A Tight Framelet Algorithm for Color Image De-Noising

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Procedia

Engineering

Procedia Engineering 00 (2011) 000–000 www.elsevier.com/locate/procedia

2011 International Conference on Advances in Engineering

A tight framelet algorithm for color image de-noising

Zemin Cai

a

, Chuliang Wei

b,

a

*

ab_{Department of Electronic Engineering, College of Engineering, Shantou University, 515063 Shantou, P.R. China}

Abstract

In recent years, there is a growing interest in the research of color image de-noising algorithms. A tight framelet is a wavelet frame constructed by multiresolution analysis. The properties of tight framelet and its applications on gray image denoising have been well studied. Our goal is to explore the power of tight framelet to de-noise color images. We analyze the relationship between gray images and color images. The frame-based gray image de-nosing scheme is extended to color image de-noising. Experimental results of color image de-noising show that the new method has a better performance than AMF approach.

Keywords: Color Image De-noising; Tight Framelet; Adaptive Median Filter

1. Introduction

Digital images are often distorted or degraded by noise. The classic image de-noising problem is modeled as follows:

I_noisy = +I n (1) where

I

_noisyis a measured noisy image,

I

is an original image to be recovered, and

n

is an additive zero-mean white and homogeneous Gaussian noise. The aim of image de-noising algorithms is to find an estimate of the ideal image

I

from the noisy image

I

_noisyusing prior information on noise and/or the ideal image. If color image de-noising problem is concerned,

I

_noisy,

I

and

n

all consist of three channels.

The most popular de-noisinging methods separate the ideal image

I

from the noise

n

in the noisy image

I

_noisybased on an assumption that frequency spectrums of the ideal image distribute in a low frequency band. For example, wavelet-based de-noising methods in [1] are carried out by thresholding wavelet coefficients(high frequency information) in the wavelet domain. A framelet is a tight wavelet

* Corresponding author Tel.: +86-754-82903349

E-mail address: [email protected]. Open access under CC BY-NC-ND license.

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frame constructed via multiresolution analysis. It has the advantages of both wavelets and frames. Framelet-based gray image de-nosing scheme has been proposed in [2].

A color image has three channels: red, green, and blue. These three channels roughly follow the color receptore in the human eyes. RGB color space is used mostly in computer displays and image scanners. Each channel of a color image is the grayscale image of the same size as a color image, which is made of just one of these primary colors. Because of this, we can apply the gray image de-noising scheme to each color channel independently and finally the three denoised channels are merged.

2. Tight framelet and gray image de-noising

In this section, we review the tight framelet theory briefly and refer the readers to Ref. 2, 3, 4 for detailed descriptions. Let

A

∈ ¡

K N× be a matrix and the tight framelet system also denoted by

A

if for any arbitrary

f ∈ ¡

N, 2 , 2 g f f g ∈ =

∑

A (2) Its synthesis operator

_A

*

:

K

_→

N_{can be defined as}

* _,

{ }

g _g g c c g _∈ ∈ =

∑

　c= cg _A A A (3) From the definitions above, we can draw a conclusion that

A

is a tight frame if and only if

A A

*

=

I

. And when

ker

A

*

≠

{ }

0

, the tight framelet system is redundant and

AA

*

≠

I

, where

I

is an identity matrix.

There are many choices for the design of tight frame

A

similar to wavelet transformation. For example, it can be generated from the spline wavelet frame filters by using the unitary extension principle, as described in Ref. 2.

From the given spline wavelet frame filters

h

_i, a tight frame system

A

can be designed. Let

h

be one of these filters, [ ( ), ( 1), , ( 1), (0), (1), , ( 1), ( )] h h n h n h h h h n h n = − − + ⋅⋅⋅ − ⋅⋅⋅ − 　 (4)

If Neumann boundary condition is used, then the matrix representation of

h

can be reached, _, ( ) ( 1 (2 )) 2 1 [ ] ( ) ( 1) ( 1) ( ) i j h i j h m i j i j N n H h i j h i j i j n h i j − + − − − − + ≥ − +   =_ − + + − + ≤ +  ₋  　if 　　　 if 　　　　　　　　 otherw ise

whose dimension is N-by-N. Then we can define

( ) ( 1) (1) 0 0 0 : L k k k G G G − = ⋅⋅⋅ (5) and 2 ( ) 1 1 ( ) 1 (2 1) 1 : L F L k k k k k n G G − − + −    _⋅     _⋅  =   ⋅         (6) where ₍₂( )t ₁₎

:

( )t ( )t n i j i j

G

_{+ +}

=

H

⊗

H

,

0 ≤

i j

,

≤

2 n

and

' '

⊗

is tensor product. Finally, for any integer

0

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1 1 : L F L F F L L L L L −      _{  }   = _{≡ } _ _⋅⋅⋅          A A (7)

It is easy to prove that

A

is a tight frame in

¡

Naccording to

A A

*

=

I

and * *

L L

+

L L

=

I

L L

A A

.

Gray image de-nosing problem can be modeled as

f

=

f

*

+

ε

on

Λ

and recovering

f

*in

Ω \ Λ

is the objective. Hence define the projection operator

Λ ( ), Λ ( ) 0, Ω \ Λ f i i P f i i ∈  =  _∈  　　　　 (8) where

f i

( )

is the grey level of pixel

i

. At last, the de-noising procedure can be defined as

1 0 * Λ 0 ( Λ) ₍ ₎ L T k L L k f f P f I P f +   = + − _ _   * A A (9) where

T

is a thresholding operator and

f

₀be the initial guess of the ideal image. For more information about tight framelet theory and its application on missing data recovering , readers are refered to Ref. 2.

3. Color image de-noising

Color images are digital images that include color information for each pixel. For people’s visual acceptability, it is necessary to provide three color channels for each pixel. They are interpreted as coordinates in some color space. The RGB color space is commonly used in computer displays. However, there still exist some other spaces such as YCbCr, HSV, which are often used in other contexts.

3.1. RGB color space

The RGB color model is based on the Young–Helmholtz theory of trichromatic color vision, which is developed by T. Young and H. Helmholtz in the early of nineteenth century. The name of the model comes from the initials of the three additive primary colors, red, green, and blue[5].

A color in the RGB space is described by indicating that how much of the red, green, and blue is included, which can be expressed as an RGB triplet (r, g, b). each component of one pixel can vary from zero to a defined maximum value, such as 255. If all the three components are at zero, the result is black while if all are 255, the result is the brightest representable white. Hence, an RGB triplet (r, g, b) can be used to represent the three-dimensional coordinate of the pixel of the given color within the cube as shown in figure 1. We develop the image de-noising algorithm in R, G, B channels. Finally the denoised samples are merged.

3.2. Noise models

Color image noise is random variation of color information in images, and is usually an aspect of electronic noise. Noise in imaging systems is usually either additive or multiplicative.

In many imaging systems, the noise is often modeled as an additive noise: given an original image

I

, we assume that it has been corrupted by some additive noise

n

. The denoising problem is then going to recover

I

from the data

I

_noisy

= +

I n

, as shown in formula 1.

Amplifier noise and salt-and-pepper impulse noise are the two major parts existing in images. The standard model of amplifier noise is additive, Gaussian and independent at each pixel. It is also known as Gaussian noise. Amplifier noise is independent of the signal intensity, and primarily comes from the reset noise of capacitors. Salt-and-pepper impulse noise is sometimes called salt-and-pepper noise or spike

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noise. It is a form of noise typically seen on images. Salt-and-pepper impulse noise represents itself as randomly occurring white and black pixels. An image containing salt-and-pepper impulse noise will have dark pixels in bright regions and bright pixels in dark regions[6]_{. Images corrupted by salt-and-pepper}

noise in situations where quick transients, such as faulty switching or take place. Existing effective noise reduction methods for this type of noise involves the usage of a median filter, such as adaptive median filter method (AMF). In this paper, we develop a new method using tight framelet to reduce this type of noise from color images and compare with AMF method.

Color image de-noising consists of scalar filtering and vector filtering methods. The algorithm developed in this work based on scalar filetering idea, which means the noisy color image

I

_noisy is divided into three samples firstly. And then the R, G and B channels are denoised independently. Finally, the three denoised channels are merged to form a denoised color image. We summarize the color image de-noising procedure in figure 2.

The tight framelet gray image de-noising method described in sectionis 2 (formula 9) is used to the R, G and B channels, and is named TF-based de-noising schemes. The initial guess of the ideal image

f

₀in formula 9 can be obtained using AMF method proposed in [7].

Fig.1 RGB color space;Fig.2 De-noising procedure

4. Experiments

In this section we demonstrate the results achieved by applying the color image de-noising algorithm on several test images. The objective of these experiments is to verify the ability of wiping off salt-and-pepper impulse noise from images and compare with AMF method. The test images include Baboon, Bird, Lena and PepperColor images with different image detail and edges, as shown in figure 3.

Experimental results on all four color images, including soft thresholding(TF-Soft) and hard thresholding(TF-Hard), are shown in Table 1, comparing with AMF method. Every result reported in Table 1 is an average over 5 experiments, having different realizations of the noise.

As we can see from Table 1, TF-based methods have a better performance than traditional AMF method. Averaging the results in this table and we can find that the PSNR value of AMF method is

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25.1833dB. For the methods of TF-Hard and TF-Soft, an average of 29.6459dB and 29.0214dB are abtained, giving about 4.5dB and 3.8dB advantage respectively. Fig.4 and Fig.5 shows the result of the proposed algorithm and AMF algorithm for the image "Baboon", at 60% and 90% noise respectively.

Fig.3 Test images Fig.4 Example of the de-noising Fig.5 Example of the de-noising results for the image 'Baboon' results for the image 'Baboon'

at 60% noise at 90% noise

5. Conclusions

In this paper we have studied the problem of color image denoising and described an effective method using tight framelet.

A tight framelet is a wavelet frame constructed by multiresolution analysis. The properties of tight framelet and its applications on gray image denoising have been well studied. The contribution of this work is to explore the power of tight framelet to de-noise color images. A noisy color image is divided into three samples and then the R, G and B channels are denoised independently. The three denoised channels are merged to form a denoised color image finally. Experimental results show that the new method has a better performance than AMF approach.

Acknowledgements

This project is supported by Guangdong Natural Science Foundation(S2011040004129) and the Youth Foundation of Shantou University, China(YR10003).

References

[1]D.L. Donoho, ‘De-noising by soft thresholding’, IEEE Transactions on Information Theory, Vol. 41, No.3, pp.613-627, 1995. [2]Jian-Feng Cai, Raymond H. Chan, Lixin Shen, and Zuowei Shen, ‘Convergence analysis of tight framelet approach for missing data recovery’, Adv. Comput. Math, pp.87-113, 2009.

[3]I. Daubechies, ‘Ten lectures on wavelets’, volume 61 of CBMS Conference Series in Applied Mathematics, SIAM, Philadelphia, 1992.

[4]A. Rou, and Z. Shen, ‘Affine system in L2(Rd): the analysis of the analysis operator,’ Journal of Functional Analysis, Vol.

148, pp. 408-447, 1997.

[5]G. Wyszecki, and W. S. Stiles, ‘Color science: concepts and methods, quantitative data and formulae,’ 2nd edition, Wiley-Interscience, August, 2000.

[6]R. C. Gonzalez, and R. E. Woods, ‘Digital image processing,’ Pearson Prenctice Hall, 2007.

[7]H. Hwang, and R. A. Haddad, ‘Adaptive median filters: new algorithm and results,’ IEEE