Image De-nosing Based on Non-subsampled Contourlet Transform Domain in Multi-bessel K Form Model

This study proposes a new image de-nosing algorithm based on Non-Subsampled Contourlet Transform (NSCT) domain in multi-Bessel k form model. Firstly, the noisy image is decomposed into a set of multi-scale and multidirectional frequency sub-bands by NSCT, according to BKF model to scale coefficient of intra-scale and inter-scale processing, fully considering correlation of internal and external scale. Lastly, the estimated coefficients are updated according to inverse non-subsampled Contourlet transformation is performed to get de-noised image. Experimental results show that out algorithm better than the other algorithms in peak signal-to-noise ratio, structural similarity and visual quality.


INTRODUCTION
Image inevitably by noise pollution in the process of acquisition and transmission, noise have reduced the image resolution.Therefore, how to remove the noise effectively become the image processing of classic problem.Scholars put forward many wavelet processing image de-noising methods, for example, There is global threshold wavelet shrinkage de-noising method (Donoho, 1995), but this method makes the processed images too vague because of fixed threshold.Crouse et al. (1998) put forward hidden markov tree model de-noising algorithm, the algorithms' time complexity too high and noise can't achieve effective purify, Chang et al. (2000) is proposed based on generalized Gaussian distribution model Bayes Shrink de-noising algorithm, But lost too much high frequency coefficient lead to the de-noised effect is not ideal.With the limitation of the wavelet transform (lack of multidirectional selectivity and the sparse solution) is becoming increasingly obvious.So Multi-scale geometric analysis method appears, Contourlet transform is the most representative one (Do and Vetterli, 2005), the base function has multi-directional selectivity, can adaptive show optimal image.But as a result of Contourlet itself does not have translation invariance, so on the basis of the Contourlet appeared the Non-subsampled Contourlet Transform will be used in image de-noising and obtain better effects (Cunha et al., 2006).
Other image de-noising algorithms only consider intra-scale or inter-scale coefficients correlation usually result in de-noising effect is not very good.BKF model fully considering correlation of internal and external scale, Therefore, this study will be the Non-subsampled Contourlet transform and multiple BKF model combined, proposed a new image de-nosing algorithm based on Non-Subsampled Contourlet Transform (NSCT) domain in multi-Bessel k form model.

NON-SUBSAMPLED CONTOURLET TRANSFORM
Do put forward Contourlet Transform (CT) (Do and Vetterli, 2005), the transform using Laplacian Pyramid decomposition (LP) and Directional Filter group (DFB) realize a multi-resolution, multi-direction and multi-scale image representation method, But, Contourlet Transform (CT) in LP decomposition interlaced every column and every row subsampled on the image lead to Contourlet transform does not have the characteristics of translation invariance.Then, Cunha et al. (2006) put forward Non-Subsampled Contourlet Transform (NSCT), It is a multi-resolution, multi-scale, has the translation invariance redundant transformations compare to CT, This transformation the sampling to filter, to the signal filtering again.NSCT implementation by two major steps: • Use the Non-subsampled pyramid filter to image multi-scale decomposition to a low-pass sub-band and a band-pass sub-band.In order to realize multistage decomposition structure, only the low frequency sub-band continue to iterative filter, finally the image decompose into a low-pass subband and multiple band-pass sub-band.• At each level the Non-subsampled pyramid bandpass sub-bands direction decomposition with the Non-subsampled direction filter.The above two steps to complete the NSCT image decomposition.

Multi-BKF model:
Assume that an image g by zero mean, variance is ߪ ଶ Gaussian white noise n, f is the coefficients by noise.So we have the following equation: After NSCT transformation we get: where, Y = Wf S = Wg N = Wn W = The Non-subsampled coutourlet transform operator The spherically-contoured zero-mean ddimensional BKF density can be written as: is the modified Bessel function of the second kind and c and p are the scale and shape parameters.S = (u i , S i , S p i ), S i are estimate coefficients and S p i is S i father, u i are the center of the S i 3×3 window eight wavelet coefficients.So we get: 2 2 2 ( ) For any intra-scale wavelet coefficients y i there is a vector y has: Therefore, the MAP estimator is: This estimator can be computed by successive substitution, namely, ‫‖ݏ‖‬ ሺାଵሻ = ݂ ‫‖ݏ‖‬ ሺሻ obtain:

THE PROPOSED ALGORITHM
Due to the NSCT not the orthogonal transformation resulted in different directions sub-bands the noise variance is not equal, so in this study using Carlo method estimated in each sub corresponding noise NSCT coefficients variance The following are the main steps: Step 1 : Compute four levels Non Contourlet Transform (NSCT) of noisy image.
Step 2 : For each direction high frequency coefficients and combined neighborhood coefficients and coefficients, using ( 4) and ( 9).
Step 4 : Compute the inverse contourlet transform by estimated coefficients, get removed noise image.

SIMULATION RESULTS
In the simulation experiment, zero mean variance Gaussian white noise is added to Lena and Barbara image and we test our proposed algorithm on these images.This study first compare our de-noising method with db8 wavelet hard threshold, Do proposed Contourlet Transform (CT) with hard threshold and bayes risk minimum threshold method (BayesShrink) (Chang et al., 2000) and (1999) proposed LAWML (5×5), In order to give an objective comparison with other approaches, Peak Signal Noise Ratio (PSNR), structural similarity et al., 2004) and visual effect are using as performance analysis.PSNR is defined by: 10 256 20 log ( ) PSNR MSE

=
And MSE is given by: 2 , ,  2 1 1 1 ( ) where, X, Y : The original noisy image and de-N 2 : Image size we give out the PSNR and de-noising image in From Table 1 to 3 show that our proposed in this study better than other method and result has obvious advantages in PSNR and SSIM, From the Fig. 3 and 4, de-noising image based on this method that keep more detail information and visual effect is better.Compared with HT and C increased about 3 db, compared with App.Sci. Eng. Technol., 6(18): 3400-3403, 2013 3402 ALGORITHM Due to the NSCT not the orthogonal transformation bands the noise variance is not equal, so in this study using Monte method estimated in each sub-band corresponding noise NSCT coefficients variance ߪ ଶ (k).
frequency sub-band combined with its and the father using ( 4) and ( 9).
‫ܭ‬ሻ using Monte coefficient estimate coefficients using Non-subsampled transform by estimated coefficients,

RESULTS
In the simulation experiment, zero mean ߪ ଶ variance Gaussian white noise is added to 512×512 image and we test our proposed his study first compare our method with db8 wavelet hard threshold, Do (CT) with hard threshold and bayes risk minimum threshold method and Mihcak et al. , In order to give an objective comparison with other approaches, Peak structural similarity (Zhou and visual effect are using as performance -noised image image in Table 1. 3 show that our proposed method in this study better than other method and de-noising result has obvious advantages in PSNR and SSIM, image based on this method that keep more detail information and visual effect is better.Compared with HT and CT, PSNR with Bayes Shrink

Fig. 1 :
Fig. 1: Scheme of NSCT transform The center of the y i neighborhood coefficient ‫ݕ‬ : y i father With reference to (1), Maximum a Posteriori estimation (MAP) theory.‫ݏ‬ሺ‫ݕ‬ሻ = arg ‫ݔܽ݉‬ ‫‬ ௦|௬ ሺ‫ݕ|ݏ‬ሻ which is equivalent to: ˆ( ) arg max[lg( ()) lg( ( ))] literature(Khazron and Selesnick,  2008)  the second term can be computed as: The current estimated high frequency sub-bands Var (X) : X variance Kurt (X) : The kurtosis of a BKF random variable X d : Equal to the vector y dimension

Fig. 4 :
Fig. 4: Results of various de-noising methods of Barbara CONCLUSION This study proposes a new image algorithm based on Non-subsampled Transform domain in multi-Bessel k form model. First, using Non-subsampled contourlet transform advantages of translation invariance and direction, then using multi-Bessel k form correlation of interna scale coefficients.No matter in PSNR, SSIM and visual

Table 1 :
PSNR values of de-noised Lena images for different variance

Table 3 :
SSIM values of de-noised Lena, Barbara images for different variance PSNR is better.Because of this study fully considering correlation of internal and external scale, Bayes Shrink only considering the relationship scale coefficients, so our method comprehensive, consequently to achieve higher PSNR, SSIM and also keep more details.