The SURE-LET approach to image denoising.

We propose a new approach to image denoising, based on the image-domain minimization of an estimate of the mean squared error--Stein's unbiased risk estimate (SURE). Unlike most existing denoising algorithms, using the SURE makes it needless to hypothesize a statistical model for the noiseless image. A key point of our approach is that, although the (nonlinear) processing is performed in a transformed domain--typically, an undecimated discrete wavelet transform, but we also address nonorthonormal transforms--this minimization is performed in the image domain. Indeed, we demonstrate that, when the transform is a "tight" frame (an undecimated wavelet transform using orthonormal filters), separate subband minimization yields substantially worse results. In order for our approach to be viable, we add another principle, that the denoising process can be expressed as a linear combination of elementary denoising processes--linear expansion of thresholds (LET). Armed with the SURE and LET principles, we show that a denoising algorithm merely amounts to solving a linear system of equations which is obviously fast and efficient. Quite remarkably, the very competitive results obtained by performing a simple threshold (image-domain SURE optimized) on the undecimated Haar wavelet coefficients show that the SURE-LET principle has a huge potential.     

Analytic Wavelet Packets—Combining the Dual-Tree Approach With Wavelet Packets for Signal Analysis and Filtering

Real wavelets (or real basis functions in general) suffer from two major disadvantages: shift-variance and oscillating coefficients. This paper provides a review of these problems as well as the undecimated wavelet transform which solves the problem of shift-variance. However, undecimated transforms have very high computational costs and still have oscillating coefficients. The work of Kingsbury is summarized afterwards. They have presented a dual-tree complex wavelet transform (DTCWT) which is nearly shift-invariant and has smooth coefficients. Subsequently, a filter swapping scheme is developed in order to create complex wavelet packets with analytic basis functions. Finally, the effectiveness of the proposed method is demonstrated by five examples.      

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.     

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     

冗余轮廓波变换的构造及其在SAR图像降斑中的应用

构造了由非抽样塔式分解和方向滤波器组实现的冗余轮廓波变换。文中利用McClellan变换设计非抽样塔式分解中满足精确重构条件的圆对称滤波器组。利用冗余轮廓波变换系数的自适应局部统计模型及最大后验概率法对SAR图像进行降斑处理,并与基于平稳小波和轮廓波变换的降斑算法进行比较。结果表明,提出的算法能有效地去除散斑噪声,并且具有更强的边缘保持能力。     

Fusion of multifocus images by lattice structures

Image fusion methods based on multiscale transform (MST) suffer from high computational load due to the use of fast Fourier transforms (ffts) in the lowpass and highpass filtering steps. Lifting wavelet scheme which is based on second generation wavelets has been proposed as a solution to this issue. Lifting Wavelet Transform (LWT) is composed of split, prediction and update operations all implemented in the spatial domain using multiplications and additions, thus computation time is highly reduced. Since image fusion performance benefits from undecimated transform, it has later been extended to Stationary Lifting Wavelet Transform (SLWT). In this paper, we propose to use the lattice filter for the MST analysis step. Lattice filter is composed of analysis and synthesis parts where simultaneous lowpass and highpass operations are performed in spatial domain with the help of additions/multiplications and delay operations, in a recursive structure which increases robustness to noise. Since the original filter is designed for the undecimated case, we have developed undecimated lattice structures, and applied them to the fusion of multifocus images. Fusion results and evaluation metrics show that the proposed method has better performance especially with noisy images while having similar computational load with LSWT based fusion method.     

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.     

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。     

非抽样双树复小波域的BPP-NMF图像融合

提出了一种非抽样双树复小波变换（UDT-CWT）与基于块主元旋转的非负矩阵分解（BPP-NMF）相结合的多聚焦图像融合算法。利用UDT-CWT具有完美的平移不变性及良好的方向选择性,首先对图像进行多尺度、多方向分解并得到低频子带和高频子带系数;然后对低频子带系数采用块主元旋转的非负矩阵分解的融合策略,高频系数则选用高斯加权区域能量与区域标准差一致性选择的融合准则。最后对融合后的系数进行UDT-CWT逆变换得到重构图像。选用多组多聚焦图像进行融合并对融合结果进行主观视觉、客观方面的评价。试验结果表明,该融合算法不仅具有良好的视觉效果,同时在客观评价指标也优于一般的融合策略,验证了该算法的有效性。     

Detection of transients in 1/f noise with the undecimated discretewavelet transform

A method based on pattern matching in the undecimated discrete wavelet transform domain is introduced for the detection of a known transient with unknown parameters in the presence of 1/f noise. Maxima tracking techniques are used to reduce the computational complexity of the matching procedure by an order of magnitude, with minimal performance impact     

基于小波域的模糊化区域竞争模型的图像分割方法

本文中提出了一种基于小波变换的模糊化区域竞 争图像分割算法。该算法利用小波变 换在纹理分类 的优越性,在模糊化区域竞争算法的框架下对图像实施非抽样小波变换,建立起基于小波域 的统计特征参 数,并根据此特征参数构造能量函数。在算法求解过程中,通过采用快速迭代算法加快算法 求解的速度。 本文提出的图像分割算法可以有效地降低信噪比的敏感性,有效地解决模糊图像分割问题。 实验结果表明, 本文提出的分割方法在分割质量和速度上比经典的主动轮廓模型和模糊化区域竟争分割算法 更具有优越性。     

Context-driven fusion of high spatial and spectral resolution images based on oversampled multiresolution analysis

This paper compares two general and formal solutions to the problem of fusion of multispectral images with high-resolution panchromatic observations. The former exploits the undecimated discrete wavelet transform, which is an octave bandpass representation achieved from a conventional discrete wavelet transform by omitting all decimators and upsampling the wavelet filter bank. The latter relies on the generalized Laplacian pyramid, which is another oversampled structure obtained by recursively subtracting from an image an expanded decimated lowpass version. Both the methods selectively perform spatial-frequencies spectrum substitution from an image to another. In both schemes, context dependency is exploited by thresholding the local correlation coefficient between the images to be merged, to avoid injection of spatial details that are not likely to occur in the target image. Unlike other multiscale fusion schemes, both the present decompositions are not critically subsampled, thus avoiding possible impairments in the fused images, due to missing cancellation of aliasing terms. Results are presented and discussed on SPOT data.     

Wavelet transform methods for object detection and recovery

We show that a biorthogonal spline wavelet closely approximates the prewhitening matched filter for detecting Gaussian objects in Markov noise. The filterbank implementation of the wavelet transform acts as a hierarchy of such detectors operating at discrete object scales. If the object to be detected is Gaussian and its scale happens to coincide with one of those computed by the wavelet transform, and if the background noise is truly Markov, then optimum detection is realized by thresholding the appropriate subband image. In reality, the Gaussian may be a rather coarse approximation of the object, and the background noise may deviate from the Markov assumption. In this case, we may view the wavelet decomposition as a means for computing an orthogonal feature set for input to a classifier. We use a supervised linear classifier applied to feature vectors comprised of samples taken from the subbands of an N-octave, undecimated wavelet transform. The resulting map of test statistic values indicates the presence and location of objects. The object itself is reconstructed by using the test statistic to emphasize wavelet subbands, followed by computing the inverse wavelet transform. We show two contrasting applications of the wavelets-based object recovery algorithm. For detecting microcalcifications in digitized mammograms, the object and noise models closely match the real image data, and the multiscale matched filter paradigm is highly appropriate. The second application, extracting ship outlines in noisy forward-looking infrared images, is presented as a case where good results are achieved despite the data models being less well matched to the assumptions of the algorithm.     

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.     

Fusion of multi-focus images with registration inaccuracies

Image fusion combines complementary information for several input images. To obtain useful information from two misaligned images, registration is required. A hybrid textural registration-based multi-focus image fusion scheme is proposed. The Gabor filtering with specific frequency and orientation is used to extract different texture features from the image. The resulting Gabor-filtered images are then aligned using existing affine transformation. The proposed registration scheme yields superior performance as compared to affine registration, as Gabor transform extracts all the features. The fusion is performed using undecimated dual-tree complex wavelet transform. The quantitative and qualitative analysis of the proposed scheme outperforms existing image fusion schemes.     

Wavelet transforms for detecting microcalcifications in mammograms

Clusters of fine, granular microcalcifications in mammograms may be an early sign of disease. Individual grains are difficult to detect and segment due to size and shape variability and because the background mammogram texture is typically inhomogeneous. The authors develop a 2-stage method based on wavelet transforms for detecting and segmenting calcifications. The first stage is based on an undecimated wavelet transform, which is simply the conventional filter bank implementation without downsampling, so that the low-low (LL), low-high (LH), high-low (HL), and high-high (HH) sub-bands remain at full size. Detection takes place in HH and the combination LH+HL. Four octaves are computed with 2 inter-octave voices for finer scale resolution. By appropriate selection of the wavelet basis the detection of microcalcifications in the relevant size range can be nearly optimized. In fact, the filters which transform the input image into HH and LH+HL are closely related to prewhitening matched filters for detecting Gaussian objects (idealized microcalcifications) in 2 common forms of Markov (background) noise. The second stage is designed to overcome the limitations of the simplistic Gaussian assumption and provides an accurate segmentation of calcification boundaries. Detected pixel sites in HH and LH+HL are dilated then weighted before computing the inverse wavelet transform. Individual microcalcifications are greatly enhanced in the output image, to the point where straightforward thresholding can be applied to segment them. FROG curves are computed from tests using a freely distributed database of digitized mammograms.     

Fusion of multispectral and panchromatic images using improved IHS and PCA mergers based on wavelet decomposition

Since Chavez proposed the highpass filtering procedure to fuse multispectral and panchromatic images, several fusion methods have been developed based on the same principle: to extract from the panchromatic image spatial detail information to later inject it into the multispectral one. In this paper, we present new fusion alternatives based on the same concept, using the multiresolution wavelet decomposition to execute the detail extraction phase and the intensity-hue-saturation (IHS) and principal component analysis (PCA) procedures to inject the spatial detail of the panchromatic image into the multispectral one. The multiresolution wavelet decomposition has been performed using both decimated and undecimated algorithms and the resulting merged images compared both spectral and spatially. These fusion methods, as well as standard IHS-, PCA-, and wavelet-based methods have been used to merge Systeme Pour l'Observation de la Terre (SPOT) 4 XI and SPOT 4 M images with a ratio 4:1. We have estimated the validity of each fusion method by analyzing, visually and quantitatively, the quality of the resulting fused images. The methodological approaches proposed in this paper result in merged images with improved quality with respect to those obtained by standard IHS, PCA, and standard wavelet-based fusion methods. For both proposed fusion methods, better results are obtained when an undecimated algorithm is used to perform the multiresolution wavelet decomposition.     

Noise reduction using wavelet cycle spinning: analysis of useful periodicities in the z-transform domain

Cycle spinning (CS) and a’trous algorithms are different implementations of the undecimated wavelet transform (UWT). Both algorithms can be used for UWT and even though the resulting wavelet coefficients are different, they keep a correspondence. This paper describes an analysis of the CS algorithm performed in the z-transform domain, showing the similarities and differences with the a’trous implementation. CS generates more wavelet coefficients than a’trous, but the number of significative and different coefficients is the same in both cases because of the occurrence of a periodic repetition in CS coefficients. Mathematical expressions for the relationship between CS and a’trous coefficients and for CS coefficient periodicities are provided in the z-transform domain. In some wavelet denoising applications, periodicities (present in the coefficients of the CS procedure) can also be found in the performance measure of the processed signals. In particular, in ultrasonic CS denoising applications, periodicities have been appreciated in the signal-to-noise ratio (SNR) of the ultrasonic denoised signals. These periodicities can be used to optimize the number of CS coefficients for an efficient implementation. Two examples showing the periodicities in the SNR are included. A selection of several reduced sets of CS wavelet coefficients has been utilized in the examples, and the SNRs resulting after denoising are analyzed.     

Multiple particle tracking in 3-D+t microscopy: method and application to the tracking of endocytosed quantum dots.

We propose a method to detect and track multiple moving biological spot-like particles showing different kinds of dynamics in image sequences acquired through multidimensional fluorescence microscopy. It enables the extraction and analysis of information such as number, position, speed, movement, and diffusion phases of, e.g., endosomal particles. The method consists of several stages. After a detection stage performed by a three-dimensional (3-D) undecimated wavelet transform, we compute, for each detected spot, several predictions of its future state in the next frame. This is accomplished thanks to an interacting multiple model (IMM) algorithm which includes several models corresponding to different biologically realistic movement types. Tracks are constructed, thereafter, by a data association algorithm based on the maximization of the likelihood of each IMM. The last stage consists of updating the IMM filters in order to compute final estimations for the present image and to improve predictions for the next image. The performances of the method are validated on synthetic image data and used to characterize the 3-D movement of endocytic vesicles containing quantum dots.     

Noise reduction in rhythmic and multitrial biosignals with applications to event-related potentials

A new noise reduction algorithm is presented for signals displaying repeated patterns or multiple trials. Each pattern is stored in a matrix, forming a set of events, which is termed multievent signal. Each event is considered as an affine transform of a basic template signal that allows for time scaling and shifting. Wavelet transforms, decimated and undecimated, are applied to each event. Noise reduction on the set of coefficients of the transformed events is applied using either wavelet de- noising or principal component analysis (PCA) noise reduction methodologies. The method does not require any manual selection of coefficients. Nonstationary multievent synthetic signals are employed to demonstrate the performance of the method using normalized mean square error against classical wavelet and PCA based algorithms. The new method shows a significant improvement in low SNRs (typically <0 dB). On the experimental side, evoked potentials in a visual oddball paradigm are used. The reduced-noise visual oddball event-related potentials reveal gradual changes in morphology from trial to trial (especially for N1-P2 and N2-P3 waves at Fz), which can be hypothetically linked to attention or decision processes. The new noise reduction method is, thus, shown to be particularly suited for recovering single-event features in non- stationary low SNR multievent contexts.