Segmentation of Pulmonary Nodule Images Using 1-Norm Minimization

Total variation minimization (in the 1-norm) has edge preserving and enhancing properties which make it suitable for image segmentation. We present Image Simplification, a new formulation and algorithm for image segmentation. We illustrate the edge enhancing properties of 1-norm total variation minimization in a discrete setting by giving exact solutions to the problem for piecewise constant functions in the presence of noise. In this case, edges can be exactly recovered if the noise is sufficiently small. After optimization, segmentation is completed using edge detection. We find that our image segmentation approach yields good results when applied to the segmentation of pulmonary nodules.     

Speckle noise removal in ultrasound images by first- and second-order total variation

Speckle noise contamination is a common issue in ultrasound imaging system. Due to the edge-preserving feature, total variation (TV) regularization-based techniques have been extensively utilized for speckle noise removal. However, TV regularization sometimes causes staircase artifacts as it favors solutions that are piecewise constant. In this paper, we propose a new model to overcome this deficiency. In this model, the regularization term is represented by a combination of total variation and high-order total variation, while the data fidelity term is depicted by a generalized Kullback-Leibler divergence. The proposed model can be efficiently solved by alternating direction method with multipliers (ADMM). Compared with some state-of-the-art methods, the proposed method achieves higher quality in terms of the peak signal to noise ratio (PSNR) and the structural similarity index (SSIM). Numerical experiments demonstrate that our method can remove speckle noise efficiently while suppress staircase effects on both synthetic images and real ultrasound images.     

Domain decomposition methods with graph cuts algorithms for total variation minimization

Recently, graph cuts algorithms have been used to solve variational image restoration problems, especially for noise removal and segmentation. Compared to time-marching PDE methods, graph cuts based methods are more efficient and able to obtain the global minimizer. However, for high resolution and large-scale images, the cost of both memory and computational time increases dramatically. In this paper, we combine the domain decomposition method and the graph cuts algorithm for solving the total variation minimizations with L ¹ and L ² fidelity term. Numerous numerical experiments on large-scale data demonstrate the proposed algorithm yield good results in terms of computational time and memory usage.     

Adaptive Segmentation Model for Images with Intensity Inhomogeneity based on Local Neighborhood Contrast

Segmentation of images with intensity inhomogeneity is a significant task in the field of image processing, especially in medical image processing and analysis. Some local region-based models work well on handling intensity inhomogeneity, but they are always sensitive to contour initialization and high noise. In this paper, we present an adaptive segmentation model for images with intensity inhomogeneity in the form of partial differential equation. Firstly, a global intensity fitting term and a local intensity fitting term are constructed by employing the global and local image information, respectively. Secondly, a tradeoff function is defined to adjust adaptively the weight between two fitting terms, which is based on the neighborhood contrast of image pixel. Finally, a weighted regularization term related to local entropy is used to ensure the smoothness of evolution curve. Meanwhile, a distance regularization term is added for stable level set evolution. Experimental results show that the proposed model without initial contour can segment inhomogeneous images stably and effectively, which thereby avoiding the influence of contour initialization on segmentation results. Besides, the proposed model works better on noise images comparing with two relevant segmentation models.     

广义全变差正则化图像去噪模型研究

本文研究了全变差正则化模型在图像去噪过程中易产生阶梯效应的问题,依据图像的局部结构特利用联合高斯滤波器和边缘检测算子的方法,构建了广义全变差正则化图像去噪模型,获得了在消除噪声的同时能够保留图像边缘细节和纹理信息的结果.实验结果表明,广义全变差正则化模型在平滑噪声的同时能够保留图像的边缘轮廓等细节信息,得到的复原图像在峰值信噪比、平均结构相似度和主观视觉效果方面均有所提高.     

Image segmentation by level set evolution with region consistency constraint

Image segmentation is a key and fundamental problem in image processing, computer graphics, and computer vision. Level set based method for image segmentation is used widely for its topology flexibility and proper mathematical formulation. However, poor performance of existing level set models on noisy images and weak boundary limit its application in image segmentation. In this paper, we present a region consistency constraint term to measure the regional consistency on both sides of the boundary, this term defines the boundary of the image within a range, and hence increases the stability of the level set model. The term can make existing level set models significantly improve the efficiency of the algorithms on segmenting images with noise and weak boundary. Furthermore, this constraint term can make edge-based level set model overcome the defect of sensitivity to the initial contour. The experimental results show that our algorithm is efficient for image segmentation and outperform the existing state-of-art methods regarding images with noise and weak boundary.     

A multi-frame super-resolution based on new variational data fidelity term

The main idea of multi-frame super-resolution (SR) algorithm is to recover a single high-resolution (HR) image from a sequence of low resolution ones of the same scene. Since the restoration step of super-resolution algorithms is always an ill-posed problem, the choice of the fidelity term and the regularization are always crucial. In this paper, we propose a new variational SR framework based on an automatic selection of the weighting parameter that control the balance between the L¹ and L² fidelity terms, which handle different type of noise distributions. Concerning the regularization, we use the combined total variation (TV) and the total variation of the first derivatives (TV²) model with a new implementation of the Primal-dual algorithm to solve the corresponding discretized problem. The obtained results are compared with some competitive algorithms and confirm that the proposed method has much benefices over the others in avoiding some undesirable artifacts.     

Adaptive total variation regularization based scheme for Poisson noise removal

To better preserve the edge features, this paper investigates an adaptive total variation regularization based variational model for removing Poisson noise. This edge‐preserving scheme comprises a spatially adaptive diffusivity coefficient, which adjusts the diffusion strength automatically. Compared with the classical total variation based one, numerical simulations distinctly indicate the superiority of our proposed strategy in maintaining the small details while denoising Poissonian image. Copyright © 2012 John Wiley & Sons, Ltd.     

A Retinex modulated piecewise constant variational model for image segmentation and bias correction

In this paper, we propose a novel Retinex induced piecewise constant variational model for simultaneous segmentation of images with intensity inhomogeneity and bias correction. Firstly, we obtain an additive model by decomposing the original image into a smooth bias component and a structure part based on the Retinex theory. Secondly, the structure part can be modeled by the piecewise constant variational model and thus deduced a new data fidelity term. Finally, we formulate a new energy functional by incorporating the data fidelity term into the level set framework and introducing a GL-regularizer to the level set function and a smooth regularizer to model the bias component. Based on the alternating minimization algorithm and the operator splitting method, we present a numerical scheme to solve the minimization problem efficiently. Experimental results on images from diverse modalities demonstrate the competitive performances of the proposed model and algorithm over other representative methods in term of efficiency and robustness.     

求解分段常数图象分割模型的一个快速算法

针对Xue-ChengTai等提出的分段常数图象分割模型,我们提出了一个新的快速求解算法。通过引进一个函数来选择模型中的正则化参数β的值,并判断在迭代过程中何时求解不含惩罚项的泛函F。此函数的引入有效地加速了算法的收敛速度。结合原始-对偶Newton方法来求解总变差最小化问题。数值试验表明新算法具有很快的收敛速度与良好的分割效果,且算法对初始值的要求不高。     

基于PICCS的非局部TGV能谱CT重建算法

能谱CT将宽谱划分为窄谱,导致通道内光子数目明显减少,加大了噪声影响,故从噪声投影中重建出高质量图像是能谱CT的一个研究热点.传统全变分(total variational,TV)容易造成重建图像中出现块状伪影等问题,总广义全变分(total generalized variation,TGV)算法可以逼近任意阶函数,再结合非局部均值算法的思想,同时考虑到不同能谱通道下重建图像的相关性,将高质量全能谱重建图像作为先验图像指导能谱CT重建,提出了基于先验图像约束压缩感知(prior image constrained compressed sensing,PICCS)的非局部TGV重建算法.实验结果表明,所提算法在抑制噪声的同时能够有效复原图像细节及边缘信息,且收敛速度快.     

A variational approach for restoring images corrupted by noisy blur kernels and additive noise

In this paper, we study a deblurring algorithm for distorted images by random impulse response. We propose and develop a convex optimization model to recover the underlying image and the blurring function simultaneously. The objective function is composed of 3 terms: the data‐fitting term between the observed image and the product of the estimated blurring function and the estimated image, the squared difference between the estimated blurring function and its mean, and the total variation regularization term for the estimated image. We theoretically show that under some mild conditions, the resulting objective function can be convex in which the global minimum value is unique. The numerical results confirm that the peak‐to‐signal‐noise‐ratio and structural similarity of the restored images by the proposed algorithm are the best when the proposed objective function is convex. We also present a proximal alternating minimization scheme to solve the resulting minimization problem. Numerical examples are presented to demonstrate the effectiveness of the proposed model and the efficiency of the numerical scheme.     

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.     

低分辨率医学图像的多图谱分割方法

基于多图谱的图像分割方法因其分割精度高和鲁棒性强,在医学图像分割领域被广泛研究,主要包含图像配准和标签融合两个步骤.目前对多图谱分割方法的研究通常都是在图谱图像和待分割目标图像具有相同分辨率的情况下展开的.然而,由于受图像采集时间,采集设备等影响,临床实践中采集的影像大多是低分辨率数据,使得目前在影像研究中广泛使用的方法无法有效应用于临床实践.因此,针对这一问题,我们结合图像超分辨率恢复方法,提出了精确鲁棒的低分辨率医学图像的多图谱分割方法,实验结果显示提出的方法显著地提高了多图谱分割方法的分割精度.     

Asymptotics of generalized derangements

An adaptive wavelet-based method is proposed for solving TV(total variation)–Allen–Cahn type models for multi-phase image segmentation. The adaptive algorithm integrates (i) grid adaptation based on a threshold of the sparse wavelet representation of the locally-structured solution; and (ii) effective finite difference on irregular stencils. The compactly supported interpolating-type wavelets enjoy very fast wavelet transforms, and act as a piecewise constant function filter. These lead to fairly sparse computational grids, and relax the stiffness of the nonlinear PDEs. Equipped with this algorithm, the proposed sharp interface model becomes very effective for multi-phase image segmentation. This method is also applied to image restoration and similar advantages are observed.     

Image restoration with a high-order total variation minimization method

In this paper, we propose a fast and efficient way to restore blurred and noisy images with a high-order total variation minimization technique. The proposed method is based on an alternating technique for image deblurring and denoising. It starts by finding an approximate image using a Tikhonov regularization method. This corresponds to a deblurring process with possible artifacts and noise remaining. In the denoising step, a high-order total variation algorithm is used to remove noise in the deblurred image. We see that the edges in the restored image can be preserved quite well and the staircase effect is reduced effectively in the proposed algorithm. We also discuss the convergence of the proposed regularization method. Some numerical results show that the proposed method gives restored images of higher quality than some existing total variation restoration methods such as the fast TV method and the modified TV method with the lagged diffusivity fixed-point iteration.     

Image denoising using the Gaussian curvature of the image surface

A number of high‐order variational models for image denoising have been proposed within the last few years. The main motivation behind these models is to fix problems such as the staircase effect and the loss of image contrast that the classical Rudin–Osher–Fatemi model [Leonid I. Rudin, Stanley Osher and Emad Fatemi, Nonlinear total variation based noise removal algorithms, Physica D 60 (1992), pp. 259–268] and others also based on the gradient of the image do have. In this work, we propose a new variational model for image denoising based on the Gaussian curvature of the image surface of a given image. We analytically study the proposed model to show why it preserves image contrast, recovers sharp edges, does not transform piecewise smooth functions into piecewise constant functions and is also able to preserve corners. In addition, we also provide two fast solvers for its numerical realization. Numerical experiments are shown to illustrate the good performance of the algorithms and test results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1066–1089, 2016     

On a New Diffeomorphic Multi-Modality Image Registration Model and Its Convergent Gauss-Newton Solver

In this work, we propose a new variational model for multi-modal image registration and present an efficient numerical implementation. The model minimizes a new functional based on using reformulated normalized gradients of the images as the fidelity term and higher-order derivatives as the regularizer. A key feature of the model is its ability of guaranteeing a diffeomorphic transformation which is achieved by a control term motivated by the quasi-conformal map and Beltrami coefficient. The existence of the solution of this model is established. To solve the model numerically, we design a Gauss-Newton method to solve the resulting discrete optimization problem and prove its convergence; a multilevel technique is employed to speed up the initialization and avoid likely local minima of the underlying functional. Finally, numerical experiments demonstrate that this new model can deliver good performances for multi-modal image registration and simultaneously generate an accurate diffeomorphic transformation.     

Nonlocal patches based Gaussian mixture model for image inpainting

We consider the inpainting problem for noisy images. It is very challenge to suppress noise when image inpainting is processed. An image patches based nonlocal variational method is proposed to simultaneously inpainting and denoising in this paper. Our approach is developed on an assumption that the small image patches should be obeyed a distribution which can be described by a high dimension Gaussian Mixture Model. By a maximum a posteriori (MAP) estimation, we formulate a new regularization term according to the log-likelihood function of the mixture model. To optimize this regularization term efficiently, we adopt the idea of the Expectation Maximization (EM) algorithm. In which, the expectation step can give an adaptive weighting function which can be regarded as a nonlocal connections among pixels. Using this fact, we built a framework for non-local image inpainting under noise. Moreover, we mathematically prove the existence of minimizer for the proposed inpainting model. By using a splitting algorithm, the proposed model are able to realize image inpainting and denoising simultaneously. Numerical results show that the proposed method can produce impressive reconstructed results when the inpainting region is rather large.     

A primal-dual active-set algorithm for bilaterally constrained total variation deblurring and piecewise constant Mumford-Shah segmentation problems

In this paper, we propose a fast primal-dual algorithm for solving bilaterally constrained total variation minimization problems which subsume the bilaterally constrained total variation image deblurring model and the two-phase piecewise constant Mumford-Shah image segmentation model. The presence of the bilateral constraints makes the optimality conditions of the primal-dual problem semi-smooth which can be solved by a semi-smooth Newton’s method superlinearly. But the linear system to solve at each iteration is very large and difficult to precondition. Using a primal-dual active-set strategy, we reduce the linear system to a much smaller and better structured one so that it can be solved efficiently by conjugate gradient with an approximate inverse preconditioner. Locally superlinear convergence results are derived for the proposed algorithm. Numerical experiments are also provided for both deblurring and segmentation problems. In particular, for the deblurring problem, we show that the addition of the bilateral constraints to the total variation model improves the quality of the solutions.