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Image denoising is an important step in eliminating any noise impact in any image transmission process. Recently we presented two approaches for Bivariate based image denoising. They were Double Density Discrete Wavelet Transform (DD DWT) and Double Density Dual Tree Complex Wavelet Transform (DD CWT). In both techniques we decomposed noisy images with either DD DWT or DD CWT decompositions and then applied the Bivariate based denoising technique for noise removal. In this paper we propose an adaptive hybrid technique for Bivariate based image denoising that is based on the synthesis of DD-DWT bands or DD-CWT bands but with different weights, to deliver enhanced image features with less denoising impact especially around image edges, which is the most effected by noisy transmission channels. This proposed technique has been also enhanced by edge sharpening and Eigen analysis, as two separate stages. Simulation result comparisons have been performed between the proposed hybrid band adaptive DD-DWT and DD-CWT technique and the two primary techniques DD-DWT, DD- CWT, as well as other superior literature techniques such the original bivariate denoising technique with both original Complex Wavelet Transform and Double Density decompositions. This work in specific compares between Double Density DWT and Double Density CWT decompositions, proposes new filter design that suits each of them and proposes a hybrid technique between as will be shown.

There has always been an amount of unavoidable noise contaminated in any communication channel. Eliminating or avoiding this noise amount is inevitable. Hence, denoising became crucial to improve quality and eliminate degradations of received data in any receiver system.

Several image denoising algorithms were proposed during the last two decades. They range from the frequency-based filtering techniques [

Typically, in wavelet-based techniques, the signal/image is represented in frequency domain as a few numbers of large coefficients as well as large number of almost zero coefficients. The adopted denoising strategy is based on thresholding small coefficients, where the noise effect is high, while keeping or modifying large coefficients, where the noise effect is low. In [

A Bivariate Shrinkage technique was first adopted in wavelet image denoising in [

Double Density DWT wavelet

The one-dimensional double-density 1-D DD-DWT, first presented in [

The proposed design to achieve a 3-channel perfect reconstruction filter bank set is detailed as follows: Let H 0 ( z ) be an N-tap low pass filter with K zeros at z = − 1 . These K zeros are placed to insure smoothness. We then would want to construct two N-tap high pass filters H 1 ( z ) & H 2 ( z ) that satisfy the following perfect reconstruction (PR) and Alias Free (AF) conditions:

PR (condition):

H 0 ( z ) H 0 ( z − 1 ) + H 1 ( z ) H 1 ( z − 1 ) + H 2 ( z ) H 2 ( z − 1 ) = C z − ( N − 1 ) , (1)

where C is a constant

AC (condition):

H 0 ( − z ) H 0 ( z − 1 ) + H 1 ( − z ) H 1 ( z − 1 ) + H 2 ( − z ) H 2 ( z − 1 ) = 0 (2)

We propose to satisfy these two conditions in two steps:

In step 1: For a given H 0 ( z ) , assume an arbitrary H 1 ( z ) , then scale H 1 ( z ) such that its norm is less than 2 − | H 0 ( z ) | 2 2 . We then construct H 2 ( z ) to meet AC condition using a root finding technique as follows:

Let

Y ( z ) = H 0 ( − z ) H 0 ( z − 1 ) + H 1 ( − z ) H 1 ( z − 1 ) = − Y a ( − z ) Y a ( z − 1 ) . (3)

This means that Y ( z ) has roots at z k & ( − 1 z k ), k = 1 , 2 , ⋯ , N − 1 .

Let

H 2 ( z ) = z − ( N − 1 ) Y a ( − z − 1 ) .

where Y a ( z ) is constructed through root grouping, either by maximum phase, minimum phase or mixed phase factorization. This would satisfy AC condition.

In step 2: we update H 1 ( z ) as follows,

H 1 u ( z ) ∗ H 1 u ( z − 1 ) = Y 1 ( z ) = 2 = C z − ( N − 1 ) − H 0 ( z ) H 0 ( z − 1 ) − H 2 ( z ) H 2 ( z − 1 )

This would allow us to obtain H 1 u ( z ) using a root finding technique of Y 1 ( z ) again either using maximum, minimum or mixed phase factorization. Then we update H 1 ( z ) = H 1 u ( z ) and reiterate step 1 until conversion is obtained. It is worth mentioning that, while the filter H 0 ( z ) has K zeros at z = − 1 ; Both H 1 ( z ) & H 2 ( z ) must has M zeros at z = 1 where M ≤ K . Finally, because of the factorization of Y ( z ) , the construction of H 2 ( z ) and subsequently H 1 ( z ) is non-unique.

Example: In this example, we choose a 6-tap minimal-length Debauches low-pass filter H 0 ( z ) with K = 4 . In our simulation, we choose M = 2 to construct an arbitrary H 1 ( z ) , length 4. We express H 1 ( z ) = ( 1 − z − 1 ) M X 1 ( z ) and similarly for H 2 ( z ) . The unknown parameters X 1 ( z ) X 2 ( z ) proceeds as described above. For this arbitrary H 1 ( z ) , Y ( z ) is factorized as in Equation (3). Only six iteration steps are needed to obtain the exact solution. Design coefficients of the upper tree H 0 , H 1 , H 2 are listed below in

We note here that the 2-D DD-DWT still suffers from phase ambiguities that is typical in any basic Discrete wavelet Transform DWT. The next utilized Complex Wavelet Transform CWT has been proposed to overcome this phase ambiguity issue.

Double Density CWT Wavelet

In this section we present our novel 2-D DD-CWT filter bank structure. We first note that the design of the regular CWT filter structure bank H 0 u , H 0 d , which is the low pass filter of the upper and lower (down) trees must be a half band filter as noted in [

It has been shown that to fulfill this Hilbert pair relation; the low-pass filters { h 0 , u ( n ) , h 0 , d ( n ) } of the upper and lower tree satisfy the following constraint

h 0 , d ( n ) = h 0 , u ( n − 0.5 ) ⇒ ψ d ( t ) = H { ψ u ( t ) }

where ψ u ( t ) & ψ d ( t ) are the wavelet functions of the upper and lower trees, respectively and H denotes Hilbert transform. Moreover, this condition also implies ϕ d ( n ) = ϕ u ( n − 0.5 ) where ϕ ( n ) represents the corresponding discrete scaling functions. This half sample shift implies that in case of multi-level decomposition, integer translates of ϕ d ( n ) should lie midway between integer translates ϕ u ( n ) . Thus in order to ensure that the upper and lower wavelet functions form a Hilbert pair at every decomposition level 1 , ⋯ , r , the first stage and the succeeding stages filters of the upper and lower should be chosen to satisfy the condition [

h 0 , d 1 ( n ) = h 0 , u 1 ( n − 1 ) ,

h 0 , d j ( n ) = h 0 , u j ( n − 0.5 ) , j = 2 , 3 , ⋯ , r (4)

where r is no. of decomposition levels and j is the stage index, as the lattice tree structure in

H_{0} | H_{1} | H_{2} |
---|---|---|

−0.0083 | −0.0001 | 0.0641 |

−0.0244 | 0.0005 | −0.0246 |

0.2342 | −0.0011 | −0.0546 |

0.7445 | −0.0004 | −0.0321 |

0.6052 | 0.0049 | 0.0581 |

−0.0402 | −0.0036 | 0.0208 |

−0.1440 | −0.0031 | −0.0456 |

0.0341 | 0.0028 | −0.0314 |

In [

For our proposed double density CWT filter structure, the proposed one-dimensional DD CWT is designed as follows:

Construct H 0 u ( z ) & H 0 d ( z ) to satisfy the half band property and group delay constraints as follows:

Assume H u ( z ) to be an N-tap FIR, N = even, of the form

H u ( z ) = α 0 + α 1 z − 1 + α 2 z − 2 + ⋯ + α N − 1 z − ( N − 1 )

where the unknown α’s are determined to satisfy the following two conditions:

1) The function R ( z ) = H u ( z ) H u ( z − 1 ) must be a half band function as in PR systems.

2) The group delay of the rational function F ( z ) = H d ( z ) H u ( z ) = z − ( N − 1 ) H u ( z − 1 ) H u ( z ) should approximate the delay τ = 1 for the first level of the decomposition bank and 0.5 for the succeeding stages in a least squares sense over a fraction r x _{ }of_{ }the pass-band of the low pass filter H u ( z ) , respectively [

Then we construct the filters [ H 1 u , H 2 u ] of the upper tree according to the following analysis:

H 1 u ( z ) H 1 u ( z − 1 ) + H 2 u ( z ) H 2 u ( z − 1 ) = H 0 u ( − z ) H 0 u ( − z − 1 ) H 1 u ( − z ) H 1 u ( z − 1 ) + H 2 u ( − z ) H 2 u ( z − 1 ) = − H 0 u ( − z ) H 0 u ( z − 1 ) (5)

similar relations are applied for H 1 d ( z ) , H 2 d ( z ) . Hence, the design of H 1 u ( z ) , H 2 u ( z ) proceeds as follows

1) Construct H 0 ( z ) , from regular filter design [

2) Take an initial H 1 ( z ) , that is scaled such that its frequency response is less than that of H 0 ( − z ) at all z point;

3) Construct H 2 ( z ) from the root finding and perfect reconstruction property, Equation (4);

4) Check the alias free property, and update H 1 ( z ) ;

5) Reiterate until conversion is obtained.

Design coefficients of H 0 ( z ) & H 1 ( z ) & H 2 ( z ) , τ = 0.5 , and 1, for the upper and lower tree, respectively, are listed.

This shows that DD-CWT is designed as a double density 3-channel decomposition for each of H 0 u , H 0 d , i.e. the low pass filter of the upper and lower (down) trees.

This design of a 1-D DD-CWT filter along with its coefficients along with the 1-D DD-DWT design methodology presented in previous section is our primary contribution of this work (

H_{0} | H_{1} | H_{2} * 10^{−3 } |
---|---|---|

0.2318 | 0.019 | 0.3722 |

−0.095 | −0.114 | −0.5272 |

0.1219 | 0.2385 | 0.048 |

0.897 | −0.2411 | 0.244 |

0.1215 | 0.187 | −0.106 |

−0.0949 | −0.2504 | −0.2397 |

0.2319 | 0.2626 | 0.139 |

−0.0001 | −0.1016 | 0.0696 |

H_{0} | H_{1} | H_{2} * 10^{−3} |
---|---|---|

−0.0051 | 0.019 | 0.5001 |

−0.091 | −0.114 | −0.7084 |

0.2196 | 0.2385 | 0.0645 |

0.8051 | −0.2411 | 0.3279 |

0.5476 | 0.187 | −0.1424 |

−0.0123 | −0.2504 | −0.3221 |

−0.055 | 0.2626 | 0.1868 |

0.0052 | −0.1016 | 0.0936 |

The 2-D Double-Density Complex Wavelet Transform i.e. 2-D DD-CWT is consequently implemented by applying the filters H 0 ( z ) & H 1 ( z ) & H 2 ( z ) of H 0 u , first to the rows, then to the columns of an image, for every stage. This would result in nine 2-D sub bands, for every stage, where one of them is the 2-D lowpass scaling filter, and the other eight make up remaining 2-D wavelet filters. For a 3 stage 2-D DD-CWT decomposition we would have a resulting total of 25 2-D subbands as will be detailed in the simulation result section.

Classical wavelet based denoising techniques, are based on thresholding wavelet coefficients. Different techniques are used to determine these thresholding levels. They are mainly based on thresholding the wavelet decomposition of the noisy image at every sub band by a specific thresholding parameter. The problem can be formulated as: Given the noisy wavelet coefficient w n , it is required to recover the clean wavelet coefficient w where w n = w + n , n is the associated independent noise. This is a Maximum Likelihood Estimation (MLE) problem, as solution can be found as,

w = max ( p ( w | w n ) ) = max ( p ( n ) p ( w ) ) , p is the pdf distribution. In case of zero mean Gaussian noise, p ( n ) can be formulated in terms of its variance σ n 2 . In this case, the variance can be estimated using the empirical formula σ n 2 = median ( w n ) / 0.6745 .

As far as w, it has been observed that the pdf of the wavelet coefficients of natural images approximates Laplacian distribution, [

In [

p ( w 1 , w 2 ) = 1 2 π σ 2 e − 3 σ w 1 2 + w 2 2

where w 1 , w 2 are the parent and children wavelet coefficients, respectively.

In this paper, we estimate the variance σ 2 = min ( σ 1 2 , σ 2 2 ) , where σ 1 2 and σ 2 2 are variances of w 1 and w 2 , respectively. Thus, in order to de-noise a noisy image, and in view of the assumption of near circular joint pdf distribution between adjacent scale wavelet coefficients, the thresholded children coefficient w n 2 t h is given by, [

w n 2 t h = w n 2 ⋅ ( w n 1 2 + w n 2 2 − 3 σ n 2 σ ) + w n 1 2 + w n 2 2

where the soft thresholding function ( x ) + = x if x > 0 and zero otherwise.

Thus, the proposed de-noising scheme amounts to thresholding the wavelet coefficients of the real and imaginary wavelets of the upper and lower trees. In this scheme, the noise variance σ n 2 is accurately estimated through estimating the pdf of the detail coefficients of the first level wavelet decomposition of the noisy image. The de-noising scheme is summarized as follows:

1) For a prescribed number of decomposition levels n and a prescribed number of vanishing moments K, determine the first and succeeding stages filters of the upper and lower trees of the dual tree DWT, as described in section 2.

2) Initially, for the first scale of the upper and lower trees, evaluate the real and imaginary parts of the complex wavelets w n 2 and its adjacent parent w n 1 at the coarser scale. Interpolate by 2 the parent coefficient w n 1 .

3) In order to estimate σ , σ 1 2 and σ 2 2 have to be estimated. They are estimated as the peak powers E 1 , E 2 of the coefficients w n 1 , w n 2 . Then, σ 1 2 = E 1 − σ n 2 , σ 2 2 = E 2 − σ n 2 , and σ 2 = min ( σ 1 2 , σ 2 2 ) . Threshold the children’s coefficients.

Repeat steps 2,3 until all children coefficients are scanned. In the last scale, threshold the parent coefficient w n 1 as well.

In this paper, we apply this bivariate shrinkage technique to denoise 2D DD-DWT and 2D DD-CWT wavelet packets. Bivariate shrinkage technique is based on strong dependency between noisy children wavelet coefficients x 1 n and their corresponding noisy parent coefficients x 2 n at coarser scale.

p x = p ( x 1 , x 2 | x 1 n , x 2 n ) (6)

where x 1 , x 2 denote the clean wavelet ceofficients.

Using Bays rule; it turns out that

p x = p ( n 1 , n 2 ) ∗ p ( x 1 , x 2 ) (7)

where n 1 = x 1 n − x 1 , n 2 = x 2 n − x 2 are two independent Gaussian noise with variance σ n 2 .

In order to obtain a closed form solution for x 1 , x 2 we express the joint p ( x 1 , x 2 ) using the empirical formulation as in [

p ( x 1 , x 2 ) = 3 2 π σ 2 ∗ e − 3 σ x 1 2 + x 2 2 (8)

where σ 2 = σ 1 2 ∗ σ 2 2 ; σ 1 2 , σ 2 2 denote the variances of x 1 , x 2 respectively. This is verified by the near circular performance of the noiseless joint pdf distribution shown in

x k = ( ( x k n ) 2 + ( x k + 1 n ) 2 − 3 σ n 2 σ ) + x k 2 + x k + 1 2 ⋅ x k n (9)

We note here that all DD-CWT or DD-DWT decompositions presented in the work adopts this bivariate denoising scheme between its parent and children bands.

In this section we propose two scenarios to enhance the performance of denoising, then we summarize how to achieve the highest performance by fusing different bands from different decompositions structures, i.e. CWT, DWT, and combine them with optimized factors to synthesize an enhanced denoising image.

We here propose to enhance the noisy image edges by processing it through a Laplacian 2-D filter with an optimized factor and add the result to the original noisy image. This Laplacian 2-D filter would have a sharpening factor that is also optimized. The output image from this Edge sharpening stage would be as the following equation:

w updated = α L a p ( w ) + w

where w is original noisy CWT or DWT wavelet band, w updated represents the final band coefficients, and α is the optimization factor.

We here propose to decompose the noisy image DD-CWT or DD-DWT with a singular value decomposition. In this decomposition we produce a diagonal matrix D, with nonnegative diagonal elements in decreasing order, and unitary matrices U and V, all of the same size of the input image band, according to following equation

[ U ⋅ D ⋅ V t ] = S V D ( w )

where w is original noisy CWT or DWT wavelet band, D is the diagonal matrix that will be rescaled in decreasing order, and w updated is reconstructed according to the following synthesis equation

U t D V = w updated

We then select the highest D matrix diagonal element and scale it to an updated denoised value and then reconstruct the wavelet band. This updated denoised value is obtained through some empirical analysis and also by denoised learning scenarios.

Finally in this section we proposed to decompose the denoised enhanced images with either DD-CWT or DD-DWT by a further DD-CWT structure for each image, where each band in each denoised image is multiplied by an optimized factor and then synthesize this CWT decomposition. This optimization process that is performed for all the CWT bank factors and is aimed at reducing edge energy in the final output denoised image. CWT optimization is performed according to the following equation.

w hybridoptimized = α 1 w 1 + α 2 w 2 + ⋯ + α i w i + ⋯ + α last w last

where w hybridoptimized is final hybrid optimized wavelet band and each i original band is multiplied with an α i multiplication factor.

We note here that this further denoising enhancement by a factorized optimized DD-CWT structure would represent a hybrid denoising technique that combines merits of both DD-DWT and DD-CWT in a fusion manner, especially if it was originally denoised with DD-DWT. The α i factors would make this factor adaptive as we can select some high density bands to get higher weight in the final denoised image. This fusion hybrid technique would achieve significant improvement in the final PSNR or SSIM values as will be shown in next section. We justify this improvement to the optimization process that selects the highest energy bands.

In this section we first show the performance of 2D DD-DWT denoising after explaining its experimental procedure. We then show the performance of 2D DD-CWT that in some circumstances achieves better PSNR results. Then we show how each of our proposed two enhancement scenarios, Edge sharpening and Eigen Analysis, would enhance the final denoised image. We finally illustrate how our proposed hybrid DWT and CWT fusion denoising methodology would achieve the most enhancement performance.

We note here that the enhancement in denoising is measured in terms of reduced energy around edges. This was also compared with PSNR values of the output denoised image compared with original clean image, before any noise attack. This PSNR calculation with the clean image could be unpractical in many denoising applications where the clean image is not available for comparisons, but it is only mentioned just to verify that enhancement in PSNR values is consistent with edge energies minimizations that is performed in all proposed techniques.

In these simulations; σ n 2 is estimated using the pdf technique of the first wavelet detail coefficient as described in [

The proposed filter design H 0 ( z ) has K = 4 zeros at z = − 1 ; H 1 ( z ) is constructed with M = 2 zeros at z = 1 using the maximum phase technique.

For space limitations, we have two 256 × 256 Cameraman & Lena images contaminated with zero mean Gaussian noise AWGN with different variance σ n 2 . The number of decomposition levels is 3 for either DWT or CWT. Figures 7-10 shows samples of our Denoising results, with a noisy image example σ n 2 = 0.1 , 0.05, or 0.2.

The proposed 2D-DD DWT Bivariate denoising technique is implemented on 2D noisy images as follows:

1) Given the 2D-DD-DWT filters H 0 ( z ) , H 1 ( z ) and H 2 ( z ) as in section 2; decompose the noisy image I n through wavelet packet structure using these filters.

2) For each of the 8 decomposed wavelet coefficients in all sub bands levels x k n , k = 1 , 2 , ⋯ , J levels; apply Bivariate shrinkage technique between x k n and its corresponding at coarse scale x k + 1 n . Expand x k + 1 n by 2 to have the same size as x k n . Estimate σ 1 2 , σ 2 2 for both x k , x k + 1 .

3) Reconstruct the denoised image I ^ as in section 2.

The proposed 2D-DD CWT Bivariate denoising technique is implemented on 2D noisy images as follows:

4) Decompose the input Noisy image by its rows through the proposed 2-D DD CWT decomposition structure in

5) Repeat step one but through columns of the Noisy image.

6) The Bivariate shrinkage technique proposed in section 3 is applied between each scaling subband and its children for all stages.

7) Reconstruct the denoised image by rows and columns in a reverse manner of the decomposition.

Results with Edge sharpening

Noisy | DD DWT proposed | DD DWT after Sharpening | DD CWT proposed | DD CWT after Sharpening | |
---|---|---|---|---|---|

σ n 2 = 0.25 | 9.1025 | 17.70 | 17.90 | 15.95 | 17.65 |

σ n 2 = 0.2 | 9.588 | 18.36 | 18.12 | 16.41 | 17.31 |

σ n 2 = 0.15 | 10.345 | 19.12 | 19.85 | 17.18 | 19.01 |

σ n 2 = 0.1 | 11.530 | 20.45 | 20.83 | 18.29 | 19.2 |

Results with Eigen Analysis

Noisy | DD DWT proposed | DD DWT after Eigen Analysis | DD CWT proposed | DD CWT after Eigen Analysis | |
---|---|---|---|---|---|

σ n 2 = 0.25 | 9.1025 | 17.70 | 17.35 | 15.95 | 16.31 |

σ n 2 = 0.2 | 9.588 | 18.36 | 19.34 | 16.41 | 17.01 |

σ n 2 = 0.15 | 10.345 | 19.12 | 19.87 | 17.18 | 17.75 |

σ n 2 = 0.1 | 11.530 | 20.45 | 20.99 | 18.29 | 18.98 |

In our final hybrid DD-DWT and DD-CWT fusion technique we decomposed the image first through either DD-DWT or DD-CWT decomposition for denoising, then we further enhance the denoising by a factorized optimized DD-CWT process. Subband multiplication factors are our main optimization variables in our simulation results as in

Noisy | DD DWT proposed | DD DWT [ | DD CWT proposed | CWT [ | |
---|---|---|---|---|---|

σ n 2 = 0.25 | 9.1025 | 17.70 | 16.90 | 15.95 | 15.65 |

σ n 2 = 0.2 | 9.588 | 18.36 | 17.12 | 16.41 | 16.31 |

σ n 2 = 0.15 | 10.345 | 19.12 | 17.85 | 17.18 | 17.01 |

σ n 2 = 0.1 | 11.530 | 20.45 | 18.83 | 18.29 | 18.2 |

Noisy | DD DWT proposed | DD DWT [ | DD CWT proposed | CWT [ | |
---|---|---|---|---|---|

σ n 2 = 0.25 | 9.126 | 18.34 | 16.68 | 16.04 | 16.55 |

σ n 2 = 0.2 | 9.598 | 18.95 | 17.20 | 16.34 | 16.44 |

σ n 2 = 0.15 | 10.332 | 19.83 | 17.96 | 17.30 | 17.24 |

σ n 2 = 0.1 | 11.541 | 20.89 | 18.56 | 17.98 | 17.67 |

Noisy | DD DWT proposed | DD DWT after Hybrid optimization | DD CWT proposed | DD CWT after Hybrid optimization | |
---|---|---|---|---|---|

σ n 2 = 0.25 | 9.1025 | 17.70 | 18.01 | 15.95 | 16.13 |

σ n 2 = 0.2 | 9.588 | 18.36 | 19.98 | 16.41 | 17.91 |

σ n 2 = 0.15 | 10.345 | 19.12 | 20.77 | 17.18 | 18.01 |

σ n 2 = 0.1 | 11.530 | 20.45 | 20.83 | 18.29 | 19.23 |

In this paper, we proposed in more details the usage of DWT or CWT decompositions in image denoising with the adoption of Double density analysis. We proposed two methodologies to enhance these decompositions for either DWT or CWT scenarios. We also presented an adaptive hybrid technique for Bivariate based image denoising that is based on the synthesis of DD-DWT bands or DD-CWT bands but with different weights, to deliver enhanced image features with less denoising impacts. Simulation results have shown that the DD_DWT bivariate shrinkage achieves the best performance of all the denoising schemes considered. From Equations (3), (4) it is clear that there is a plenty of solutions for H_{2}(z) that satisfy the alias free conditions, yet simulations have shown that the maximum phase solution yields the optimum denoising performance.

This work has been funded mainly from Alexander Von Humboldt foundation, Germany, 2019. Early results of this work was presented in [

The authors declare no conflicts of interest regarding the publication of this paper.

Fahmy, G., Fahmy, M.F. and Fahmy, O. (2023) Adaptive Hybrid Bivariate Double Density Discrete and Complex Wavelet for Image Denoising. Journal of Computer and Communications, 11, 39-56. https://doi.org/10.4236/jcc.2023.112004