Image Restoration by Lifting‐Based Wavelet Domain E‐Median Filter

In this paper, we propose a method of applying a lifting‐based wavelet domain e‐median filter (LBWDEMF) for image restoration. LBWDEMF helps in reducing the number of computations. An e‐median filter is a type of modified median filter that processes each pixel of the output of a standard median filter in a binary manner, keeping the output of the median filter unchanged or replacing it with the original pixel value. Binary decision‐making is controlled by comparing the absolute difference of the median filter output and the original image to a preset threshold. In addition, the advantage of LBWDEMF is that probabilities of encountering root images are spread over sub‐band images, and therefore the e‐median filter is unlikely to encounter root images at an early stage of iterations and generates a better result as iteration increases. The proposed method transforms an image into the wavelet domain using lifting‐based wavelet filters, then applies an e‐median filter in the wavelet domain, transforms the result into the spatial domain, and finally goes through one spatial domain e‐median filter to produce the final restored image. Moreover, in order to validate the effectiveness of the proposed method we compare the result obtained using the proposed method to those using a spatial domain median filter (SDMF), spatial domain e‐median filter (SDEMF), and wavelet thresholding method. Experimental results show that the proposed method is superior to SDMF, SDEMF, and wavelet thresholding in terms of image restoration.     

基于贝叶斯估计的小波阈值图像降噪方法

提出一种新的基于贝叶斯估计的小波收缩阈值的图像降噪方法,该方法是通过最小Bayes风险的方法对图像小波变换后的小波系数进行估计,这种对小波系数的估计不仅与子带的方向和层次有关,而且与小波系数的大小有关.试验结果表明该方法比一般小波收缩阈值方法的降噪效果要好;还表明在峰值信噪比较低时该方法的降噪效果比Wiener滤波差,当峰值信噪比较高时该方法的降噪效果比Wiener滤波好.     

Subband Weighting With Pixel Connectivity for 3-D Wavelet Coding

Performing optimal bit-allocation with 3-D wavelet coding methods is difficult because energy is not conserved after applying the motion-compensated temporal filtering (MCTF) process and the spatial wavelet transform. The problem cannot be solved by extending the 2-D wavelet coefficients weighting method directly and then applying the result to 3-D wavelet coefficients, since this approach does not consider the complicated pixel connectivity that results from the lifting-based MCTF process. In this paper, we propose a novel weighting method, which takes account of the pixel connectivity, to solve the problem and derive the effect of the quantization error of a subband on the reconstruction error of a group of pictures. We employ the proposed method on a ${hbox {2-D}}+{rm t}$ structure with different temporal filters, namely the 5-3 filter and the 9-7 filter. Experiments on various coding parameters and sequences show that the proposed approach improves the bit-allocation performance over that obtained by using the weightings derived without considering the pixel connectivity in the MCTF process.      

基于贝叶斯估计的小波阈值图像降噪方法

提出一种新的基于贝叶斯估计的小波收缩阈值的图像降噪方法，该方法是通过最小Bayes风险的方法对图像小波变换后的小波系数进行估计，这种对小波系数的估计不仅与子带的方向和层次有关，而且与小波系数的大小有关。试验结果该方法比一般小波收缩阈值方法的降噪效果要好；还表明在峰值信噪比较低时该方法的降噪效果比Wiener滤波差，当峰值信噪比较高时该方法的降噪效果比Wiener滤波好。     

A new multiscale Bayesian algorithm for speckle reduction in medical ultrasound images

This paper introduces a new multiscale speckle reduction method based on the extraction of wavelet interscale dependencies to visually enhance the medical ultrasound images and improve clinical diagnosis. The logarithm of the image is first transformed to the oriented dual-tree complex wavelet domain. It is then shown that the adjacent subband coefficients of the log-transformed ultrasound image can be successfully modeled using the general form of bivariate isotropic stable distributions, while the speckle coefficients can be approximated using a zero-mean bivariate Gaussian model. Using these statistical models, we design a new discrete bivariate Bayesian estimator based on minimizing the mean square error (MSE). To assess the performance of the proposed method, four image quality metrics, namely signal-to-noise ratio, MSE, coefficient of correlation, and edge preservation index, were computed on 80 medical ultrasound images. Moreover, a visual evaluation was carried out by two medical experts. The numerical results indicated that the new method outperforms the standard spatial despeckling filters, homomorphic Wiener filter, and new multiscale speckle reduction methods based on generalized Gaussian and symmetric alpha-stable priors.     

Smooth wavelets, transform coding, and Markov-1 processes

This paper compares the energy compacting properties of unitary transforms from transform coders and two-band paraunitary filter banks from subband coders using a cost criterion that is proposed. Stationary processes for which paraunitary filters have better energy compaction than unitary filters are denoted as subband optimal, and all subband optimal processes are analytically characterized for the case of length-4 filters. It is shown analytically for length-4 filters and empirically for longer-length filters that Markov-1 processes are subband optimal and that the Daubechies (1988) maximally smooth wavelet sequences achieve better energy compaction than the best unitary filters for Markov-1 inputs     

Invertible temporal subband/wavelet filter banks with half-pixel-accurate motion compensation

Three-dimensional (3-D) subband/wavelet coding with motion compensation has been demonstrated to be an efficient technique for video coding applications in some recent research works. When motion compensation is performed with half-pixel accuracy, images need to be interpolated in both temporal subband analysis and synthesis stages. The resulting subband filter banks developed in these former algorithms were not invertible due to image interpolation. In this paper, an invertible temporal analysis/synthesis system with half-pixel-accurate motion compensation is presented. We look at temporal decomposition of image sequences as a kind of down-conversion of the sampling lattices. The earlier motion-compensated (MC) interlaced/progressive scan conversion scheme is extended for temporal subband analysis/synthesis. The proposed subband/wavelet filter banks allow perfect reconstruction of the decomposed video signal while retaining high energy compaction of subband transforms. The invertible filter banks are then utilized in our 3-D subband video coder. This video coding system does not contain the temporal DPCM loop employed in the conventional hybrid coder and the earlier MC 3-D subband coders. The experimental results show a significant PSNR improvement by the proposed method. The generalization of our algorithm for MC temporal filtering at arbitrary subpixel accuracy is also discussed.     

基于整数小波变换的Wiener插值算法

小波变换中高低分辨率子带之间的相似性使得利用小波变换进行图像插值的方法成为可能.根据数字图像信号的特点,分析了小波变化后各个子带信号的特点,提出了基于整数小波变换的Wiener插值算法.用Wiener自适应滤波器训练得到插值滤波系数,同时结合既符合图像性质又能减小运算量的整数双正交小波基对图像插值.结果得到较高的信噪比和较好的主观视觉效果.平均峰值信噪比比传统的双线性插值法提升了2.4dB.     

基于对偶树复数小波变换的邻域自适应的图像降噪

该文提出一种新的基于对偶树复数小波变换的邻域自适应的图像降噪方法,它是现存的NeighShrink降噪方法的改进。该文运用Stein的无偏风险估计,在小波域每一个子带为NeighShrink方法确定一个最优的阈值和邻域窗口,并将NeighShrink方法从正交的小波变换推广到对偶树复数小波变换。实验结果证实,该文方法比当前基于小波的最具竞争力的图像降噪方法取得了更好的降噪效果。     

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.     

Fast spatial combinative lifting algorithm of wavelet transformusing the 9/7 filter for image block compression

A new fast spatial combinative lifting algorithm (SCLA) of the wavelet transform using the 9/7 filter for image block compression is proposed. In comparison with its lifting-based implementation, the number of multiplications is reduced by a ratio of 5/12 and the speed of implementation of the wavelet transform is increased     

Multivariate statistical modeling for image denoising using wavelet transforms

Recently a variety of efficient image denoising methods using wavelet transforms have been proposed by many researchers. In this paper, we derive the general estimation rule in the wavelet domain to obtain the denoised coefficients from the noisy image based on the multivariate statistical theory. The multivariate distributions of the original clean image can be estimated empirically from a sample image set. We define a parametric multivariate generalized Gaussian distribution (MGGD) model which closely fits the sample distribution. Multivariate model makes it possible to exploit the dependency between the estimated wavelet coefficients and their neighbours or other coefficients in different subbands. Also it can be shown that some of the existing methods based on statistical modeling are subsets of our multivariate approach. Our method could achieve high quality image denoising. Among the existing image denoising methods using the same type of wavelet (Daubechies 8) filter, our results produce the highest peak signal-to-noise ratio (PSNR).     

Subband coding of images using asymmetrical filter banks

We address the problem of the choice of subband filters in the context of image coding. The ringing effects that occur in subband-based compression schemes are the major unpleasant distortions. A new set of two-band filter banks suitable for image coding applications is presented. The basic properties of these filters are linear phase, perfect reconstruction, asymmetric length, and maximum regularity. The better overall performances compared to the classical QMF subband filters are explained. The asymmetry of the filter lengths results in a better compaction of the energy, especially in the highpass subbands. Moreover, the quantization error is reduced due to the short lowpass synthesis filter. The undesirable ringing effect is considerably reduced due to the good step response of the synthesis lowpass filter. The proposed design takes into account the statistics of natural images and the effect of quantization errors in the reconstructed images, which explains the better coding performance.     

Construction of optimal subband coders using optimized and optimalquantizers

A method is presented for the optimization of arbitrary quantizers by use of a compensating postfilter. It is shown that the resulting optimized quantizers fit the model of a linear time-invariant filter followed by additive noise uncorrelated with the input which also characterizes the optimal (Lloyd-Max) quantizers. On the basis of this model, an expression for the variance of the error of a subband coder using optimized quantizers is explicitly determined. Given analysis filters which statistically separate the subbands, it is shown that this variance is minimized if these synthesis filters are chosen, which would achieve perfect reconstruction in lossless coding. The globally optimum filter bank, minimizing the coder error variance, is further obtained by proper choice of its analysis filters. A novel method for the determination of optimal bit allocation to subbands of the filter banks with optimized quantizers is also developed. The results are evaluated experimentally by comparison of the optimum uniformly split subband image coding scheme to classical logarithmically-split filter bank (wavelet) coding methods.     

利用二维非分离型嵌入式子波系数的零树图象编码

本文讨论利用二维非分离型滤渡器组的嵌入式零树子波(Embedded Warder Zerotree，简称EWZ)图象编码。文中讨论了二维非分离型滤波器组完全重构的充要条件，还讨论了子带图象的分割和五点式抽样的问题。实验表明嵌入式零树编码方法可以用在非分离型二维子波变换情况，在低比特率时有很好的巅值信噪比PSNR。     

Weighted Adaptive Lifting-Based Wavelet Transform for Image Coding

In this paper, a new weighted adaptive lifting (WAL)-based wavelet transform is presented. The proposed WAL approach is designed to solve the problems existing in the previous adaptive directional lifting (ADL) approach, such as mismatch between the predict and update steps, interpolation favoring only horizontal or vertical direction, and invariant interpolation filter coefficients for all images. The main contribution of the proposed approach consists of two parts: one is the improved weighted lifting, which maintains the consistency between the predict and update steps as far as possible and preserves the perfect reconstruction at the same time; another is the directional adaptive interpolation, which improves the orientation property of the interpolated image and adapts to statistical property of each image. Experimental results show that the proposed WAL-based wavelet transform for image coding outperforms the conventional lifting-based wavelet transform up to 3.06 dB in PSNR and significant improvement in subjective quality is also observed. Compared with the ADL-based wavelet transform, up to 1.22-dB improvement in PSNR is reported.     

一种小波域与空域相结合的图像滤波方法

利用小波阈值去噪方法和传统空间域Lee滤波的特点,提出了一种图像去噪的的组合滤波方案。首先在小波域对图像阈值去噪,得到预去噪图像;再在空间域上利用自适应Wiener滤波器进一步提高恢复图像的精度。为了保证小波域和空间域两种算法之间的匹配,对预去噪图像中残留噪声的分布进行了研究,对其噪声方差估计做了改进,提出了一种估计噪声方差的近似最优公式。仿真实验表明,与单独的在小波域或空域去噪相比,该方法的均方误差和信噪比指标均得到了改善。     

Multifocus and multispectral image fusion based on pixel significance using discrete cosine harmonic wavelet transform

The energy compaction and multiresolution properties of wavelets have made the image fusion successful in combining important features such as edges and textures from source images without introducing any artifacts for context enhancement and situational awareness. The wavelet transform is visualized as a convolution of wavelet filter coefficients with the image under consideration and is computationally intensive. The advent of lifting-based wavelets has reduced the computations but at the cost of visual quality and performance of the fused image. To retain the visual quality and performance of the fused image with reduced computations, a discrete cosine harmonic wavelet (DCHWT)-based image fusion is proposed. The performance of DCHWT is compared with both convolution and lifting-based image fusion approaches. It is found that the performance of DCHWT is similar to convolution-based wavelets and superior/similar to lifting-based wavelets. Also, the computational complexity (in terms of additions and multiplications) of the proposed method scores over convolution-based wavelets and is competitive to lifting-based wavelets.     

Adaptive directional lifting-based wavelet transform for image coding.

We present a novel 2-D wavelet transform scheme of adaptive directional lifting (ADL) in image coding. Instead of alternately applying horizontal and vertical lifting, as in present practice, ADL performs lifting-based prediction in local windows in the direction of high pixel correlation. Hence, it adapts far better to the image orientation features in local windows. The ADL transform is achieved by existing 1-D wavelets and is seamlessly integrated into the global wavelet transform. The predicting and updating signals of ADL can be derived even at the fractional pixel precision level to achieve high directional resolution, while still maintaining perfect reconstruction. To enhance the ADL performance, a rate-distortion optimized directional segmentation scheme is also proposed to form and code a hierarchical image partition adapting to local features. Experimental results show that the proposed ADL-based image coding technique outperforms JPEG 2000 in both PSNR and visual quality, with the improvement up to 2.0 dB on images with rich orientation features.     

Lossless image compression with projection-based and adaptive reversible integer wavelet transforms

Reversible integer wavelet transforms are increasingly popular in lossless image compression, as evidenced by their use in the recently developed JPEG2000 image coding standard. In this paper, a projection-based technique is presented for decreasing the first-order entropy of transform coefficients and improving the lossless compression performance of reversible integer wavelet transforms. The projection technique is developed and used to predict a wavelet transform coefficient as a linear combination of other wavelet transform coefficients. It yields optimal fixed prediction steps for lifting-based wavelet transforms and unifies many wavelet-based lossless image compression results found in the literature. Additionally, the projection technique is used in an adaptive prediction scheme that varies the final prediction step of the lifting-based transform based on a modeling context. Compared to current fixed and adaptive lifting-based transforms, the projection technique produces improved reversible integer wavelet transforms with superior lossless compression performance. It also provides a generalized framework that explains and unifies many previous results in wavelet-based lossless image compression.