Optimal FPGA implementation of CL multiwavelets architecture for signal denoising application

Wavelet transform is considered one of the efficient transforms of this decade for real time signal processing. Due to implementation constraints scalar wavelets do not possess the properties such as compact support, regularity, orthogonality and symmetry, which are desirable qualities to provide a good signal to noise ratio (SNR) in case of signal denoising. This leads to the evolution of the new dimension of wavelet called ‘multiwavelets’, which possess more than one scaling and wavelet filters. The architecture implementation of multiwavelets is an emerging area of research. In real time, the signals are in scalar form, which demands the processing architecture to be scalar. But the conventional Donovan Geronimo Hardin Massopust (DGHM) and Chui-Lian (CL) multiwavelets are vectored and are also unbalanced. In this article, the vectored multiwavelet transforms are converted into a scalar form and its architecture is implemented in FPGA (Field Programmable Gate Array) for signal denoising application. The architecture is compared with DGHM multiwavelets architecture in terms of several objective and performance measures. The CL multiwavelets architecture is further optimised for best performance by using DSP48Es. The results show that CL multiwavelet architecture is suited better for the signal denoising application.     

Balanced multiwavelet bases based on symmetric FIR filters

This paper describes a basic difference between multiwavelets and scalar wavelets that explains, without using zero moment properties, why certain complications arise in the implementation of discrete multiwavelet transforms. Assuming we wish to avoid the use of prefilters in implementing the discrete multiwavelet transform, it is suggested that the behavior of the iterated filter bank associated with a multiwavelet basis of multiplicity r is more fully revealed by an expanded set of r² scaling functions φ_i,j. This paper also introduces new K-balanced orthogonal multiwavelet bases based on symmetric FIR filters. The nonlinear design equations arising in this work are solved using the Grobner basis. The minimal-length K-balanced multiwavelet bases based on even-length symmetric FIR filters are better behaved than those based on odd-length symmetric FIR filters, as illustrated by special relations they satisfy and by examples constructed     

The application of multiwavelet filterbanks to image processing

Multiwavelets are a new addition to the body of wavelet theory. Realizable as matrix-valued filterbanks leading to wavelet bases, multiwavelets offer simultaneous orthogonality, symmetry, and short support, which is not possible with scalar two-channel wavelet systems. After reviewing this theory, we examine the use of multiwavelets in a filterbank setting for discrete-time signal and image processing. Multiwavelets differ from scalar wavelet systems in requiring two or more input streams to the multiwavelet filterbank. We describe two methods (repeated row and approximation/deapproximation) for obtaining such a vector input stream from a one-dimensional (1-D) signal. Algorithms for symmetric extension of signals at boundaries are then developed, and naturally integrated with approximation-based preprocessing. We describe an additional algorithm for multiwavelet processing of two-dimensional (2-D) signals, two rows at a time, and develop a new family of multiwavelets (the constrained pairs) that is well-suited to this approach. This suite of novel techniques is then applied to two basic signal processing problems, denoising via wavelet-shrinkage, and data compression. After developing the approach via model problems in one dimension, we apply multiwavelet processing to images, frequently obtaining performance superior to the comparable scalar wavelet transform.     

多小波:理论和应用

多小波是单小波的推广．普通的单小波不可能同时具有对称、正交、有限支撑等性质;而多小波放松了对单小波函数的限制,可以同时拥有这些特性,将会有更广阔的应用前景．本文首先介绍多小波的基本理论,然后举例说明在使用多小波中会碰到的预处理问题及其解决的方法．     

Wavelet transforms for vector fields using omnidirectionally balanced multiwavelets

Vector wavelet transforms for vector-valued fields can be implemented directly from multiwavelets; however, existing multiwavelets offer surprisingly poor performance for transforms in vector-valued signal-processing applications. In this paper, the reason for this performance failure is identified, and a remedy is proposed. A multiwavelet design criterion known as omnidirectional balancing is introduced to extend to vector transforms the balancing philosophy previously proposed for multiwavelet-based scalar-signal expansion. It is shown that the straightforward implementation of a vector wavelet transform, namely, the application of a scalar transform to each vector component independently, is a special case of an omnidirectionally balanced vector wavelet transform in which filter-coefficient matrices are constrained to be diagonal. Additionally, a family of symmetric-antisymmetric multiwavelets is designed according to the omnidirectional-balancing criterion. In empirical results for a vector-field compression system, it is observed that the performance of vector wavelet transforms derived from these omnidirectionally-balanced symmetric-antisymmetric multiwavelets is far superior to that of transforms implemented via other multiwavelets and can exceed that of diagonal transforms derived from popular scalar wavelets.     

Generalized Discrete Multiwavelet Transform With Embedded Orthogonal Symmetric Prefilter Bank

Prefilters are generally applied to the discrete multiwavelet transform (DMWT) for processing scalar signals. To fully utilize the benefit offered by DMWT, it is important to have the prefilter designed appropriately so as to preserve the important properties of multiwavelets. To this end, we have recently shown that it is possible to have the prefilter designed to be maximally decimated, yet preserve the linear phase and orthogonal properties as well as the approximation power of multiwavelets. However, such design requires the point of symmetry of each channel of the prefilter to match with the scaling functions of the target multiwavelet system. It can be very difficult to find a compatible filter bank structure; and in some cases, such filter structure simply does not exist, e.g., for multiwavelets of multiplicity 2. In this paper, we suggest a new DMWT structure in which the prefilter is combined with the first stage of DMWT. The advantage of the new structure is twofold. First, since the prefiltering stage is embedded into DMWT, the computational complexity can be greatly reduced. Experimental results show that an over 20% saving in arithmetic operations can be achieved comparing with the traditional DMWT realizations. Second, the new structure provides additional design freedom that allows the resulting prefilters to be maximally decimated, orthogonal and symmetric even for multiwavelets of low multiplicity. We evaluated the new DMWT structure in terms of computational complexity, energy compaction ratio as well as the compression performance when applying to a VQ based image coding system. Satisfactory results are obtained in all cases comparing with the traditional approaches.     

Multiwavelet prefilters. II. Optimal orthogonal prefilters

For pt.I see IEEE Trans. Circuits Syst. II, vol.45, p.1106-12 (1998). Prefiltering a given discrete signal has been shown to be an essential and necessary step in applications using unbalanced multiwavelets. In this paper, we develop two methods to obtain optimal second-order approximation preserving prefilters for a given orthogonal multiwavelet basis. These procedures use the prefilter construction introduced in Hardin and Roach (1998). The first prefilter optimization scheme exploits the Taylor series expansion of the prefilter combined with the multiwavelet. The second one is achieved by minimizing the energy compaction ratio (ECR) of the wavelet coefficients for an experimentally determined average input spectrum. We use both methods to find prefilters for the cases of the DGHM and Chui-Lian (CL) multiwavelets. We then compare experimental results using these filters in an image compression scheme. Additionally, using the DGHM multiwavelet with the optimal prefilters from the first scheme, we find that quadratic input signals are annihilated by the high-pass portion of the filter bank at the first level of decomposition.     

基于不同预处理方法的多小波暂态信号去噪

在介绍多小波基本理论的基础上,探讨了多小波的不同预处理方法并对多小波滤波器响应产生的影响进行了比较.通过对噪声信号的多小波变换分析,设计基于多小波变换的去噪方法.最后通过大量的仿真工作,对不同预处理方法的多小波与传统小波的电力系统故障暂态信号去噪效果进行了深入分析,结果表明:预处理方法的选择是影响多小波去噪效果的关键因素,若选择合适的预处理方法,利用多小波对暂态信号进行去噪,可以获得比传统小波更好的去噪效果.     

基于多小波分析的图像融合算法

从电磁波频谱带宽的角度,提出了图像融合的一种物理解释。介绍了多小波变换的一般理论,描述了多小波图像融合的方法。用GHM多小波、CL多小波及Db4单小波对可见光与红外图像进行了融合。利用平均熵作为判定准则,比较了GHM多小波、CL多小波及Db4单小波的融合效果,实验结果表明CL多小波算法效果最好。     

New image compression techniques using multiwavelets andmultiwavelet packets

Advances in wavelet transforms and quantization methods have produced algorithms capable of surpassing the existing image compression standards like the Joint Photographic Experts Group (JPEG) algorithm. For best performance in image compression, wavelet transforms require filters that combine a number of desirable properties, such as orthogonality and symmetry. However, the design possibilities for wavelets are limited because they cannot simultaneously possess all of the desirable properties. The relatively new field of multiwavelets shows promise in obviating some of the limitations of wavelets. Multiwavelets offer more design options and are able to combine several desirable transform features. The few previously published results of multiwavelet-based image compression have mostly fallen short of the performance enjoyed by the current wavelet algorithms. This paper presents new multiwavelet transform and quantization methods and introduces multiwavelet packets. Extensive experimental results demonstrate that our techniques exhibit performance equal to, or in several cases superior to, the current wavelet filters.     

Design of prefilters for discrete multiwavelet transforms

The pyramid algorithm for computing single wavelet transform coefficients is well known. The pyramid algorithm can be implemented by using tree-structured multirate filter banks. The authors propose a general algorithm to compute multiwavelet transform coefficients by adding proper premultirate filter banks before the vector filter banks that generate multiwavelets. The proposed algorithm can be thought of as a discrete vector-valued wavelet transform for certain discrete-time vector-valued signals. The proposed algorithm can be also thought of as a discrete multiwavelet transform for discrete-time signals. The authors then present some numerical experiments to illustrate the performance of the algorithm, which indicates that the energy compaction for discrete multiwavelet transforms may be better than the one for conventional discrete wavelet transforms     

对称—反对称多小波滤波器组的参数化构造及其在图像压缩中的应用

本文给出了一类对称—反对称多小波滤波器组参数化设计方法,并在此基础上提出了新的多小波零树编码方案.实验表明,与常用的9-7单小波相比,我们所设计的双正交对称—反对称多小波可使编码性能有较大幅度的改善.     

Fusion algorithm for multisensor images based on discrete multiwavelet transform

The authors review the notion of multiwavelets and describe the use of the discrete multiwavelet transform (DMWT) in image fusion processing. Multiwavelets are extensions from scalar wavelets, and have several advantages in comparison with scalar wavelets. Multiwavelet analysis can offer more precise image analysis than wavelet multiresolution analysis. A novel fusion algorithm is presented for multisensor images based on the discrete multiwavelet transform that can be performed at pixel level. After the registering of source images, a pyramid for each source image can be obtained by applying decomposition with multiwavelets in each level. The multiwavelet decomposition coefficients of the input images are appropriately merged and a new fused image is obtained by reconstructing the fused multiwavelet coefficients. This image fusion algorithm may be used to combine images from multisensors to obtain a single composite with extended information content. The results of experiments indicate that this image fusion algorithm can provide a more satisfactory fusion outcome.     

多小波变换及其在信号滤波中的应用

多小波拥有对称性、正交性、有限支撑等这些信号处理中十分重要的特性,从而弥补了单小波的不足,具有更为广阔的应用前景.本文从滤波的角度,重点分析并设计了多小波预滤波与多小波分解系数处理方法.仿真表明多小波分析方法的优越性.     

Construction of Biorthgonal Multiwavelets

The theory of multiwavelets was initiated by Goodman et al [1].Then Geronimo et al constructed the GHM multiwavelet with multiplicity 2 using fractal interpolation[2]. As we know, there are symmetric smooth orthogonal multiwavelet bases while there is no symmetric continuous orthogonal scalar wavelet[3].In recent years, the papers about construction of biorthogonal multiwavelets have arised more and more, see Refs.[4~11]. In this paper, the methods of construction and dual functions are giv…     

Adaptive multiwavelet initialization

The multiwavelet concept uses a set of scaling functions and a set of wavelets to generate an orthogonal multichannel multiresolution pyramid decomposition for finite energy signals. The multiple scaling functions extract the lowpass information of the input data set, and the multiwavelets extract the bandpass information. When the lowpass filters associated with the scaling functions are cascaded, a multiresolution pyramid decomposition/reconstruction system is formed with each pyramid convolution operator having several inputs and several outputs. However, there is only one data set available to initialize this process. This paper addresses the question of how to best modify the data set using prefilters so that its decomposition contains the most useful information for the chosen application. The “best” prefilters are determined by the minimization of the energy of preselected decomposition components. The resulting decomposition is, therefore, signal adaptive, and under appropriate conditions, perfect reconstruction of the input data set can be achieved with a proper postfiltering. The detailed discussions in the paper are focussed on the two-wavelet and two-scaling function case; the general multiwavelet case is addressed at the end of the paper. A compression example is provided to demonstrate the performance of the optimally initialized multiwavelet method and to compare it with a single wavelet method and another multiwavelet initialization method proposed by other authors     

Symmetric prefilters for multiwavelets

When applying discrete multiwavelets, prefiltering is necessary because the initial multiscaling coefficients cannot be trivially derived from the samples of scalar signals. There have been many studies on the design of prefilters, and one main approach is to use a superfunction. The idea is to construct a low-pass function from the multiscaling functions that inherits their approximation power for scalar signals. However, none of the existing prefilters give linear phase combined filters, which is important for many practical applications. The authors analyse the conditions on which the prefilters and the combined filters are symmetric. A method is proposed for the design of good multiwavelet prefilters that allow the superfunction to be symmetric, satisfying the Strang-Fix conditions and the resulting combined filters are linear phase. Design examples using DGHM and Chui-Lian multiwavelets are given.     

具有无边界失真的多小波

多小波是近几年小波理论研究的一个重要方向。该文综述了多小波的重要性质,利用正交性和对称性构造了一个支集在[0,1]上具有精确重构特性和二阶逼近性的多小波,其最大特点是无边界失真效应;经平衡后,有更好的低通和高通特性,不用预滤波。实验结果表明重构效果比单小波好。     

Armlets and balanced multiwavelets: flipping filter construction

In the scalar-valued setting, it is well-known that the two-scale sequences {q/sub k/} of Daubechies orthogonal wavelets can be given explicitly by the two-scale sequences {p/sub k/} of their corresponding orthogonal scaling functions, such as q/sub k/=(-1)/sup k/p/sub 1-k/. However, due to the noncommutativity of matrix multiplication, there is little such development in the multiwavelet literature to express the two-scale matrix sequence {Q/sub k/} of an orthogonal multiwavelet in terms of the two-scale matrix sequence {P/sub k/} of its corresponding scaling function vector. This paper, in part, is devoted to this study for the setting of orthogonal multiwavelets of dimension r=2. In particular, the two lowpass filters are flipping filters, whereas the two highpass filters are linear phase. These results will be applied to constructing both a family of the most recently introduced notion of armlet of order n and a family of n-balanced orthogonal multiwavelets.     

Boundary filter design for multiwavelets

When designing boundary filters for multiwavelets, perfect reconstruction and moment conditions are two of the most important criteria to be considered. Since the filter coefficients of multiwavelets are expressed in the form of matrices, the moment conditions for multiwavelets do not relate to the filter coefficients in a simple way as for scalar wavelets. The authors first formulate the moment conditions of multiwavelets and then use them as the criteria for the optimisation of the boundary multifilters. They then apply the proposed optimisation scheme to the design of boundary multifilters for CL and GHM multiwavelets. The optimised boundary filters possess the desired perfect reconstruction property and vanishing moments that allow them to reduce the boundary artefact by up to 75% as compared with the traditional approach.