Image Denoising Using Redundant FinerDirectional Wavelet TransformShrishail S. GajbharDA-IICT, Gandhinagar,

Gujarat, IndiaEmail: shrishail gajbhar@daiict.ac.in

Manjunath V. JoshiDA-IICT, Gandhinagar,

Gujarat, IndiaEmail: mv joshi@daiict.ac.in

Abstract—In this paper, we propose two designs of redundantfiner directional wavelet transform (FiDWT) and explain itsapplication to image denoising. 2-channel perfect reconstruction(PR) checkerboard-shaped filter bank (CSFB) is at the core ofthe designs. The 2-channel CSFB, uses 2-D nonseparable analysisand synthesis filter responses without downsampling/upsamplingmatrices resulting in redundancy factor of 2. Both these designshave two lowpass and six highpass directional subbands. Thehard-thresholding results for image denoising using proposeddesigns clearly shows improvement in PSNR as well as visualquality of the denoised images. Using the Bayes least squares-Gaussian scale mixture (BLS-GSM), a current state-of-the-artwavelet-based image denoising technique with the proposed twotimes redundant FiDWT design indicates encouraging results ontextural images with much less computational cost.

Index Terms—Wavelet transform, multidimensional filterbanks, image denoising, BLS-GSM.

I. INTRODUCTION

Discrete wavelet transform (DWT) has proven to be success-ful in case of image denoising under orthogonality of waveletbases and additive white Gaussian noise (AWGN) assumption[1]. This success can be attributed to its multiscale sparserepresentation of signal along with the decorrelation propertyi.e., separating noise and the signal.

However, performance of the DWT-based image denois-ing methods gets limited by its shift sensitivity and poordirectional selectivity. Shift sensitivity is caused by inherentaliasing due to the downsampling operations in the analysisbranches. Shift sensitivity causes variations in energy distri-bution across the inter-scale DWT coefficients [2]. Traditionalseparable way of obtaining 2-D DWT makes it suitable torepresent image features in horizontal and vertical directionsonly. It also suffers from poor diagonal orientation selectivitydue to subband mixing problem i.e., image features orientedat 45◦ and 135◦ are mixed in diagonal (HH) subband at eachscale. Although DWT is a powerful tool for representing point-wise singularities in images it fails to differentiate geometricfeatures in images oriented at arbitrary direction.

The problem of shift sensitivity/translation variance is com-pletely removed by undecimated wavelet transform (UWT)implemented efficiently using “algorithme a trous” [1]. Al-though redundancy of the transform is increased, it minimizespseudo-Gibbs phenomena at edges providing considerableimprovement in PSNR as well as visual quality in case of

image denoising [3]. To improve the directionality, severalredundant and non-redundant multiscale geometric transforms(MGTs) have been proposed in the literature. In [4], authorsrepresent directional image features using directional filterbanks (DFBs) by dividing the 2-D frequency spectra withwedge-shaped partitions. Although DFBs do not provide mul-tiscale representation, they have been key in widely recognizedcontourlet family [5] and hybrid wavelets and directional filterbanks (HWD) transform in [6]. A comprehensive coverage ofpreviously proposed MGTs is given in [5, 7].

Apart from the numerous families of MGTs using com-pletely different basis from wavelets, traditional DWT andUWT are still attractive for image processing. Along withthe fast and efficient implementations of these transforms,numerous algorithms are proposed on their basis, so the newdesigns using them can benefit from these algorithms [6].

The present work 1 has been motivated by the work in [8–10] to design new redundant directional filter banks suitablefor image denoising application. In [8], Lu and Do pro-posed the critically sampled finer directional wavelet transform(FiDWT) using an additional stage of analysis CSFB filters onthe 3 highpass subbands of DWT (LH, HL and HH) to get onelowpass and 6 highpass directional subbands (75◦, 105◦, 15◦,165◦, 45◦ and 135◦). In [9], aliasing phenomena inherent inFiDWT is explained and undecimated FiDWT (UFiDWT) isproposed. Both FiDWT and UFiDWT are not shift invarianttransforms due to the presence of downsamplers in the designto have same redundancy as that of DWT and UWT. In [10],shift invariant nonsubsampled contourlet transform (NSCT)is proposed using undecimated, 2-channel, 2-D, nonseparablefilter bank structure at its core. Such a design is useful forconstructing shift invariant transforms providing flexibility todesign frequency selective nonseparable filters with ease.

In this paper we propose two designs, redundant FiDWTi.e., RFiDWT with redundancy factor of 2, a less shift invariantdesign than DWT and FiDWT. Here, transform has overall twowavelet decomposition images, each one having one lowpassand three highpass directional subabands similar to DWTrepresentation, hence adaptive to DWT-based algorithms. Thesecond design redundant UFiDWT (RUFiDWT) is completely

1This work is supported by Department of Science and Technology (DST),Govt. of India (Grant No: NRDMS/11/1586/2009/Phase-II).

shift invariant with redundancy factor of 2× (3J + 1), whereJ is number of decomposition levels.

II. PROPOSED DESIGNS OF FINER DIRECTIONAL WT

A. 2-channel PR CSFB and design of filter responses

Figure 1 shows 2-channel, 2-D, perfect reconstruction andundecimated version of checkerboard-shaped filter bank usedin both the designs proposed here.

X XH0csfb

H1csfb

F0csfb

F1csfb

Fig. 1. Checkerboard-shaped filter bank used in the proposed designs.

The analysis CSFB filters Hcsfb0 and Hcsfb

1 split the com-plete input signal spectra X into two diagonally quadrant pass-bands, giving two output images having directional featuresdepending on passband support of the filters. By using 2-D z-transform parameter as z = (z0, z1), the input/output relationsin z-transform domain can be given as,

X(z) =

[1∑

i=0

Hcsfbi (z)F csfb

i (z)

]X(z) = T (z)X(z) (1)

For perfect reconstruction, T (z) should be 1 i.e., it shouldsatisfy the Bezout identity [10]. Analysis and synthesischeckerboard-shaped filter responses satisfying Bezout iden-tity are obtained from 2-D nonseparable finite impulse re-sponse (FIR) filters designed using transformation of variables(TROV) technique given in [11]. It is a simple and flexiblemapping technique equivalent to the generalized McClellantransformation that designs filter responses of different shapesand sampling lattices. The responses used here are designedwith 1-D Cohen-Daubechies-Feauveau CDF 9-7 filter. A 2-D transformation kernel is designed by truncating the 2-Dideal impulse response by multiplying it with the 2-D windowobtained from 1-D Kaiser window having parameters N = 7and β = 4.5. Figure 2 shows the analysis lowpass and highpasscheckerboard-shaped filters designed with these parameters.The 2-D ideal impulse response for CSFB filters is given by

hCS(k1, k2) = sinc

[k1π

2

]sinc

[k2π

2

]cos

[(k1 + k2)π

2

](2)

The 2-D impulse response coefficients of Hcsfbi and F csfb

i

where i = 0, 1 should be divided by√

2 to get perfectreconstruction in filter bank structure shown in figure 1.

B. Design of RFiDWT

Figure 3 shows, the analysis bank of the proposed redun-dant finer directional wavelet transform (RFiDWT) with theredundancy factor of 2.

(a) (b)

Fig. 2. Checkerboard-shaped filter responses (a) analysis lowpass Hcsfb0 (b)

analysis highpass Hcsfb1 .

XinH0csfb

H1csfb

LL

LH

LH

HL HL

LH

LH

LLHL HL

DWT

DWT

HH

HH HH

HH

HH HH

HH HH

1

1

2

2

3

3

4

4

5

5

6

6

L1

L1

L2

L2

FiDWT

FiDWT

Xd1

Xd2

Fig. 3. Proposed 2X redundant FiDWT design.

In RFiDWT, the analysis part of the CSFB is followed bycritically sampled DWT hence the overall redundancy of thetransform is 2. The transform has 2 lowpass subbands and 6highpass directional subbands having orientation selectivity of15◦, 45◦, 75◦, 105◦, 135◦ and 165◦.

In case of DWT, H1Di (zi) and F 1D

i (zi) where i = 0, 1denotes 1-D z-transforms of the analysis and synthesis waveletfilters then, 2-D z-transform of resulting LL, LH, HL and HHsubbands obtained by the tensor product of 1-D wavelet filterscan be written as,

LL(z) = (H1D0 (z0)⊗H1D

0 (z1)) (3)LH(z) = (H1D

0 (z0)⊗H1D1 (z1)) (4)

HL(z) = (H1D1 (z0)⊗H1D

0 (z1)) (5)HH(z) = (H1D

1 (z0)⊗H1D1 (z1)) (6)

Thus, resultant lowpass and directional subbands in the pro-posed RFiDWT can be obtained as,

L1(z) = Hcsfb0 (z)LL(z), L2(z) = Hcsfb

1 (z)LL(z)

D1(z) = Hcsfb0 (z)LH(z), D2(z) = Hcsfb

1 (z)LH(z)

D3(z) = Hcsfb0 (z)HL(z), D4(z) = Hcsfb

1 (z)HL(z)

D5(z) = Hcsfb0 (z)HH(z), D6(z) = Hcsfb

1 (z)HH(z)

Figure 4 shows filter responses of lowpass and highpasssubbands of DWT and 6 directional subbands of the proposedRFiDWT design and figure 5 displays the, 1 level decomposi-tion of the cameraman image using DWT, critically sampledFiDWT and proposed RFiDWT. .

RFiDWT has standard DWT representation with eachwavelet coefficient image having one lowpass subband and3 directional highpass subbands. Such representation is ad-vantageous since many sophisticated denoising algorithms areproposed with standard DWT can be easily used, one suchexample using RFiDWT is given in section III(B).

(a) (b)

Fig. 4. Frequency responses of (a) LL, LH, HL and HH subbands of DWT(b) two lowpass and 6 directional subbands of RFiDWT.

(a) (b) (c) (d)

Fig. 5. 1 level decomposition of cameraman image using (a) DWT (b)FiDWT. (c) RFiDWT: Xd1 (d) RFiDWT: Xd2.

C. Design of RUFiDWT

Similarly to the RFiDWT, complete shift invariant designcan be obtained by replacing DWT as shown in figure 3by UWT. With this, we get the redundant undecimated finerdirectional wavelet transform i.e., RUFiDWT. It has samedirectionality as in the case of RFiDWT, while each subbandimage size is same as that of original image. The redundancyof this transform can be given as 2× (3J + 1), where J is thenumber of levels of decomposition.

III. EXPERIMENTAL RESULTS

In all experiments, we have used the CSFB filters designedwith procedure and parameters given in section II(A). The sizeof the analysis and synthesis lowpass CSFB filters is 57 ×57 and 43 × 43, respectively. Due to this high order of thefilters and sharp frequency roll-off characteristics the aliasingphenomena inherent in FiDWT is suppressed to some extent.Three levels of decomposition of all transforms is used. Sym8wavelet filter is used for all DWT and UWT decompositions.Additive white Gaussian noise of standard deviation σ is addedto the original image in order to test the performance of ourproposed designs on noisy images.

A. Image denoising using hard thresholding

To verify the denoising ability of the proposed design, wecompare the image denoising results using hard threshold-ing method [12] using the transforms DWT, UWT, FiDWT,UFiDWT, RFiDWT and RUFiDWT. A value of T = 3σ isused for thresholding, while σ is estimated from the Donoho’srobust MAD (median absolute deviation) estimator from thefirst scale HH subband of DWT.

Figure 6 shows hard thresholding results on the part ofBarbara image containing oriented texture for AWGN of

(a) (b) (c)

(d) (e) (f)

(g) (h)

Fig. 6. Image denoising using hard-thresholding (a) Original image (b)noisy image with σ = 30, PSNR = 18.72 dB (c) DWT, PSNR = 22.05 dB(d) UWT, PSNR = 23.51 dB (e) FiDWT, PSNR = 22.15 dB (f) UFiDWT,PSNR = 24.18 dB (g) proposed RFiDWT, PSNR = 23.95 dB (h) proposedRUFiDWT, PSNR = 24.67 dB.

18 19 20 21 22 23 24 2522

23

24

25

26

27

28

29

30

PSNR (dB) for different noise levels (σ = 15 to 30) on part of Barbara image

PS

NR

(d

B)

of

de

no

ise

d im

ag

e f

or

ea

ch

no

ise

le

ve

l

DWT

UWT

FiDWT

UFiDWT

Proposed RFiDWT

Proposed RUFIDWT

Fig. 7. PSNR (dB) comparison of mentioned transforms under differentnoisy levels on the image shown in 6(a).

standard deviation σ = 30. Figure 7 shows PSNR comparisonof the mentioned transforms under different noise levels.

Proposed design RFiDWT shows improvement in denoisingperformance over DWT, UWT and FiDWT, while RUFiDWTdesign performs better in terms of PSNR value and haveless visual artifacts in the denoised image while preservingthe oriented textural features. Table I shows comparison ofPSNR, SSIM[13] and FSIM[14] values of the hard-thresholdeddenoised images using mentioned transforms on two widelyused images. For both of the images, proposed designs main-tain PSNR improvement while preserving geometrical featuresfrom the original image (clear from the better SSIM and FSIMvalues). From figures 6, 7 and table I, it is validated that,increased redundancy, directionality and frequency selectivityof the filters in the transform designs leads to improvementin image denoising performance in terms of PSNR as well asvisual quality.

TABLE IPSNR/SSIM/FSIM COMPARISON OF IMAGE DENOISING PERFORMANCE

OF MENTIONED TRANSFORMS USING HARD THRESHOLDING METHOD

σ 10 30 50Input

PSNR/SSIM/FSIM 28.13 0.72 0.96 18.59 0.35 0.81 14.15 0.21 0.70

Transform Barbara 512×512DWT 29.83 0.86 0.96 23.97 0.65 0.89 21.81 0.54 0.85UWT 31.96 0.90 0.98 25.47 0.71 0.92 22.92 0.58 0.88

FiDWT 29.71 0.85 0.95 24.34 0.65 0.88 22.00 0.53 0.84UFiDWT 32.63 0.91 0.98 26.40 0.74 0.93 23.35 0.59 0.89RFiDWT 32.19 0.90 0.97 26.03 0.72 0.92 23.10 0.58 0.87

RUFiDWT 33.01 0.92 0.98 26.81 0.76 0.93 23.79 0.62 0.89

σ 10 30 50Input

PSNR/SSIM/FSIM 28.13 0.61 0.95 18.59 0.22 0.79 14.15 0.12 0.67

Transform Lena 512×512DWT 32.40 0.85 0.97 27.03 0.70 0.90 24.52 0.62 0.86UWT 34.14 0.88 0.98 28.61 0.74 0.93 25.70 0.63 0.89

FiDWT 32.00 0.84 0.96 26.79 0.67 0.89 24.33 0.57 0.85UFiDWT 34.34 0.88 0.98 28.75 0.73 0.94 25.71 0.60 0.89RFiDWT 33.93 0.88 0.98 28.36 0.72 0.92 25.40 0.60 0.87

RUFiDWT 34.74 0.89 0.98 29.36 0.76 0.94 26.38 0.65 0.90

B. Image denoising using RFiDWT and BLS-GSM

BLS-GSM [15] is a state-of-the-art wavelet-based imagedenoising technique. MATLAB code for the same is availableon authors website (decsai.ugr.es/ javier/denoise), providingfunctionality to use different wavelets in orthogonal, undeci-mated and fully steerable pyramid mode. Best denoising re-sults are obtained with the fully steerable pyramid which has 8directions with redundancy factor of 18.6. In our case, we haveused this method in orthogonal decomposition mode separatelyon the two RFiDWT transformed images (e.g., figures 5(c) and5(d)). Figure 8 shows comparison between denoised imagesusing fully steerable pyramid and RFiDWT using BLS-GSMdenoising method. It is clear that, the proposed RFiDWT hasslightly better results than full steerable pyramids using BLS-GSM for textural part of the image, due to better frequencyselectivity of the filters. The lines on the cloth are more clearlyvisible in proposed approach when compared to the othermethod. The directional texture features in denoised imagesare better preserved using RFiDWT. For an N ×N image, wehave numerical complexity O(N2) for RFiDWT/BLS-GSM,much less than O(N2 log2N) as in the case of [15].

IV. CONCLUSION

We have presented the two redundant transform designswith enriched directionality using an additional stage prior tothe traditional decimated and undecimated wavelet transformsand explained their use for image denoising application. Bothdesigns have better adaptability to the oriented features in theunderlying image, since the filter bank construction enables usto design the filters with better frequency selectivity, therebymaintaining denoising and visual artifacts suppression trade-off. Image denoising using RFiDWT with BLS-GSM showsencouraging results for oriented textural images with muchless computational cost.

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