Robust anisotropic diffusion

Relations between anisotropic diffusion and robust statistics are described in this paper. Specifically, we show that anisotropic diffusion can be seen as a robust estimation procedure that estimates a piecewise smooth image from a noisy input image. The "edge-stopping" function in the anisotropic diffusion equation is closely related to the error norm and influence function in the robust estimation framework. This connection leads to a new "edge-stopping" function based on Tukey's biweight robust estimator that preserves sharper boundaries than previous formulations and improves the automatic stopping of the diffusion. The robust statistical interpretation also provides a means for detecting the boundaries (edges) between the piecewise smooth regions in an image that has been smoothed with anisotropic diffusion. Additionally, we derive a relationship between anisotropic diffusion and regularization with line processes. Adding constraints on the spatial organization of the line processes allows us to develop new anisotropic diffusion equations that result in a qualitative improvement in the continuity of edges.     

Speckle reducing anisotropic diffusion

This paper provides the derivation of speckle reducing anisotropic diffusion (SRAD), a diffusion method tailored to ultrasonic and radar imaging applications. SRAD is the edge-sensitive diffusion for speckled images, in the same way that conventional anisotropic diffusion is the edge-sensitive diffusion for images corrupted with additive noise. We first show that the Lee and Frost filters can be cast as partial differential equations, and then we derive SRAD by allowing edge-sensitive anisotropic diffusion within this context. Just as the Lee (1980, 1981, 1986) and Frost (1982) filters utilize the coefficient of variation in adaptive filtering, SRAD exploits the instantaneous coefficient of variation, which is shown to be a function of the local gradient magnitude and Laplacian operators. We validate the new algorithm using both synthetic and real linear scan ultrasonic imagery of the carotid artery. We also demonstrate the algorithm performance with real SAR data. The performance measures obtained by means of computer simulation of carotid artery images are compared with three existing speckle reduction schemes. In the presence of speckle noise, speckle reducing anisotropic diffusion excels over the traditional speckle removal filters and over the conventional anisotropic diffusion method in terms of mean preservation, variance reduction, and edge localization.     

GVF-based anisotropic diffusion models.

In this paper, the gradient vector flow fields are introduced in image restoration. Within the context of flow fields, the shock filter, mean curvature flow, and Perona-Malik equation are reformulated. Many advantages over the original models can be obtained; these include numerical stability, large capture range, and high-order derivative estimation. In addition, a fairing process is introduced in the anisotropic diffusion, which contains a fourth-order derivative and is reformulated as the intrinsic Laplacian of curvature under the level set framework. By applying this fairing process, the shape boundaries will become more apparent. In order to overcome numerical errors, the intrinsic Laplacian of curvature is computed from the gradient vector flow fields instead of the observed images.     

Oriented speckle reducing anisotropic diffusion.

Ultrasound imaging systems provide the clinician with noninvasive, low-cost, and real-time images that can help them in diagnosis, planning, and therapy. However, although the human eye is able to derive the meaningful information from these images, automatic processing is very difficult due to noise and artifacts present in the image. The speckle reducing anisotropic diffusion filter was recently proposed to adapt the anisotropic diffusion filter to the characteristics of the speckle noise present in the ultrasound images and to facilitate automatic processing of images. We analyze the properties of the numerical scheme associated with this filter, using a semi-explicit scheme. We then extend the filter to a matrix anisotropic diffusion, allowing different levels of filtering across the image contours and in the principal curvature directions. We also show a relation between the local directional variance of the image intensity and the local geometry of the image, which can justify the choice of the gradient and the principal curvature directions as a basis for the diffusion matrix. Finally, different filtering techniques are compared on a 2-D synthetic image with two different levels of multiplicative noise and on a 3-D synthetic image of a Y-junction, and the new filter is applied on a 3-D real ultrasound image of the liver.     

Well-posed anisotropic diffusion for image denoising

A nonlinear iterative smoothing filter based on a second-order partial differential equation is introduced. It smooths out the image according to an anisotropic diffusion process. The approach is based on a smooth approximation of the total variation (TV) functional which overcomes the non-differentiability of the TV functional at the origin. In particular, the authors perform linear smoothing over smooth areas but selective smoothing over candidate edges. By relating the smoothing parameter to the time step, they arrive at a CFL condition which guarantees the causality of the discrete scheme. This allows the adoption of higher time discretisation steps, while ensuring the absence of artefacts deriving from the non-smooth behaviour of the TV functional at the origin. In particular, it is shown that the proposed approach avoids the typical staircase effects in smooth areas which occur in the standard time-marching TV scheme     

Processing textured surfaces via anisotropic geometric diffusion

A multiscale method in surface processing is presented which carries over image processing methodology based on nonlinear diffusion equations to the fairing of noisy, textured, parametric surfaces. The aim is to smooth noisy, triangulated surfaces and accompanying noisy textures-as they are delivered by new scanning technology-while enhancing geometric and texture features. For an initial textured surface a fairing method is described which simultaneously processes the texture and the surface. Considering an appropriate coupling of the two smoothing processes one can take advantage of the frequently present strong correlation between edge features in the texture and on the surface edges. The method is based on an anisotropic curvature evolution of the surface itself and an anisotropic diffusion on the processed surface applied to the texture. Here, the involved diffusion tensors depends on a regularized shape operator of the evolving surface and on regularized texture gradients. A spatial finite element discretization on arbitrary unstructured triangular grids and a semi-implicit finite difference discretization in time are the building blocks of the corresponding numerical algorithm. A normal projection is applied to the discrete propagation velocity to avoid tangential drifting in the surface evolution. Different applications underline the efficiency and flexibility of the presented surface processing tool.     

Image recovery using the anisotropic diffusion equation

A new approach for image recovery using the anisotropic diffusion equation is developed which is based on the first derivative of the signal in time embedded in family of images with different scales. The diffusion coefficient is determined as a function of the gradient of the signal convolved with a symmetric exponential filter. A new discrete realization is developed for the simultaneous removal of noise and preservation of edges.     

一种改进的各向异性扩散超声图像去噪算法

针对传统的各向异性扩散算法中扩散系数函数的平滑效果不好,扩散过程中扩散门限K的选取依靠经验确定,扩散过程对图像细节保护不足的问题,提出了一种改进的各向异性扩散算法。介绍了几种当前比较典型的各向异性扩散去噪算法;在典型算法分析的基础上提出了一种基于自适应中值滤波的改进扩散模型;根据扩散系数应满足的3个条件及经典的扩散系数函数,提出了改进的扩散模型中的改进扩散系数函数;提出了一种扩散门限K的自适应选取的方法。通过在改进的扩散模型中使用改进的扩散系数函数并结合扩散门限K的自适应选取,对超声图像进行去噪。实验结果表明,所提算法优于PM模型、Catte模型、王常虹算法等,去噪后图像的FOM值比PM模型高出3.34%,PSNR值比PM模型高出0.250 6。该算法在去除散斑噪声的同时有效保护了图像的细节及边缘,有助于医务人员对患病区域的准确诊断。     

Fractional-order anisotropic diffusion for image denoising.

This paper introduces a new class of fractional-order anisotropic diffusion equations for noise removal. These equations are Euler-Lagrange equations of a cost functional which is an increasing function of the absolute value of the fractional derivative of the image intensity function, so the proposed equations can be seen as generalizations of second-order and fourth-order anisotropic diffusion equations. We use the discrete Fourier transform to implement the numerical algorithm and give an iterative scheme in the frequency domain. It is one important aspect of the algorithm that it considers the input image as a periodic image. To overcome this problem, we use a folded algorithm by extending the image symmetrically about its borders. Finally, we list various numerical results on denoising real images. Experiments show that the proposed fractional-order anisotropic diffusion equations yield good visual effects and better signal-to-noise ratio.     

Behavioral analysis of anisotropic diffusion in image processing

In this paper, we analyze the behavior of the anisotropic diffusion model of Perona and Malik (1990). The main idea is to express the anisotropic diffusion equation as coming from a certain optimization problem, so its behavior can be analyzed based on the shape of the corresponding energy surface. We show that anisotropic diffusion is the steepest descent method for solving an energy minimization problem. It is demonstrated that an anisotropic diffusion is well posed when there exists a unique global minimum for the energy functional and that the ill posedness of a certain anisotropic diffusion is caused by the fact that its energy functional has an infinite number of global minima that are dense in the image space. We give a sufficient condition for an anisotropic diffusion to be well posed and a sufficient and necessary condition for it to be ill posed due to the dense global minima. The mechanism of smoothing and edge enhancement of anisotropic diffusion is illustrated through a particular orthogonal decomposition of the diffusion operator into two parts: one that diffuses tangentially to the edges and therefore acts as an anisotropic smoothing operator, and the other that flows normally to the edges and thus acts as an enhancement operator.     

A new noise erosion operator for anisotropic diffusion

A noise erosion operator based on partial differential equation (PDE) is introduced, which has an excellent ability of noise removal and edge preservation for two-dimensional (2D) gradient data. The operator is applied to estimate a new diffusion coefficient. Experimental results demonstrate that anisotropic diffusion based on this new erosion operator can efficiently reduce noise and sharpen object boundaries.     

基于各向异性扩散方程的图像对比度增强方法

讨论了光学图像中同时存在噪声与模糊时的对比度增强问题.构造了一种基于边缘定向扩散的各向异性非线性扩散方程来作为图像的光滑约束项,并根据模糊后的图像在边缘处相对清晰图像具有较大误差的事实,构造增强图像与原图像之间的非均匀逼近项,将此两项通过正则化参数联系起来,得到了一种图像对比度增强的正则化模型,并利用Euler方程将该模型转换成一种可以快速求解的各向异性非线性扩散模型.该模型能在抑制噪声的同时增强图像的边缘,在模型的解算上也不存在大型矩阵的存储与运算问题.数值计算结果表明,新方法适合于多种形式的模糊和不同程度的噪声污染,相对现有方法具有更好的边缘锐化能力和更高的清晰度,峰值信噪比比现有方法提高了1～2 dB,边缘保护指数也比现有方法有较大提高.     

Combined curvelet shrinkage and nonlinear anisotropic diffusion.

In this paper, a diffusion-based curvelet shrinkage is proposed for discontinuity-preserving denoising using a combination of a new tight frame of curvelets with a nonlinear diffusion scheme. In order to suppress the pseudo-Gibbs and curvelet-like artifacts, the conventional shrinkage results are further processed by a projected total variation diffusion, in which only the insignificant curvelet coefficients or high-frequency part of the signal are changed by use of a constrained projection. Numerical experiments from piecewise-smooth to textured images show good performances of the proposed method to recover the shape of edges and important detailed components, in comparison to some existing methods.     

Modes in anisotropic optical waveguides formed by diffusion

Solutions of the wave equation are obtained for planar guides whose dielectric constant ε is anisotropic and decreases exponentially in the direction perpendicular to that of wave propagation. Although for TE modes the solutions are quite similar to those obtained earlier for exponentially varying but isotropic ε, there are some singificant differences for TM modes. The condition for single-mode operation and the number of modes of such guides are given in terms of the exponential decay rate, the difference between the surface and bulk values of ε, and the wavelength. The results are illustrated for the cases of Se-diffused CdS and out-diffused LiNbO3.     

Image denoising via an adaptive weighted anisotropic diffusion

Multidimensional Systems and Signal Processing - This paper introduces an adaptive weighted anisotropic diffusion model for image denoising. A simple but efficient patch-based diffusivity function...     

Enhancement of X-ray angiogram images by adaptive anisotropic diffusion

A new angiogram image enhancement algorithm based on anisotropic diffusion is proposed. This novel method can adaptively choose the conductance parameter, which has great significance on the whole diffusion. Experimental results show that this new method is more effective compared with the original anisotropic diffusion.     

Robust anisotropic diffusion filter via robust spatial gradient estimation

Multidimensional Systems and Signal Processing - This paper aims to develop a robust anisotropic diffusion filter associated with a robust spatial gradient estimator for simultaneously removing the...     

结合Contourlet变换的各向异性扩散图像去噪

提出了一种结合Contourlet变换的各向异性扩散图像去噪算法.结合Contourlet变换和各向异性扩散的优点,通过Contourlet变换得到图像不同尺度的Contourlet系数矩阵,实现了建立在对图像多尺度分析基础上新的各向异性扩散算法.实验表明,新算法能够获得良好的视觉效果并且有效地提高了去噪图像的PSNR值,同时避免了传统的各向异性扩散出现的阶梯效应,更好地保留了图像的纹理和细节.     

基于模糊各向异性扩散的骨科CT图像去噪

针对X射线散射、量化噪声、电器噪声在人体骨骼CT图像中存在噪声污染等问题,提出一种基于模糊各向异性扩散的骨科CT图像去噪方法,可有效的降低骨骼图像的噪声。一幅清晰的CT图像对于骨折部位的确定与骨折程度的判断有着重要的意义。故要使骨骼图像细节明显,需要对骨骼图像降低噪声。实验表明该方法能够求得一个较为合理的梯度阈值,在除去图像噪声的同时能够较好的保留图像的边缘和细节。     

Flux-based anisotropic diffusion applied to enhancement of 3-D angiogram

We present a new approach to anisotropic diffusion based on a multidirectional diffusion flux. The diffusion flux is decomposed in an orthogonal basis, effectively enabling enhancement of contours as well as diffusion along the contours. To this end, we have selected a three-dimensional basis that depicts the directions of principal curvature and has an interesting interpretation in the context of the vessels. The diffusion function associated to each vector of the basis depends on the first-order derivative of the intensity in this direction, instead of the traditional norm of the smoothed gradient. Accordingly, we present the results of a restoration of computed tomography data of the liver.