Non-separable Wavelet Filter Banks Research Articles

The lifting factorization of 2D 4-channel nonseparable wavelet transforms

One-dimensional (1D) wavelet transform was lifted successfully using the division with remainder of univariate Laurent polynomials. However, division with remainder does not exist in the case of a bivariate polynomial. Thus, it makes sense that the polyphase matrix of a two-dimensional (2D) nonseparable wavelet transform cannot be decomposed into the lifting format using the same solution as that of the 1D wavelet transform. In this research, we present a new lifting factorization method of two-dimensional four-channel (2D 4-channel) nonseparable wavelet filter banks. According to the uniform constructing format of high-dimensional multivariate wavelet filter banks, the general form of 2D 4-channel nonseparable wavelet filter bank is given. With these filter banks, the polyphase matrix of 2D 4-channel nonseparable wavelet transform is found and proven. Then, we present the lifting factorization of the polyphase matrix and some examples are demonstrated in the factorization procedures. Finally, the lifting performances of the proposed method are analyzed. This lifting method factorizes the polyphase matrix into the product of a series of unit lower left triangular numerical matrices, unit upper right triangular numerical matrices, diagonal numerical matrices, and diagonal polynomial matrices whose elements on the diagonal line are 1, x, y, and xy. The original filter banks that all leading principal minors of the numerical matrices are not equal to zero can be factorized into lifting format. The proposed method transforms the lifting factorization of the polyphase matrix into the decompositions of the numerical matrices without Euclidian division. Thereby, only multiplication and addition operations are performed with no Fourier transformation involved. When compared with the lifting method of the tensor product lifting wavelet transform and the contourlet transform, the proposed lifting method can extract more edge information of images. The computational complexity of the original 2D 4-channel nonseparable wavelet transform for image decomposition is N+1 times as much as that of the proposed lifting factorization method and the original wavelet transform is accelerated. Furthermore, the proposed lifting factorization method is faster than the conventional 2D 4-channel nonseparable wavelet transform based on Fourier transformation theory and convolution operation when the size of each filter in the latter is greater than 4(N+1). The proposed lifting factorization has better sparsity than that of its original 2D 4-channel nonseparable wavelet transform and other typical 2D 4-channel nonseparable wavelet transforms.

Prime Coset Sum: A Systematic Method for Designing Multi-D Wavelet Filter Banks With Fast Algorithms

As constructing multi-D wavelets remains a challenging problem, we propose a new method called prime coset sum to construct multi-D wavelets. Our method provides a systematic way to construct multi-D non-separable wavelet filter banks from two 1-D low-pass filters, with one of which being interpolatory. Our method has many important features including the following: 1) it works for any spatial dimension, and any prime scalar dilation; 2) the vanishing moments of the multi-D wavelet filter banks are guaranteed by certain properties of the initial 1-D low-pass filters, and furthermore; 3) the resulting multi-D wavelet filter banks are associated with fast algorithms that are faster than the existing fast tensor product algorithms.

Multi-D Wavelet Filter Bank Design Using Quillen-Suslin Theorem for Laurent Polynomials

In this paper we present a new approach for constructing the wavelet filter bank. Our approach enables constructing nonseparable multidimensional non-redundant wavelet filter banks with FIR filters using the Quillen-Suslin Theorem for Laurent polynomials. Our construction method presents some advantages over the traditional methods of multidimensional wavelet filter bank design. First, it works for any spatial dimension and for any sampling matrix. Second, it does not require the initial lowpass filters to satisfy any additional assumption such as interpolatory condition. Third, it provides an algorithm for constructing a wavelet filter bank from a single lowpass filter so that its vanishing moments are at least as many as the accuracy number of the lowpass filter.

基于IHS和非下采样三通道不可分小波的多光谱图像融合

A new multispectral image fusion method based on intensity-hue-saturation (IHS) transform and three-channel non-separable wavelets whose dilation matrix is [2,1;-1,1] is proposed. The multi-resolution decompositions of the intensity of multispectral image and panchromatic image are performed in nonsubsampled mode using the three-channel non-separable wavelet filter bank. The approximation images and the detail images of the multi-resolution pyramids are fused. The experimental results show that this method has good visual effect. The performance outperforms the IHS fusion method, the discrete wavelet transform (DWT) fusion method, and the IHS-DWT fusion method in preserving spectral quality and high spatial resolution information.

Texture image retrieval based on non-tensor product wavelet filter banks

In this paper, we present a novel method, which uses non-separable wavelet filter banks, to extract the features of texture images for texture image retrieval. Compared to traditional tensor product wavelets (such as db wavelets), our new method can capture more direction and edge information of texture images, which is highly valuable to reflect the essential properties of the texture images. Experiments show that the proposed method is satisfactory and can achieve better retrieval accuracies than db wavelets.

Image denoising by using nonseparable wavelet filters and two-dimensional principal component analysis

In this paper, we propose an image denoising method based on nonseparable wavelet filter banks and two-dimensional principal component analysis (2D-PCA). Conventional wavelet domain processing techniques are based on modifying the coefficients of separable wavelet transform of an image. In general, separable wavelet filters have limited capability of capturing the directional information. In contrast, nonseparable wavelet filters contain the basis elements oriented at a variety of directions and different filter banks capture the different directional features of an image. Furthermore, we identify the patterns from the noisy image by using the 2D-PCA. In comparison to the prevalent denoising algorithms, our proposed algorithm features no complex preprocessing. Furthermore, we can adjust the wavelet coefficients by a threshold according to the denoising results. We apply our proposed technique to some benchmark images with white noise. Experimental results show that our new technique achieves both good visual quality and a high peak signal-to-noise ratio for the denoised images.

Image fusion method based on nonseparable wavelets

This paper presents an image fusion method based on a new class of wavelet - nonseparable wavelet with compact support, linear phase, orthogonality, and dilation matrix $\left( {{\begin{array}{*{20}c} 2 \hfill & 0 \hfill \\ 0 \hfill & 2 \hfill \\ \end{array}}} \right)$ . We first construct a 6 x 6 nonseparable wavelet filter bank. Using these filters the images involved are decomposed into nonseparable wavelet pyramids. Then the following fusion algorithm is proposed: for the low-frequency part, we select the average of the low-frequency subimages from both sensors. For every high-frequency subimage of each level, we select the absolute value of each pixel of the high-frequency subimage to form a new subimage, and the variance of each image patch over a 3 x 3 window in the new subimages is computed as an activity measurement. If the variance of the 3 x 3 window in one new subimage is greater than the variance of the corresponding 3 x 3 window in the other new subimage, then the center pixel value of the 3 x 3 window is selected as a new pixel value of the fused image. A new fused image is then reconstructed. The performance of the method is evaluated using entropy, root mean square error, and peak-to-peak signal-to-noise ratio. The experimental results show that this method has good vision effect. Because the nonseparable wavelet transform can extract more details from source images, all the features in the source images can be seen in the fused image, and the fused image can extract more information from the source images.

IMAGE FUSION METHOD BASED ON SHORT SUPPORT SYMMETRIC NON-SEPARABLE WAVELET

In this paper, image fusion method based on a new class of wavelet — non-separable wavelet with compactly supported, linear phase, orthogonal and dilation matrix [Formula: see text] is presented. We first construct a non-separable wavelet filter bank. Using these filters, the images involved are decomposed into wavelet pyramids. Then the following fusion algorithm was proposed: for low-frequency part, the average value is selected for new pixel value, For the three high-frequency parts of each level, the standard deviation of each image patch over 3×3 window in the high-frequency sub-images is computed as activity measurement. If the standard deviation of the area 3×3 window is bigger than the standard deviation of the corresponding 3×3 window in the other high-frequency sub-image. The center pixel values of the area window that the weighted area energy is bigger are selected. Otherwise the weighted value of the pixel is computed. Then a new fused image is reconstructed. The performance of the method is evaluated using the entropy, cross-entropy, fusion symmetry, root mean square error and peak-to-peak signal-to-noise ratio. The experiment results show that the non-separable wavelet fusion method proposed in this paper is very close to the performance of the Haar separable wavelet fusion method.