A generic approach to diffusion filtering of matrix-fields

Summary Diffusion tensor magnetic resonance imaging, is a image acquisition method, that provides matrix- valued data, so-called matrix fields. Hence image processing tools for the filtering and analysis of these data types are in demand. In this article, we propose a generic framework that allows us to find the matrix-valued counterparts of the Perona–Malik PDEs with various diffusivity functions. To this end we extend the notion of derivatives and associated differential operators to matrix fields of symmetric matrices by adopting an operator-algebraic point of view. In order to solve these novel matrix-valued PDEs successfully we develop truly matrix-valued analogs to numerical solution schemes of the scalar setting. Numerical experiments performed on both synthetic and real world data substantiate the effectiveness of our novel matrix-valued Perona–Malik diffusion filters. The Dutch Organization NWO is gratefully acknowledged for financial support. The German Organization DFG is gratefully acknowledged for financial support.     

Active contour model for simultaneous MR image segmentation and denoising

In this paper, a new region-based active contour model is proposed for magnetic resonance image segmentation and denoising based on the global minimization framework and level set evolution. A new region fitting energy based on Nadaraya–Watson estimator and local image information is defined to enforce the curve evolution. By this improved region fitting term, the images with noise and intensity un-uniformity can be segmented and denoised. Inspired by the Perona–Malik diffusion equation, an edge-preserving regularization term is defined through the duality formulation to penalize the length of region boundaries. By this new regularization term, the edge information is utilized to improve the contour?s ability of capturing the edge and remaining smooth during the evolution. The energy functional of the proposed model is minimized by an efficient dual algorithm avoiding the inefficiency of the gradient descent method. Experiments on medical images demonstrate the proposed model provides a hybrid way to perform image segmentation and image denoising simultaneously.     

Regularization aspects in continuous-time model identification

This paper presents an analysis of some regularization aspects in continuous-time model identification. The study particulary focuses on linear filter methods and shows that filtering the data before estimating their derivatives corresponds to a regularized signal derivative estimation by minimizing a compound criterion whose expression is given explicitly. A new structure based on a null phase filter corresponding to a true regularization filter is proposed and allows to discuss the filter phase effects on parameter estimation by comparing its performances with those of the Poisson filter-based methods. Based on this analysis, a formulation of continuous-time model identification as a joint system input-output signal and model parameter estimation is suggested. In this framework, two linear filter methods are interpreted and a compound criterion is proposed in which the regularization is ensured by a model fitting measure, resulting in a new regularization filter structure for signal estimation.     

能量泛函正则化模型在图像恢复中的应用分析

目的 能量泛函正则化模型是图像恢复研究的热点。为使更多工程领域的研究者对正则化技术进行探索和应用,推动不适定问题的研究,对能量泛函正则化模型的进展进行了分析。方法 首先建立图像整体坐标与局部坐标的关系,分析图像恢复正则化模型的基本原理,给出并证明正则化模型各向同性与各向异性扩散定理。然后结合函数空间、图像分解和紧框架,评述能量泛函正则化模型国内外发展现状,并对正则化模型解的适定性进行分析。结果 推导出图像恢复正则化模型扩散基本原理,给出正则化模型通用表达式,讨论正则化模型存在的问题及未来的发展方向。结论 正则化技术在解决图像恢复、修复等反问题起着重要作用。目前,国内外学者对该问题的研究取得了一些成果,但许多理论问题有待进一步研究。     

自适应非局部数据保真项和双边总变分的图像去噪模型

针对常见去噪方法容易造成特定区域过度平滑、奇异结构残余噪声以及产生阶梯效应和对比度损失等问题,提出一种自适应非局部数据保真项和双边总变分的图像去噪模型,建立了自适应非局部正则化能量泛函和相应的变分框架。首先,对噪声图像利用自适应权值的非局部均值求得数据拟合项;其次,引入双边总变分正则化项,利用正则化系数来适度平衡数据拟合项和正则化项的影响;最后,通过能量函数最小化对不同的噪声统计快速求得最优解,从而达到降低残余噪声并纠正过度平滑的目的。通过理论分析和针对模拟噪声图像与真实噪声图像的实验结果表明,所提出的图像去噪模型能够较好地处理具有不同统计特性的图像噪声,与自适应非局部均值滤波去噪相比,所提算法的峰值信噪比（PSNR）值最多可以得到0.6 dB的改善;与全变分正则化图像去噪算法比较,所提算法的主观视觉效果明显更好,在去噪的同时图像纹理和边缘等细节信息保护得更好,PSNR值最多可以提高10 dB,而多尺度结构相似性度（MS-SSIM）指标可以提升0.3。因此,所提出的图像去噪模型可以在理论上更好地探讨如何合理处理噪声和图像内容本身的高频细节信息,在视频和图像分辨率提升等领域也具有良好的实际应用价值。     

An Adaptive Approach for the Segmentation and the TV-Filtering in the Optic Flow Estimation

We consider a variational model for the joint total variation filtering (TV) and the segmentation in the optic flow estimation. The model is based on a functional with a spatially varying regularization parameter to solve such an ill-posed problem. We present an adaptive approach based on a posteriori error indicators which allows us to select locally the optimal values of the diffusion coefficient in the functional. We show that the adaptive process applied to the linear part of the functional (with respect to the optic flow variable) fulfills the segmentation objective. Moreover, this adaptive approach provides an approximation of the Mumford–Shah functional in the sense of the \(\Gamma \)-convergence of a family of discrete energies. The simultaneous filtering and segmentation are achieved within this approach with accuracy and a reduced number of degrees of freedom, which improves each task to obtain a reliable optic flow estimation. We present some numerical simulations to show the performances of the method for the segmentation and the simultaneous segmentation-filtering.     

Robust anisotropic diffusion to produce enhanced statistical parametric map from noisy fMRI

This paper presents a new, simple, and elegant technique to obtain enhanced statistical parametric maps (SPMs) from noisy functional magnetic resonance imaging (fMRI) data. This technique is based on the robust anisotropic diffusion (RAD), a technique normally used as an edge-preserving filter. A direct application of the RAD to the fMRI data does not work, because in this case RAD would perform an edge-preserving filtering of the fMRI structural information, instead of enhancing its functional information. The RAD can be applied directly to SPM but, in this case, only a small improvement of the SPM quality can be achieved, because the originating fMRI is not taken into account. To overcome these difficulties, we propose to estimate the SPM from the noisy fMRI, compute the diffusion coefficients in the SPM space, and then perform the diffusion in the structural information-removed fMRI data using the coefficients previously computed. These steps are iterated until convergence. We have tested the new technique in both simulated and real fMRI images, yielding surprisingly sharp and noiseless SPMs with increased statistical significance. We also describe how to automatically estimate an appropriate scale parameter.     

基于各向异性全变分的迭代滤波算法

空间邻近度和像素值相似度的双边滤波(BF)器在滤波时,由于其值域滤波核系数的计 算易受到噪声的干扰,在噪声水平较大时,直接使用噪声图像来指导核函数权值计算的方案不可行。 为此,提出一种结合各向异性全变分和BF 的图像去噪算法,将各向异性全变分算法与BF 算法相结 合,首先利用各向异性全变分算法对噪声图像进行处理,得到一幅边缘结构信息较为丰富的结果图 像,接着将该结果图像作为BF 算法的引导图像来指导值域滤波核系数的计算,为保证算法的稳定 性,对上述过程进行迭代处理。此外,为提高各向异性全变分算法的计算效率,引入了Split Bregman 迭代算法进行加速处理。实验表明,该算法能在较好去噪的同时,保留较多的边缘结构信息。     

Delay-dependent – filter design for stochastic time-delay systems

This paper is concerned with the problem of – filter design for a class of stochastic time-delay systems. A delay-dependent sufficient condition is presented, which guarantees the existence of a linear filter ensuring that the filtering error system is stochastically asymptotically stable and its – performance satisfies a prescribed level. A desired filter can be constructed by solving certain linear matrix inequalities. A numerical example is given to demonstrate the effectiveness of the proposed method.     

Filtering of Linear Partially Observe Stochastic Systems: The Fuzzy Logic Approach

In this paper we consider the filtering problem for a class of partially observed systems governed by linear stochastic differential equations. Using Takagi–Sugeno linear fuzzy model and assuming membership functions of Gaussian type, we have proposed minimum unbiased linear fuzzy filter, driven by the observed process, with the help of which the system states can be estimated from the observed data. Further, using calculus of variation, we have developed a set of necessary conditions of optimality on the basis of which filter parameters can be determined. Finally, numerical simulations are presented to illustrate the effectiveness of the proposed filter.     

A coarsening finite element strategy in image selective smoothing

We introduce the coarsening of finite element computational grid strategy into the method proposed by Kačur and Mikula in [11] for the numerical solution of the Perona–Malik model modified in the sense of Catté, Lions, Morel and Coll. It improves the efficiency of the method, while the solution tends to be more flat during the selective smoothing process applied to the image. The numerical approximation consists of Rothe's method in time and linear finite elements on unstructured adaptively coarsened meshes in space. Received: 2 September 1996 / Accepted: 24 February 1997     

The Perona–Malik Method as an Edge Pruning Algorithm

The Perona–Malik equation (PM), in the continuum limit, is interpreted as the gradient flow for a functional, corresponding to the reconstruction of an image with edges with non-zero thickness. This result is based on an image model (u,Γ) where Γ is an edge set, and u is a slowly-varying function. PM simplifies the image by reducing the jump across each component of Γ, resulting in an automatic edge pruning procedure. The initial-value problem thus defined is well-posed, but practically stable only for small times: it leads to a semi-group with exponential growth. The rigorous analysis gives a mathematical basis for empirical observations, including edge localization and the need to use a small number of iterations. The variational formulation enables an easy comparison with earlier methods.

Regularization, Scale-Space, and Edge Detection Filters

Computational vision often needs to deal with derivatives ofdigital images. Such derivatives are not intrinsic properties ofdigital data; a paradigm is required to make them well-defined.Normally, a linear filtering is applied. This can be formulated interms of scale-space, functional minimization, or edge detectionfilters. The main emphasis of this paper is to connect these theoriesin order to gain insight in their similarities and differences. We donot want, in this paper, to take part in any discussion of how edgedetection must be performed, but will only link some of the current theories. We take regularization (or functional minimization) as astarting point, and show that it boils down to Gaussian scale-space ifwe require scale invariance and a semi-group constraint to besatisfied. This regularization implies the minimization of afunctional containing terms up to infinite order of differentiation.If the functional is truncated at second order, the Canny-Deriche filter arises. It is also shown that higher dimensional regularizationboils down to a rotated version of the one dimensional case, whenCartesian invariance is imposed and the image is vanishing at theborders. This means that the results from 1D regularization can beeasily generalized to higher dimensions. Finally we show how anefficient implementation of regularization of order n can be made byrecursive filtering using 2n multiplications and additions peroutput element without introducing any approximation.     

Admissible factorizations of Hankel operators induce well-posed linear systems

One of the basic axioms of a well-posed linear system says that the Hankel operator of the input–output map of the system factors into the product of the input map and the output map. Here we prove the converse: every factorization of the Hankel operator of a bounded causal time-invariant map from L² to L² which satisfies a certain admissibility condition induces a stable well-posed linear system. In particular, there is a one-to-one correspondence between the set of all minimal stable well-posed realizations of a given stable causal time-invariant input–output map (or equivalently, of a given H^∞ transfer function) and all minimal stable admissible factorizations of the Hankel operator of this input–output map.     

基于量测迭代更新集合卡尔曼滤波的机动目标跟踪算法

在机动目标跟踪中,用于模型辨识和状态估计的非线性滤波器的合理选择和优化是提升滤波精度的关键.融合量测迭代更新集合卡尔曼滤波和交互式多模型(interacting multiple models,IMM)方法,本文提出了基于量测迭代更新集合卡尔曼滤波的机动目标跟踪算法.通过迭代更新思想的引入构建了一种量测迭代更新下集合卡尔曼滤波的实现结构,并将其作为IMM的模型滤波器实现对于目标运动模式和状态的辨识与估计.针对算法结合过程中滤波精度和计算量的平衡,设计了用于输入交互环节的状态估计样本,同时简化输入交互环节和输出交互环节中滤波误差协方差矩阵的交互过程.理论分析和仿真结果验证了算法的可行性和有效性.     

Local and Nonlocal Discrete Regularization on Weighted Graphs for Image and Mesh Processing

We propose a discrete regularization framework on weighted graphs of arbitrary topology, which unifies local and nonlocal processing of images, meshes, and more generally discrete data. The approach considers the problem as a variational one, which consists in minimizing a weighted sum of two energy terms: a regularization one that uses the discrete p-Dirichlet form, and an approximation one. The proposed model is parametrized by the degree p of regularity, by the graph structure and by the weight function. The minimization solution leads to a family of simple linear and nonlinear processing methods. In particular, this family includes the exact expression or the discrete version of several neighborhood filters, such as the bilateral and the nonlocal means filter. In the context of images, local and nonlocal regularizations, based on the total variation models, are the continuous analog of the proposed model. Indirectly and naturally, it provides a discrete extension of these regularization methods for any discrete data or functions.     

Morphological Amoebas Are Self-snakes

This paper is concerned with amoeba median filtering, a structure-adaptive morphological image filter. It has been introduced by Lerallut et al. in a discrete formulation. Experimental evidence shows that iterated amoeba median filtering leads to segmentation-like results that are similar to those obtained by self-snakes, an image filter based on a partial differential equation. We establish this correspondence by analysing a space-continuous formulation of iterated amoeba median filtering. We prove that in the limit of vanishing radius of the structuring elements, iterated amoeba median filtering indeed approximates the partial differential equation of self-snakes. This result holds true under very general assumptions on the metric used to construct the amoebas. We present experiments with discrete iterated amoeba median filtering that confirm qualitative and quantitative predictions of our analysis.     

Hierarchical bayesian models for regularisation in sequential learning

We show that a hierarchical Bayesian modeling approach allows us to perform regularization in sequential learning. We identify three inference levels within this hierarchy: model selection, parameter estimation, and noise estimation. In environments where data arrive sequentially, techniques such as cross validation to achieve regularization or model selection are not possible. The Bayesian approach, with extended Kalman filtering at the parameter estimation level, allows for regularization within a minimum variance framework. A multilayer perceptron is used to generate the extended Kalman filter nonlinear measurements mapping. We describe several algorithms at the noise estimation level that allow us to implement on-line regularization. We also show the theoretical links between adaptive noise estimation in extended Kalman filtering, multiple adaptive learning rates, and multiple smoothing regularization coefficients.     

Stereo matching using iterative reliable disparity map expansion in the color–spatial–disparity space

In this paper, we propose a new stereo matching algorithm using an iterated graph cuts and mean shift filtering technique. Our algorithm estimates the disparity map progressively through the following two steps. In the first step, with a previously estimated RDM (reliable disparity map) that consists of sparse ground control points, an updated dense disparity map is constructed through a RDM constrained energy minimization framework that can cope with occlusion. The graph cuts technique is employed for the solution of the proposed energy model. In the second step, more accurate and denser RDM is estimated through the disparity crosschecking technique and the mean shift filtering in the CSD (color–spatial–disparity) space. The proposed algorithm expands the reliable disparities in RDM repeatedly through the above two steps until it converges. Experimental results on the standard data set demonstrate that the proposed algorithm achieves comparable performance to the state-of-the-arts, and gives excellent results especially in the areas such as the disparity discontinuous boundaries and occluded regions, where the conventional methods usually suffer.     

Locally adaptive image denoising by a statistical multiresolution criterion

It is shown how to choose the smoothing parameter in image denoising by a statistical multiresolution criterion, both globally and locally. Using inhomogeneous diffusion and total variation regularization as examples for localized regularization schemes, an efficient method for locally adaptive image denoising is presented. As expected, the smoothing parameter serves as an edge detector in this framework. Numerical examples together with applications in confocal microscopy illustrate the usefulness of the approach.