Construction of parametric biorthogonal wavelet filter banks with two parameters for image coding

We had presented a simple technique, which is based on the theory of Diophantine equation, for parametrization of popular biorthogonal wavelet filter banks (BWFBs) having the linear phase and arbitrary multiplicity of vanishing moments (VMs), and constructed a type of parametric BWFBs with one free parameter [15]. Here we generalize this technique to the case of two parameters, and construct a type of parametric BWFBs with two free parameters. The closed-form parameter expressions of the BWFBs are derived, with which any two-parameter family of BWFBs having preassigned VMs can be constructed, and six families, i.e., 9/11, 10/10, 13/11, 10/14, 17/11, and 10/18 families, are considered here. Two parameters provide two degrees of freedom to optimize the resulting BWFBs with respect to other criteria. In particular, in each family, three specific rational-coefficient BWFBs with attractive features are obtained by adjusting the parameters: the first is not only very close to a quadrature mirror filter (QMF) bank, but has optimum coding gain; the second possesses characteristics that are close to the irrational BWFB with maximum VMs by Cohen et al.; and the last which has binary coefficients can realize a multiplication-free discrete wavelet transform. In addition, two BWFBs are systematically verified to exhibit performance competitive to several state-of-the-art BWFBs for image compression, and yet require lower computational costs. This work was supported by the Natural Science Foundation of Jiangsu province, China under Grant 07KJD520005.     

17/11双正交小波的优化设计及其对图像压缩性能的分析

许多适合于图像编码的小波,如CDF-9/7,Winger-17/11(W-17/11),Villasenor-6/10和10/18(V-6/10和V-10/18)小波,其滤波器系数是无理数,需要用无限的计算精度实现对应的离散小波变换(DWT)。该文给出了一种参数化构造17/11双正交小波组的简便方法:首先把小波合成滤波器表示为用两个自由参数表达的三角多项式,然后把双正交小波的精确重构条件归结为一个线性方程组,最后求解此方程组得到对应的小波分解滤波器,从而得到了17/11双正交小波滤波器的参数表达式。通过调整表达式中的自由参数,可以随意构造具有所需特征的17/11线性相位小波滤波器。作为构造实例,构造出一种新的有理系数17/11双正交小波滤波器,它具有优化的编码增益。实验表明:其压缩性能与W-17/11和V-10/18小波滤波器相当,优于CDF-9/7和V-6/10小波滤波器。     

Numerical solutions for orthogonal wavelet filters by Newton method

The wavelet transform has recently generated much interest in applied mathematics, signal processing and image coding. Mallat (1989) used the concept of the function space as a bridge to link the wavelet transform and multiresolution analysis. Daubechies (1990) added regularity conditions to find 2N, 2N10, tap coefficients for orthogonal wavelet filters. Owing to the difficulty of finding their closed solutions for large N a numerical method called the Newton method is proposed. We constructed the orthogonal wavelet filter with 2N-tap coefficients by N linear equations and N nonlinear equations. The 2N-tap, 2N10, coefficients we found are very consistent with those of Daubechies. Also, the method can be used to find the orthogonal wavelet filter with N-tap coefficients for N>10.     

Biorthogonal nearly coiflet wavelets for image compression

Filter bank design for wavelet compression is crucial; careful design enables superior quality for broad classes of images. The Bernstein basis for frequency-domain construction of biorthogonal nearly coiflet (BNC) wavelet bases forms a unified design framework for high-performance medium-length filters. A common filter bandwidth is characteristic of widely favoured BNC filter pairs: the classical CDF 9/7, the Villasenor 6/10, and the Villasenor 10/18. Based on this observation, we construct previously unknown BNC 17/11 and BNC 16/8 wavelet filters. Key filter-quality evaluation metrics, due to Villasenor, demonstrate these filters to be well suited for image compression. Also studied are the biorthogonal coiflet 17/11 (half-band), 18/10 and 10/6 filter pairs, which have not previously been formally evaluated for image coding. Simulation results confirm that the BNC 17/11 and BNC 16/8 wavelet bases are outstanding for compression of natural and medical images, and particularly for images with significant high-frequency detail, such as fingerprints. The BNC 17/11 pair recommends itself for international standardization for the compression of still images; the BNC 16/8 pair for high-quality compression of production quality video. Experimental evidence suggests biorthogonal filters achieve good compression if, subject to a filter bandwidth constraint, maximum vanishing moments are obtained for a given filter support.     

Popular biorthogonal wavelet filters via a lifting scheme and its application in image compression

A technique using a lifting scheme is presented for constructing compactly supported wavelets whose coefficients are composed of free variables locating in an interval. An efficient approach-based wavelet for image compression is developed by selecting the coefficients of the 9-7 wavelet filter and associated lifting scheme. Furthermore, the rationalised coefficients wavelet filter that can be implemented with simple integer arithmetic is achieved and its characteristic is close to the well known original irrational coefficients 9-7 wavelet filters developed by A. Cohen et al. (Commun. Pure Appl. Maths., vol.45, no.1, p.485-560, 1992). To reduce the computational cost of image coding applications further, an acceleration technique is proposed for the lifting steps. Software and hardware simulations show that the new method has very low complexity, and simultaneously preserves the high quality of the compressed image.     

A new class of biorthogonal wavelet systems for image transformcoding

We construct general biorthogonal Coifman wavelet systems, a new class of compactly supported biorthogonal wavelet systems with vanishing moments equally distributed for a scaling function and wavelet pair. A time-domain design method is employed and closed-form expressions for the impulse responses and the frequency responses of the corresponding dual filters are derived. The resulting filter coefficients are all dyadic fractions, which is an attractive feature in the realization of multiplication-free discrete wavelet transform. Even-ordered systems in this family are symmetric, which correspond to linear-phase dual filters. In particular, three filterbanks (FBs) in this family are systematically verified to have competitive compression potential to the 9-7 tap biorthogonal wavelet FB by Cohen et al. (1992), which is currently the most widely used one in the field of wavelet transform coding. In addition, the proposed FB's have much smaller computational complexity in terms of floating-point operations required in transformation, and therefore indicate a better tradeoff between compression performance and computational complexity.     

A New Design Method of 9-7 Biorthogonal Filter Banks Based on Odd Harmonic Function

The floating-point implementation of a CDF-9-7 wavelet transform with irrational coefficients on a resource limited hardware platform is a challenging task. This paper presents a new design method of 9-7 biorthogonal wavelet filter bank (FB) based on classical Fourier theory, the so-called odd harmonic function (OHF) analysis. Three types of binary rational 9-7 biorthogonal wavelet FBs have been derived, considering vanishing moments in addition to the rationality of filter coefficients. The extensive experiments for the implementation of the new design on the SPIHT (Set Partitioning In Hierarchical Trees) platform have been conducted and the results show that the performance of the proposed new biorthogonal FBs is equal to, or in several cases outperforms the, CDF-9-7 FB.     

基于Bubble函数的子波构造

本文首次系统地探讨基于 Ｂｕｂｂｌｅ函数的子波构造原理和方法。通过选取不同形状参数和时移因子,可以得到许多具有不同侧抑制性能的子波函数．文中给出了相应的Ｍａｌｌａｔ塔式分解快速算法所用的滤波器组系数计算公式与设计条件。     

Estimation of 2-D noisy fractional Brownian motion and itsapplications using wavelets

The two-dimensional (2-D) fractional Brownian motion (fBm) model is useful in describing natural scenes and textures. Most fractal estimation algorithms for 2-D isotropic fBm images are simple extensions of the one-dimensional (1-D) fBm estimation method. This method does not perform well when the image size is small (say, 32x32). We propose a new algorithm that estimates the fractal parameter from the decay of the variance of the wavelet coefficients across scales. Our method places no restriction on the wavelets. Also, it provides a robust parameter estimation for small noisy fractal images. For image denoising, a Wiener filter is constructed by our algorithm using the estimated parameters and is then applied to the noisy wavelet coefficients at each scale. We show that the averaged power spectrum of the denoised image is isotropic and is a nearly 1/f process. The performance of our algorithm is shown by numerical simulation for both the fractal parameter and the image estimation. Applications to coastline detection and texture segmentation in a noisy environment are also demonstrated.     

Estimation of fractal signals using wavelets and filter banks

A filter bank design based on orthonormal wavelets and equipped with a multiscale Wiener filter was recently proposed for signal restoration and for signal smoothing of 1/f family of fractal signals corrupted by external noise. The conclusions obtained in these papers are based on the following simplificative hypotheses: (1) The wavelet transformation is a whitening filter, and (2) the approximation term of the wavelet expansion can be avoided when the number of octaves in the multiresolution analysis is large enough. In this paper, we show that the estimation of 1/f processes in noise can be improved avoiding these two hypotheses. Explicit expressions of the mean-square error are given, and numerical comparisons with previous results are shown