Generalized smoothing splines and the optimal discretization of the Wiener filter

We introduce an extended class of cardinal L/sup */L-splines, where L is a pseudo-differential operator satisfying some admissibility conditions. We show that the L/sup */L-spline signal interpolation problem is well posed and that its solution is the unique minimizer of the spline energy functional /spl par/Ls/spl par//sub L2//sup 2/, subject to the interpolation constraint. Next, we consider the corresponding regularized least squares estimation problem, which is more appropriate for dealing with noisy data. The criterion to be minimized is the sum of a quadratic data term, which forces the solution to be close to the input samples, and a "smoothness" term that privileges solutions with small spline energies. Here, too, we find that the optimal solution, among all possible functions, is a cardinal L/sup */L-spline. We show that this smoothing spline estimator has a stable representation in a B-spline-like basis and that its coefficients can be computed by digital filtering of the input signal. We describe an efficient recursive filtering algorithm that is applicable whenever the transfer function of L is rational (which corresponds to the case of exponential splines). We justify these algorithms statistically by establishing an equivalence between L/sup */L smoothing splines and the minimum mean square error (MMSE) estimation of a stationary signal corrupted by white Gaussian noise. In this model-based formulation, the optimum operator L is the whitening filter of the process, and the regularization parameter is proportional to the noise variance. Thus, the proposed formalism yields the optimal discretization of the classical Wiener filter, together with a fast recursive algorithm. It extends the standard Wiener solution by providing the optimal interpolation space. We also present a Bayesian interpretation of the algorithm.     

A stochastic framework for recursive computation of spline functions--Part I: Interpolating splines

The method for exploiting stochastic smoothing techniques to develop dynamical recursive algorithms for the deterministic problem of d interpolation (optimal curve fitting) is shown. A reproducing kernel Hilbert space approach is used to develop an explicit correspondence between spline interpolation and linear least-squares smoothing of a particular zero-mean random process. This random process is shown to be the output of a white-noise-driven dynamical system whose parameters and initial conditions are fixed by the functional form chosen for the spline. A recursive algorithm is then derived for this (nonstandard) smoothing problem, and thus also for the original spline interpolation problem.     

基于广义B样条滤波器的图像边缘检测

为了更准确有效地提取图像边缘,提出一种基于广义B样条数字滤波器的边缘检测算法.首先由线性微分方程推导出广义B样条的一般形式.其后,利用广义B样条函数组建了边缘检测微分算子模板,该模板继承了广义B样条尺度因子α,通过调节α,可以改变边缘算子滤波器的幅值特性和带通特性,进而获得边缘表征的最佳效果.而后,结合变分公式和广义B样条函数构造广义B样条光滑滤波器,该滤波器实现了在光滑滤波意义下的直接样条变换,与边缘检测模板算子配合使用可提高边缘检测的抗噪能力.实验证明：该边缘检测算法能有效地检测出图像边缘,无论是边缘提取效果还是抗干扰能力都优于传统的差分算子.     

Optimization of linear filters under power-spectral-densitystabilization

Geometric-mean filters compose a family of filters indexed by a parameter k varying between 0 and 1. They have been used to provide frequency-based filtering that mitigates the noise suppression of the optimal-linear Wiener filter in the blurred-signal-plus-noise model. For k=0 and k=1, the geometric-mean filter gives the inverse filter and the Wiener filter for the model, respectively. The geometric-mean for k=1/2 has previously been derived as the optimal linear filter for the model under power-spectral-density (PSD) equalization. This constraint requires the PSD of the filtered signal to be equal to the PSD of the uncorrupted signal that it estimates. This paper defines the notion of PSD stabilization, in which the PSD of the restored signal is equal to a predetermined function times the PSD of the uncorrupted signal. A particular parameterized stabilization function yields the geometric-mean family as the optimal linear filter for the model under PSD stabilization. Relative to unconstrained optimization, geometric-means are suboptimal; however, we consider a parameterized model for which the noise is such that the geometric-mean filters provide optimal linear filtering. In the altered signal-plus-noise model for which the geometric-mean is optimal, the blur is the same as the original model in which the geometric-mean is defined, but the noise PSD is a function of the Fourier transform of the blur and the PSD of the original noise. Since the altered model depends on k, we consider a robustness question: what kind of suboptimality results from applying the geometric-mean for k₁ to the model fur which the geometric-mean for k₂ is optimal?     

Image data compression using cubic convolution spline interpolation

A new cubic convolution spline interpolation (CCSI )for both one-dimensional (1-D) and two-dimensional (2-D) signals is developed in order to subsample signal and image compression data. The CCSI yields a very accurate algorithm for smoothing. It is also shown that this new and fast smoothing filter for CCSI can be used with the JPEG standard to design an improved JPEG encoder-decoder for a high compression ratio.     

Adaptive restoration of textured images with mixed spectra

We consider the adaptive restoration of inhomogeneous textured images, where the individual regions are modeled using a Wold-like decomposition. A generalized Wiener filter is developed to accommodate mixed spectra, and unsupervised restoration is achieved by using the expectation-maximization (EM) algorithm to estimate the degradation parameters. This algorithm yields superior results when compared with supervised Wiener filtering using autoregressive (AR) image models.     

基于HSV空间的彩色监控视频前景提取方法

提出了一种有效的HSV空间视频序列图像背景去除方法.首先,将彩色视频分解为彩色图像帧序列,然后将图像进行高斯滤波,做平滑处理,再利用surendra背景更新算法动态更新背景图像.将当前帧像素矩阵与背景矩阵做差分,通过比较差分矩阵来确定当前帧中前景的区域.实验结果表明,此方法成功地减除了彩色帧图像的背景,较好地保留了前景...     

LFM信号的一种最优滤波算法

本文提出了一种基于分数阶傅立叶变换的LFM信号的最优滤波算法.首先由线性最小均方误差估计的正交条件出发,得到了连续分数阶傅立叶域上的等效Wiener滤波算子的求解方法;在此基础上,进一步给出了滤波算子的离散化算法.分析及数值仿真的结果表明,这一算法不仅在性能上接近普通的Wiener滤波器,而且计算简单,便于实现.     

Design of linear equalizers optimized for the structural similarity index.

We propose an algorithm for designing linear equalizers that maximize the structural similarity (SSIM) index between the reference and restored signals. The SSIM index has enjoyed considerable application in the evaluation of image processing algorithms. Algorithms, however, have not been designed yet to explicitly optimize for this measure. The design of such an algorithm is nontrivial due to the nonconvex nature of the distortion measure. In this paper, we reformulate the nonconvex problem as a quasi-convex optimization problem, which admits a tractable solution. We compute the optimal solution in near closed form, with complexity of the resulting algorithm comparable to complexity of the linear minimum mean squared error (MMSE) solution, independent of the number of filter taps. To demonstrate the usefulness of the proposed algorithm, it is applied to restore images that have been blurred and corrupted with additive white gaussian noise. As a special case, we consider blur-free image denoising. In each case, its performance is compared to a locally adaptive linear MSE-optimal filter. We show that the images denoised and restored using the SSIM-optimal filter have higher SSIM index, and superior perceptual quality than those restored using the MSE-optimal adaptive linear filter. Through these results, we demonstrate that a) designing image processing algorithms, and, in particular, denoising and restoration-type algorithms, can yield significant gains over existing (in particular, linear MMSE-based) algorithms by optimizing them for perceptual distortion measures, and b) these gains may be obtained without significant increase in the computational complexity of the algorithm.     

Interactive Natural Image Segmentation via Spline Regression

This paper presents an interactive algorithm for segmentation of natural images. The task is formulated as a problem of spline regression, in which the spline is derived in Sobolev space and has a form of a combination of linear and Green's functions. Besides its nonlinear representation capability, one advantage of this spline in usage is that, once it has been constructed, no parameters need to be tuned to data. We define this spline on the user specified foreground and background pixels, and solve its parameters (the combination coefficients of functions) from a group of linear equations. To speed up spline construction, K-means clustering algorithm is employed to cluster the user specified pixels. By taking the cluster centers as representatives, this spline can be easily constructed. The foreground object is finally cut out from its background via spline interpolation. The computational complexity of the proposed algorithm is linear in the number of the pixels to be segmented. Experiments on diverse natural images, with comparison to existing algorithms, illustrate the validity of our method.     

多通道ARMA信号的三种多传感器信息融合Wiener滤波器

应用现代时间序列分析方法,基于ARMA新息模型、白噪声估值器和观测预报器,对带白色观测噪声的多通道ARMA信号,在线性最小方差最优信息融合准则下,提出了统一的和通用的按矩阵加权、按标量加权和按对角阵加权的多传感器信息融合Wiener滤波器,可统一处理滤波、平滑和预报问题．提出了计算局部估计误差方差和协方差的公式,它们被用于计算最优加权．同单传感器情形相比,可提高滤波精度．一个目标跟踪仿真例子说明了其有效性,且说明了三种加权融合滤波器的精度无显著差异,因而利用按标量加权融合滤波器以轻微的精度损失提供一种快速融合估计算法,便于实时应用．     

A transform domain SVD filter for suppression of muscle noise artefacts in exercise ECG's

The proposed filter assumes the noisy electrocardiography (ECG) to be modeled as a signal of deterministic nature, corrupted by additive muscle noise artefact. The muscle noise component is treated to be stationary with known second-order characteristics. Since noise-free ECG is shown to possess a narrow-band structure in discrete cosine transform (DCT) domain and the second-order statistical properties of the additive noise component is preserved due to the orthogonality property of DCT, noise abatement is easily accomplished via subspace decomposition in the transform domain. The subspace decomposition is performed using singular value decomposition (SVD). The order of the transform domain SVD filter required to achieve the desired degree of noise abatement is compared to that of a suboptimal Wiener filter using DCT. Since the Wiener filter assumes both the signal and noise structures to be statistical, with a priori known second-order characteristics, it yields a biased estimate of the ECG beat as compared to the SVD filter for a given value of mean-square error (mse). The filter order required for performing the subspace smoothing is shown to exceed a certain minimal value for which the mse profile of the SVD filter follows the minimum-mean-quare error (mmse) performance warranted by the suboptimal Wiener filter. The effective filter order required for reproducing clinically significant features in the noisy ECG is then set by an upper bound derived by means of a finite precision linear perturbation model. A significant advantage resulting from the application of the proposed SVD filter lies in its ability to perform noise suppression independently on a single lead ECG record with only a limited number of data samples.     

H optimal approximation for causal spline interpolation

In this paper, we give a causal solution to the problem of spline interpolation using H^∞ optimal approximation. Generally speaking, spline interpolation requires filtering the whole sampled data, the past and the future, to reconstruct the inter-sample values. This leads to non-causality of the filter, and this becomes a critical issue for real-time applications.Our objective here is to derive a causal system which approximates spline interpolation by H^∞ optimization for the filter. The advantage of H^∞ optimization is that it can address uncertainty in the input signals to be interpolated in design, and hence the optimized system has robustness property against signal uncertainty.We give a closed-form solution to the H^∞ optimization in the case of the cubic splines. For higher-order splines, the optimal filter can be effectively solved by a numerical computation. We also show that the optimal FIR (finite impulse response) filter can be designed by an LMI (linear matrix inequality), which can also be effectively solved numerically. A design example is presented to illustrate the result.     

小波变换域的局部自适应Wiener滤波器设计方法研究

小波阈值去噪方法被广泛地用在信号去噪中,这个方法在很多信号空间上是近似最优的。但在MSE意义上最优的信号估计是Wiener滤波器,鉴于传统小波变换域Wiener滤波器的缺点,本文设计了小波域局部自适应Wiener滤波器。实验仿真验证本方法具有较好的去噪效果。     

A rapid hybrid algorithm for image restoration combining parametric Wiener filtering and wave atom transform

Image restoration refers to removal or minimization of known degradations in an image. This includes de-blurring images degraded by the limitations of sensors or source of captures in addition to noise filtering and correction of geometric distortion due to sensors. There are several classical image restoration methods such as Wiener filtering. To find an estimate of the original image, Wiener filter requires the prior knowledge of the degradation phenomenon, the blurred image and the statistical properties of the noise process. In this work, we propose a new rapid and blind algorithm for image restoration that does not require a priori knowledge of the noise distribution. The degraded image is first de-convoluted in Fourier space by parametric Wiener filtering, and then, it is smoothed by the wave atom transform after setting the threshold to its coefficients. Experiment results are significant and show the efficiency of our algorithm compared with other techniques in use.     

Restricted structure optimal linear estimators

The restricted structure optimal deconvolution filtering, smoothing and prediction problem for multivariable, discrete-time linear signal processing problems is considered. A new class of discrete-time optimal linear estimators is introduced that directly minimises a minimum variance criterion but where the structure is prespecified to have a relatively simple form. The resulting estimator can be of much lower order than a Kalman or Wiener estimator and it minimises the estimation error variance, subject to the constraint referred to above. The numerical optimisation algorithm is simple to implement and the full-order optimal solutions are available as a by-product of the analysis. Moreover, the restricted structure solution may be used to compute both IIR and FIR estimators. A weighted H/sub 2/ cost-function is minimised, where the dynamic weighting function can be chosen for robustness improvement. The signal and noise sources can be correlated and the signal channel dynamics can be included in the system model. The algorithm enables low-order optimal estimators to be computed that directly minimise the cost index. The main technical advance is in the pre-processing, which enables the expanded cost expression to be simplified considerably before the numerical solution is obtained. The optimisation provides a direct minimisation over the unknown parameters for the particular estimator structure chosen. This should provide advantages over the simple approximation of a high-order optimal estimator. The results are demonstrated in the estimation of a signal heavily contaminated by both coloured and white noise.     

Symmetric alpha-stable filter theory

Symmetric α-stable (SαS) processes are used to model infinite-variance impulsive noise. In general, Wiener filter theory is not meaningful in (SαS) environments because the expectations may be unbounded. To develop a theory for linear finite impulse response systems with independent identically distributed (SαS) inputs, we propose median orthogonality as a linear filter criterion, derive a generalized Wiener-Hopf solution equation, and show a sufficient condition for a filter to achieve the criterion. For non-Gaussian (SαS) densities, zero-forcing least-mean-squares is the only well-known filter that satisfies the criterion, but others can be designed. We present a second algorithm and simulations showing that both converge to the generalized Wiener-Hopf solution     

Information fusion Wiener filter for the multisensor multichannel ARMA signals with time-delayed measurements

For the multisensor multichannel autoregressive moving average (ARMA) signals with time-delayed measurements, a measurement transformation approach is presented, which transforms the equivalent state space model with measurement delays into the state space model without measurement delays, and then using the Kalman filtering method, under the linear minimum variance optimal weighted fusion rules, three distributed optimal fusion Wiener filters weighted by matrices, diagonal matrices and scalars are presented, respectively, which can handle the fused filtering, prediction and smoothing problems. They are locally optimal and globally suboptimal. The accuracy of the fuser is higher than that of each local signal estimator. In order to compute the optimal weights, the formulae of computing the cross-covariances among local signal estimation errors are given. A Monte Carlo simulation example for the three-sensor target tracking system with time-delayed measurements shows their effectiveness.     

Propagated guided image filtering for edge-preserving smoothing

This paper proposes an edge-preserving smoothing filtering algorithm based on guided image filter (GF). GF is a well-known edge-preserving smoothing filter, but is ineffective in certain cases. The proposed GF enhancement provides a better solution for various noise levels associated with image degradation. In addition, halo artifacts, the main drawback of GF, are well suppressed using the proposed method. In our proposal, linear GF coefficients are updated sequentially in the spatial domain by using a new cost function, whose solution is a weighted average of the neighboring coefficients. The weights are determined differently depending on whether the pixels belong to the edge region, and become zero when a neighborhood pixel is located within a region separated from the center pixel. This propagation procedure is executed twice (from upper-left to lower-right, and vice versa) to obtain noise-free edges. Finally, the filtering output is computed using the updated coefficient values. The experimental results indicate that the proposed algorithm preserves edges better than the existing algorithms, while reducing halo artifacts even in highly noisy images. In addition, the algorithm is less sensitive to user parameters compared to GF and other modified GF algorithms.