High-order balanced M-band multiwavelet packet transform-based remote sensing image denoising

Transcription

1 Wang et a. EURASIP Journa on Advances in Signa Processing (2016) 2016:10 DOI /s RESEARCH High-order baanced M-band mutiwaveet packet transform-based remote sensing image denoising Open Access Haijiang Wang 1,2, Jingpu Wang 3*, Hai Jin 2, Lihong Li 1, Dongi Jia 1, Yongqiang Ma 1, Yongmei Liu 4, Fuqi Yao 5 and Qinke Yang 4 Abstract This artice proposes high-order baanced muti-band mutiwaveet packet transforms for denoising remote sensing images. First, properties of severa waveet transforms and their reationships are anayzed. The artice then presents theoretica principes and a fast agorithm for constructing high-order baanced muti-band mutiwaveet packet transforms. The remote sensing image denoising method based on this transform scheme is then described, and its utiity is demonstrated by iustrative resuts of its appication to denoise remote sensing images. The method provides cear improvements in denoising quaity, due to the baanced order or band number, consistenty outperforming traditiona waveet transform-based methods in terms of both visua quaity and evauation indicators. The method aso incurs reasonabe computationa costs compared with the traditiona methods. Keywords: Remote sensing image, Denoising, Mutiwaveet packet transform, Muti-band, Baanced 1 Introduction Remote sensing imaging has become a powerfu technique for exporing and obtaining knowedge of numerous phenomena. However, during acquisition and transmission processes, the images are often contaminated by noise, which impairs their visua quaity and imits the precision of subsequent processing steps, such as cassification, target detection, and environmenta monitoring. Thus, remote sensing image denoising appications have attracted growing interest. Transform domain denoising methods have shown remarkabe success in the ast decade. An assumption typicay underying these methods is that signa can be sparsey represented in the transform domain. Hence, by preserving the few high-magnitude transform coefficients that convey most of the true signa energy and discarding the rest, which are mainy due to noise, the true signa can be effectivey estimated. The sparsity of the representation depends on both the transform used and the true signa s properties [1, 2]. * Correspondence: [email protected] 3 Schoo of Resources and Environmenta Engineering, Ludong University, Yantai , Peope s Repubic of China Fu ist of author information is avaiabe at the end of the artice The waveet transform (i.e., 2-band scaar waveet transform) can provide good sparsity for spatiay ocaized detais, and a number of advanced denoising methods based on them have been deveoped [2 4]. For exampe, the waveet threshoding approach popuarized by Donoho is now widey used in scientific and engineering appications [2]. The best image denoising systems incude fiters with symmetric and compact-support properties, which can effectivey extract features and eiminate artifacts. Unfortunatey, however, no nontrivia, symmetric, compact-support, orthogona scaar waveet transforms are avaiabe [1, 5]. Mutiwaveet transforms (i.e., 2-band mutiwaveet transforms or 2-band MWTs) have severa advantages over scaar waveet transforms, because they can simutaneousy possess a the above properties [5, 6]. Since these properties are highy significant in image processing, mutiwaveets have attracted considerabe research interest and shown superior denoising performance over scaar waveets in various studies [6 10] (see the diagram about the deveopment of waveet transforms in Fig. 1). Sveinsson et a. [11] and Wang et a. [12] appied 2-band MWT in remote sensing image denoising and found that they generay outperform 2-band scaar waveet transforms 2016 Wang et a. Open Access This artice is distributed under the terms of the Creative Commons Attribution 4.0 Internationa License ( which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the origina author(s) and the source, provide a ink to the Creative Commons icense, and indicate if changes were made.

2 Wang et a. EURASIP Journa on Advances in Signa Processing (2016) 2016:10 Page 2 of 14 Fig. 1 Deveopment of waveet transforms and reationships among different kinds of waveet transform schemes. The transform schemes marked with ight green coor are compared with our proposed scheme (with ight bue coor) in our evauation experiment of denoising performance in both visua quaity and objective evauation. Further improvements of 2-band MWT coud potentiay be obtained by taking into account the properties of the signa to be anayzed, but a the cited transforms use the unchangeabe transform pattern for diverse images. That is, as iustrated in Fig. 2a, the decomposition at each decomposition eve is ony appied to the ow-frequency component of signas and does nothing to the other, reativey high-frequency components, athough this partitioning is not suitabe for a signas with different behaviors across the frequency domain [1, 13]. Mutiwaveet packet transforms (i.e., 2-band mutiwaveet packet transforms or 2-band MWPT), as one extension of 2-band MWT, provide an effective mean to seect a suitabe decomposition pattern corresponding to an anayzed signa. As shown in Fig. 2b, a 2-band MWPT offers a finer frequency domain partition than a 2-band MWT, especiay in the high-frequency domain. Hence, numerous subsets corresponding to different mutiwaveet bases (or packets) can be found in its partition point set, one of which wi match the properties of the anayzed signas better than a the others. The 2-band MWPT based on this subset (the best packet) can provide a better sparse representation of the signa than those based on other subsets (incuding the 2-band MWT, as the subset it uses is just one of these subsets) [1, 13]. Martin et a. [14] and Wang [15] introduced 2-band MWPTs to image compression and texture segmentation, respectivey, and showed that they exhibit performance generay superior to 2-band MWT. Liu et a. proposed a choice agorithm of the best mutiwaveet packet and found that the 2-MWPT based on the agorithm can generay obtain better denoising resut than 2-band MWT at the same condition [16]. Deveoping a 2-band MWT scheme into its packet version is an effective way to improve performance (as shown in Fig. 1), however, the probem of baanced or high-order baanced must be addressed when using them in practica appications. The baanced order ρ of a mutiwaveet system corresponds to its abiity to represent images sparsey [17, 18]. Recent studies show that the 2-band MWTs based on baanced (ρ = 1) mutiwaveets (i.e., baanced 2-band MWT, as shown in Fig. 1) consistenty outperform those based on unbaanced (ρ = 0) counterparts (i.e., 2-band MWT) in image denoising [19 22]. Aso, baanced 2-band MWPT schemes derived by deveoping the baanced 2-band MWT into packet versions aso consistenty outperform (unbaanced) 2-band MWPT in seismic data compression and denoising [23]. However, there have been no pubished indepth studies on the reationship between the baanced order ρ and sparse representation abiity of baanced (ρ = 1) or high-order baanced (ρ > 1) 2-band MWPT (or even MWT). More importanty, athough the 2-band MWPTs overcome many shortcomings of waveet transforms, they Fig. 2 The idea frequency domain portioning patterns offered by different mutiwaveet transforms with normaized highest frequency, taking M = 3 and k =3(k refers to a decomposition eve) as an exampe. a For a 2-band mutiwaveet transform, the idea partition points are ony in the set {1/2, 1/2 2,,1/2 k }. b For a 2-band mutiwaveet packet transform, they are in the set {1/2 k,2/2 k,,2 k 1/2 k }. c For an M-band mutiwaveet transform, they are in the set k i¼1 1=Mi ; 2=M i ; ; M 1=M i. d For an M-band mutiwaveet packet transform, they are extended to the set {1/M k,2/m k,, M k 1/M k }

3 Wang et a. EURASIP Journa on Advances in Signa Processing (2016) 2016:10 Page 3 of 14 Fig. 3 Library of M-band mutiwaveet packet basis functions, in 3-eve decomposition retain the 2-band weakness of the atter, especiay in spatia-frequency tiing, which has triggered great interest in their extension [1]. As another extension of 2-band MWT, M-band (M Z and M > 2) MWTs provide greater fexibiity in spatiafrequency tiing and more robust sparse representation [1, 24] (see Fig. 1). As shown in Fig. 2c, the idea partition point set of an M-band MWT is denser than that of a 2-band counterpart (see Fig. 2a). M-band MWT has been shown to outperform 2-band MWT in terms of both visua quaity and objective indicators in image fusion [25]. Simiary, highorder baanced M-band MWTs reportedy outperform (unbaanced) M-band MWT in image compression [26]. Potentiay, a high-order baanced M-band MWT may aso be further improved (ike a 2-band counterpart) by using a packet version(i.e.,anm-bandmwpt,asshowninfig.1). As shown by the theoretica partition pattern of an M-band MWPT in Fig. 2d, it may provide a finer partitioning pattern and better matching subset (or more effective representation) for an anayzed signa, reative to a 2-band counterpart. M-band MWPTs may offer great potentia for image processing, but substantia extension of both their fundamenta theory and convenient methodoogy are required. Thus, here we present basic principes of, and fast agorithms for, highorder baanced M-band MWPT by deveoping the highorder baanced M-band MWT presented in [26], according to key theory of deveoping baanced 2-band MWT into packet versions presented in [23]. We aso evauate their performance in remote sensing image denoising in comparison with the 2-band MWPT presented in [14 16], the baanced 2-band MWPT in [23], the M- band MWT in [25], and the high-order baanced M-band MWT in [26] (see Fig. 1). Moreover, we systematicay anayze the impact of the baanced order ρ and band number M on the sparse representation abiity of MWPT. The rest of the paper is organized as foows. In Section 2, we anayze the basic principes and fast agorithm of highorder baanced M-band MWPT based on the reative theory of M-band MWT. In Section 3, we present the denoising method based on the proposed transform, and evauate the denoising performance of the method by using both synthetic and rea noisy remote sensing images in comparison with the method based on the transforms recenty proposed. We aso anayze the impacts of the baanced order ρ and band number M on their denoising performance. We summarize and discuss the work in Section 4. 2 High-order baanced M-band mutiwaveet packet transforms Cassica waveet transforms cannot provide many key properties simutaneousy, as they estabish a mutiresoution anaysis frame using a singe scae function [1, 5, 6]. In contrast, mutiwaveet transforms use r (r Z and r > 1) scae functions (r-mutipicity) for this work, thus reducing the number of constraint conditions and increasing the freedom in design. The r scae functions are denoted in vector form, i.e., Φ 0 (t)=[ϕ 0,0, ϕ 1,0,, ϕ r 1,0 ]. An M-band r-mutipicity mutiwaveet system has M-1 waveet function vectors denoted as Φ i (t)=[ϕ 0,i, ϕ 1,i,, ϕ r 1,i ](1 i < M). If (and ony if) such a system is orthogona, the Φ i (t) meets the foowing condition [24, 26], Fig. 4 Decomposition patterns of different M-band mutiwaveet packet transforms (taking M = 2 and k = 3 as an exampe). The pattern in (a) corresponds to the standard M-band mutiwaveet decomposition scheme and that in (d) corresponds to a kind of mutiwaveet decomposition scheme that presents the finest frequency-domain partition. The patterns in (b-c) correspond to two other possibe decomposition schemes respectivey

4 Wang et a. EURASIP Journa on Advances in Signa Processing (2016) 2016:10 Page 4 of 14 Fig. 5 The decomposition and reconstruction of the M-band mutiwaveet packet transform p Φ i ðxþ ¼ ffiffiffiffiffi X M P n Z iðnþφ 0 ðmx nþ: ð1þ Here, P i (n) is a vector fiter bank of r r and n Z (the same as beow). We can get the foowing M-band mutiwaveet transform agorithm after deveoping the corresponding agorithm of the cassica waveet transform (i.e., Maat agorithm) [24, 26], C ðkþ1þ p i ðnþ ¼ ffiffiffiffiffi X M P m Z iðmþc ðkþ 0 ðmn þ mþ; ð2þ C ðkþ p 0 ðmþ ¼ ffiffiffiffiffi X X M i Z n Z PT i ðnþc ðkþ1þ i ðmn þ mþ: ð3þ Here, m, n Z and both are shift parameters, whie C k i is an r-dimensiona vector (the same as beow). A mutiwaveet system is said to be baanced of order ρ if its owpass and highpass fiters preserve and cance, respectivey, a the monomia poynomia signas of order ess than ρ. The order of a mutiwaveet system corresponds to its abiity to effectivey represent the information of the macroscae change trend and oca textures of a signa [17, 18]. If P i (n) meets the constraint condition corresponding to each property above (e.g., ρ-order baanced, orthogonaity), the baanced or highorder baanced M-band MWPT coud be constructed by deveoping the above M-band mutiwaveet transform agorithm. Their basic principes and fast agorithms are present as foows. 2.1 Basic principes Let Ψ i (t)=φ i (t) (0 i < M), and define p Ψ Mþi ¼ ffiffiffiffiffi X M P n Z iðnþψ ðmx nþ: ð4þ M-fod rescaing and transation of these functions yied a function ibrary χ ¼fΨ ðkþ ¼ M k=2 Ψ ðm k x nþg (0 < M k ). As shown in Fig. 3, the ibrary can be viewed in terms of an M-fod compete tree, and the th function at the kth decomposition eve Ψ ðkþ yieds M functions at the k +1 eve, i.e., Ψ ðkþ1þ Mþi (0 i <M). The ibrary is overcompete, and many compete orthogona basis sets can be found by propery seecting different subsets in the ibrary with an appropriate parameter set {k, }.. A compete orthogona basis set corresponds to a kind of mutiwaveet packet transform scheme, a subset of the ibrary and a parameter set {k, }.. Each subset can be viewed in terms of an M-fod tree structure. Exampes of possibe basis sets are shown in the trees in Fig. 4. The tree in Fig. 4a seects the subset of the ibrary with the parameter set {1 <M,0 k <3} {0 <M,k = 3} and corresponds to the standard M-band mutiwaveet decomposition scheme [5, 6, 24]. The tree in Fig. 4d seects the subset with the parameter set {0 <M k 1, k =3}andcorresponds to a kind of mutiwaveet decomposition scheme that presents the finest frequency domain partition. Fig. 6 Reationship of the transform domain coefficient at different decomposition eves in the M-band mutiwaveet packet transform, taking k =3as an exampe

5 Wang et a. EURASIP Journa on Advances in Signa Processing (2016) 2016:10 Page 5 of 14 Fig. 7 Iustrative resuts of (1-eve) decomposition by the M-band mutiwaveet packet transform of a remote sensing image, with M = 2 (a), 3(b), and 4 (c) For any anayzed dataset, there is a basis set (the best basis set) that can best represent the spatia and frequency domain information in the dataset, and the corresponding best mutiwaveet packet transform provides the most effective representation of the dataset (denoted as MWPT at beow). The best basis set is found by using a cost function searching agorithm that seeks the best subset for some appication in a set with tree structure [1, 13]. For exampe, a searching agorithm that uses information entropy as a cost function can find the most informative basis subset and consistenty perform effectivey in image denoising [1, 13]. Since the overcompete ibrary of the basis sets generated by the high-order baanced M-band mutiwaveet packet transform can aso be viewed in terms of a tree structure, we used entropic cost function searching agorithms to find both the best basis set corresponding to test datasets and for estabishing the searching agorithm. 2.2 Fast agorithm Defining D ð0þ i ¼ C ð0þ i ð0 i < MÞ; a fast agorithm for decomposition of the M-band mutiwaveet packet transform is shown in Eq. (5). Reconstructions are the inverse process of decompositions, and its fast agorithm is shown in Eq. (6). Based on the expression, its cear procedures in Z transform domain are iustrated in Fig. 5, where P i (z) is the Z transformation of P i (n). D ðkþ1þ Mþi D ðkþ p ðnþ ¼ ffiffiffiffiffi X M P m Z iðmþd ðkþ ðmn þ mþ ð5þ p ðmþ ¼ ffiffiffiffiffi X M 1 X M P i¼0 n Z iðnþd ðkþ1þ Mþi ðmn þ mþ ð6þ Using the decomposition agorithm above, one can obtain a compete M-fod tree incuding information on a Fig. 8 a Origina noiseess remote sensing test image and b a cose-up of (a) regarding the region surrounded by the rectange in (a). The noisy images and denoising resuts reevant to the part are shown in Figs. 9, 10, 11, and 12 for cear comparison

6 Wang et a. EURASIP Journa on Advances in Signa Processing (2016) 2016:10 Page 6 of 14 its nodes D ðkþ ð1 < MkÞ; as shown in Fig. 6. D ðkþ are mutiwaveet domain coefficients containing spatia information of the incuded frequency bands (see Fig. 2d). After decompositions, we can use the cost function-searching n o agorithm to find the best basis set D ðkþ among a the possibe basis sets, which can represent the spatia and frequency domain information of the anayzed dataset more effectivey than a other basis sets. Use of this best basis set-based mutiwaveet packet transform in image processing appications shoud provide optima quaity resuts. Information on the root node D ðþ 0 0 in the tree iustrated in Fig. 6 can be obtained by r-dimensiona vectorization of the anayzed datasets. Using the baanced or high-order baanced M-band mutiwaveet system, this vectorization procedure can be simpified as an r-fod downsamping process, demonstrated as foows. Defining the anayzed signa as S =[s 1, s 2, s N 1 ] T,S shoud be periodicay extended unti N = N/r r if N mod r 0. Then, the extended S shoud be vectorized as an r (N/r) matrix, h i T; i.e., D ð0þ 0 ¼ D ð0þ 0 ð0þ; D ð0þ 0 ð1þ; ; D ðþ 0 ðþ 0 0 ðr 1Þ where D 0 ðþ¼ j s 0 rþj ; s 1 rþj ; ; s ðn=r 1Þ rþj and 0 j < r. After estabishing D ð0þ 0 with the vectorization process, D ðkþ at each eve can be cacuated by repeatedy performing the decomposition process in Eq. (5). Figure 7 shows iustrative resuts of decomposition by the M-band mutiwaveet 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 a the transforms are orthogona. As shown in Fig. 5, the fast agorithm of the proposed high-order baanced M-band MWPT mainy consists of convoutions and subsamping processes and is very simiar to the Maat agorithm generay used in cassica waveet transforms. Fig. 9 Cose-ups of the same part of the noisy image (a) and reevant denoising resuts obtained by different methods (b e), when mixing the origina image in Fig. 8a with pattern A noise. The part covered is that shown in Fig. 8, the same beow

7 Wang et a. EURASIP Journa on Advances in Signa Processing (2016) 2016:10 Page 7 of 14 3 MWPT-based remote sensing image denoising 3.1 Denoising agorithm Many previousy proposed waveet transform-based denoising approaches can be introduced to MWPTbased remote sensing image denoising, notaby the waveet shrinkage method proposed by Donoho [2] and widey advocated in ater studies [6, 10, 19, 22]. We appied this simpe denoising scheme (rather than more compex shrinkage schemes) to compare the denoising performance of our transform and other traditiona waveet transforms. First, we appied the proposed MWPT for mutiscae decomposition of noisy images then used soft threshoding to shrink the transform domain coefficients [2]. The threshod (universay) used was T ¼ σ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 nðnþ,wheren is p the number of pixes in the origina image, and σ is the variance of an additive noise, which may be unknown. If so, we used a robust median estimator ^σ of σ, computed from the mutiwaveet coefficients of high-pass subband at scae k =1asshowninEq.(7). Finay, the proposed MWPT was appied for reconstruction using the shrunk coefficients. hn oi ^σ ¼ median D ð1þ =0:6745; 1 < M ð7þ 3.2 Denoising performance comparison Extensive simuations were carried out using both synthetic and rea noisy remote sensing images to investigate the performance of the proposed high-order baanced MWPT-based denoising method. The high-order baanced M-band mutiwaveet 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 recenty reevant works, incuding the 2-band MWPT proposed by Liu et a. [16] (MWPT with M = 2 and ρ = 0), the baanced 2-band MWPT proposed by He et a. [23] (MWPT with M = 2 and ρ >0),andtheM-band MWT proposed by Ren et a. [25]. Using the agorithms in Fig. 10 Cose-ups of the same part of the noisy image (a) and reevant denoising resuts (b e), when mixing the origina image in Fig. 8a with pattern B noise

8 Wang et a. EURASIP Journa on Advances in Signa Processing (2016) 2016:10 Page 8 of 14 Section 2, the M-band MWT was deveoped into M-band MWPT (MWPT with M > 2 and ρ =0) here for a fair comparison. The denoising strategies appied in these approaches are the same as those described in Section 3.1, apart from minor differences in the transforms per se Synthetic noisy data-based anaysis Various remote sensing images were used as origina noiseess images for generating the synthetic noisy images. One of the test images is shown in Fig. 8a. This image, coected from the panchromatic band of the SPOT-5 sateite, covers an area in Yantai, China. As seen, it incudes rich textures and edges. Figure 8b dispays a cose-up of the image and shows the detais of its textures. To test the denoising performance to different patterns of noise attack, four kinds of noise are mixed with the origina images as foows. Pattern A: mutipicative noise. Mutipicative noise is added to the origina noiseess image I with the equation J = I + V I, where V is uniformy distributed random noise with zero mean and variance 0.1. Pattern B: sat and pepper noise. Sat and pepper noise is added to the origina noiseess image I with the noise density equas to 0.2. Pattern C: Gaussian white noise. Gaussian white noise is added to the origina noiseess image I with zero mean and variance 5. Pattern D: Poisson noise. Poisson noise is generated from the image itsef instead of adding artificia noise to the origina noiseess image I. Resuts obtained with a the test images are simiar. Thus, resuts obtained using the image in Fig. 8a are presented here to iustrate the denoising performance of the test methods. In the foowing figures, we dispay the denoising resuts of MWPT-based methods with typica cases of (M,ρ), incuding those with (M,ρ) equa to (2,0), (4,0), (2,3), and (4,2) (as proposed in [16, 23, 25] and this work). Fig. 11 Cose-ups of the same part of the noisy image (a) and reevant denoising resuts (b e), when mixing the origina image in Fig. 8a with pattern C noise

9 Wang et a. EURASIP Journa on Advances in Signa Processing (2016) 2016:10 Page 9 of 14 Fig. 12 Cose-ups of the same part of the noisy image (a) and reevant denoising resuts (b e), when mixing the origina image in Fig. 8a with pattern D noise A. Visua quaity anaysis As it is difficut to distinguish differences among the denoising images obtained using the tested methods at entire-image scae, we wi consider the enarged 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 cose-ups when the origina image is subjected to pattern A noise. As dispayed in Fig. 9a, the image to be denoised is seriousy contaminated by the mutipicative noise, compared with the Tabe 1 Denoising indicator resuts obtained by appying a the test methods to denoise the above used image corrupted by different noise patterns Noise patterns Indicators Noisy image Method of Liu et a. Method of Ren et a. Method of He et a. Method of our proposed method Pattern A PSNR SSIM Pattern B PSNR SSIM Pattern C PSNR SSIM Pattern D PSNR SSIM

10 Wang et a. EURASIP Journa on Advances in Signa Processing (2016) 2016:10 Page 10 of 14 noiseess counterpart in Fig. 8b. Figure 9b, c shows that athough the methods of Liu et a. and Ren et a. reduce the noise, both bur some of the edges in Fig. 9a (e.g., those in the area arrowed). The method of He et a. preserves edges reativey we but at the same time introduces artifacts (especiay where arrowed), as shown in Fig. 9d. By contrast, the proposed method with high M and ρ (i.e., (4,2)) desiraby suppresses the noise, whie we preserving the edges and textures in the noisy image, as shown in Fig. 9e. Figure 10 shows the resuts for images contaminated by pattern B noise. It can be seen that a of the methods effectivey suppress the sat and pepper noise shown in Fig. 10a. However, our proposed method more desiraby retains some fine structures in the noisy image, in comparison with the other test methods. The resuts for images contaminated by pattern C noise are shown in Fig. 11. The method of Liu et a. suppresses the Gaussian white noise in the smooth area we, but some edges and textures are somewhat oversmoothed (e.g., where arrowed). The method of Ren et a. aso undesiraby preserves some edges, as shown in Fig. 11c. Figure 11d shows that the method of He et a. retains edge information we 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 we (see Fig. 11e). The proposed method aso shows desirabe performance in suppressing Poisson noise. As shown in Fig. 12, there are cear differences in resuts provided by the tested methods, athough they a improve the quaity of the noisy image dispayed in Fig. 12a we. Figure 12b shows that the method of Liu et a. burs some edges in Fig. 12a. The method of Ren et a. desiraby reduces the noise but somewhat oversmooths some edges and textures (e.g., those arrowed). The method of He et a. retains the fine structures we but introduces some artifacts, whie as shown in Fig. 12e, our method with high M and ρ provides sighty better resuts. B. Quantitative anaysis Fig. 13 Origina test noisy image (a) and the denoising images obtained by using a the test methods (b e). The part surrounded by the rectange in a is enarged in Fig. 14 for cear comparison

11 Wang et a. EURASIP Journa on Advances in Signa Processing (2016) 2016:10 Page 11 of 14 Fig. 14 Cose-ups of the same part of the noisy image (a) and reevant denoising resuts (b e). The area covered is that shown in Fig. 13a. The districts surrounded by red (regions 1 and 2) and yeow (regions 3 and 4) rectanges are seected as homogeneous and heterogeneous regions for ENL cacuation, respectivey In order to anayze the tested methods denoising effects quantitativey, we cacuated peak signa-to-noise ratio (PSNR) and structura simiarity (SSIM) vaues to assess from pixe- and structure-eve fideity aspects, respectivey. SSIM is an image quaity assessment index based on the human vision system and indicates degrees to which structura information in noisy image has been retained [27]. It is given by: SSIMðx; yþ ¼ 2μ x μ y þ C 1 2σ xy þ C 2 ð8þ μ 2 x þ μ2 y þ C 1 σ 2 x þ σ2 y þ C 2 where μ x and μ y represent the average gray vaues of the noiseess reference image x and the denoising image y, respectivey, σ x and σ y represent the variances of x and y, respectivey, and σ xy represents the covariance between x Tabe 2 ENL vaues obtained by a the test methods, using the data in Fig. 14 Region patterns Regions Size ENL of origina image and denoising resuts Origina image Method of He et a. Method of He et a. Method of He et a. Homogeneous region Region 1 (awn) Region 2 (roadbed) Heterogeneous region Region 3 (pane) Region 4 (buiding) Method of our proposed method

12 Wang et a. EURASIP Journa on Advances in Signa Processing (2016) 2016:10 Page 12 of 14 Tabe 3 PSNR and SSIM vaues obtained by MWPT-based methods with different M and ρ parameters, when using the test image in Fig. 8a and mixing Gaussian white noise with σ =10 Methods PSNR SSIM ρ =0 ρ =1 ρ =2 ρ =3 ρ =0 ρ =1 ρ =2 ρ =3 MWPT with M = MWPT with M = MWPT with M = and y. The symbos C 1 and C 2 are two constants which are used to prevent unstabe resuts when either μ 2 x þ μ2 y or σ 2 x þ σ2 y very cose to zero. Tabe 1 presents PSNR and SSIM vaues obtained by appying a the test methods to denoise the above used image corrupted by different noise patterns. For every pattern, our proposed method consistenty provides better resuts (higher PSNR and SSIM vaues) than the other test methods Rea noisy data-based anaysis Synthetic aperture radar (SAR) images are inherenty affected by mutipicative specke noise, which generay affects the basic textures of SAR imagery [28]. Hence, we aso chose rea noisy SAR images without adding artificia noise to evauate the performance of the proposed method. One of the test images was shown in Fig. 13a, covering an area of Beijing Airport, China, coected by airborne radars and provided by Chinese Academy of Science. As shown, it covers many ground objects and rich textures. Figure 13b e dispays the denoising resuts obtained by appying a the test methods to the image shown in Fig. 13a. As can be seen, a the methods effectivey reduce the specke noise in the origina image, especiay in the smooth area (e.g., the surface of the runway). Figure 14 shows cose-ups of the same part of the image obtained using each method for cear comparison. A of the MWPT-based methods ceary suppress specke we, but those with ow M or ρ oversmooth images and thus bur many features (e.g., in the arrowed area). By contrast, the proposed 2-order baanced 4-band MWPTbased method preserves more structura detais in the origina image. Since noise-free reference images were not avaiabe, equivaent number of ooks (ENL) vaues of homogeneous regions were cacuated. ENL is often appied to characterize the smoothing effect of denoising methods, given by: ENLðÞ¼μ x 2 x =σ x ð9þ 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 vaue wi simpy refect the degree to which the denoising method suppressed the noise in the region. However, for a heterogeneous region, the ENL vaue shoud not be too high, otherwise the method may oversmooth structura information regarding the region. ENL vaues were cacuated for two homogeneous regions (1 and 2 in Fig. 14a) and two heterogeneous regions (3 and 4) shown in Fig. 14b e. The resuts, isted in Tabe 2, ceary show that the proposed method provides the highest ENL vaue for each homogeneous region and modest vaues for each heterogeneous region, demonstrating its superiority for both noise suppression and texture preservation. The resuts of the quantitative evauation are aso consistent with the visua assessment presented above. 3.3 Infuence of the M, ρ, and k The impacts of the band number M, the baanced order ρ, and the decomposition eve k on denoising performance when using the MWPT-based method can be summarized in more detai as foows. Tabe 4 PSNR and SSIM vaues obtained by MWPT-based methods with different M, ρ, and k parameters, when using the test image in Fig. 8a and mixing Gaussian white noise with σ =10 Methods PSNR SSIM k =1 k =2 k =3 k =4 k =5 k =1 k =2 k =3 k =4 k =5 MWPT with (M,ρ) = (2,3) MWPT with (M,ρ) = (3,2) MWPT with (M,ρ) = (4,2)

13 Wang et a. EURASIP Journa on Advances in Signa Processing (2016) 2016:10 Page 13 of 14 Tabe 5 Computationa time (seconds) required by different MWPT-based methods, using the data in Fig. 8a Size of Computationa times required by different methods image Method of Liu et a. Method of Ren et a. Method of He et a. The proposed method (1)For every M vaue between 2 and 4, as the baanced order ρ rises within an appropriate range, the PSNR andssimprovidedbythemwpt-basedmethod increase, as iustrated in Tabe 3 for the dataset considered above. These resuts ceary confirm the superiority of the high-order baanced M-band MWPT for preserving image features. (2)As M rises from 2 to 4, the resuts of the MWPTbased method improve at each baanced order ρ. Tabe 3 shows the improvement when appied to the dataset in Fig. 8a. This further corroborates the advantages of M-band over 2-band mutiwaveet systems. (3)The MWPT-based method provides optima resuts (at a given ρ and M) when the decomposition eve k is appropriate (3 or 4 in 3-band cases, and 2 or 3 in 4-band cases). This is iustrated by the PSNR and SSIM vaues in Tabe 4 for the images in Fig. 8a, showing that for severa typica M and ρ cases, 3-band and 4-band MWPTs provide optima indicator vaues when k = Computationa compexity anaysis As the proposed transform is orthogona, it adds itte compexity and the computationa costs of the proposed method are simiar to those of methods based on traditiona mutiwaveets (e.g., GHM mutiwaveet). Moreover, the appropriate decomposition eves for M > 2 cases are aways ess than that for the M = 2 case reported in Section 3.2, which further reduces the computationa compexity of theproposedm-bandmwpt-basedmethod.theseassertions were verified by the average computationa times in a MATLAB environment using a workstation with an Inte(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 high-order baanced M-band MWPT are sighty onger than those of other MWPT-based methods but sti reasonabe (Tabe 5). 4 Concusions This work shows that using an appropriate higher baanced order or band number improves the denoising performance of mutiwaveet packet transform-based methods at itte additiona computationa cost. The high-order baanced M-band mutiwaveet packet transform is aso an effective scheme for other image processing tasks, such as texture anaysis and edge extraction. This is because with an appropriate higher baanced order, thetransformschemeprovidesmoreeffectivesparseimage representation, a higher band number provides more fexibe spatia-frequency domain partitioning, and the most suitabe basis set for an anayzed signa can be seected for the mutiwaveet packet transform scheme. We used a simpe denoising strategy to evauatetheperformance of the proposed transform scheme. To further improve its performance, other approaches that are more suitabe for an M-band mutiwaveet system shoud aso be appied (e.g., taking into account the different frequency behaviors of the waveets in a mutiwaveet system in fusion rue seection). Competing interests The authors decare that they have no competing interests. Acknowedgements This work was supported by the Natura Science Foundation of Hebei Province, China (Grant No. D ) and the Nationa Natura Science Foundation of China (Grant Nos and ), and aso supported by the Natura Science Foundation of Hebei Province, China (Grant No. F ), the Hebei Education Department, China (Grant Nos. ZD and ZD ), the Ministry of Agricuture s Specia Funds for Scientific Study on Pubic Causes, China (Grant No ) and the Nationa Natura Science Foundation of China (Grant Nos and ). The author woud ike to thank I. W. Seesnick of Poytechnic Institute of New York University for many usefu questions and comments. Author detais 1 Schoo of Information and Eectrica Engineering, Hebei University of Engineering, Handan , Peope s Repubic of China. 2 Schoo of Eectronic and Information Engineering, Bayin Guoeng Vocationa and Technica Coege, Economic Deveopment Zone, Kora Xinjiang Uygur autonomous region, Peope s Repubic of China. 3 Schoo of Resources and Environmenta Engineering, Ludong University, Yantai , Peope s Repubic of China. 4 Coege of Urban and Environmenta Science, Northwest University, Xi an , Peope s Repubic of China. 5 Institute of Agricutura Water Conservancy, Changjiang River Scientific Research Institute, Wuhan , Peope s Repubic of China. Received: 27 March 2015 Accepted: 17 December 2015 References 1. I Daubechies, Ten ectures on waveets (SIAM, Phiadephia, 1992), pp DL Donoho, De-noising by soft-threshoding. IEEE T. Inform. Theory 41(3), (1995) 3. H Xie, LE Pierce, FT Uaby, SAR specke reduction using waveet denoising and Markov random fied modeing. IEEE T. Geosci. 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