Due to the lack of translation invariance of the contourlet transform, we employ a cycle-spinning-based technique to develop translation invariant contourlet de- ...
The Contourlet Transform for Image De-noising Using Cycle Spinning Ramin Eslami and Hayder Radha ECE Department, Michigan State University, East Lansing, Michigan, USA Emails: {eslamira, radha}@msu.edu , Tel: 517-432-9958.
Abstract- We propose a new method for image de-noising based on the contourlet transform, which has been recently introduced. Image de-noising by means of the contourlet transform introduces many visual artifacts due to the Gibbs-like phenomena. Due to the lack of translation invariance of the contourlet transform, we employ a cycle-spinning-based technique to develop translation invariant contourlet de-noising scheme. This scheme achieves enhanced estimation results for images that are corrupted with additive Gaussian noise over a wide range of noise variance. Our experiments show that the proposed approach outperforms the translation invariant wavelets both visually and in terms of the PSNR values, especially for the images that include mostly fine textures and contours.
I.
• LP
• DFB
Image
INTRODUCTION
Recently, there has been a growing awareness to the observation that wavelets may not be the best choice for representing natural images. This observation is due to the fact that wavelets are blind to the smoothness along the edges commonly found in images. Hence, recently, some new transforms have been introduced to take advantage of this property. The curvelet [5] and contourlet transforms [3] are examples of two new transforms with a similar structure, which are developed to sparsely represent natural images. Both of these geometrical transforms offer the two important features of anisotropy scaling law and directionality. Starck, Candes, and Donoho [5] used the curvelet transform for image de-noising by using a translation invariant wavelet as the first stage of the curvelet transform. Do and Vetterli [2][3] utilized a double filter banks structure to develop the contourlet transform and used it for some non-linear approximation and de-noising experiments. In this work, we propose a new approach for image de-noising that is based on the contourlet transform. To compensate for the lack of translation invariance property of the contourlet transform, we apply the principle of cycle spinning [1] to contourlets. Our simulation results show significant improvements when compared to previous methods. In particular, we compared the proposed method to the wavelet transform using cycle spinning (WTCS), which is one of the best image de-noising techniques. Our approach achieved better visual results and outperformed the WT-CS, especially for images comprising mostly detailed and fine textures.
Fig. 1. A flow graph of the contourlet transform. The image is first decomposed into subbands by the Laplacian pyramid and then each detail image is analyzed by the directional filter banks.
II. A.
METHOD
The Contourlet Transform
Fig. 1 shows a flow graph of the contourlet transform. It consists of two major stages: the subband decomposition and the directional transform. At the first stage, we used Laplacian pyramid (LP), and for the second one we used directional filter banks (DFB). Fig. 2 shows an example of the frequency decomposition achieved by the DFB. Quincunx filter banks are the building blocks of the DFB. We used the fan filters designed by Phoong, Kim, Vaidyanathan, and Ansari [4] with support size of (23, 23) and (45, 45) for the quincunx filter banks in the DFB stage. The FIR half-band filter used for constructing fan filters is designed using the “remez” function in MATLAB. Fig. 3 shows the frequency responses of the designed fan filter pair of the quincunx filter banks. To decrease artifacts due to the Gibbs-like phenomenon in the DFB stage, we move downsampling and resampling to the end of the synthesis part and to the beginning of the analysis part, using the Nobel identities [2]. Fig. 4 depicts the contourlet coefficients of the Boats image using 3 LP levels and 8 directions at the finest level.
ω 4
3
2
(π , π )
1
1 8
5 6
7
7
6
8
ω
2
5 1
2
3
4
( −π , −π )
Fig. 2. An example of the directional filter bank frequency partitioning.
Fig. 4. The contourlet transform of the Boats image using 3 LP levels and 8 directions at the finest level. For better visualization, the transform coefficients are clipped between 0 and 15.
III.
Fig. 3. The frequency response of fan filter pair with support size of (23, 23) and (45, 45) used in the simulation.
B. Cycle Spinning Cycle spinning for de-noising is a simple yet efficient method that can be applied to a shift variant transform for signal de-noising. For a shift variant transform T, operating on a noisy image x = s + noise , where s is the original image, we denote the 2-D circulant shift by Si , j and the threshold operator by θ . Now, if the following procedure is applied:
sˆ =
1 K1 K 2
K1 , K 2
∑S
(T (θ [ T ( Si , j ( x )) ])) , −1
−i ,− j
i =1, j =1
where ( K1 , K 2 ) are the maximum number of shifts, one would expect an improvement for the estimation sˆ compared to the de-noised image without cycle spinning. For the wavelet K
transform, if the input image is of size (N, N) and N = 2 , after K shifts in each direction, the transform output repeats and so the maximum numbers of shifts will be K in each direction. Since the contourlet transform is a shift variant transform, we can apply the same approach to the contourlet transform. Similar to wavelets, if one decomposes an image of size (N, N) using the contourlet transform, the maximum number of decomposition levels in the LP stage will be K, and therefore, the maximum number of shifts are (K, K) in the row and column directions. We applied this procedure to contourlets and achieved superior performance in our denoising experiments as briefly demonstrated in the next section.
NUMERICAL EXPERIMENTS
To test our algorithm, we selected four images of size 512x512: Barbara, GoldHill, Mandrill, and Peppers. We used four approaches for our experiments: the contourlet transform (CT), the wavelet transform (WT), and the translation invariant wavelet transform (WT-CS) in addition to the proposed method based on the contourlet transform using cycle spinning (CT-CS). We used biorthogonal Daubechies 79 wavelet transform and the same wavelet filters for the LP stage of the contourlet transform. For the contourlet transform, we used 6 LP levels and 32 directions at the finest level. We added zero-mean Gaussian noise to the images and applied the above de-noising methods using a simple hard-thresholding to the noisy images. We set the thresholds to some values so that we obtain best PSNR values of the de-noised images. Fig. 5 shows the PSNR values of the de-noised images versus a range of the input noise. Except at few points for one image, the CT-CS outperforms the other methods. In particular, only for the Peppers image, which is a piecewise smooth image, and hence it is a “wavelet-friendly” image, the WT-CS performs almost the same as the CT-CS at a range of the input noise power. However, in case of the images containing mostly textures and contours such as the Barbara image, the CT-CS yields significant improvements up to 1.5 db over the WT-CS. To visually compare the estimated images, we show the de-noised images of the GoldHill image where sigma = 20 in Fig. 6. As seen, the result of the contourlet transform shows many visual artifacts that are due to the Gibbs-like phenomena; however we could reduce these artifacts to a large extent by using cycle spinning. In addition to the superior PSNR values that are achieved using the proposed method, the estimated images by the CT-CS (Fig. 6 and Fig. 7) are visually better (sometimes significantly better). As can be seen, significantly more levels of detail and texture are retrieved by the proposed scheme. Fig. 8 is another example that clearly shows this advantage for the CT-CS. Moreover, most of the visual artifacts due to the Gibbs-like phenomena in the CT denoising are reduced by using cycle spinning.
GoldHill
Barbara
30
30 WT CT WT-CS CT-CS
28
26
PSNR-db
PSNR-db
28
24
26
24
22
22
20 20
40 60 standard deviation
20 20
80
40 60 standard deviation
80
Peppers
Mandrill
32
26 WT CT WT-CS CT-CS
25 24
WT CT W T-CS CT-CS
30 28
23
PSNR-db
PSNR-db
WT CT WT-CS CT-CS
22 21
26 24
20
22
19 18 20
40 60 standard deviation
80
20 20
40 60 standard deviation
80
Fig. 5. The PSNR values of the de-noised images versus the standard deviation of noise for the de-noising experiments.
IV.
REFERENCES
CONCLUSIONS
We proposed an efficient method for image de-noising. We utilized the cycle spinning algorithm in developing a translation invariant contourlet-based de-noising. Our experiment results clearly demonstrated the capability of the proposed scheme in image de-noising experiments especially for those images possessing detailed textures. By using this approach, we could eliminate most of the visual artifacts resulting from the contourlet transform de-noising process. This approach outperforms the translation invariant wavelet de-noising (that is among the best image de-noising methods) both visually and in terms of PSNR values.
[1]
R. R. Coifman and D. L. Donoho, “Translation Invariant Denoising,” in Wavelets and Statistics, Springer Lecture Notes in Statistics 103, New York, Springer-Verlag, pp. 125-150, 1994.
[2]
M. N. Do, Directional multiresolution image representa- tions. PhD thesis, EPFL, Lausanne, Switzerland, Dec. 2001.
[3]
M. N. Do and M. Vetterli, “Contourlets: A Directional Multiresolution Image Representation,” in Proc. of IEEE Intl. Conf. on Image Processing, Rochester, September 2002.
[4]
S. M. Phoong, C. W. Kim, P. P. Vaidyanathan, and R. Ansari, “A new class of two-channel biorthogonal filter banks and wavelet bases,” IEEE trans. Signal Processing, vol. 43, no. 3, pp. 649-665, Mar. 1995.
[5]
J. L. Starck, E. J. Candes, and D. L. Donoho, “The curvelet transform for image denoising,” IEEE trans. Image Processing, vol. 11, pp. 670-684, Jun. 2002.
The GoldHill Image
GoldHill + Noise (sigma = 20)
PSNR = 22.18
De-noised Image Using WT
PSNR = 26.91
De-noised Image Using WT-CS
PSNR = 28.87
De-noised Image Using CT
PSNR = 27.24
De-noised Image Using CT-CS
PSNR = 29.39
Fig. 6. De-noising experiment with the GoldHill image corrupted with a Gaussian noise of sigma = 20.
De-noised Image Using WT-CS
De-noised Image Using CT-CS
Fig. 7. Parts of the de-noised GoldHill images magnified from Fig. 6.
Original Image
Wavelet Transform
Wavelets + Cycle Spinning
PSNR = 22.54
PSNR = 24.09
Noisy Image, sigma = 40
Contourlet Transform
Contourlets + Cycle Spinning
PSNR = 16.39
PSNR = 23.39
PSNR = 25.52
Fig. 8. De-noising experiment with the Barbara image corrupted with a Gaussian noise of sigma = 40 (Small parts of the images are depicted).
