ARTICLE IN PRESS. Signal Processing

Transcription

1 Signal Processing 88 (2008) Contents lists available at ScienceDirect Signal Processing journal homepage: Construction of 2-D directional filter bank by cascading checkerboard-shaped filter pair and CMFB Zuo-feng Zhou a,b,, Yuan-yuan Cheng a, Peng-lang Shui a a National Laboratory of Radar Signal Processing, Xidian University, Xi an , PR China b Xi an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi an, , PR China article info Article history: Received 21 November 2007 Received in revised form 20 March 2008 Accepted 22 April 2008 Available online 3 May 2008 Keywords: Cosine modulated filter bank (CMFB) Checkerboard-shaped filter pair (CSFP) Directional filter bank (DFB) Directional-frequency selectivity Image denoising abstract In this paper, we propose a new approach to construct a 2-dimensional (2-D) directional filter bank (DFB) by cascading a 2-D nonseparable checkerboard-shaped filter pair and 2-D separable cosine modulated filter bank (CMFB). Similar to diagonal subbands in 2-D separable wavelets, most of the subbands in 2-D separable CMFBs, tensor products of two 1-D CMFBs, are poor in directional selectivity due to the fact that the frequency supports of most of the subband filters are concentrated along two different directions. To improve the directional selectivity, we propose a new DFB to realize the subband decomposition. First, a checkerboard-shaped filter pair is used to decompose an input image into two images containing different directional information of the original image. Next, a 2-D separable CMFB is applied to each of the two images for directional decomposition. The new DFB is easy in design and has merits: low redundancy ratio and fine directionalfrequency tiling. As its application, the BLS-GSM algorithm for image denoising is extended to use the new DFBs. Experimental results show that the proposed DFB achieves better denoising performance than the methods using other DFBs for images of abundant textures. & 2008 Elsevier B.V. All rights reserved. 1. Introduction Directional information is important in many areas of image processing, such as denoising, compression, edge detection and feature extraction. During the past decade, a mass of efforts have been made to find efficient directional representations of natural images, and various directional filter banks (DFBs) have been designed [1 9]. Two-dimensional (2-D) separable wavelets are the easiest choice and have been successfully applied in image restoration and compression. However, 2-D separable wavelets are suited for representation of point singularities but unsuited for representation of line-singularities Corresponding author at: National Laboratory of Radar Signal Processing, Xidian University, Xi an , PR China. Tel.: addresses: ever817@126.com, plshui@xidian.edu.cn (Z.-f. Zhou). such as edges and textures in images, partly owing to their poor directional selectivity. This deficiency of 2-D separable wavelets promotes fast growing in DFBs. There exist many directional multiscale transforms, including the steerable pyramid [3], DFB [4], contourlet [5,6], DT-CWT [7], complex wavelet [8], direction-let [9], etc. The DFBs were originally proposed by Bamberger and Smith in [4] and subsequently improved by several authors [1,10,11]. It is critically sampled and implemented by cascading two-channel diamond-shaped and parallelogram filter banks with wedge-shaped frequency partitions. Cascading the Laplacian pyramid (LP) [12] and the wedge-shaped DFB [4] generates the contourlet transform [5]. The other effective approach is to construct 2-D DFBs from 2-D separable wavelets, including the 2-D dual-tree complex wavelets [7] and the mapping-based complex wavelets [8]. The 2-D DT-CWT is the tensor product of the two 1-D DT-CWT constructed from 1-D approximate Hilbert pairs of wavelet bases [13,14]. The 2-D /$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi: /j.sigpro

2 Z.-f. Zhou et al. / Signal Processing 88 (2008) mapping-based complex wavelet transforms extract directional information of real-valued natural images in the two stages. A real-valued image is first decomposed into two real-valued images containing different directional information. Next, 2-D separable DWTs are applied to each of the two images to extract directional information. The 2-D mapping-based CWTs have six oriented subbands at each scale. In this paper, following the two-stage scheme in [8], we construct new DFBs by cascading checkerboardshaped filter pair (CSFP) and 2-D separable cosine modulated filter banks (CMFBs) that are the tensor products of two 1-D CMFBs. 1-D CMFBs are mature in theory and design algorithms [15 21] and there are many available design algorithms and numerical examples. Thus, the proposed scheme is a simple but efficient approach to construct DFBs. Similar to 2-D separable wavelets, 2-D separable CMFBs are poor in directional selectivity due to the fact that most of the subband filter s support is along the two different directions in the 2-D frequency plane. In order to improve their directional selectivity, we design a CSFP that consists of two realvalued FIR energy-complimentary filters. One is used to extract the information of a natural image in the first and third quadrants in the frequency domain, while the other is used to extract the information in the second and fourth quadrants. The CSFP followed by the 2-D separable CMFB constitutes a new 2-D DFB with a redundant ratio of only 2. The new DFBs are very efficient to represent vibrating patterns of different directions and frequencies in natural images. As its application, we use the new 2-D DFB instead of the steerable pyramids in the BLS-GSM algorithm [26] for image denoising. The experimental results show that the BLS-GSM algorithm using the proposed 2-D DFBs significantly improves the denoising performance for natural images with abundant textures. This paper is organized as follows. Section 2 gives the structure of the new DFB by cascading CSFP and 2-D separable CMFBs. Section 3 reviews the design methods of CMFBs and gives the design algorithm for the CSFP. In Section 4, the BLS-GSM image denoising algorithm using the proposed DFBs is given, and the experimental results and performance comparison are made. Finally, we conclude our paper. 2. Structure of new directional filter banks Many image processing tasks, such as restoration, compression and features extraction, rely on efficient representations of image features such as smooth contours, edges and textures. These image details often exhibit different directions and vibrating frequencies. Various DFBs were developed for representation of these image details. The DFBs include the steerable pyramid [3], critically sampled DFBs [4], Contourlet [5], 2-D DT-CWTs [7] and the 2-D mapping-based complex wavelets [8]. The directional selectivity of a DFB is closely related to their 2-D frequency tiling. As shown in Fig. 1, several different types of DFBs provide different frequency tiling. The 2-D separable wavelets have three oriented subbands at each scale, the contourlet transform has adjustable oriented subbands at each scale, the critically sampled DFBs have Fig. 1. Frequency tiling of the existing DFBs: (a) 2-D separable wavelets; (b) contourlet transform; (c) critically sampled DFBs; (d) 2-D DT-CWTs and mapping-based CWTs.

3 2502 Z.-f. Zhou et al. / Signal Processing 88 (2008) six oriented subbands at each scale, and DT-CWTs and mapping-based CWTs also have six oriented subbands at each scale. It is known that M-band filter banks provide finer frequency tiling than 2-band wavelets. Among M-band filter banks, the CMFBs are simple in design and efficient in implementation. The 2-D separable CMFBs consist of the tensor product of two 1-D CMFBs. As shown in Fig. 2, the tensor product of a four-channel CMFB segments the 2-D frequency plane into 16 subbands. Unfortunately, the fine frequency tiling does not bring satisfactory directional selectivity. Similar to the diagonal subbands in 2-D separable wavelets, the energy of most subband filters in the 2-D CMFB is concentrated along the two different directions aa[0,p/2] and a+p/2 in the 2-D frequency plane. Enlightened by the mapping-based CWTs [8], we attempt to use a pair of real-valued 2-D checkerboard-shaped filters to separate the frequency content of an input image in the first and third quadrants from that in the second and fourth quadrants. And then, the two images are input to 2-D CMFB to extract the directional information of the original image. The two-stage scheme corresponds to a new DFB which is obtained by cascading the 2-D CSFP and Fig. 2. Frequency tiling of 2-D separable CMFB generated from a 1-D four-channel CMFB. a 2-D CMFB. The structure of the new DFB is shown in Fig. 3, where G i (o 1,o 2 ), i ¼ 0,1 are the CSFP, and 2-D M 2 - channel CMFBs are represented using 2-D filters H 2D j(o 1,o 2 ),j ¼ 0,1,y,M 2 1. As shown in Fig. 3, we give the structure and the 2-D frequency tiling of the DFB by cascading the CSFP and 2-D CMFB, where the CSFP serves for a coarse frequency partition of an image and the CMFB for extraction of fine directional information of the image. From Fig. 1, it can be seen that most directional multi-scale image representation tools mentioned above have the same directional resolution for lower and higher frequency components in the 2-D frequency plane. Taking the 2-D wavelet as an example, it has only two directions, the horizontal and vertical directions, at each scale. But the proposed DFB has different directional resolutions for lower and higher frequency components; that is, there is finer directional resolution for higher frequency components while coarser directional resolution for lower frequency components. From Figs. 2 and 4, except for the low frequency component h 00, it can be seen that the proposed DFB with M ¼ 4 has 14, 10 and 6 directions from the highest frequency components to the lowest frequency components, respectively. The 1-D CMFBs have been widely used in signal processing owing to their efficiency in computation and easy in design [15 20]. All the analysis and synthesis filters of a 1-D CMFB are modulated versions of one or two lowpass prototype filters. A 1-D CMFB with a single prototype filter p(n) and a system delay D is described as follows: h m ðnþ ¼2pðnÞ cos p M ðm þ 1=2Þðn p D=2Þþð 1Þm, 4 f m ðnþ ¼2pðnÞ cos p M ðm þ 1=2Þðn p D=2Þ ð 1Þm, (1) 4 for n ¼ 0,1,?,N and m ¼ 0,1,?, M 1, where h m (n), m ¼ 0,1,?, M 1 are the analysis filters and f m (m) ¼ 0,1,?, M 1 are synthesis filters. Similar to the 2-D separable wavelet, the 2-D separable CMFB is constructed from the tensor product of the two 1-D CMFBs. Without loss of generality, we assume that the same 1-D CMFBs are used for the horizontal and vertical directions. The 2-D separable CMFB is composed of M 2 analysis filters and M 2 Fig. 3. Structure of the proposed 2-D DFB.

4 Z.-f. Zhou et al. / Signal Processing 88 (2008) Fig. 4. Frequency tiling of the CSFP and the new DFB with M ¼ 4. (a) G 0 (o 1, o 2 ), (b) G 1 (o 1, o 2 ), (c) half subbands of the new DFB and (d) another half subbands of the new DFB. synthesis filters as follows: h 2D m ðn x; n y Þ¼h mx ðn x Þh my ðn y Þ; m ¼ Mm x þ m y, f 2e m ðn x; n y Þ¼f mx ðn x Þf my ðn y Þ; m x ; m y ¼ 0; 1;...; M 1. (2) Since each filter in a 2-D CMFB is variable-separable, image decomposition based on the 2-D separable CMFB can be fast implemented by sequentially applying 1-D CMFB to the rows and columns of an image. Moreover, when 1-D CMFB is critically sampled, the associated 2-D CMFB is also critically sampled. In terms of the structure in Fig. 3, the analysis filter bank of the new DFB is composed of 2M 2 filters as follows: G 0 ðo x ; o y ÞH 2D m ðo x; o y Þ; G 1 ðo x ; o y ÞH 2e m ðo x; o y Þ, m ¼ 0; 1;...; M 2 1, (3) and the synthesis filter bank is composed of G 0 ðo x ; o y ÞF 2D m ðo x; o y Þ; G 1 ðo x ; o y ÞF 2D m ðo x; o y Þ; m ¼ 0; 1;...; M 2 1. (4) Therefore, when the 2-D CMFB is critically sampled, the new DFB has a redundancy ratio of two. The properties of the new DFB are determined by both the CSFP and CMFB. To ensure the perfect reconstruction of the DFB, the CSFP must be energy-complementary, that is, G 0 ðo x ; o y Þ 2 þ G 1 ðo x ; o y Þ 2 ¼ 1. (5) As above-mentioned, most of the subband filters in a 2- D separable CMFB lack the directional selectivity because the frequency supports of these filters locate in the first third quadrants and the second fourth quadrants. In order to improve the directional selectivity, the CSFP is required to extract the frequency content of an image in the first-third quadrants and the frequency content in the second-fourth quadrants. Typically, the CSFP consists of an ideal checker-board-shaped filter pair, and the two filters have the frequency responses ( G ideal 0 ðo x ; o y Þ¼ 1; o xo y X0; 0; otherwise; (6) G ideal 1 ðo x ; o y Þ¼1 G ideal 0 ðo x ; o y Þ: The ideal CSFP is equivalent to separating the frequency content of an image in the first third quadrants and the frequency content in the second fourth quadrants via the 2-D discrete Fourier transform (DFT). If the ideal CSFP is used in Fig. 3, the corresponding DFB has an ideal frequency tiling and directional selectivity as shown in Fig. 4. However, it is well-known that Fourier transform suffers from poor spatial localization. Since images are typical spatial nonstationary signals, the CSFP is desired to preserve the local features in images when it is used to split an image into two images containing the frequency contents in the first third quadrants and the second fourth quadrants, respectively. In the next section, we will investigate the design of FIR CSFP with good spatial localization. 3. Design of FIR checkerboard-shaped filter pair The CSFP and CMFB are the two important parts of the new 2-D DFB. The CMFBs have been thoroughly

5 2504 Z.-f. Zhou et al. / Signal Processing 88 (2008) investigated in theory and design [15,18 20], and thus there exist several algorithms for CMFB design and a number of examples available for the DFB design. In this paper, the linear-phase critically sampled 1-D CMFBs are available for our design. In what follows, we deal with the design of CSFP. Fig. 4(a) and (b) is the ideal frequency tiling of the desired CSFP. The ideal frequency tiling can be achieved by the simple 2-D DFT. First, a real-valued grayscale image is transferred to the 2-D frequency domain by the 2-D DFT. Then the two images are obtained from its frequency contents in the first-third quadrants and the second fourth quadrants by the inverse DFT. Since 2-D DFT lacks the spatial localization, the corresponding CSFP heavily blurred the image details though the ideal frequency tiling is achieved. As shown in Fig. 5(a) (c), the test image Lena is split into two images via the 2-D DFT. It can be seen that the details such as edges and textures in the two obtained images are heavily blurred. Therefore, the CSFP is required to good spatial localization and have the frequency response close to the ideal frequency tiling in Fig. 4(a) and (b). Here, the two filters in the CSFP are required to be FIR and symmetric with respect to the origin so as to assure good spatial localization, that is, the filter s coefficients satisfy Geometrically, the frequency response of G 1 (o x,o y )is equivalent to anticlockwise rotation of the frequency response of G 0 (o x,o y ) with 901. Thus, G 1 (o x,o y ) has support in the second fourth quadrants, while G 0 (o x,o y ) has supports in the first third quadrants. The design of the CSFP is reduced to design a single symmetric and FIR filter g 0 (n x,n y ). Considering the periodicity of the frequency response and the desired passband, we specify the stopband, passband and transition band of the filter g 0 (n x,n y ) as shown in Fig. 6, where the stopband region is D 2 S D4, the passband region is D 1 S D3, the residual part is taken as the transition band and the parameters D 1, D 2 are used to adjust the width of the transition band. g 0 ð n x ; n y Þ¼g 0 ðn x ; n y Þ; g 1 ð n x ; n y Þ¼g 1 ðn x ; n y Þ. (7) In this way, the frequency responses of the two filters are two-variable cosine polynomials. Further, for the sake of simplification, we assume that the two filters also satisfy G 1 ðo x ; o y Þ¼G 0 ðo y ; o x Þ; g 1 ðn x ; n y Þ¼g 0 ðn y ; n x Þ. (8) Fig. 6. The stopband, passband and transition band of the filter g 0 (n x, n y ). Fig. 5. (a) Lena image, (b) and (c) the obtained two images by the 2-D DFT, and (d) and (f) their edges using the classical Canny edge detector [30].

6 Z.-f. Zhou et al. / Signal Processing 88 (2008) Combining the energy-complementary constraint with the passband flatness and stopband attenuation, design of the symmetric FIR filter g 0 (n x,n y ) can be transferred to the following optimization problem: ( ) min g 0 ðnx;nyþ R ðg 0 ðo x ; o y Þ 1Þ 2 do x do y þ R G 2 0 ðo x; o y Þ do x do y D 1[D 3 D 2[D 4 s:t: G 2 0 ðox; oyþþg2 0 ðoy; oxþ 1: (9) In terms of the symmetry of the filter, the frequency response is written as G 0 ðo x ; o y Þ¼g 0 ð0; 0Þþ2 X n2o g 0 ðn x ; n y Þ cosðn x o x þ n y o y Þ, O ¼fðn x ; n y Þ : ðn y 40Þ and ðn x 40; n y ¼ 0Þg. (10) In Eq. (10), the objective function can be simplified to a quadratic function of the filter s coefficients while the constraint is relatively complicated. By expanding G 2 0(o x,o y ) into the two-variable cosine polynomial and contrasting the coefficients of the polynomial, the constraint can be transferred into a set of quadratic equalities of the filter s coefficients. As a result, the filter can be obtained by solving a nonlinear programming with a quadratic objective function and a set of quadratic equality constraints. Nonlinear programming is sensitive to the initial point. Here, the truncated filter of the ideal CSFP is taken as the initial point in solving the nonlinear programming. For example, when the support of the filter is a squared region whose center is at the origin and D 1 ¼ 2D 2 ¼ 6p/25, the obtained filter and the frequency responses of the corresponding CBFP are plotted in Fig. 7(a) and (b). It can be seen that the frequency responses of the CSFP are closed to the desired frequency responses in Fig. 4(a) and (b). Particularly, the two filters are linear phase and FIR and provide good spatial localization. As shown in Fig. 8(a) and (b), the good spatial localization of the CSFP significantly reduces the detailed blurring of the obtained two images. According to the frequency tiling of the CSFP, the image decomposition based on it realizes the classification of different directional information of the original image, where the first image contains the edges whose directions are in [ p/2,0] while the second image contains the edges whose directions are in [0,p/2]. Only stronger edges are displayed in Fig. 8(c) and (d). Additionally, it can also be seen that the horizontal and vertical edges in the original image do not appear in Fig. 8(c) and (d), which is because the two frequency axes just locate in the transition bands of the two filters and thus the horizontal and vertical edges are blurred by the filtering process. By the FIR CSFP, an image is decomposed into two images that contain different details in the original image. The CSFP substantially performs a coarse classification of image details. Further, the 2-D separable CMFB is applied to the two images from the CSFP to decompose them into multiple oriented subbands. In this way, a directional decomposition of the image is obtained. When the CMFB is critically sampling, the decomposition has a low redundancy of only two. In Fig. 9, we give the directional decomposition of the image Lena by the proposed directional filter bank. It can be seen that the edges of different directions in the image are classified into different oriented subbands. However, since the proposed DFB is redundant it is not suited for image compression, and thus we consider its application in image denoising in the next section. 4. Application of the new DFBs in image denoising 4.1. BLS-GSM algorithm using the proposed DFB Various DFBs have been used to improve the image denoising performance [22 29]. The overall performance of the transform-based image denoising algorithms mainly depends on two factors: the efficiency of the underlying transform and statistical modeling of the coefficients in the transform domain. The efficiency of a transform indicates the capability for image details such as edges and texture to be represented using a small number of coefficients of large magnitude. Since most of the filtering rules suppress or withdraw coefficients of small magnitude while retaining or slightly reducing coefficients of large magnitude, image details will be preserved well in denoised images when the details are conveyed by coefficients of large magnitude. The DFBs can represent better image details than the 2-D separable Fig. 7. Frequency responses of the two linear phase filters in the designed CSFP and the spatial support in squared region.

7 2506 Z.-f. Zhou et al. / Signal Processing 88 (2008) Fig. 8. Two images from the linear phase FIR CSFP and their capability for detail classification, where the edges are extracted by the classical Canny detector [30]. wavelets owing to their finer directional selectivity. The statistical modeling of the coefficients determines the filtering rules in transform domain and influences the final denoising performance. Images are spatial nonstationary and the coefficients in each oriented subband also exhibit spatial nonstationary and high correlation of adjacent coefficients. A good statistical model can capture the dependency of adjacent coefficients and thus can instruct us to develop a good filtering rule. In [26], the steerable pyramid is used to decompose a noisy image into oriented subbands at multiple scales and the Gaussian scale mixture (GSM) model is used to describe correlations of adjacent coefficients of the image in space and scale. Based on the statistical model, the BLS- GSM denoising algorithm was developed, which is one of the existing best image denoising algorithms in terms of peak-signal-to-noise-ratio (PSNR). The proposed DFB is a new directional multi-scale image representation tool which has the similar properties to the steerable pyramid, such as redundancy, orientation, etc. The coefficients by the proposed DFB decomposition also exhibit the strongly leptokurtotic behavior of the marginal densities and the dependency between local coefficients amplitudes. The GSM model is an effective model which can account both the non-gaussian marginal behavior of the transformdomain coefficients and the strong correlation between the amplitudes of neighbor coefficients. Though it is primarily applied to the steerable pyramid, it also remains valid for other DFBs, for example 2-D wavelet and contourlet. In this paper, we empirically use the GSM model to characterize the non-gaussian marginal behavior and the spatial correlation of the coefficients in a squared neighborhood of each oriented subband by the new DFB decomposition. Let x(i,j) ¼ s(i,j)+w(i,j), 1pi,jpN be a noisy grayscale image. S(i,j) and w(i,j) are the noise-less image and zeromean additive white Gaussian noise with known variance, respectively. By the new DFB decomposition, the noisy image is decomposed into noisy subband coefficients: d x g ði; jþ ¼ds g ði; jþþdw g ði; jþ; g 2 G (11) where G is the index set of all the oriented subbands by the new DFB decomposition. Let v x (i,j), v s (i,j) and v w (i,j) be the column vector consisting of the noisy coefficients, noise-less coefficients and noise s coefficients in a neighborhood of the reference pixel (i,j), respectively. In the GSM model, the noise-less vector v s is modeled as a random vector p v s ¼ ffiffi z u (12)

8 Z.-f. Zhou et al. / Signal Processing 88 (2008) Fig. 9. Directional decomposition of the image Lena using the proposed directional filter bank, where (a) contains a lowpass approximation and 15 oriented subbands of the image in Fig. 8(a) and (b) contains that of the image in Fig. 8(b). where z and u are the independent positive random multipliers satisfying E(z) ¼ 1 and zero-mean Gaussian random vector with covariance matrix C u, respectively. The covariance matrix C u describes the correlation structure of adjacent coefficients and the random multiplier z reflects inhomogeneous energy distribution in the oriented subband. Then, the probability density p(v s )isa mixture of Gaussians Z pðv s Þ¼ pðv s zþpðzþ dz: (13) The covariance matrix Ĉ s can be estimated from the sampling covariance matrix and noise covariance matrix by Ĉ s ¼ C x C w. Based on the widely used Jeffery s prior on the random variable z, p(z)p1/z, the noise-less vector v s is estimated by Bayesian least squared estimator Z ^v s ¼ z ^C s ðz ^C s þ C w Þ 1 v x pðzjv x Þ dz. (14) Finally, the denoised image is reconstructed by the inverse new DFB decomposition. The details of the denoising algorithm refer to Ref. [26]. In the experiments in the next subsection, 3 3 squared neighborhood is available in the BLS-GSM algorithm and other parameters of the algorithm are identical with the algorithm in [26] Experimental results In order to evaluate the performance of the algorithm using the proposed DFB, we use 8-bit grayscale images, 512 by 512, Lena, Barbara, Boats and 256 by 256 Texture image [31]. The Lena and Boats are composed of smooth regions and curve-type edges and the Barbara and Texture contain a mass of inhomogeneous textures. These images are added computer-generated zero-mean white Gaussian noise of variance s 2 to verify the performance of the proposed image denoising algorithms. In our algorithm, the DFB is generated from the CSFP in Fig. 7 and a 1-D linear phase critically sampled CMFB with M ¼ 6, N ¼ 48. The generated 2-D separable CMFB is nearly perfect reconstruction, but the reconstruction PSNR is higher than 47 db for Gaussian white noise samples and thus is not enough influence on the denoising performance. In this way, the directional decompositions have 70 oriented subbands and two lowpass subbands (not processed in denoising). In the experiments, the output PSNR is the average of 20 independent tests for each noise level and test image. The output PSNRs of our algorithm and several recent state-of-the-art algorithms are listed in Table 1, where the BLS-GSM algorithm in [26] is from the Web site, From Table 1, it can be seen that, for the test images Barbara and Texture with abundant inhomogeneous textures, the proposed algorithm provides better denoising performance than the BLS-GSM algorithm [26], which is owing to the fact that the bases in the proposed 2-D DFB can efficiently represent long oscillatory patterns in images such as textures while the steerable pyramid is suited for representing edges of different orientations. In order to illustrate the visual effect, a local texture region of Barbara image is illustrated in Fig. 10. It can be seen that the proposed DFB preserves better the texture structures in the region. For the images composed of smooth regions and edges, for example, Lena and Boats, the proposed algorithm is poorer in performance than the BLS-GSM algorithm [26]. But the redundant ratio of the proposed DFB is much smaller than the steerable pyramid in the BLS-GSM algorithm and the non-subsampled contourlet

9 2508 Z.-f. Zhou et al. / Signal Processing 88 (2008) Table 1 Performance comparison of the proposed image denoising algorithm and other image denoising algorithms Image s DLWFDW [29] DT-CWT [25] NSCT-LAS [28] BLS-GSM [26] Proposed Redundancy ratio E Barbara Texture Lena Boats Fig. 10. Zoomed texture regions: (a) the original Barbara image, (b) the noisy image (s ¼ 20), (c) denoised image by the BLS-GSM [26] (the PSNR in the zoomed region is db) and (d) denoised image using the proposed DFB (the PSNR in the zoomed region is db).

10 Z.-f. Zhou et al. / Signal Processing 88 (2008) Fig. 11. Hat region: (a) the original Lena image, (b) the noisy Lena image (s ¼ 20), (c) the denoised image by the BLS-GSM [26] (the PSNR in the zoomed region is db) and (d) denoised image using the proposed image denoising algorithm (the PSNR in the zoomed region is db). transform in the NSCT-LSA algorithm [28]. In fact, even if the output PSNR of the algorithm using the proposed DFB is poorer, the denoised image still exhibits better capability to preserve textures. For example, for Lena image when s ¼ 20, though the denoised image using the proposed DFB has only output PSNR ¼ db, which is 0.36 db lower than the output PSNR of the BLS-GSM algorithm, the denoised image using the proposed DFB retains clearer textures in the hat region of the Lena than does the denoised image by the BLS-GSM algorithm, as shown in Fig. 11(c) and (d). The results in Figs. 10 and 11 imply that it is a possible approach to improve the denoising performance of image with abundant textures that the steerable pyramid is combined with the proposed DFB based on the texture segmentation of noisy images. In other words, the smooth and edge region of the denoised image using the steerable pyramid and the texture region of the denoised image using the proposed DFB can be composed into a denoised image of high quality. The numerical complexity of the proposed algorithm consists of two parts. One is the computational cost of the new DFB decomposition and the other comes from the BLS-GSM filtering. For a given image size, the computational cost of the BLS-GSM filtering is directly proportional to the redundancy ratio of the decompositions. The decomposition using the steerable pyramid [26] has a redundancy ratio of 18.6, while the new DFB has only redundancy ratio of 2. Additionally, for a given image with size N N, the numerical complexity of the new DFB decomposition is O(N 2 ) while the steerable pyramid decomposition used in Ref. [26] is O(N 2 log 2 N). On the whole, in the computational complexity our algorithm is lower than the algorithm in Ref. [26]. 5. Conclusion In this paper, a new 2-D DFB is proposed by cascading a CSFP and a 2-D separable CMFB. The new DFBs provide fine directional-frequency selectivity and are of low redundancy ratio. The new DFB can replace the steerable pyramid in the BLS-GSM algorithm for grayscale image denoising. The experimental results show that the new DFBs can obtain better denoising performance for grayscale images with abundant textures and retain better texture information in denoised images than does the BLS-GSM algorithm using the steerable pyramid. Additionally, the low redundancy ratio of the new DFB brings the denoising algorithm of low computational complexity. References [1] S.I. Park, M.J. Smith, R.M. Mersereau, Improved structure of maximally decimated directional filter banks for spatial image analysis, IEEE Trans. Image Process. 13 (11) (2004) [2] S. Mallat, A Wavelet Tour of Signal Processing, second ed., Academic, New York, 1999.

11 2510 Z.-f. Zhou et al. / Signal Processing 88 (2008) [3] W.T. Freeman, E.H. Adelson, The design and use of steerable filters, IEEE Trans. Patt. Anal. Mach. Intell. 13 (9) (1991) [4] R.H. Bamberger, M.J.T. Smith, A filter bank for directional decomposition of images: theory and design, IEEE Trans. Signal Process. 40 (4) (1992) [5] M. Do, M. Vetterli, The contourlet transforms: an efficient directional multi-resolution image representation, IEEE Trans. Image Process. 14 (12) (2005) [6] D.D.-Y. Po, M.N. Do, Directional multiscale modeling of images using the contourlet transforms, IEEE Trans. Image Process. 15 (6) (2006) [7] I.W. Selesnick, R.G. Baraniuk, N.G. Kingsbury, The dual-tree complex wavelet transform, IEEE Signal Process. Mag. 22 (6) (2005) [8] F.C.A. Fernandes, R.L.C. Spaendonck, C.S. Burrus, Multidimensional, mapping-based complex wavelet transforms, IEEE Trans. Image Process. 14 (1) (2005) [9] V. Velisavljevic, B.B. Lozano, M. Vetterli, P.L. Dragotti, Directionlets: anisotropic multidirectional representation with separable filtering, IEEE Trans. Image Process. 15 (7) (2006) [10] T.T. Nguyen, S. Oraintara, A class of multiresolution directional filter banks, IEEE Trans. Image Process. 55 (3) (2007) [11] T.T. Nguyen, S. Oraintara, Multiresolution directional filter banks: theory, design, and application, IEEE Trans. Signal Process. 53 (10) (2005) [12] M.N. Do, M. Vetterli, Framing pyramids, IEEE Trans. Image Process. 51 (9) (2003) [13] I.W. Selenick, The design of approximate Hilbert transform pairs of wavelet bases, IEEE Trans. Signal Process. 50 (5) (2002) [14] I.W. Selenick, The double-density dual-tree discrete wavelet transform, IEEE Trans. Signal Process. 52 (5) (2004) [15] W.-S. Lu, T. Saramaki, R. Bregovic, Design of practically perfectreconstruction cosine-modulated filter banks: a second-order cone programming approach, IEEE Trans. Circuits Syst. I: Regular Pap. 51 (3) (2004) [16] R. Bregovic, T. Saramaki, A systematic technique for designing linear-phase FIR prototype filters for perfect-reconstruction cosinemodulated and modified DFT filterbanks, IEEE Trans. Signal Process. 53 (8) (2005) [17] H. Xu, W.-S. Lu, A. Antoniou, Efficient iterative design method for cosine-modulated QMF banks, IEEE Trans. Signal Process. 44 (7) (1996) [18] Z.-J. Zhang, Design of cosine modulated filter banks using iterative Lagrange multiplier method, in: IEEE International Symposium on Microwave, Antenna, Propagation, 2005, pp [19] P.N. Heller, T. Karp, T.Q. Nguyen, A general formulation of modulated filter banks, IEEE Trans. Signal Process. 47 (4) (1999) [20] T. Karp, A. Mertins, G. Schuller, Efficient biorthogonal cosine-modulated filter banks, Signal Process. 81 (5) (2001) [21] W.-S. Lu, A. Andreas, H. Xu, A direct method for the design of 2-D nonseparable filter banks, IEEE Trans. Circuits Syst. II: Analog Digital Signal Process 45 (8) (1998) [22] M.K. Mihc-ak, I. Kozinsev, K. Ramchandran, P. Moulin, Lowcomplexity image denoising based on statistical modeling of wavelet coefficients, IEEE Signal Process. Lett. 7 (6) (1999) [23] S.G. Chang, B. Yu, M. Vetterli, Spatially adaptive wavelet thresholding with context modeling for image denoising, IEEE Trans. Image Process. 9 (9) (2000) [24] G. Fan, X.G. Xia, Image denoising using local contextual hidden Marcov model in the wavelet domain, IEEE Signal Process. Lett. 8 (5) (2001) [25] L. Sendur, I.W. Selesnick, Bivariate shrinkage with local variance estimation, IEEE Signal Procss. Lett. 9 (12) (2002) [26] J. Portilla, V. Strela, M. Wainwright, E.P. Simoncelli, Image denoising using scale mixtures of Gaussians in the wavelet domain, IEEE Trans. Image Process. 12 (11) (2003) [27] R. Eslami, H. Radha, Translation-invariant contourlet transform and its application to image denoising, IEEE Trans. Image Process. 15 (11) (2006) [28] A.L. Cunha, J.-P. Zhou, M.N. Do, The nonsubsampled contourlet transform: theory, design, and applications, IEEE Trans. Image Process. 15 (10) (2006) [29] P.-L. Shui, Image denoising algorithm via doubly local Wiener filtering with directional windows in wavelet domain, IEEE Signal Process. Lett. 12 (10) (2005) [30] J. Canny, A computational approach to edge detection, IEEE Trans. Patt. Anal. Mach. Intell. 8 (6) (1986) [31] P.L. Shui, Z.F. Zhou, J.X. Li, Image denoising algorithm via best wavelet packer base using Wiener cost function, IET Image Process. 1 (3) (2007)