Denoising by Averaging Reconstructed Images: Using Singularity Function Analysis

Masumeh Shafiee

Islamic Azad University – Central Tehran Branch

Tehran, Iran masumehshafiee@yahoo.com

Mohammad Reza Karami Faculty of Electrical & Computer Engineering Babol University of Technology, P.O.Box 484 .IRAN

mkarami@nit.ac.ir

Kaveh Kangarloo Islamic Azad University – Central

Tehran Branch Tehran, Iran

k_kangarloo@yahoo.com

Abstract—A newfound method of denoising that based on Averaging Reconstructed Image (AVREC), is used. The approach was proposed on signals, approximately about last decade, Since 2004. In definition (procedure), first of all, we divide the spectrum of noisy image into several images that can be then, reconstructed with 2-D Singularity Function Analysis (SFA) model. Among this mathematical model, each matrix or a discrete set of data, represents as a weighted sum of singularity functions. In image denoising field, this technique, rebuilt all lost high frequencies parameters that are essential. Illustrate each new image, as a sum of noise-free image and the small noise. So on, we can then, denoise image by averaging reconstructed ones. Both theoretical and experimental results on standard gray-scale images, confirm the advantages (benefits) of this approach as an applicable method of denoising.

Keywords— Image Processing; denoising; reconstruction; SFA(Singularity Function Analysis)

I. INTRODUCTION In face with low SNR images (we’re dealing with, white

Gaussian noise of zero mean), the importance of denoising is much more than the other noisy cases. Many researchers , try to solve this problem and improve the lost quality of picture. Lot of investigations were done, to gain high quality noise-free image. Although, we work on frequency domain, a very simple and efficient way of completing the non defined data is Zero Filling Images (ZFI) Such an operation did not create any new information, but simply cover image pixels. Some other methods such as sparse representation [5], and Hilbert-Huang transform [6] were effective, but in most of them, some high frequency components were lost. So we focus on this problem and try a new method of denoising to solve that. For achieving this goal, we use 2-D Singularity Function Analysis (SFA).

1-D SFA is a mathematical technique. For the first time, Jianhua Luo, Yuemin Zhu (2004) defined that method, to reconstructing MR Images. [1] The method handle frequential content through asymmetric truncation of k-space, so maximum data can be obtained from a limited quantity of information. As a result, data gathering time clearly decrease while achieving a significantly improved image quality. After

that, in 2005, they Ft. Patrick Clarysse, and Isabelle Magnin, tested it, as an application of SFA, to correct of Bias Field in MR Images. [2] One of it’s advantage is simplicity. It does’nt nead any initialization, or a prior models, nor image preprocessing , such as brain masking or pre-segmentation. It is fully automatic and does not require any user interaction. After, in 2009 and 2012 they expanded and developed this it into two dimension, in usage of denoising of MR images. [3][4]

And now, in this paper, we inquire that if is it effective in noisy standard gray-scale images (such as Barbara, lena ,etc) or not. We prove this way to obtain the highest SNR. At first, divide the spectrum of the image into different parts. Each image then reconstructed using a 2-D SFA model (2-D singular function is shown in figure 1). By assuming each of the thus images as the sum of the same noise-free image and a different smaller noise, the denoising is get through averaging reconstructed images (AVREC). We improve the results in 2 steps. First of all: change the size of filter’s window to gain the best results. In second step defined a best window length, for finding singular points. The results show that in compare with ZFI, median and wavelet[7] we have higher SNR and lower error in small noise SNR.

II. THEORY OF SINGULARITY FUNCTION ANALYSIS

It is acceptable that under specific conditions, a continuous function can be expressed as a linear functional of trigonometric function series, wavelet function series, or power function series. In addition, a discrete function can be represented as a linear functional of some discrete function series. Let )( jf assign as a discrete signal. A general interpretation of it, with discrete basis function series )( jBk

is as bellow:

(1)

1

01,...,1,0),()( N

k kk NjjBajf

The main focus, is on de definition of )( jBk. Because we

can analyze, filtering, estimation, reconstruction, compression,

2013 8th Iranian Conference on Machine Vision and Image Processing (MVIP)

978-1-4673-6184-2/13/$31.00 ©2013 IEEE 279

denoising and visualization of )( jf , just when, we know what is )( jBk

.

For example, if it is express as complex exponentials, (1) turns to the well-known Fourier series model that is famous and common used in signal or image processing. Different

)( jBkmay lead to different ways. In the present study, we

choose “singularity functions”, to explain )( jf in this definition.

Explanation of Singular Point (SP): Location k integer is called a SP of discrete signal )( jf , j integer , if in j=k we have 0)1()( jfjf

Definition of Singularity Function Space (SFS): Discrete function space is called SFS, where is the singularity function (SF) if it proves the next three conditions:

1) )( jWk is k shift function of kernel function )(0 jW which

has only one singular point at location 0;

2) )( jWk has only one singular point at location k ;

3) uk, integer , there exists )]([)(ˆ jWDFTjW kk , where DFT[.] is Discrete Fourier transform operator, )(ˆ uWk

the singularity spectrum function (SSF), and ukuWk ,),(ˆ integer the singularity spectrum function space (SSFS).

Theorem 1: SFS is self-contained, i.e., any discrete function can be shown as linear functional of singularity functions in SFS.

Theorem 2: Note that, )]([)(ˆ jfDFTuf . )(ˆ uf can be expressed as a linear functional of { ukuWk ,),(ˆ integer } in SSFS.

Conclusion: If a discrete signal has only Q singular points, it can be defined as the weighted sum of Q singularity functions in SFS and its spectrum function can be explain by the weighted sum of Q singularity spectrum functions in SSFS. That is

1

01,...,1,0,0),()( Q

k kkk NjajWajf (2)

1

0)(ˆ)( Q

k kk uWajf (3)

Where )( jWk is the singularity function having its

singular point at k , and ka is the weighting coefficient.

Let me give you an example: Imagine a discrete function like {2,2,2,4,4,5,5,8,8,8,0}. As you see, we have 11 components. We have 4 SP in points k=3,5,7,10 and the weights are

ka =2,1,3,-8 . Now, we can show that series in new form:

)(8)(3)()(32)( 1075 jWjWjWjWjf , j=0,1,…,10

Now, you know the main idea. It is the time that, we can explain 2-D SFA (figure 1). It demonstrates as this:

),(),( iiii yyxxyyxxw (4)

When

i

ii xx

xxxx

,0,1

)( (5)

And

i

ii yy

yyyyu

p,0,1

)( (6)

Fig. 1: Demonstrate of a 2-D singular function at singular point (64, 32).

So (2) becomes to:

),(),(1

ii

Q

ii yyxxwayxf

(7)

SPs are: ),(),...,,(),,( 2211 QQ yxyxyx that their weights or singular degree are: kaaa ,...,, 21

By Discrete Fourier transform of last equation, we will have:

QiyyxxwDFTkkW iiyxi ,...,2,1)],(),([),( (8)

And (7) turn to:

Q

i yxiiyx kkWakkf1

),(),( (9)

Equation (7)&(9) defined 2-D SFA. Consider ),( yxg as an observed image. We can assume that:

1,..,1,0,),,(),(),( Myxyxnyxfyxg (10)

Where ),( yxf is noise-free image and ),( yxn means noise. Notice that images are MM . We transform (10) to spectral domain. So

1,...,1,0,),,(),(),( MkkkkNkkFkkG yxyxyxyk (11)

We take different part of ),( yk kkG as Ω then reconstruct full image from that part with 2-D singular function analysis. In

280

each section the rest of spectrum is called . In other word , indicator function of Ω and . is expresses as :

),(,0

),(,1)(

yx

yx

yx kk

kkkkI (12)

and

),(,0

),(,1)(

yx

yx

yxkk

kkkkI (13)

So

1),(),( yxyx kkIkkI (14)

And observed image can be defined as:

)],(),(),(),([

)],([),(

yxyxyxyx

yx

kkGkkIkkGkkIIDFT

kkGIDFTyxg

(15)

Now for finding SPs we fined diffrence image in y-direction. The points that it’s value is greater than a given threshold, is SP. The greater threshold may lead to false negatives of singular points and the smaller one has more risk of false positives in singular points. In this moment, for reconstructing and denoising, we can replace with

Q

i yxii kkWa1

),( so (15) changes to:

)],(),([

)],(),([),(

yxyx

yxyxr

kkGkkIIDFT

kkGkkIIDFTyxg (16)

That ),( yxgr is a reconstructed image.

)],(),([ yxyx kkGkkIIDFT represents the image denoised by the filter ),( yx kkI and the rest of it approximates the components

of ),( yxf corresponding to the spectral data in . It is to be noted that the second term represents the image components recovered by the 2-D SFA, which have been lost due to the spectral truncation by the filter ),( yx kkI

Q

i

Q

iyxiiyxyxiniyx

Q

iifi

yx

Q

iinifi

Q

i yxii

kkWakkFkkWakkWa

kkWaakkWa

1 11

11

),(),(),(),(

),()(),( (17)

Relation (17) shows that, due to the noise, the recovered spectral data ),( yx kkF is not exactly equal to the noise-free spectral data to which an error spectral data term

Q

i yxii kkWa1

),( is added that represents spectral data of the noise at the Q singular points of ),( yxf .

Put (17) to (16) and denoting (11), we have:

Q

iiini

Q

iyxiniyxyxyx

Q

iyxIniyxyx

yxyxyx

yx

Q

iiiyxyxyx

r

yyxxwayxiyxnyxiyxf

kkWakkIkkIkkFIDFT

kkWakkFkkI

kkNkkFkkIIDFT

kkWakkIkkGkkIIDFT

yxg

1

1

1

1

),(*),(),(*),(),(

)],(),(),(),([

))],(),()(,(

)),(),()(,([

)],(),()],(),([

),(

(18)

Q

iyxiniyx kkWakkI

1

)],(),( is so smaller than the other part. So we

can pass the third part up. Finally we can said:

),(*),(),(),( yxnyxiyxfyxgr (19)

Attention that, in choosing Ωs we should be carefull about two main conceptions:

1. low-frequency components have higher SNR and play an important role in the accurate reconstruction, so , they should be kept. If we lost them and did’nt mention to this essential factor in determining of parts, we will face with great reconstruction errors.

2. The union of different partial domains covers the complete spectral domain even if they are overlapped in some areas.

After all, according to these terms, we designate four partial area, Ω1, Ω2 , Ω3 and Ω, that are shown in figure 2.

Fig 2 : AVREQ denoising algorithm

281

As you seen, first of all,we transfoem noisy image to k-space(frequency domain), then select diffrent parts of that. At the third step reconstruct each picture with 2-D SFA. Finally , average four new images to have noise free image.

III. EXPERIMENTS AND RESULTS

As we said before, in first step we lose some frequencies with a filter, then try to regain them with SFA model. Too find Singular Points (SPs) we can use each rows or column of image, or both of them. The results show that, we obtained better SNR by 2-D scope. But it takes much more time than 1-D just for about 0.1dB difference in SNR. So in this paper as [3],[4] we decide to find SPs, Just in y direction because in most of our images the finally SNR after denoising is better and most of the pointes in x direction, were in y direction too.

To have the best results, change our Ωs length. As we see in figure 3, we have maximum SNR in about 0.53n (when image is n*n). We choose 0.535n, as a length of fourier’s window.

Fig 3 : Effect of changing the filter’s window length

Now it is a time for deciding a way to determining SPs. At first we should have chosen a threshold. We consider 0.1 of maximum difference in each column. To improve the results we explain another theory. In this way we divide a picture into smaller ones, with windows from 44 to 128128 pixels. Finally, the results show that, considering window of

3232 pixels is a good idea.

Fig 4 : Effect of changing the window for finding SPs

Note that, to verifying the threshold coefficient, we sweep the results in two ranges. 1: from 0.01 to 0.1 2: from 0.01 to 0.5. Comparing the new SNRs, shows that, the best answer for best SNR in according to time saving, is 0.1.

We compare our way with two other common famous denoing methods. One of them is 33 median filter, and the other one is wavelet transform. The wavelet denoising consists of first using the “ddencmp” function for calculating default values (the threshold THR, the soft or hard thresholding SORH, the approximation coefficients thresholding KEEPAPP) for denoising, and then, employing the “wdencmp” function for obtaining the denoised image. The best results are obtained with the following settings: “sym8” for wavelet function, and the values of THR4.0 , s, and 1 for THR, SORH, and KEEPAPP, respectively.

You can see the outcoming SNR, based on SNR of applied noise, in following tables. Denoised images and their errors are shown above them, too.

Fig (a)Main image,(b)Noisy image (zero mean white Gaussian noise with SNR5) (c)AVREQ with SFA(d)AVREQ with ZFI (e)Median (f)Wavelet(g) AVREQ with ZFI error(h)median error(i)wavelet error (j)AVREC with SFA error

TABLE1: Denoised image SNR(AVEREQ, SFA analysis for whole image)

Image peppers fingerprint House Boat Lena Barbara 5 dB 18.96 16.69 19.74 19.60 20.34 19.11

10 dB 21.05 17.72 22.56 22.26 23.30 21.19 15 dB 21.90 18.27 24.00 23.48 25.17 22.12

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TABLE 2: Denoised image SNR (AVEREQ, SFA analysis with window)

Image peppers fingerprint House Boat Lena Barbara 5 dB 19.07 16.84 19.89 19.71 20.39 19.18 10 dB 21.21 18.00 22.74 22.40 23.42 21.26 15 dB 22.35 18.48 24.23 23.79 25.32 22.25

TABLE3: Denoised image SNR (AVEREQ, ZFI)

Image peppers fingerprint House Boat Lena Barbara 5 dB 15.1362 14.0474 15.2563 15.2773 15.4723 15.3297 10 dB 18.8479 16.9102 19.4537 19.4287 19.6484 19.0958 15 dB 21.6638 18.5175 22.8370 22.7589 23.3034 21.9808

TABLE 3: Denoised image SNR, obtained with “wavelet” method

Image peppers fingerprint House Boat Lena Barbara 5 dB 16.25 16.36 16.49 16.53 16.79 16.22

10 dB 20.14 20.18 21.30 20.63 21.26 19.81 15 dB 22.86 22.81 24.83 23.56 25.11 22.12

TABLE: Denoised image SNR, obtained with “median” method

Image peppers fingerprint House Boat Lena Barbara 5 dB 15.14 14.05 15.26 15.28 15.47 15.33 10 dB 18.85 16.91 19.45 19.43 19.65 19.82 15 dB 21.66 18.52 22.84 22.76 23.30 21.98

IV. CONCLUSION

As you see in above tables, in lower SNRs, AVREC method is more applicable than others. The reson is hidden in the base of our idea. We fined points with highest jump of pixel value. And suppose that are noisy point.Now, imagine a picture with higher SNR. In these ones we have more false negative in finding SPs.

The efficiency of this method, depends on different factors. The main one is filtering window or in other words Ωs. As we

all know, most of real image’s frequency domains are asymmetric, so we should pick out the areas with some overlap. But not so big that few spectral data of the initial noisy image are removed. The SNR decrease when the filter length is increase.(Fig.3) Becouse the lager domain may lead to a larger chage in frequency band width and this has a negative influence in output SNR.

ACKNOWLEDGMENT

The author would like to thanks M.Sc. F.Najafi, M.Sc. F.Geraiili and, especially, M.Sc. M.Salehi for their unaffected helps.

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