A new variational model for removal of combined additive and multiplicative noise and a fast algorithm for its numerical approximation

Variational image restoration models for both additive and multiplicative noise (MN) removal are rarely encountered in the literature. This paper proposes a new variational model and a fast algorithm for its numerical approximation to remove independent additive and MN from digital images. Two previous works by L. Rudin, S. Osher, and E. Fatemi [Nonlinear total variation based noise removal algorithms, Phys. D 60 (1992), pp. 259–268] and Z. Jin and X. Yang [Analysis of a new variational model for multiplicative noise removal, J. Math. Anal. Appl. 362 (2010), pp. 415–426] are used to develop the new model. As a result, developing a fast numerical algorithm is difficult because the associated Euler–Lagrange equation is highly nonlinear and standard unilevel iterative methods are not appropriate. To this end, we develop an efficient nonlinear multigrid approach via a robust fixed-point smoother. Numerical tests using both synthetic and realistic images not only confirm that our new model delivers quality results but also that the proposed numerical algorithm allows a very fast numerical realization of the model.     

Edge preserving mixed noise removal

To faithfully recover the clean images corrupted by additive white Gaussian noise (AWGN) and impulse noise (IN), a novel edge preserving image denoising algorithm is proposed. The low- and high-frequency components of the image are restored separately. The high-frequency components of the images are restored based on nonlocal self-similarity (NSS) learning from natural images. An energy minimization function is developed to combine the low- and high-frequency components into one model. Experiments demonstrate that the proposed method outperforms existing mixture noise removal methods in peak signal-to-noise ratio (PSNR), edges preservation and visual performance.

     

保持泊松噪声图像细节的快速变分去噪算法

去除医学、天文图像中的泊松噪声一直是人们关注的热点问题之一。在充分分析泊松去噪[α]-Le模型的基础上结合交替方向乘子（ADMM）算法,给出该模型一基于框式约束的快速求解算法,并证明了该算法的收敛性。数值实验结果表明,该算法在去噪的同时,不仅能很好地保留图像中的边缘及小细节特征,还能大幅提高运算效率。     

A new cluster based adaptive fuzzy switching median filter for impulse noise removal

A new clustering technique based on most allied directional neighbors is proposed to suppress low and high-density impulse noise from digital images. Most allied neighbors exhibit a vital role in estimation as well restoration of appropriate gray level value of corrupted pixels. In first phase, most allied directional neighbors, i.e., pixels directly attached to central pixel and the directional pixels (horizontal, vertical and two diagonal directions) next to attached pixels in the processing window are partitioned into two equal size clusters based on gradient values. Cluster with a minimum sum of gradient values (most similar neighbors) and the one with relatively large gradient values are passed to fuzzy inference system to infer the current pixel to be noisy-free, edge or a noisy. In second phase, a switching technique opts one of the three options depending upon fuzzy membership degrees and local information to restore the corrupted pixel value. A non-parametric approach based on local information for dynamic threshold setting using fuzzy logic makes the proposed filter computationally effective and adaptive to process a large number of images without user-defined parameters. The proposed algorithm is simple to implement and simulation results based on well know quantitative measures indicate the supremacy of the proposed filter for random-valued impulse noise as well as salt and peppers noise.     

基于模糊逻辑的脉冲噪声自适应滤除

为了去除具有多个随机值的普通脉冲噪声,提出一种基于模糊逻辑的自适应去噪方法。首先,利用像素邻域梯度并结合模糊逻辑相关理论进行噪声点初步检测;然后,通过噪声直方图统计对噪声点作二次检测,排除误判的噪声点;最后,保留非噪声点,对噪声点进行自适应加权均值滤波,滤除噪声。仿真结果表明,该算法可以取得优于传统方法的去噪效果,并且能很好地保护图像的边缘和纹理信息。     

基于粗集的图像混合噪声滤波算法

实际应用中图像常受到不同类型噪声的同时干扰,为解决混合噪声图像的去噪问题,提出了一种基于粗集的图像滤波算法.该算法将粗集理论应用到图像滤波中,利用粗集理论的等价关系分离出不同噪声点及非噪声点,根据图像被污染的情况,选取不同的去噪策略对噪声进行滤除.实验结果表明,该算法在抑制混合噪声时性能优于传统中值滤波、均值滤波等方法,同时也能较好地保护细节信息.     

去除乘性噪声的分数阶变分模型及算法

目的 针对变分方法在去除图像乘性噪声时易产生“阶梯效应”的问题,分析研究了几种经典乘性去噪变分模型的特性和相关性,在此基础上考虑到分数阶微分的频率特性,提出一种用于去除乘性Gamma噪声的分数阶凸变分模型。方法 提出的分数阶凸变分模型是经典I-divergence变分模型的分数阶扩展。基于对偶理论,提出一种用于求解该模型的分数阶原始对偶算法。并且基于鞍点理论,分析了确保算法收敛的参数取值范围。结果 实验中从频域角度分析并验证了提出的分数阶变分模型较经典的一阶变分模型能够有效缓解“阶梯效应”现象,更好地保持图像的中频纹理和高频边缘信息。同时提出的分数阶原始对偶数值算法能有效收敛,且收敛速度较快。结论 本文提出了一种去除图像乘性噪声的分数阶变分模型,该模型可采用一种基于预解式的原始对偶算法求解。实验结果表明,提出的模型能有效改善图像的视觉效果,采用的数值算法能有效快速收敛。     

Tropical algebra based adaptive filter for noise removal in digital image

Multimedia Tools and Applications - The concept of the tropical algebra was first introduced to solve problems in mathematical economy such as optimization and approximation problems. In this...     

一种变分自适应中值滤波算法

针对自适应中值和变分滤波方法脉冲噪声去除能力的不足,提出了一种新的变分自适应中值滤波方法。首先采用自适应中值滤波器对脉冲噪点进行标识,然后对标识的噪点构建由逼近条件和边缘保持正则化条件构成的代价函数,通过变分方法对代价函数寻优求解,对噪点进行恢复。最后进行了仿真试验,并与标准中值滤波,开关中值滤波,自适应中值滤波,和变分滤波方法进行了比较。试验结果表明,在信噪比和细节保留方面明显优于上述滤波方法,可以有效去除高达90%的脉冲噪声。     

Dissimilar pixel counting based impulse detector for two-phase mixed noise removal

Mixed noise is a challenging noise model due to its statistical complexity. A new two-phase denoising method based on an impulse detector using dissimilar pixel counting is proposed in this paper. This method consists of two stages: detection and filtering. For the detection phase, average difference scheme is proposed to distinguish whether two neighboring pixels are similar or not, and then the number of dissimilar pixels is compared with a threshold to locate the outlier point in noisy image. An iterative framework is used for detection accuracy with the least numbers of iteration. For the filtering phase, an extended trilateral filter is used to remove the mixture of Gaussian and impulse noise, which are treated differently depending on the guidance matrix from the detection phase. Extensive experimental results demonstrate that the proposed method exhibits better noise detection capability and outperforms many existing two-phase mixed noise removal methods in both quantitative evaluation and visual quality.

     

结合自适应核回归和全变差的乘性噪声去除

为了更好地去除图像中的乘性噪声,提出一个新的三阶段乘性噪声去除算法。第一阶段在图像的对数域用自适应的掌舵核回归(SKR)对图像进行去噪处理;第二阶段用全变差(TV)方法对第一阶段处理的结果进行补充处理;第三阶段通过指数变换和误差纠偏,把图像变回到真实的图像域。新方法具有掌舵核回归与全变差两种方法的优点,实验结果证明了其去除乘性噪声的有效性。     

Multiplicative noise removal via adaptive learned dictionaries and TV regularization

Multiplicative noise removal is a key issue in image processing problem. While a large amount of literature on this subject are total variation (TV)-based and wavelet-based methods, recently sparse representation of images has shown to be efficient approach for image restoration. TV regularization is efficient to restore cartoon images while dictionaries are well adapted to textures and some tricky structures. Following this idea, in this paper, we propose an approach that combines the advantages of sparse representation over dictionary learning and TV regularization method. The method is proposed to solve multiplicative noise removal problem by minimizing the energy functional, which is composed of the data-fidelity term, a sparse representation prior over adaptive learned dictionaries, and TV regularization term. The optimization problem can be efficiently solved by the split Bregman algorithm. Experimental results validate that the proposed model has a superior performance than many recent methods, in terms of peak signal-to-noise ratio, mean absolute-deviation error, mean structure similarity, and subjective visual quality.     

含脉冲噪声模糊图像复原的自适应二阶变分模型 *

针对经典的基于L1数据保真项的总变分图像复原模型易导致阶梯效应和损失图像重要细节的缺陷,提出了一种基于L1数据保真项的二阶总广义变分(Total Generalized Variation, TGV)图像复原模型。为进一步提升含脉冲噪声模糊图像复原质量,在二阶TGV图像复原模型中引入边缘检测算子,使其在图像边缘区域减弱扩散,较好地保护图像边缘特征;在图像平滑区域增强扩散,有效地消除脉冲噪声和抑制阶梯效应。为稳定地复原降质图像,采用交替方向乘子法求解二阶变分模型。实验结果表明,提出的图像复原模型在消除噪声和模糊的同时,能成功抑制阶梯效应并保留图像的边缘结构特征。相比经典的图像复原模型,新模型在信噪比、相对误差和结构相似度等方面均取得了较好的图像复原效果。     

自适应分数阶变分去泊松噪声新模型

目的 针对传统总变分方法在去除泊松噪声时容易出现“阶梯效应”和图像边缘模糊的问题,提出了一种基于分数阶变分的自适应去泊松噪声新模型。方法 首先新模型在分析了泊松噪声分布特点的基础上导出了非凸自适应正则项,它能够根据图像不同区域的特点自适应地调节正则项系数,以达到保持图像边缘的目的。然后,新模型利用分数阶离散微分向量能够结合更多图像信息的特点,将正则项中的一阶离散微分向量替换为分数阶离散微分向量,以此来达到抑制“阶梯效应”的目的。对于新模型的求解,结合交替迭代法和加权原始-对偶法提出了一种高效的数值解法。结果 新模型明显优于传统总变分去泊松噪声模型,在有效抑制“阶梯效应”的同时图像边缘也得到了较好地保护,以经典的Peppers图片为例,新模型相比于传统模型,峰值信噪比（PSNR）由28.98 dB提高到了30.24 dB,图像结构相似度（SSIM）由0.77提高到了0.87。另外,所提的数值解法具有收敛速度快、复杂度低的特点,收敛时间从偏微分方程、Chambolle投影等传统数值解法的0.5 s与0.1 s缩短至0.056 s。结论 实验结果表明,所提模型与数值解法的可行性,模型与数值解法在主要客观评价指标和图像视觉效果方面均优于传统的变分去泊松噪声模型,且模型与数值解法具有较好的普适性。但是模型中分数阶的阶次选取有待进一步优化。     

A mixed variational formulation for the navier-stokes problem with hyper-dissipation

A mixed variational formulation is used to solve the stationary Navier-Stokes equations with hyper dissipation. In this formulation, the laplacian of the velocity, the velocity and the pressure are the most relevant unknowns. For the linear case, the existence and uniqueness results for this mixed formulation are proved. Then, numerical results are presented for the nonlinear case.     

A mixed noise removal algorithm based on multi-fidelity modeling with nonsmooth and nonconvex regularization

Multimedia Tools and Applications - In this article, we propose a mixed-noise removal model which incorporates with a nonsmooth and nonconvex regularizer. To solve this model, a multistage convex...     

A mixed finite element for shell model with free edge boundary conditions Part 1. The mixed variational formulation

After a very brief recall on general shell theory, we construct a mixed variational formulation based on the introduction of a new unknown: the rotation of the normal to the medium surface. In Koiter shell theory (for instance), this rotation can be expressed with respect to the three components of the displacement field of the medium surface and their derivatives. The Lagrange multiplier corresponding to this relation (known as the Kirchhoff-Love kinematical assumption), is also introduced as an independent unknown. There are two main difficulties: one is due to the differential geometry of surfaces and is rather technical; the other is to define correctly the dual space for the Kirchhoff-Love relation. The difficulty is similar to the one met in the characterization of the dual space of the Sobolev space: H¹(ω) (ω being the medium surface of the shell), for which a boundary component appears except for clamped shells which is a very restrictive situation.     

自适应线性预测图像椒盐噪声去除方法

针对灰度图像中的椒盐噪声,提出了一种基于自适应权值与线性预测方法相结合的噪声去除方法。用椒盐噪声模型确定图像中的噪声点,以及噪声点所在滤波窗口内的噪声密度,在密度较小时利用设计的权值函数进行加权平均计算,以达到去噪声的目的;在噪声密度较大时,利用线性预测方法预测噪声点所在位置的灰度值,以实现去噪。对于非噪声点像素则不做处理,较好地保持了图像的细节。实验结果表明,与同类方法相比,此方法有良好的去噪性能。     

A model for radar images and its application to adaptive digital filtering of multiplicative noise

Standard image processing techniques which are used to enhance noncoherent optically produced images are not applicable to radar images due to the coherent nature of the radar imaging process. A model for the radar imaging process is derived in this paper and a method for smoothing noisy radar images is also presented. The imaging model shows that the radar image is corrupted by multiplicative noise. The model leads to the functional form of an optimum (minimum MSE) filter for smoothing radar images. By using locally estimated parameter values the filter is made adaptive so that it provides minimum MSE estimates inside homogeneous areas of an image while preserving the edge structure. It is shown that the filter can be easily implemented in the spatial domain and is computationally efficient. The performance of the adaptive filter is compared (qualitatively and quantitatively) with several standard filters using real and simulated radar images.     

A heat mixed variational theorem for thermoelastic multilayered composites

This paper presents for thermoelastic media the analogue of the Reissner’s mixed variational theorem (RMVT), initially developed for elastic media, by adding the transverse thermal field–temperature increment relation as a constraint via a Lagrange multiplier. The latter is shown to be the transverse (normal) heat flux, which continuity at the laminated composite interfaces can now be fulfilled in a natural way as is the case for transverse stresses in the RMVT. Hence, the resulting new heat mixed variational theorem (HMVT) will be well suited for modelling thermoelastic layered composites. Mixed coupled thermoelastic constitutive equations to be used in conjunction with the HMVT are also presented. Finally, discussions are made on the application of the HMVT to the quasi-static and steady-state classical cases leading to uncoupled heat transfer and thermal-stress analyses.