A new framework for complex wavelet transforms

Although the discrete wavelet transform (DWT) is a powerful tool for signal and image processing, it has three serious disadvantages: shift sensitivity, poor directionality, and lack of phase information. To overcome these disadvantages, we introduce two-stage mapping-based complex wavelet transforms that consist of a mapping onto a complex function space followed by a DWT of the complex mapping. Unlike other popular transforms that also mitigate DWT shortcomings, the decoupled implementation of our transforms has two important advantages. First, the controllable redundancy of the mapping stage offers a balance between degree of shift sensitivity and transform redundancy. This allows us to create a directional, non-redundant, complex wavelet transform with potential benefits for image coding systems. To the best of our knowledge, no other complex wavelet transform is simultaneously directional and non-redundant. The second advantage of our approach is the flexibility to use any DWT in the transform implementation. As an example, we can exploit this flexibility to create the complex double-density DWT (CDDWT): a shift-insensitive, directional, complex wavelet transform with a low redundancy of (3/sup m/-1/2/sup m/-1) in m dimensions. To the best of our knowledge, no other transform achieves all these properties at a lower redundancy.     

Multidimensional orthogonal FM transforms

The present a novel class of multidimensional orthogonal FM transforms. The analysis suggests a novel signal-adaptive FM transform possessing interesting energy compaction properties. We show that the proposed signal-adaptive FM transform produces point spectra for multidimensional signals with uniformly distributed samples. This suggests that the proposed transform is suitable for energy compaction and subsequent coding of broadband signals and images that locally exhibit significant level diversity. We illustrate these concepts with simulation experiments.     

基于复小波变换的地震资料角度域信息提取

局部角度域属性是地震资料非常重要的属性之一,应用地震波在局部角度域的传播特性,提取地震属性,分析介质的物性参数.通过对三雏地震数据的深层切片处理获取局部角度域属性,提出采用基于映射的复小波变换对二维地震资料两步算法,提出局部角度域信息的解决方案.     

基于复小波变换的地震资料角度域信息提取

局部角度域属性是地震资料非常重要的属性之一,应用地震波在局部角度域的传播特性,提取地震属性,分析介质的物性参数。通过对三维地震数据的深层切片处理获取局部角度域属性,提出采用基于映射的复小波变换对二维地震资料两步算法．提出局部角度域信息的解决方案。     

On sampling theorem, wavelets, and wavelet transforms

The classical Shannon sampling theorem has resulted in many applications and generalizations. From a multiresolution point of view, it provides the sine scaling function. In this case, for a band-limited signal, its wavelet series transform (WST) coefficients below a certain resolution level can be exactly obtained from the samples with a sampling rate higher than the Nyquist rate. The authors study the properties of cardinal orthogonal scaling functions (COSF), which provide the standard sampling theorem in multiresolution spaces with scaling functions as interpolants. They show that COSF with compact support have and only have one possibility which is the Haar pulse. They present a family of COSF with exponential decay, which are generalizations of the Haar function. With these COSF, an application is the computation of WST coefficients of a signal by the Mallat (1989) algorithm. They present some numerical comparisons for different scaling functions to illustrate the advantage of COSF. For signals which are not in multiresolution spaces, they estimate the aliasing error in the sampling theorem by using uniform samples     

Multidimensional fourier transforms by systolic architectures

In this paper, time-efficient systolic and semi-systolic architectures for the implementation of multidimensional DFTs are proposed that allow modularity and easy expansibility while keeping throughput independent of the dimension of the DFT. We shall call an array semi-systolic if the input data is to be preloaded into every cell of the array or if the output data can be calculated not only in the boundary cells of the array. DFT algorithms are represented by FORTRAN-like code, in order to explicitly display the suggested rotational transformations in the index space. After the transformation, multidimensional DFT algorithm can be mapped onto a systolic structure. The area-time² complexity of the proposed design is within logk factor of the lower bound for ak-point DFT (AT ²=(k ² log² k)), i.e. equals 0(k ² log³ k)=0(N ^2M log³ N) fork=N ^2M point DFT;A is the area complexity andT is the system throughput.     

Symmetric-extension-compatible reversible integer-to-integer wavelet transforms

Symmetric extension is explored as a means for constructing nonexpansive reversible integer-to-integer (ITI) wavelet transforms for finite-length signals. Two families of reversible ITI wavelet transforms are introduced, and their constituent transforms are shown to be compatible with symmetric extension. One of these families is then studied in detail, and several interesting results concerning its member transforms are presented. In addition, some new reversible ITI structures are derived that are useful in conjunction with techniques like symmetric extension. Last, the relationship between symmetric extension and per-lifting-step extension is explored, and some new results are obtained in this regard.     

Multidimensional chirp algorithms for computing Fourier transforms

Continuous versions of the multidimensional chirp algorithms compute the function G(y)=F(My), where F(y) is the Fourier transform of a function f(x) of a vector variable x and M is an invertible matrix. Discrete versions of the algorithms compute values of F over the lattice L(2)=ML(1 ) from values of f over a lattice L(1), where L(2) need not contain the lattice reciprocal to L(1). If M is symmetric, the algorithms are multidimensional versions of the Bluestein chirp algorithm, which employs two pointwise multiplication operations (PMOs) and one convolution operation (CO). The discrete version may be efficiently implemented using fast algorithms to compute the convolutions. If M is not symmetric, three modifications are required. First, the Fourier transform is factored as the product of two Fresnel transforms. Second, the matrix M is factored as M=AB, where A and B are symmetric matrices. Third, the Fresnel transforms are modified by the matrices A and B and each modified transform is factored into a product of two PMOs and one CO.     

Time-varying lapped transforms and wavelet packets

The perfect reconstruction conditions for a time-varying lapped transform (paraunitary filter bank) are developed through the factorization of the transform matrix into sparse factors. A general formulation is presented allowing one to switch between two paraunitary filter banks. However, the extended lapped transform (ELT) is often used as an example. Furthermore, an adaptive wavelet packet is developed employing a time varying, tree association of ELTs. In all cases perfect reconstruction is inherently assured     

Discrete inverses for nonorthogonal wavelet transforms

Discrete nonorthogonal wavelet transforms play an important role in signal processing by offering finer resolution in time and scale than their orthogonal counterparts. The standard inversion procedure for such transforms is a finite expansion in terms of the analyzing wavelet. While this approximation works quite well for many signals, it fails to achieve good accuracy or requires an excessive number of scales for others. This paper proposes several algorithms that provide more adequate inversion and compares them in the case of Morlet wavelets. In the process, both practical and theoretical issues for the inversion of nonorthogonal wavelet transforms are discussed     

Parallel pipeline implementation of wavelet transforms

Wavelet transforms have been one of the important signal processing developments in the last decade, especially for applications such as time-frequency analysis, data compression, segmentation and vision. Although several efficient implementations of wavelet transforms have been derived, their computational burden is still considerable. The paper describes two generic parallel implementations of wavelet transforms, based on the pipeline processor farming methodology, which have the potential to achieve real-time performance. Results show that the parallel implementation of the oversampled wavelet transform achieves virtually linear speedup, while the parallel implementation of the discrete wavelet transform (DWT) also outperforms the sequential version, provided that the filter order is large. The DWT parallelisation performance improves with increasing data length and filter order, while the frequency-domain implementation performance is independent of wavelet filter order. Parallel pipeline implementations are currently suitable for processing multidimensional images with data length at least 512 pixels     

VLSI architectures for discrete wavelet transforms

A folded architecture and a digit-serial architecture are proposed for implementation of one- and two-dimensional discrete wavelet transforms. In the one-dimensional folded architecture, the computations of all wavelet levels are folded to the same low-pass and high-pass filters. The number of registers in the folded architecture is minimized by the use of a generalized life time analysis. The converter units are synthesized with a minimum number of registers using forward-backward allocation. The advantage of the folded architecture is low latency and its drawbacks are increased hardware area, less than 100% hardware utilization, and the complex routing and interconnection required by the converters used. These drawbacks are eliminated in the alternate digit-serial architecture at the expense of an increase in the system latency and some constraints on the wordlength. In latency-critical applications, the use of the folded architecture is suggested. If latency is not so critical, the digit-serial architecture should be used. The use of a combined folded and digit-serial architecture is proposed for implementation of two-dimensional discrete wavelet transforms     

Rotation-invariant multiresolution texture analysis using radon and wavelet transforms.

A new rotation-invariant texture-analysis technique using Radon and wavelet transforms is proposed. This technique utilizes the Radon transform to convert the rotation to translation and then applies a translation-invariant wavelet transform to the result to extract texture features. A kappa-nearest neighbors classifier is employed to classify texture patterns. A method to find the optimal number of projections for the Radon transform is proposed. It is shown that the extracted features generate an efficient orthogonal feature space. It is also shown that the proposed features extract both of the local and directional information of the texture patterns. The proposed method is robust to additive white noise as a result of summing pixel values to generate projections in the Radon transform step. To test and evaluate the method, we employed several sets of textures along with different wavelet bases. Experimental results show the superiority of the proposed method and its robustness to additive white noise in comparison with some recent texture-analysis methods.     

Time-domain design of low delay wavelet transforms

The authors use a time-domain design methodology previously used for perfect reconstruction filter banks to design wavelet prototypes which may be used to generate the discrete wavelet transform (DWT). An advantage of this method is the significant reduction in the inherent delay, and consequent reduction in the number of multiplications required to implement the DWT. It is proposed that these low delay wavelets will find applications in time critical systems.<>     

The coding gain of multiplexed wavelet transforms

Evangelista (1994, 1993) introduced the multiplexed wavelet transform (MWT) and pointed out its potential applications to the analysis, synthesis, processing and coding of pseudo-periodic signals such as voiced speech and music. Coders based on the MWT have been shown to outperform the conventional subband coders when a reliable pitch parameter can be extracted from data. In this paper, we investigate the effects of uniform quantization of the MWT coefficients. We compare the performance of the new coders with that of WT, block-DCT, and KLT coders in terms of the coding gain achieved when optimal bit allocation schemes are adopted     

基于小波变换的多尺度SSIM算法

提出一种基于小波变换的多尺度结构相似度的图像质量评价(WWMS-SSIM)算法。首先对原始图像和失真图像分别进行小波变换,在对应的低频子带图像中比较亮度和相关度差异,在对应高频子带图像中比较对比度、相关度和结构差异,然后赋予不同子带图像不同的视觉加权值,最后得到归一化的图像质量评价值。通过对不同失真类型图像以及Live数据库图像的评价,证明了提出的算法(WWMS-SSIM)相对均方误差、峰值信噪比、结构相似度以及改进的结构相似度评价算法更符合人类的主观视觉感受。     

Architectures for wavelet transforms: A survey

Wavelet transforms have proven to be useful tools for several applications, including signal analysis, signal compression and numerical analysis. This paper surveys the VLSI architectures that have been proposed for computing the Discrete and Continuous Wavelet Transforms for 1-D and 2-D signals. The architectures are based upon on-line versions of the wavelet transform algorithms. These architectures support single chip implementations and are optimal with respect to both area and time under the word-serial model.     

Optimization of integer wavelet transforms based on difference correlation structures.

In this paper, a novel lifting integer wavelet transform based on difference correlation structure (DCCS-LIWT) is proposed. First, we establish a relationship between the performance of a linear predictor and the difference correlations of an image. The obtained results provide a theoretical foundation for the following construction of the optimal lifting filters. Then, the optimal prediction lifting coefficients in the sense of least-square prediction error are derived. DCCS-LIWT puts heavy emphasis on image inherent dependence. A distinct feature of this method is the use of the variance-normalized autocorrelation function of the difference image to construct a linear predictor and adapt the predictor to varying image sources. The proposed scheme also allows respective calculations of the lifting filters for the horizontal and vertical orientations. Experimental evaluation shows that the proposed method produces better results than the other well-known integer transforms for the lossless image compression.