions.

A new selective segmentation model for texture images and applications to medical images

Abstract Segmentation of texture images is an active area of image processing. Selective segmentation is the process of extracting a region of interest (ROI) in the image. In this paper, we propose a new model for selective segmentation of texture images, by incorporating geometrical constraints after smoothing the texture in image. The proposed model uses L0 norm for smoothing of the texture and Badshah–Chen energy with local Gaussian kernel data fitting for selective segmentation. The proposed model is minimized to get gradient descent through Euler Lagrange's equation, which is then discretized through finite differences and the corresponding difference equation is solved by using additive operator splitting method. Experimental results of the proposed model are compared with the existing selective segmentation models and related models which are not based on selective segmentation. The model is further tested on medical images.

Fast Von Mises strain imaging on ultrasound carotid vessel wall by flow driven diffusion method.

The elasticity of the vessel wall is important for the clinical identification of rupture-risks. The Von Mises strain can be a potential index for the indication of carotid vessel pathologies. In this paper, a fast clinically applicable real-time algorithm from time-sequence of B-mode carotid images is developed. Due to the compression induced by the normal cardiac pulsation, tissue motion occurs radially and non-rigidly. To obtain an accurate motion field, we developed a variational functional integrating the optical flow equation and an anisotropic regularizer, and designed a diffusion tensor to encourage coherence diffusion. The motion field is smoothed along the desired motion flow orientation. A fast, additive operator splitting scheme, which is ten times faster than the conventional discrete method, is used for the numerical implementation. To demonstrate the efficiency of the proposed approach, finite element models are set up for normal and pathological carotid vessel walls. The results indicate that the proposed diffusion approach obtains an accurate smooth and continuous motion field and greatly improves the follow up strain estimation using a fast differential strain filter. Furthermore, our approach using the Von Mises strain imaging on clinical ultrasound images of the carotid artery is validated. Participants above 65-years in age suffering from different stages of atherosclerosis in their carotid artery are selected. The results are evaluated by an experienced physician. The evaluation results demonstrate that the Von Mises strain has a good correspondence to the presence of certain suspicious areas in the B-mode images. The proposed method is therefore clinically applicable for the real-time Von Mises strain imaging of carotid vessel walls, and can be of great value as a complementary method to B-mode image for the clinical identification of the risk of plaque vulnerability.

An Efficient Multi-Scale Local Binary Fitting-Based Level Set Method for Inhomogeneous Image Segmentation

An efficient level set model based on multiscale local binary fitting (MLBF) is proposed for image segmentation. By introducing multiscale idea into the LBF model, the proposed MLBF model can effectively and efficiently segment images with intensity inhomogeneity. In addition, by adding a reaction diffusion term into the level set evolution (LSE) equation, the regularization of the level set function (LSF) can be achieved, thus completely eliminating the time-consuming reinitialization process. In the implementation phase, in order to greatly improve the efficiency of the numerical solution of the level set segmentation model, we introduce three strategies: The first is the additive operator splitting (AOS) solver which is used for breaking the restrictions on time step; the second is the salient target detection mechanism which is used to achieve full automatic initialization of the LSE process; the third is the sparse filed method (SFM) which is used to restrict the groups of pixels that need to be updated in a small strip region. Under the combined effect of these three strategies, the proposed model achieves very high execution efficiency in the following aspects: contour location accuracy, speed of evolution convergence, robustness against initial contour position, and robustness against noise interference.

Combined Shearlet Shrinkage and Total Variation Minimization for Image Denoising

In this paper, a TV-based shearlet shrinkage is proposed for discontinuity-preserving denoising using a combination of shearlet with a total variation scheme. For TV denoising numerical procedure, we use two approaches. In the first approach, we apply semi-implicit method for total variation. To solve Euler–Lagrange equation associated with total variation, we use additive operator splitting (AOS) scheme. This approach has good effect on suppressing the pseudo-Gibbs and shearlet-like artifacts and is very efficient for reducing iterations. In the second approach, we use Split Bregman iteration method. This techniques converges very quickly and in combine by shearlet shrinkage produce good results.

A splitting algorithm for a novel regularization of Perona-Malik and application to image restoration

In this paper, we focus on a numerical method of a problem called the Perona-Malik inequality which we use for image denoising. This model is obtained as the limit of the Perona-Malik model and the p-Laplacian operator with p→∞. In Atlas et al., (Nonlinear Anal. Real World Appl 18:57–68, 2014), the authors have proved the existence and uniqueness of the solution of the proposed model. However, in their work, they used the explicit numerical scheme for approximated problem which is strongly dependent to the parameter p. To overcome this, we use in this work an efficient algorithm which is a combination of the classical additive operator splitting and a nonlinear relaxation algorithm. At last, we have presented the experimental results in image filtering show, which demonstrate the efficiency and effectiveness of our algorithm and finally, we have compared it with the previous scheme presented in Atlas et al., (Nonlinear Anal. Real World Appl 18:57–68, 2014).

On convergent finite difference schemes for variational–PDE-based image processing

We study an adaptive anisotropic Huber functional based image restoration scheme. By using a combination of L2-L1 regularization functions, an adaptive Huber functional based energy minimization model provides denoising with edge preservation in noisy digital images. We study a convergent finite difference scheme based on continuous piecewise linear functions and use a variable splitting scheme, namely the Split Bregman, to obtain the discrete minimizer. Experimental results are given in image denoising and comparison with additive operator splitting, dual fixed point, and projected gradient schemes illustrate that the best convergence rates are obtained for our algorithm.