The analysis and design of two-dimensional nearly-orthogonal symmetric wavelet filter banks

The design and analysis of two-channel two-dimensional (2D) nonseparable nearly-orthogonal symmetric wavelet filter banks with quincunx decimation is studied. The basic idea is to impose multiple zeros at the aliasing frequency to a symmetric filter and minimize the deviation of the filter satisfying the orthogonal condition to obtain a nearly-orthogonal FIR filter bank. Since multiple zeros are imposed, a scaling function may be generated from the minimized filter. With this filter, a semi-orthogonal filter bank is constructed. Methods for analyzing the correlation of the semi-orthogonal filter banks are proposed. The integer translates of the wavelet and scaling function are nearly-orthogonal. The integer translates of the wavelet at different scale are completely orthogonal. The semi-orthogonal filter bank can be efficiently implemented using the corresponding nearly-orthogonal FIR filter bank.     

Semi-orthogonal versus orthogonal wavelet basis sets for solvingintegral equations

The two categories of wavelets, orthogonal and semi-orthogonal, are compared and it is shown that the semi-orthogonal wavelet is best suited for integral equation applications. The Battle-Lemarie orthogonal wavelet and the spline generated semi-orthogonal wavelet are each used to solve for the current distribution on an infinite strip illuminated by a transverse magnetic (TM) plane wave and a straight thin wire illuminated by a normally incident plane wave. The grounds for comparison are accuracy in characterizing the current, matrix sparsity, computation time, and singularity extraction capability. The limitations and advantages of solving integral equations with each of the two wavelet categories are discussed     

基于球面Haar小波和卷积神经网络的飞行员虹膜识别

虹膜识别面临两个重要的问题：一是如何精细分解与重构虹膜球面图像;二是如何识别虹膜图特征。虹膜表面几何位置信息是一种重要的信号,传统的虹膜识别通常使用虹膜图像的平面特征,然而人的眼睛是一种球体,从平面图像难以提取到虹膜球体的几何特征。针对平面特征容易出现虹膜纹理的扭曲和失真等问题,该文建议一种正交对称的球面Haar小波(OSSHW)基,对球面虹膜信号进行多尺度分解与重构,获得更精细的虹膜曲面几何特征,同时对比球谐函数和半正交或正交球面Haar小波基的虹膜球面信号特征提取能力。在此基础上,该文提出一种基于卷积神经网络(CNN)和正交对称的球面Haar小波的虹膜识别方法,它能够有效捕获虹膜球体曲面的局部精细特征,比半正交或正交球面Haar小波基具有更强的虹膜识别能力。     

基于球面Haar小波和卷积神经网络的飞行员虹膜识别

虹膜识别面临两个重要的问题：一是如何精细分解与重构虹膜球面图像;二是如何识别虹膜图特征。虹膜表面几何位置信息是一种重要的信号,传统的虹膜识别通常使用虹膜图像的平面特征,然而人的眼睛是一种球体,从平面图像难以提取到虹膜球体的几何特征。针对平面特征容易出现虹膜纹理的扭曲和失真等问题,该文建议一种正交对称的球面Haar小波(OSSHW)基,对球面虹膜信号进行多尺度分解与重构,获得更精细的虹膜曲面几何特征,同时对比球谐函数和半正交或正交球面Haar小波基的虹膜球面信号特征提取能力。在此基础上,该文提出一种基于卷积神经网络(CNN)和正交对称的球面Haar小波的虹膜识别方法,它能够有效捕获虹膜球体曲面的局部精细特征,比半正交或正交球面Haar小波基具有更强的虹膜识别能力。     

Sparsity and conditioning of impedance matrices obtained withsemi-orthogonal and bi-orthogonal wavelet bases

Wavelet and wavelet packet transforms are often used to sparsify dense matrices arising in the discretization of CEM integral equations. This paper compares orthogonal, semi-orthogonal, and bi-orthogonal wavelet and wavelet packet transforms with respect to the condition numbers, matrix sparsity, and number of iterations for the transformed systems. The best overall results are obtained with the orthogonal wavelet packet transforms that produce highly sparse matrices requiring fewest iterations. Among wavelet transforms the semi-orthogonal wavelet transforms lead to the sparsest matrices, but require too many iterations due to high condition numbers. The bi-orthogonal wavelets produce very poor sparsity and require many iterations and should not be used in these applications     

A simple method to build oversampled filter banks and tight frames.

This paper presents conditions under which the sampling lattice for a filter bank can be replaced without loss of perfect reconstruction. This is the generalization of common knowledge that removing up/downsampling will not lose perfect reconstruction. The results provide a simple way of building oversampled filter banks. If the original filter banks are orthogonal, these oversampled banks construct tight frames of l2 (Z(n)) when iterated. As an example, a quincunx lattice is used to replace the rectangular one of the standard wavelet transform. This replacement leads to a tight frame that has a higher sampling in both time and frequency. The frame transform is nearly shift invariant and has intermediate scales. An application of the transform to image fusion is also presented.     

基于具有对称性的非张量积小波图像融合方法

提出了基于一类新的小波具有紧支撑、对称性和正交性、伸缩矩阵为(20 02)的非张量积小波的图像融合新方法.首先根据非张量积小波理论,提出了一种新的二维4通道4× 4对称滤波器组的设计方法,并用此方法设计出一组具有上述性质的非张量积小波4× 4滤波器组,利用此滤波器组对参加融合的图像进行滤波;然后对低频部分采用取均值、高频部分采用基于局部窗口能量取大的融合算法对滤波后的图像进行融合;最后重构.并采用熵、交叉熵、互信息、均方根误差和峰值信噪比等指标对该方法的融合性能进行了客观评价.对可见光图像与红外图像、远红外图像与近红外图像、航空图像和卫星图像、多聚焦图像等其它多类图像的融合实验结果表明本方法有较好的融合效果,可得到无边缘失真的融合结果图像,其融合性能比采用同样融合算法的张量积Haar小波的融合方法的融合性能好.     

Factorization approach to unitary time-varying filter bank treesand wavelets

A complete factorization of all optimal (in terms of quick transition) time-varying FIR unitary filter bank tree topologies is obtained. This has applications in adaptive subband coding, tiling of the time-frequency plane and the construction of orthonormal wavelet and wavelet packet bases for the half-line and interval. For an M-channel filter bank the factorization allows one to construct entry/exit filters that allow the filter bank to be used on finite signals without distortion at the boundaries. One of the advantages of the approach is that an efficient implementation algorithm comes with the factorization. The factorization can be used to generate filter bank tree-structures where the tree topology changes over time. Explicit formulas for the transition filters are obtained for arbitrary tree transitions. The results hold for tree structures where filter banks with any number of channels or filters of any length are used. Time-varying wavelet and wavelet packet bases are also constructed using these filter bank structures. the present construction of wavelets is unique in several ways: 1) the number of entry/exit functions is equal to the number of entry/exit filters of the corresponding filter bank; 2) these functions are defined as linear combinations of the scaling functions-other methods involve infinite product constructions; 3) the functions are trivially as regular as the wavelet bases from which they are constructed     

Design of IIR orthogonal wavelet filter banks using lifting scheme

The lifting scheme is well known to be an efficient tool for constructing second generation wavelets and is often used to design a class of biorthogonal wavelet filter banks. For its efficiency, the lifting implementation has been adopted in the international standard JPEG2000. It is known that the orthogonality of wavelets is an important property for many applications. This paper presents how to implement a class of infinite-impulse-response (IIR) orthogonal wavelet filter banks by using the lifting scheme with two lifting steps. It is shown that a class of IIR orthogonal wavelet filter banks can be realized by using allpass filters in the lifting steps. Then, the design of the proposed IIR orthogonal wavelet filter banks is discussed. The designed IIR orthogonal wavelet filter banks have approximately linear phase responses. Finally, the proposed IIR orthogonal wavelet filter banks are applied to the image compression, and then the coding performance of the proposed IIR filter banks is evaluated and compared with the conventional wavelet transforms.     

Pseudolapped orthogonal transform

The pseudo-LOT, a modified version of the lapped orthogonal transform (LOT), is introduced. It can be viewed as either a block transform with overlapping basis functions or a critically-sampled multirate filter bank with nearly perfect reconstruction. A fast algorithm for the pseudo-LOT, based on the discrete cosine transform, is presented.<>     

From differential equations to the construction of new wavelet-like bases

In this paper, an approach is introduced based on differential operators to construct wavelet-like basis functions. Given a differential operator L with rational transfer function, elementary building blocks are obtained that are shifted replicates of the Green's function of L. It is shown that these can be used to specify a sequence of embedded spline spaces that admit a hierarchical exponential B-spline representation. The corresponding B-splines are entirely specified by their poles and zeros; they are compactly supported, have an explicit analytical form, and generate multiresolution Riesz bases. Moreover, they satisfy generalized refinement equations with a scale-dependent filter and lead to a representation that is dense in L/sub 2/. This allows us to specify a corresponding family of semi-orthogonal exponential spline wavelets, which provides a major extension of earlier polynomial spline constructions. These wavelets are completely characterized, and it is proven that they satisfy the following remarkable properties: 1) they are orthogonal across scales and generate Riesz bases at each resolution level; 2) they yield unconditional bases of L/sub 2/-either compactly supported (B-spline-type) or with exponential decay (orthogonal or dual-type); 3) they have N vanishing exponential moments, where N is the order of the differential operator; 4) they behave like multiresolution versions of the operator L from which they are derived; and 5) their order of approximation is (N-M), where N and M give the number of poles and zeros, respectively. Last but not least, the new wavelet-like decompositions are as computationally efficient as the classical ones. They are computed using an adapted version of Mallat's filter bank algorithm, where the filters depend on the decomposition level.     

Time-varying discrete-time signal expansions as time-varying filter banks

The time-varying discrete-time signal expansion was analysed based on the theory of time-varying filter banks in detail. A general definition of time-varying discrete-time wavelet transforms is provided. Usually, a time-varying discrete-time signal expansion can be implemented using a time-varying filter bank. Using the time-varying filter bank theory, the authors developed a useful algorithm to calculate the dual basis function in a biorthogonal time-varying discrete-time signal expansion. Example is given to show the usage of the algorithm. In the last part, the authors provide a detailed analysis of the general time-varying discrete-time wavelet transform. Some useful properties of the time-varying discrete-time wavelet transform including their proofs are given. The relationship between the tree-structured implementation and the non-uniform filter bank implementation is discussed.     

Computation of DWT via FHT-based implementation

It is well known that most wavelet transform algorithms compute sampled coefficients of the continuous wavelet transform using the filter bank structure of the discrete wavelet transform (DWT). Although this method is efficient, noticeable computational savings have been obtained through an FFT-based implementation. The authors present a fast Hartley transform (FHT)-based implementation of the filter bank and show that noticeable overall computational savings can be obtained     

BOT's based on nonuniform filter banks

Parallels between orthogonal transforms and filter banks have been drawn before. Block orthogonal transform (BOT) is a special case of orthogonal transform where a nonoverlapping window is used. We relate BOTs to filter banks. Specifically, we show that any BOT can be shown as a perfect reconstruction filter bank, and any tree-structured perfect reconstruction filter bank or any orthonormal filter bank for which no filter length exceeds its decimation factor can be shown as a BOT. We then show that all conventional BOTs map to uniform filter banks. A construction method to design a BOT from any nonuniform filter bank is presented, and finding an optimal tree structure (in the sense of transform coding gain) for a given source is also discussed. The results show that the optimal, nonuniform BOT outperforms uniform BOTs having either the same number of bands or the same size in most cases     

Speech frame recognition based on less shift sensitive wavelet filter banks

The wavelet transform possesses multi-resolution property and high localization performance; hence, it can be optimized for speech recognition. In our previous work, we show that redundant wavelet filter bank parameters work better in speech recognition task, because they are much less shift sensitive than those of critically sampled discrete wavelet transform (DWT). In this paper, three types of wavelet representations are introduced, including features based on dual-tree complex wavelet transform (DT-CWT), perceptual dual-tree complex wavelet transform, and four-channel double-density discrete wavelet transform (FCDDDWT). Then, appropriate filter values for DT-CWT and FCDDDWT are proposed. The performances of the proposed wavelet representations are compared in a phoneme recognition task using special form of the time-delay neural networks. Performance evaluations confirm that dual-tree complex wavelet filter banks outperform conventional DWT in speech recognition systems. The proposed perceptual dual-tree complex wavelet filter bank results in up to approximately 9.82 % recognition rate increase, compared to the critically sampled two-channel wavelet filter bank.     

Completeness of arbitrarily sampled discrete time wavelettransforms

An arbitrarily sampled discrete time wavelet transform is said to be complete if it is uniquely invertible, i.e., if the underlying signal can be uniquely recovered from the available samples of the wavelet transform. We develop easy-to-compute necessary and sufficient conditions and necessary but not sufficient conditions for the completeness of an arbitrarily sampled dyadic discrete time wavelet transform of a periodic signal. In particular, we provide necessary and sufficient conditions for completeness of the sampled wavelet transform when the lowpass filter associated with the dyadic wavelet filter bank has no unit circle zeros other than those at z=1. We present necessary but not sufficient conditions for completeness when the lowpass filter associated with the dyadic wavelet filter bank has arbitrary unit circle zeros. We also provide necessary and sufficient conditions for completeness of a set of samples of both the lowpass approximation to the signal and its wavelet transform. All the conditions we derive use only information about the location of the retained samples and the analyzing wavelet filter bank. They can easily be checked without explicitly computing of the rank of a matrix. Finally, we present a simple signal reconstruction procedure that can be used once we have determined the arbitrarily sampled discrete time wavelet transform is complete     

正交小波变换中边界的零延拓及其无失真恢复

在有限长信号的小波变换中,一般说来零延拓不是一种好边界处理方法,但对有些类别的小波,零延拓却能带来处理上的很大方便和收到良好的一文主要探讨了正交小波中低通滤波器ｈπ的支撑为「０,Ｍ」的小波变换的边界处理问题,提出了零延拓展是其较好的选择,我们首先利用小波变换的性质证明了一个有意义的结论：然后由这个结论证明了这类小波边界零延拓的可完全恢复性,并给出了一种无失真恢复算法,最后把这种方法与其它边界处理方     

Perceptual Audio Coding Using Sinusoidal/Optimum Wavelet Representation

A perceptual audio coder, in which each audio segment is adaptively analyzed using either a sinusoidal or an optimum wavelet basis according to the time-varying characteristics of the audio signals, has been constructed. The basis optimization is achieved by a novel switched filter bank scheme, which switches between a uniform filter bank structure (discrete cosine transform) and a non-uniform filter bank structure (discrete wavelet transform). A major artifact of the International ISO/Moving Pictures Experts Group (MPEG) audio coding standard (MPEG-I layers 1 and 2) known as pre-echo distortion which uses a uniform filter bank structure for audio signal analysis, is almost eliminated in the proposed coder. A perceptual masking model implemented using a high-resolution wavelet packet filter bank with 27 subbands, closely mimicking the critical bands of the human auditory system, is employed in this audio coder. The resulting scheme is a variable bit-rate audio coder, which provides compression ratios comparable to MPEG-I layers 1 and 2 with almost transparent quality.