Seismic Denoising Through Improved Algorithm Curvelet Transform

Seismic Denoising Through Improved Algorithm

Curvelet Transform

RENU AWASTHI

Department of Electronics & Communication Engineering Galgotias College of Engineering & Technology

Greater Noida,UP,201301,India renu.dit@gmail.com

Abstract: This paper addresses the problem of denoising the seismic data through improved curvelet transform. The curvelet transform is a multi-scale directional transform developed as an improvement over wavelet transform.To reduce the noise or to improve the quality of seismic data we have used two parameters i.e. quantitative and qualitative. For quantity we will compare peak signal to noise ratio (PSNR).Higher the PSNR better the quality of the seismic data. Forqualitywecomparevisualeffectofdata. In this paper we proposed a Curvelet Transformation based seismic data denoising, which is combined with wavelet transform and gabor filter in curvelet domain. The simulation results show that the improved curvelet transform can achieve an almost complete data reconstruction and give comparatively high PSNR which cannot be achieved by conventional methods of denoising.

Keywords: Curvelet transform, Wavelet transform, Gabor filter, Directional.

1.INTRODUCTION

The goal of exploration is to find oil and gas reservoirs by seismically imaging the earth’s reflectivity distribution. Seismic records obtained are wave front components which contain many damageable directional information about important geological substances [3].Therefore, most geophysical work involves separation of the desired information from the information that is not of interest, that is, the noise. The aim of noise filtering is to eliminate noise and its effect from the original image, while corrupting the image as little as possible. Many methods have been developed to denoise the seismic data contaminated with random noise. Some of these methods discriminate between signal and noise based on their frequency content, while, some uses technique of predictive filtering. The most recent advancement in this field is a multi scale directional transfom, called curvelet transform, proposed, in order to overcome the drawbacks of conventional transforms like wavelet transform.

In this paper we presented a new method of seismic data denoising, which is an improved method over curvelet transform and other conventional techniques.

1. Theory of curvelet transform denoising

A.Wavelet transform

Since 1990s, wavelet transform has been well used in image processing, speech recognition and seismic data denoising due to its nice localization characteristic. Wavelet do a good job in approximating signals with sharp spikes or signals having discontinuities..For function x(t), wavelet transform is defined as follow in eq. (1).

, =

√ .

∗

₍

₎

where ,isthe mother wavelet or the basis function, s is scaling (dilation or compression) function and τ _isthe

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Seismic data do not simply consist of smooth scan lines rather they consist of discontinuous curves which are located along the smooth contours. A contour let is a flexible multi resolution, local and directional image representation using contour segments. The contour lets have elongated supports at various scales, directions and aspect ratios. The above properties allow contour let to efficiently represent an image with smooth curves at multiple resolutions [15].

The contour let transform is obtained by first applying multi scale transform like wavelet for edge detection which is then followed by a local directional transform for contour segment detection.Unfortunately, contour let functions have less clear directional geometry leading to artifacts in denoising and compression.

C.Ridgelet transform

The theory of Ridgelet is developed by Emmanuel Candès in his PhD thesis in 1998. Candès showed that one could develop a system of analysis based on ridge functions Forthe2Dfunctionf(x_1,x_{2),Ridge let transform is defined as}

follows:

, , ∬ , , , ! , " " (2)

In equation (2) _{, ,} is a ridgelet wavelet and is defined as

, , , # cos ' sin -b)/a) (3)

From eq .(2) & (3),it is clear that ridgelet transform has directional property.

D.Curvelet transform

Ridgetet transform details the 2D image with a line. Butin most images the reare many curves. So we must divide image in topieces. To get a better result we de compose image into sub-band, and divide these images with different scalesintopieces. This kind of multi-level and multi scale ridge let transform is called curvelet transform [15].The m a i n s t a ge s of curvelet transform, are decomposition and image re construction. The image decomposition includes sub band decomposition, smooth partitioning, renormalization and ridge let analysis, shown in fig 2.The image reconstruction includes ridge let composition, smooth integration, renormalization and sub band reconstruction, shown in fig 3.

Fig 1.Image decomposition

Fig 2. Image reconstruction Sub band

decomposition

Smooth

partitioning Renormalization

Ridgelet analysis

Ridgelet composition

Smooth

integration Renormalization

I.Image decomposition

i)Sub band decomposition:Here the object is filtered intodifferent subbands,the formula is given as:

! → +,!, ∆ !, ∆ !, … . (4)

Here the seismic image is divided into different resolution layers.Each layer contains the details of different frequencies. Here P0denotes the low-pass filter and ∆₁, ∆₂ represents band- pass (high-pass) filters. The original image can be reconstructed from the sub-bands:

! = +, +,! + ∑ ∆ ∆ ! (5)

ii)Smooth- Partitioning: Now each sub-band image is divided using dyadic square of size 2-s x 2-s . Let Qs denote all

the dyadic squares of the grid and w be a smooth windowing function. For each square wQ is a displacement of w,

localized near Q. we multiply ∆_sf withwQ which produces a smooth dissection of the function into ‘squares’ i.e

ℎ1= 21.∆ ! (6)

The original image can then be reconstructed by smooth integration i.e.

∆ ! = ∑1∈1421. ℎ1 (7)

iii)Renormalization: Each resulting square is renormalized to unit square [0,1] x [0,1] i.e.

g

=

T

h

defined as

( )

T

f

(

x

,

x

)

2

f

(

2

x

k

,

2

x

k

)

s s

Q

=

−

−

iv) Ridgelet analysis:Each Square is analyzed via the discrete ridgelet transform which is explained in.Theridgelettransform may be achieved by computing the following steps [23]:

• Compute the 2D FFT of the image.

• Substitute the sampled values of the Fourier transform obtained on the square lattice with sampled values on a polar lattice.

• Compute the 1D inverse FFT on each angular line.

• Perform the 1D scalar wavelet transform on the resulting angular lines in order to obtain the ridgelet coefficients.

II.Image reconstruction

i)RidgeletSynthesis:

51= ∑ 689 ,7. :7 (9)

ii) Renormalization :

ℎ1= ;151 (10)

iii) Smooth Integration:

∆ ! = ∑1=1><1. ℎ1 (11)

iv) Sub band reconstruction:

! = +, +,! + ∑ ∆ ∆ ! (12)

2. Log gabor filter

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ridge structure so fi mages [Lajevardi and Hussain ,2009].

The log gabor filter expression in the log-polar Fourier domain is as shown:

? :, , @, A exp E1/2 : E :I/J8 K @ E1/2 E I,L/J8 (13)

In which (ρ,θ) both are log-polar coordinates and σρ and σθ are the angular band width and radial band width respectively (common for all the filters). The pair(ρ_k and θ_k,p) corresponds to the frequency center of the filters, where the variables and k represent the orientation and scale selection, respectively. In addition, the scheme is completed by a Gaussian low-pass filter (ρ,θ,p,k)(approximation).

II. PROPOSED ALGORITHM

A.Methodology used

The methodology used for seismic data denoising is first to apply wavelet transform on the noisy seismic data and then to introduce a log gabor filter inside the curvelet domain. The technique used will give good psnr and also the best visual quality. Through different simulation we found that our improved method exhibits better performance in both psnr (peak signal to noise Ratio) and visual effect.

B.Proposed algorithm steps

I) Take noisy seismic data Ii) Apply wavelet transform

Iii) Then apply the algorithm given below

• Take any value of N

• For i=1 to N

• Restored data=Curvelet transform(Noisy data)

• Take inverse curvelet transform of restored data

• If (i mod 3)=0

• Restored data=Wavelet transform(Restored data)

Iv) Applycurvelet transform as under: I) Sub band decomposition

• First divide the image into resolution layers

• Each layer contains details of different frequencies, which are attenuated with the help of log gabor filter. Ii) in smooth paritioning, we will dissect the layer into small portions as explainedin Eq. (6).

Iii) Renormalization is explained in Eq. (8).

Iv) ridge let analysis is taking place, which is explained by Eq (2). V) Inverse of curvelet transform is performed in the end.

C.Block diagram for proposed method

Fig 3.Block diagram of proposed method.

Input noisy seismic data

Apply wavelet transform

Apply the proposed algorithm

The aim of experiments is to make rules to estimate the performance commonly used measurement for co in the PSNR. The seismic data is s methods and the proposed method.

Fig 6.Denoised d

III.EXPERIMENTAL RESULTS

e comparision among the different multi-resolution me e of the methods used. A quality metric known as P comparision purpose. The method proposed in the paper s simulated by adding the random noise, then it is den

Fig 4.Original seismic datafig 5.Noisy seismic data

data from wavelet transformfig 7.Denoised data from contourlettrans

methods. Here we used some s PSNR is one of the most per show a noticeable change enoised by the conventional

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Fig 10.Comp

In this paper we presented a new i different denoising techiques and co analysis of the denosied seismic dat the traditional and multi-resolution quality also.

f Electronics and Computer Science Engine

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d data from curvelet transformfig 9. Denoised data from proposed me

mparision of PSNR values of denoised data from different transforms

IV. CONCLUSIONS

improved method for seismic data denoising. The aim compare them with the improved method. We did a q

ata. Simultation results proved that the improved metho on methods. The denoised data obtained from method

gineering

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method

ms.

Table 1.Comparision of PSNR value of Curvelet transform and Proposed method

S.no Variance PSNR for Curvelet

transform

PSNR for Proposed method

1

10

46.7690

47.7304

2

15

45.5897

47.3039

3

20

44.1829

46.8489

4

25

42.8949

46.2925

5

30

41.5924

45.1906

6

35

40.3172

44.9847

7

40

39.7264

44.2865

8

45

38.2331

43.0716

9

50

37.9230

42.8017

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