SURE-LET multichannel image denoising: interscale orthonormal wavelet thresholding.

We propose a vector/matrix extension of our denoising algorithm initially developed for grayscale images, in order to efficiently process multichannel (e.g., color) images. This work follows our recently published SURE-LET approach where the denoising algorithm is parameterized as a linear expansion of thresholds (LET) and optimized using Stein's unbiased risk estimate (SURE). The proposed wavelet thresholding function is pointwise and depends on the coefficients of same location in the other channels, as well as on their parents in the coarser wavelet subband. A nonredundant, orthonormal, wavelet transform is first applied to the noisy data, followed by the (subband-dependent) vector-valued thresholding of individual multichannel wavelet coefficients which are finally brought back to the image domain by inverse wavelet transform. Extensive comparisons with the state-of-the-art multiresolution image denoising algorithms indicate that despite being nonredundant, our algorithm matches the quality of the best redundant approaches, while maintaining a high computational efficiency and a low CPU/memory consumption. An online Java demo illustrates these assertions.     

Fast interscale wavelet denoising of Poisson-corrupted images

We present a fast algorithm for image restoration in the presence of Poisson noise. Our approach is based on (1) the minimization of an unbiased estimate of the MSE for Poisson noise, (2) a linear parametrization of the denoising process and (3) the preservation of Poisson statistics across scales within the Haar DWT. The minimization of the MSE estimate is performed independently in each wavelet subband, but this is equivalent to a global image-domain MSE minimization, thanks to the orthogonality of Haar wavelets. This is an important difference with standard Poisson noise-removal methods, in particular those that rely on a non-linear preprocessing of the data to stabilize the variance.Our non-redundant interscale wavelet thresholding outperforms standard variance-stabilizing schemes, even when the latter are applied in a translation-invariant setting (cycle-spinning). It also achieves a quality similar to a state-of-the-art multiscale method that was specially developed for Poisson data. Considering that the computational complexity of our method is orders of magnitude lower, it is a very competitive alternative.The proposed approach is particularly promising in the context of low signal intensities and/or large data sets. This is illustrated experimentally with the denoising of low-count fluorescence micrographs of a biological sample.     

Optimal denoising in redundant representations

Image denoising methods are often designed to minimize mean-squared error (MSE) within the subbands of a multiscale decomposition. However, most high-quality denoising results have been obtained with overcomplete representations, for which minimization of MSE in the subband domain does not guarantee optimal MSE performance in the image domain. We prove that, despite this suboptimality, the expected image-domain MSE resulting from applying estimators to subbands that are made redundant through spatial replication of basis functions (e.g., cycle spinning) is always less than or equal to that resulting from applying the same estimators to the original nonredundant representation. In addition, we show that it is possible to further exploit overcompleteness by jointly optimizing the subband estimators for image-domain MSE. We develop an extended version of Stein's unbiased risk estimate (SURE) that allows us to perform this optimization adaptively, for each observed noisy image. We demonstrate this methodology using a new class of estimator formed from linear combinations of localized "bump" functions that are applied either pointwise or on local neighborhoods of subband coefficients. We show through simulations that the performance of these estimators applied to overcomplete subbands and optimized for image-domain MSE is substantially better than that obtained when they are optimized within each subband. This performance is, in turn, substantially better than that obtained when they are optimized for use on a nonredundant representation.     

Microarray image enhancement by denoising using decimated and undecimated multiwavelet transforms

In this paper, we present a new approach to deal with the noise inherent in the microarray image processing procedure. We use the denoising capabilities of decimated and undecimated multiwavelet transforms, DMWT and UMWT respectively, for the removal of noise from microarray data. Multiwavelet transforms, with appropriate initialization, provide sparser representation of signals than wavelet transforms so that their difference from noise can be clearly identified. Also, the redundancy of the UMWT transform is particularly useful in image denoising in order to capture the salient features such as noise or transients. We compare this method with the discrete and stationary wavelet transforms, denoted by DWT and SWT, respectively, and the Wiener filter for denoising microarray images. Results show enhanced image quality using the proposed approach, especially in the undecimated case in which the results are comparable and often outperform that of the stationary wavelet transform. Both multiwavelet transforms outperform the DWT and the Wiener filter.     

A new SURE approach to image denoising: interscale orthonormal wavelet thresholding.

This paper introduces a new approach to orthonormal wavelet image denoising. Instead of postulating a statistical model for the wavelet coefficients, we directly parametrize the denoising process as a sum of elementary nonlinear processes with unknown weights. We then minimize an estimate of the mean square error between the clean image and the denoised one. The key point is that we have at our disposal a very accurate, statistically unbiased, MSE estimate--Stein's unbiased risk estimate--that depends on the noisy image alone, not on the clean one. Like the MSE, this estimate is quadratic in the unknown weights, and its minimization amounts to solving a linear system of equations. The existence of this a priori estimate makes it unnecessary to devise a specific statistical model for the wavelet coefficients. Instead, and contrary to the custom in the literature, these coefficients are not considered random anymore. We describe an interscale orthonormal wavelet thresholding algorithm based on this new approach and show its near-optimal performance--both regarding quality and CPU requirement--by comparing it with the results of three state-of-the-art nonredundant denoising algorithms on a large set of test images. An interesting fallout of this study is the development of a new, group-delay-based, parent-child prediction in a wavelet dyadic tree.     

基于小波影响锥分析的图像去噪方法

采用非抽取小波变换(UDWT),在小波影响锥(COI)分析的基础上,提出一种新的图像去噪方法,能够有效地去除脉冲噪声同时保护图像的边缘.该方法与传统小波阈值去噪法结合,可以很好地抑制高斯噪声和泊松噪声,甚至混合形式的噪声.实验结果证实了该方法的有效性.     

The undecimated wavelet decomposition and its reconstruction.

This paper describes the undecimated wavelet transform and its reconstruction. In the first part, we show the relation between two well known undecimated wavelet transforms, the standard undecimated wavelet transform and the isotropic undecimated wavelet transform. Then we present new filter banks specially designed for undecimated wavelet decompositions which have some useful properties such as being robust to ringing artifacts which appear generally in wavelet-based denoising methods. A range of examples illustrates the results.     

基于非抽取小波变换的遥感图像贝叶斯去噪

图像去噪是遥感图像处理的一个重要方面。文中基于非抽取小波变换,提出了一种贝叶斯图像去噪方法。对小波系数采用广义高斯分布建模,根据贝叶斯估计理论,得到贝叶斯收缩阈值,采用软阈值收缩去噪。实验结果表明：该去噪方法能够有效地抑制正交小波变换产生的人为干扰和伪Gibbs现象,与正交小波变换阈值去噪方法相比具有明显的优越性。     

双Haar小波变换系数的MAP估计及在图像去噪中的应用

小波变换作为一种新的工具,在信号去噪中得到了重要的应用。本文对双Haar小波变换系数,提出了MAP的估计方法,并对其在图像去噪中的应用进行了讨论。实验表明所提出的小波收缩算法与软门限方法相比较,用于图像去噪时可以给出更好的结果。     

Denoising by singularity detection

A new algorithm for noise reduction using the wavelet transform is proposed. Similar to Mallat's (1992) wavelet transform modulus maxima denoising approach, we estimate the regularity of a signal from the evolution of its wavelet transform coefficients across scales. However, we do not perform maxima detection and processing; therefore, complicated reconstruction is avoided. Instead, the local regularities of a signal are estimated by computing the sum of the modulus of its wavelet coefficients inside the corresponding “cone of influence”, and the coefficients that correspond to the regular part of the signal for reconstruction are selected. The algorithm gives an improved denoising result, as compared with the previous approaches, in terms of mean squared error and visual quality. The new denoising algorithm is also invariant to translation. It does not introduce spurious oscillations and requires very little a priori information of the signal or noise. Besides, we extend the method to two dimensions to estimate the regularity of an image by computing the sum of the modulus of its wavelet coefficients inside the so-called “directional cone of influence”. The denoising technique is applied to tomographic image reconstruction, where the improved performance of the new approach can clearly be observed     

Monte-Carlo sure: a black-box optimization of regularization parameters for general denoising algorithms

We consider the problem of optimizing the parameters of a given denoising algorithm for restoration of a signal corrupted by white Gaussian noise. To achieve this, we propose to minimize Stein's unbiased risk estimate (SURE) which provides a means of assessing the true mean-squared error (MSE) purely from the measured data without need for any knowledge about the noise-free signal. Specifically, we present a novel Monte-Carlo technique which enables the user to calculate SURE for an arbitrary denoising algorithm characterized by some specific parameter setting. Our method is a black-box approach which solely uses the response of the denoising operator to additional input noise and does not ask for any information about its functional form. This, therefore, permits the use of SURE for optimization of a wide variety of denoising algorithms. We justify our claims by presenting experimental results for SURE-based optimization of a series of popular image-denoising algorithms such as total-variation denoising, wavelet soft-thresholding, and Wiener filtering/smoothing splines. In the process, we also compare the performance of these methods. We demonstrate numerically that SURE computed using the new approach accurately predicts the true MSE for all the considered algorithms. We also show that SURE uncovers the optimal values of the parameters in all cases.     

Two denoising methods by wavelet transform

To wavelet-based noise reduction methods are discussed. First, we improve the traditional spatially selective noise filtration technique proposed by Xu et al. (1994). Second, we introduce a new threshold-based denoising algorithm based on undecimated discrete wavelet transform. Simulations and comparisons are given     

The curvelet transform for image denoising

We describe approximate digital implementations of two new mathematical transforms, namely, the ridgelet transform and the curvelet transform. Our implementations offer exact reconstruction, stability against perturbations, ease of implementation, and low computational complexity. A central tool is Fourier-domain computation of an approximate digital Radon transform. We introduce a very simple interpolation in the Fourier space which takes Cartesian samples and yields samples on a rectopolar grid, which is a pseudo-polar sampling set based on a concentric squares geometry. Despite the crudeness of our interpolation, the visual performance is surprisingly good. Our ridgelet transform applies to the Radon transform a special overcomplete wavelet pyramid whose wavelets have compact support in the frequency domain. Our curvelet transform uses our ridgelet transform as a component step, and implements curvelet subbands using a filter bank of a; trous wavelet filters. Our philosophy throughout is that transforms should be overcomplete, rather than critically sampled. We apply these digital transforms to the denoising of some standard images embedded in white noise. In the tests reported here, simple thresholding of the curvelet coefficients is very competitive with "state of the art" techniques based on wavelets, including thresholding of decimated or undecimated wavelet transforms and also including tree-based Bayesian posterior mean methods. Moreover, the curvelet reconstructions exhibit higher perceptual quality than wavelet-based reconstructions, offering visually sharper images and, in particular, higher quality recovery of edges and of faint linear and curvilinear features. Existing theory for curvelet and ridgelet transforms suggests that these new approaches can outperform wavelet methods in certain image reconstruction problems. The empirical results reported here are in encouraging agreement.     

基于小波域Curvelet变换的湍流图像去噪算法

为了提高湍流图像的空间分辨率,提出了一种基于小波域Curvelet变换(wavelet domain Curvelet transform,WDCT)的湍流图像去噪算法。该算法根据湍流退化图像噪声的统计特性,结合贝叶斯萎缩方法优化阈值选择。首先,对含噪湍流图像进行单层二维离散小波变换,接着提取高频系数并对它作快速离散Curvelet变换,最后根据贝叶斯准则估计阈值T,改进阈值的自适应选取方法,获得最优阈值,最后给出湍流图像去噪实现过程。为验证本文算法,根据客观评价标准峰值信噪比(peak signal to noise ratio,PSNR)和均方根误差(mean square error,MSE),对模拟图像和实测湍流图像进行去噪实验。与DWT-NABayesShrink算法、UWT算法相比,视觉效果更好,PSNR值分别提高7.27%和4.92%,MSE值分别降低26.3%和23.1%。本文算法得到较清晰的目标图像,对湍流退化图像去噪有一定的应用价值。     

Speckle removal from SAR images in the undecimated wavelet domain

Speckle reduction is approached as a minimum mean-square error (MMSE) filtering performed in the undecimated wavelet domain by means of an adaptive rescaling of the detail coefficients, whose amplitude is divided by the variance ratio of the noisy coefficient to the noise-free one. All the above quantities are analytically calculated from the speckled image, the variance and autocorrelation of the fading variable, and the wavelet filters only, without resorting to any model to describe the underlying backscatter. On the test image Lena corrupted by synthetic speckle, the proposed method outperforms Kuan's local linear MMSE filtering by almost 3-dB signal-to-noise ratio. When true synthetic aperture radar (SAR) images are concerned, empirical criteria based on distributions of multiscale local coefficient of variation, calculated in the undecimated wavelet domain, are introduced to mitigate the rescaling of coefficients in highly heterogeneous areas where the speckle does not obey a fully developed model, to avoid blurring strong textures and point targets. Experiments carried out on widespread test SAR images and on a speckled mosaic image, comprising synthetic shapes, textures, and details from optical images, demonstrate that the visual quality of the results is excellent in terms of both background smoothing and preservation of edge sharpness, textures, and point targets. The absence of decimation in the wavelet decomposition avoids typical impairments often introduced by critically subsampled wavelet-based denoising.     

Optimizing the multiwavelet shrinkage denoising

Denoising methods based on wavelet domain thresholding or shrinkage have been found to be effective. Recent studies reveal that multivariate shrinkage on multiwavelet transform coefficients further improves the traditional wavelet methods. It is because multiwavelet transform, with appropriate initialization, provides better representation of signals so that their difference from noise can be clearly identified. We consider the multiwavelet denoising by using multivariate shrinkage function. We first suggest a simple second-order orthogonal prefilter design method for applying multiwavelet of higher multiplicities. We then study the corresponding thresholds selection using Stein's unbiased risk estimator (SURE) for each resolution level provided that we know the noise structure. Simulation results show that higher multiplicity wavelets usually give better denoising results and the proposed threshold estimator suggests good indication for optimal thresholds.     

全变差数字滤波器与Ridgelet变换相结合 的图像去噪方法

针对Gurvelet变换采用的金字塔分解对图像细节表现的不足,我们提出利用全变差数字滤波器提取图像细节,然后对其采用基于分数阶傅立叶变换和投影－切片定理的Ridgelet变换,在变换域中由极小化极大误差准则进行阈值估计并对变换域系数进行阈值处理,以实现图像去噪.与金字塔分解相比,全变差数字滤波器能够简化图像分解并得到包含几乎所有细节的单幅图像,从而更有利于在Ridgelet域中进行降噪处理.实验结果表明,相对于Ridgelet和Curvelet变换的去噪方法,本文方法在抑制噪声的同时具有更有效的边缘保护能力,同时消除了边缘处的振荡,并且相对于Curvelet变换节省了计算.     

基于四元数小波变换域统计模型的图像去噪

图像去噪是图像处理的基本问题,四元数小波变换是1种新的多尺度分析工具.图像经四元数小波变换后,其小波系数不仅在尺度间具有相关性,而且在尺度内也具有一定的相关性.首先利用层内及层间的相关性,用非高斯分布对四元数小波系数进行建模,然后给出分类准则,把小波系数分类为重要系数和不重要系数,再用非高斯分布模型对重要系数与其邻域系...     

Image denoising with anisotropic bivariate shrinkage

This paper presents a novel image denoising algorithm based on the modeling of wavelet coefficients with an anisotropic bivariate Laplacian distribution function. The anisotropic bivariate Laplacian model not only captures the child-parent dependency between wavelet coefficients, but also fits the anisotropic property of the variances of wavelet coefficients in different scales of natural images. With this statistical model, we derive a closed-form anisotropic bivariate shrinkage function in the framework of Bayesian denoising and a new image denoising approach with local marginal variance estimation based on this newly derived shrinkage function is proposed in the discrete wavelet transform (DWT) domain. The proposed anisotropic bivariate shrinkage approach is also extended to the dual-tree complex wavelet transform (DT-CWT) domain to further improve the performance of image denoising. To take full advantage of DT-CWT, a more accurate noise variance estimator is proposed and the way the anisotropic bivariate shrinkage function applied to the magnitudes of DT-CWT coefficients is presented. Experiments were carried out in both the DWT and the DT-CWT domain to validate the effectiveness of the proposed method. Using a representative set of standard test images corrupted by additive white Gaussian noise, the simulation results show that the proposed method provides promising results and is competitive with the best wavelet-based denoising results reported in the literature both in terms of peak signal-to-noise ratio (PSNR) and in visual quality.     

Dual-tree complex wavelet hidden Markov tree model for image denoising

A new non-training complex wavelet hidden Markov tree (HMT) model, which is based on the dual-tree complex wavelet transform and a fast parameter estimation technique, is proposed for image denoising. This new model can mitigate the two problems (high computational cost and shift-variance) of the conventional wavelet HMT model simultaneously. Experiments show that the denoising approach with this new model achieves better performance than other related HMT- based image denoising algorithms.