24-08-2012, 04:22 PM
The Contourlet Transform for Image De-noising Using Cycle Spinning
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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.
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.
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.
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 7-
9 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.
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.