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Highorder balanced Mband multiwavelet packet transformbased remote sensing image denoising
EURASIP Journal on Advances in Signal Processing volume 2016, Article number: 10 (2016)
Abstract
This article proposes highorder balanced multiband multiwavelet packet transforms for denoising remote sensing images. First, properties of several wavelet transforms and their relationships are analyzed. The article then presents theoretical principles and a fast algorithm for constructing highorder balanced multiband multiwavelet packet transforms. The remote sensing image denoising method based on this transform scheme is then described, and its utility is demonstrated by illustrative results of its application to denoise remote sensing images. The method provides clear improvements in denoising quality, due to the balanced order or band number, consistently outperforming traditional wavelet transformbased methods in terms of both visual quality and evaluation indicators. The method also incurs reasonable computational costs compared with the traditional methods.
1 Introduction
Remote sensing imaging has become a powerful technique for exploring and obtaining knowledge of numerous phenomena. However, during acquisition and transmission processes, the images are often contaminated by noise, which impairs their visual quality and limits the precision of subsequent processing steps, such as classification, target detection, and environmental monitoring. Thus, remote sensing image denoising applications have attracted growing interest. Transform domain denoising methods have shown remarkable success in the last decade. An assumption typically underlying these methods is that signal can be sparsely represented in the transform domain. Hence, by preserving the few highmagnitude transform coefficients that convey most of the true signal energy and discarding the rest, which are mainly due to noise, the true signal can be effectively estimated. The sparsity of the representation depends on both the transform used and the true signal’s properties [1, 2].
The wavelet transform (i.e., 2band scalar wavelet transform) can provide good sparsity for spatially localized details, and a number of advanced denoising methods based on them have been developed [2–4]. For example, the wavelet thresholding approach popularized by Donoho is now widely used in scientific and engineering applications [2]. The best image denoising systems include filters with symmetric and compactsupport properties, which can effectively extract features and eliminate artifacts. Unfortunately, however, no nontrivial, symmetric, compactsupport, orthogonal scalar wavelet transforms are available [1, 5].
Multiwavelet transforms (i.e., 2band multiwavelet transforms or 2band MWTs) have several advantages over scalar wavelet transforms, because they can simultaneously possess all the above properties [5, 6]. Since these properties are highly significant in image processing, multiwavelets have attracted considerable research interest and shown superior denoising performance over scalar wavelets in various studies [6–10] (see the diagram about the development of wavelet transforms in Fig. 1). Sveinsson et al. [11] and Wang et al. [12] applied 2band MWT in remote sensing image denoising and found that they generally outperform 2band scalar wavelet transforms in both visual quality and objective evaluation. Further improvements of 2band MWT could potentially be obtained by taking into account the properties of the signal to be analyzed, but all the cited transforms use the unchangeable transform pattern for diverse images. That is, as illustrated in Fig. 2a, the decomposition at each decomposition level is only applied to the lowfrequency component of signals and does nothing to the other, relatively highfrequency components, although this partitioning is not suitable for all signals with different behaviors across the frequency domain [1, 13].
Multiwavelet packet transforms (i.e., 2band multiwavelet packet transforms or 2band MWPT), as one extension of 2band MWT, provide an effective mean to select a suitable decomposition pattern corresponding to an analyzed signal. As shown in Fig. 2b, a 2band MWPT offers a finer frequency domain partition than a 2band MWT, especially in the highfrequency domain. Hence, numerous subsets corresponding to different multiwavelet bases (or packets) can be found in its partition point set, one of which will match the properties of the analyzed signals better than all the others. The 2band MWPT based on this subset (the best packet) can provide a better sparse representation of the signal than those based on other subsets (including the 2band MWT, as the subset it uses is just one of these subsets) [1, 13]. Martin et al. [14] and Wang [15] introduced 2band MWPTs to image compression and texture segmentation, respectively, and showed that they exhibit performance generally superior to 2band MWT. Liu et al. proposed a choice algorithm of the best multiwavelet packet and found that the 2MWPT based on the algorithm can generally obtain better denoising result than 2band MWT at the same condition [16]. Developing a 2band MWT scheme into its “packet” version is an effective way to improve performance (as shown in Fig. 1), however, the problem of balanced or highorder balanced must be addressed when using them in practical applications.
The balanced order ρ of a multiwavelet system corresponds to its ability to represent images sparsely [17, 18]. Recent studies show that the 2band MWTs based on balanced (ρ = 1) multiwavelets (i.e., balanced 2band MWT, as shown in Fig. 1) consistently outperform those based on unbalanced (ρ = 0) counterparts (i.e., 2band MWT) in image denoising [19–22]. Also, balanced 2band MWPT schemes derived by developing the balanced 2band MWT into “packet” versions also consistently outperform (unbalanced) 2band MWPT in seismic data compression and denoising [23]. However, there have been no published indepth studies on the relationship between the balanced order ρ and sparse representation ability of balanced (ρ = 1) or highorder balanced (ρ > 1) 2band MWPT (or even MWT). More importantly, although the 2band MWPTs overcome many shortcomings of wavelet transforms, they retain the “2band” weakness of the latter, especially in spatialfrequency tiling, which has triggered great interest in their extension [1].
As another extension of 2band MWT, Mband (M ∈ ℤ and M > 2) MWTs provide greater flexibility in spatialfrequency tiling and more robust sparse representation [1, 24] (see Fig. 1). As shown in Fig. 2c, the ideal partition point set of an Mband MWT is denser than that of a 2band counterpart (see Fig. 2a). Mband MWT has been shown to outperform 2band MWT in terms of both visual quality and objective indicators in image fusion [25]. Similarly, highorder balanced Mband MWTs reportedly outperform (unbalanced) Mband MWT in image compression [26]. Potentially, a highorder balanced Mband MWT may also be further improved (like a 2band counterpart) by using a “packet” version (i.e., an Mband MWPT, as shown in Fig. 1). As shown by the theoretical partition pattern of an Mband MWPT in Fig. 2d, it may provide a finer partitioning pattern and better matching subset (or more effective representation) for an analyzed signal, relative to a 2band counterpart. Mband MWPTs may offer great potential for image processing, but substantial extension of both their fundamental theory and convenient methodology are required. Thus, here we present basic principles of, and fast algorithms for, highorder balanced Mband MWPT by developing the highorder balanced Mband MWT presented in [26], according to key theory of developing balanced 2band MWT into “packet” versions presented in [23]. We also evaluate their performance in remote sensing image denoising in comparison with the 2band MWPT presented in [14–16], the balanced 2band MWPT in [23], the Mband MWT in [25], and the highorder balanced Mband MWT in [26] (see Fig. 1). Moreover, we systematically analyze the impact of the balanced order ρ and band number M on the sparse representation ability of MWPT.
The rest of the paper is organized as follows. In Section 2, we analyze the basic principles and fast algorithm of highorder balanced Mband MWPT based on the relative theory of Mband MWT. In Section 3, we present the denoising method based on the proposed transform, and evaluate the denoising performance of the method by using both synthetic and real noisy remote sensing images in comparison with the method based on the transforms recently proposed. We also analyze the impacts of the balanced order ρ and band number M on their denoising performance. We summarize and discuss the work in Section 4.
2 Highorder balanced Mband multiwavelet packet transforms
Classical wavelet transforms cannot provide many key properties simultaneously, as they establish a multiresolution analysis frame using a single scale function [1, 5, 6]. In contrast, multiwavelet transforms use r (r ∈ ℤ and r > 1) scale functions (rmultiplicity) for this work, thus reducing the number of constraint conditions and increasing the freedom in design. The r scale functions are denoted in vector form, i.e., Φ _{0}(t) = [ϕ _{0,0}, ϕ _{1,0}, ⋯, ϕ _{ r − 1,0}]. An Mband rmultiplicity multiwavelet system has M1 wavelet function vectors denoted as Φ _{ i }(t) = [ϕ _{0,i }, ϕ _{1,i }, ⋯, ϕ _{ r − 1,i }] (1 ≤ i < M). If (and only if) such a system is orthogonal, the Φ _{ i }(t) meets the following condition [24, 26],
Here, P _{ i }(n) is a vector filter bank of r × r and n ∈ ℤ (the same as below). We can get the following Mband multiwavelet transform algorithm after developing the corresponding algorithm of the classical wavelet transform (i.e., Mallat algorithm) [24, 26],
Here, m, n ∈ ℤ and both are shift parameters, while \( {C}_i^k \) is an rdimensional vector (the same as below).
A multiwavelet system is said to be balanced of order ρ if its lowpass and highpass filters preserve and cancel, respectively, all the monomial polynomial signals of order less than ρ. The order of a multiwavelet system corresponds to its ability to effectively represent the information of the macroscale change trend and local textures of a signal [17, 18]. If P _{ i }(n) meets the constraint condition corresponding to each property above (e.g., ρorder balanced, orthogonality), the balanced or highorder balanced Mband MWPT could be constructed by developing the above Mband multiwavelet transform algorithm. Their basic principles and fast algorithms are present as follows.
2.1 Basic principles
Let Ψ _{ i }(t) = Φ _{ i }(t) (0 ≤ i < M), and define
Mfold rescaling and translation of these functions yield a function library \( \chi =\left\{{\varPsi}_l^{(k)}={\mathrm{M}}^{k/2}{\varPsi}_l\left({\mathrm{M}}^{k}xn\right)\right\} \) (0 ≤ l < M^{k}). As shown in Fig. 3, the library can be viewed in terms of an Mfold complete tree, and the lth function at the kth decomposition level \( {\varPsi}_l^{(k)} \) yields M functions at the k + 1 level, i.e., \( {\varPsi}_{\mathrm{M}l+i}^{\left(k+1\right)} \) (0 ≤ i < M).
The library is overcomplete, and many complete orthogonal basis sets can be found by properly selecting different subsets in the library with an appropriate parameter set {k, l}.. A complete orthogonal basis set corresponds to a kind of multiwavelet packet transform scheme, a subset of the library and a parameter set {k, l}.. Each subset can be viewed in terms of an Mfold tree structure. Examples of possible basis sets are shown in the trees in Fig. 4. The tree in Fig. 4a selects the subset of the library with the parameter set {1 ≤ l < M, 0 ≤ k < 3} ∪ {0 ≤ l < M, k = 3} and corresponds to the standard Mband multiwavelet decomposition scheme [5, 6, 24]. The tree in Fig. 4d selects the subset with the parameter set {0 ≤ l < M^{k − 1}, k = 3} and corresponds to a kind of multiwavelet decomposition scheme that presents the finest frequency domain partition.
For any analyzed dataset, there is a basis set (the best basis set) that can best represent the spatial and frequency domain information in the dataset, and the corresponding “best” multiwavelet packet transform provides the most effective representation of the dataset (denoted as MWPT at below). The best basis set is found by using a cost function searching algorithm that seeks the best subset for some application in a set with tree structure [1, 13]. For example, a searching algorithm that uses information entropy as a cost function can find the most informative basis subset and consistently perform effectively in image denoising [1, 13]. Since the overcomplete library of the basis sets generated by the highorder balanced Mband multiwavelet packet transform can also be viewed in terms of a tree structure, we used entropic cost function searching algorithms to find both the best basis set corresponding to test datasets and for establishing the searching algorithm.
2.2 Fast algorithm
Defining \( {D}_i^{(0)}={C}_i^{(0)}\left(0\le i<\mathrm{M}\right), \) a fast algorithm for decomposition of the Mband multiwavelet packet transform is shown in Eq. (5). Reconstructions are the inverse process of decompositions, and its fast algorithm is shown in Eq. (6). Based on the expression, its clear procedures in Z transform domain are illustrated in Fig. 5, where P _{ i }(z) is the Z transformation of P _{ i }(n).
Using the decomposition algorithm above, one can obtain a complete Mfold tree including information on all its nodes \( {D}_l^{(k)}\left(1\le l<{\mathrm{M}}^k\right), \) as shown in Fig. 6. \( {D}_l^{(k)} \) are multiwavelet domain coefficients containing spatial information of the included frequency bands (see Fig. 2d). After decompositions, we can use the cost functionsearching algorithm to find the best basis set \( \left\{{D}_l^{(k)}\right\} \) among all the possible basis sets, which can represent the spatial and frequency domain information of the analyzed dataset more effectively than all other basis sets. Use of this best basis setbased multiwavelet packet transform in image processing applications should provide optimal quality results.
Information on the root node \( {D}_0^{(0)} \) in the tree illustrated in Fig. 6 can be obtained by rdimensional vectorization of the analyzed datasets. Using the balanced or highorder balanced Mband multiwavelet system, this vectorization procedure can be simplified as an rfold downsampling process, demonstrated as follows. Defining the analyzed signal as S = [s _{1}, s _{2}, … s _{ N − 1}]^{T},S should be periodically extended until N = ⌈N/r⌉ × r if N mod r ≠ 0. Then, the extended S should be vectorized as an r × (N/r) matrix, i.e., \( {D}_0^{(0)}={\left[{D}_0^{(0)}(0),{D}_0^{(0)}(1),\dots, {D}_0^{(0)}\left(r1\right)\right]}^T, \) where \( {D}_0^{(0)}(j)=\left[{s}_{0\times r+j},{s}_{1\times r+j},\dots, {s}_{\left(N/r1\right)\times r+j}\right] \) and 0 ≤ j < r. After establishing \( {D}_0^{(0)} \) with the vectorization process, \( {D}_l^{(k)} \) at each level can be calculated by repeatedly performing the decomposition process in Eq. (5). Figure 7 shows illustrative results of decomposition by the Mband multiwavelet packet transform of a remote sensing image. As can be seen, numbers of transform domain coefficients in different cases of M are the same as all the transforms are orthogonal.
As shown in Fig. 5, the fast algorithm of the proposed highorder balanced Mband MWPT mainly consists of convolutions and subsampling processes and is very similar to the Mallat algorithm generally used in classical wavelet transforms.
3 MWPTbased remote sensing image denoising
3.1 Denoising algorithm
Many previously proposed wavelet transformbased denoising approaches can be introduced to MWPTbased remote sensing image denoising, notably the wavelet shrinkage method proposed by Donoho [2] and widely advocated in later studies [6, 10, 19, 22]. We applied this simple denoising scheme (rather than more complex shrinkage schemes) to compare the denoising performance of our transform and other traditional wavelet transforms. First, we applied the proposed MWPT for multiscale decomposition of noisy images then used soft thresholding to shrink the transform domain coefficients [2]. The threshold (universally) used was \( T=\sigma \times \sqrt{2\times \ln (N)} \), where N is the number of pixels in the original image, and σ is the variance of an additive noise, which may be unknown. If so, we used a robust median estimator \( \widehat{\sigma} \) of σ, computed from the multiwavelet coefficients of highpass subband at scale k = 1 as shown in Eq. (7). Finally, the proposed MWPT was applied for reconstruction using the shrunk coefficients.
3.2 Denoising performance comparison
Extensive simulations were carried out using both synthetic and real noisy remote sensing images to investigate the performance of the proposed highorder balanced MWPTbased denoising method. The highorder balanced Mband multiwavelet systems constructed in [26] were used here to construct our transform (i.e., MWPT with M > 2 and ρ > 0). We compared the denoising performance of our transform with those of the MWPT schemes proposed in recently relevant works, including the 2band MWPT proposed by Liu et al. [16] (MWPT with M = 2 and ρ = 0), the balanced 2band MWPT proposed by He et al. [23] (MWPT with M = 2 and ρ > 0), and the Mband MWT proposed by Ren et al. [25]. Using the algorithms in Section 2, the Mband MWT was developed into Mband MWPT (MWPT with M > 2 and ρ = 0) here for a fair comparison. The denoising strategies applied in these approaches are the same as those described in Section 3.1, apart from minor differences in the transforms per se.
3.2.1 Synthetic noisy databased analysis
Various remote sensing images were used as original noiseless images for generating the synthetic noisy images. One of the test images is shown in Fig. 8a. This image, collected from the panchromatic band of the SPOT5 satellite, covers an area in Yantai, China. As seen, it includes rich textures and edges. Figure 8b displays a closeup of the image and shows the details of its textures.
To test the denoising performance to different patterns of noise attack, four kinds of noise are mixed with the original images as follows.

Pattern A: multiplicative noise. Multiplicative noise is added to the original noiseless image I with the equation J = I + V × I, where V is uniformly distributed random noise with zero mean and variance 0.1.

Pattern B: salt and pepper noise. Salt and pepper noise is added to the original noiseless image I with the noise density equals to 0.2.

Pattern C: Gaussian white noise. Gaussian white noise is added to the original noiseless image I with zero mean and variance 5.

Pattern D: Poisson noise. Poisson noise is generated from the image itself instead of adding artificial noise to the original noiseless image I.
Results obtained with all the test images are similar. Thus, results obtained using the image in Fig. 8a are presented here to illustrate the denoising performance of the test methods. In the following figures, we display the denoising results of MWPTbased methods with typical cases of (M,ρ), including those with (M,ρ) equal to (2,0), (4,0), (2,3), and (4,2) (as proposed in [16, 23, 25] and this work).

A.
Visual quality analysis
As it is difficult to distinguish differences among the denoising images obtained using the tested methods at entireimage scale, we will consider the enlarged parts of the images obtained for the four noise patterns shown in Figs. 9, 10, 11, and 12. In each case, the part covered is that shown in Fig. 8.
Figure 9 shows comparative closeups when the original image is subjected to pattern A noise. As displayed in Fig. 9a, the image to be denoised is seriously contaminated by the multiplicative noise, compared with the noiseless counterpart in Fig. 8b. Figure 9b, c shows that although the methods of Liu et al. and Ren et al. reduce the noise, both blur some of the edges in Fig. 9a (e.g., those in the area arrowed). The method of He et al. preserves edges relatively well but at the same time introduces “artifacts” (especially where arrowed), as shown in Fig. 9d. By contrast, the proposed method with high M and ρ (i.e., (4,2)) desirably suppresses the noise, while well preserving the edges and textures in the noisy image, as shown in Fig. 9e.
Figure 10 shows the results for images contaminated by pattern B noise. It can be seen that all of the methods effectively suppress the salt and pepper noise shown in Fig. 10a. However, our proposed method more desirably retains some fine structures in the noisy image, in comparison with the other test methods.
The results for images contaminated by pattern C noise are shown in Fig. 11. The method of Liu et al. suppresses the Gaussian white noise in the smooth area well, but some edges and textures are somewhat oversmoothed (e.g., where arrowed). The method of Ren et al. also undesirably preserves some edges, as shown in Fig. 11c. Figure 11d shows that the method of He et al. retains edge information well but adds some “artifacts” (e.g., in the arrowed area). In contrast, our method both reduces the noise and preserves the edges in the noisy image well (see Fig. 11e).
The proposed method also shows desirable performance in suppressing Poisson noise. As shown in Fig. 12, there are clear differences in results provided by the tested methods, although they all improve the quality of the noisy image displayed in Fig. 12a well. Figure 12b shows that the method of Liu et al. blurs some edges in Fig. 12a. The method of Ren et al. desirably reduces the noise but somewhat oversmooths some edges and textures (e.g., those arrowed). The method of He et al. retains the fine structures well but introduces some “artifacts,” while as shown in Fig. 12e, our method with high M and ρ provides slightly better results.

B.
Quantitative analysis
In order to analyze the tested methods’ denoising effects quantitatively, we calculated peak signaltonoise ratio (PSNR) and structural similarity (SSIM) values to assess from pixel and structurelevel fidelity aspects, respectively. SSIM is an image quality assessment index based on the human vision system and indicates degrees to which structural information in noisy image has been retained [27]. It is given by:
where μ _{ x } and μ _{ y } represent the average gray values of the noiseless reference image x and the denoising image y, respectively, σ _{ x } and σ _{ y } represent the variances of x and y, respectively, and σ _{ xy } represents the covariance between x and y. The symbols C _{1} and C _{2} are two constants which are used to prevent unstable results when either \( {\mu}_x^2+{\mu}_y^2 \) or \( {\sigma}_x^2+{\sigma}_y^2 \) very close to zero.
Table 1 presents PSNR and SSIM values obtained by applying all the test methods to denoise the above used image corrupted by different noise patterns. For every pattern, our proposed method consistently provides better results (higher PSNR and SSIM values) than the other test methods.
3.2.2 Real noisy databased analysis
Synthetic aperture radar (SAR) images are inherently affected by multiplicative speckle noise, which generally affects the basic textures of SAR imagery [28]. Hence, we also chose real noisy SAR images without adding artificial noise to evaluate the performance of the proposed method. One of the test images was shown in Fig. 13a, covering an area of Beijing Airport, China, collected by airborne radars and provided by Chinese Academy of Science. As shown, it covers many ground objects and rich textures.
Figure 13b–e displays the denoising results obtained by applying all the test methods to the image shown in Fig. 13a. As can be seen, all the methods effectively reduce the speckle noise in the original image, especially in the smooth area (e.g., the surface of the runway). Figure 14 shows closeups of the same part of the image obtained using each method for clear comparison. All of the MWPTbased methods clearly suppress speckle well, but those with low M or ρ oversmooth images and thus blur many features (e.g., in the arrowed area). By contrast, the proposed 2order balanced 4band MWPTbased method preserves more structural details in the original image.
Since noisefree reference images were not available, equivalent number of looks (ENL) values of homogeneous regions were calculated. ENL is often applied to characterize the smoothing effect of denoising methods, given by:
where the average μ _{ x } and variance σ _{ x } are carried out over a target region x [3, 4, 28]. For a homogeneous region in a denoising image, the ENL value will simply reflect the degree to which the denoising method suppressed the noise in the region. However, for a heterogeneous region, the ENL value should not be too high, otherwise the method may oversmooth structural information regarding the region.
ENL values were calculated for two homogeneous regions (1 and 2 in Fig. 14a) and two heterogeneous regions (3 and 4) shown in Fig. 14b–e. The results, listed in Table 2, clearly show that the proposed method provides the highest ENL value for each homogeneous region and modest values for each heterogeneous region, demonstrating its superiority for both noise suppression and texture preservation. The results of the quantitative evaluation are also consistent with the visual assessment presented above.
3.3 Influence of the M, ρ, and k
The impacts of the band number M, the balanced order ρ, and the decomposition level k on denoising performance when using the MWPTbased method can be summarized in more detail as follows.

(1)
For every M value between 2 and 4, as the balanced order ρ rises within an appropriate range, the PSNR and SSIM provided by the MWPTbased method increase, as illustrated in Table 3 for the dataset considered above. These results clearly confirm the superiority of the highorder balanced Mband MWPT for preserving image features.

(2)
As M rises from 2 to 4, the results of the MWPTbased method improve at each balanced order ρ. Table 3 shows the improvement when applied to the dataset in Fig. 8a. This further corroborates the advantages of Mband over 2band multiwavelet systems.

(3)
The MWPTbased method provides optimal results (at a given ρ and M) when the decomposition level k is appropriate (3 or 4 in 3band cases, and 2 or 3 in 4band cases). This is illustrated by the PSNR and SSIM values in Table 4 for the images in Fig. 8a, showing that for several typical M and ρ cases, 3band and 4band MWPTs provide optimal indicator values when k = 3.
3.4 Computational complexity analysis
As the proposed transform is orthogonal, it adds little complexity and the computational costs of the proposed method are similar to those of methods based on traditional multiwavelets (e.g., GHM multiwavelet). Moreover, the appropriate decomposition levels for M > 2 cases are always less than that for the M = 2 case reported in Section 3.2, which further reduces the computational complexity of the proposed Mband MWPTbased method. These assertions were verified by the average computational times—in a MATLAB environment using a workstation with an Intel(R) Core^{TM} i5 CPU (3.2 GHz) and 4 Gb RAM—for the denoising cases reported in Section 3.2. Computation times of the denoising methods based on highorder balanced Mband MWPT are slightly longer than those of other MWPTbased methods but still reasonable (Table 5).
4 Conclusions
This work shows that using an appropriate higher balanced order or band number improves the denoising performance of multiwavelet packet transformbased methods at little additional computational cost. The highorder balanced Mband multiwavelet packet transform is also an effective scheme for other image processing tasks, such as texture analysis and edge extraction. This is because with an appropriate higher balanced order, the transform scheme provides more effective sparse image representation, a higher band number provides more flexible spatialfrequency domain partitioning, and the most suitable basis set for an analyzed signal can be selected for the multiwavelet packet transform scheme. We used a simple denoising strategy to evaluate the performance of the proposed transform scheme. To further improve its performance, other approaches that are more suitable for an Mband multiwavelet system should also be applied (e.g., taking into account the different frequency behaviors of the wavelets in a multiwavelet system in fusion rule selection).
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Acknowledgements
This work was supported by the Natural Science Foundation of Hebei Province, China (Grant No. D2015402159) and the National Natural Science Foundation of China (Grant Nos. 41330746 and 41171225), and also supported by the Natural Science Foundation of Hebei Province, China (Grant No. F2015402150), the Hebei Education Department, China (Grant Nos. ZD2014081 and ZD2015087), the Ministry of Agriculture’s Special Funds for Scientific Study on Public Causes, China (Grant No. 201203062) and the National Natural Science Foundation of China (Grant Nos. 41371274 and 51309016). The author would like to thank I. W. Selesnick of Polytechnic Institute of New York University for many useful questions and comments.
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Wang, H., Wang, J., Jin, H. et al. Highorder balanced Mband multiwavelet packet transformbased remote sensing image denoising. EURASIP J. Adv. Signal Process. 2016, 10 (2016). https://doi.org/10.1186/s1363401502987
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DOI: https://doi.org/10.1186/s1363401502987