Choosing Interpolation RBF Function in Image Filtering with the Bidimentional Empirical Modal Decomposition

Faten Ben Arfia Abdelouahed Sabri

Computer Engineering System design Laboratory (CES) Laboratory of Electronics Signals, Systems and Computers National Engineering School of Sfax: Tunisia Faculty of Sciences, Dhar El mahraz FES. Morocc

benarfia_faten@yahoo.fr abdelouahed.sabri@gmail.com

Mohamed Ben Messaoud

Laboratory of Advanced Technologies of medicine and signals (UR-ATMS)

National Engineering School of Sfax: Tunisia M.benMessaoud@enis.rnu.tn

Abstract – The data interpolation is an essential part of Bidimensional Empirical Mode Decomposition (BEMD) of an image. In the decomposition process, local maxima and minima of the image are extracted at each iteration and then interpolated to form the upper and lower envelopes, respectively. Because of the properties of radial basis function (RBF) interpolators, they are good candidates for use in BEMD. However, only one or two of the RBF interpolators have been utilized for BEMD so far. This paper employs many RBF interpolators for BEMD, compares their performances, and finds out the useful ones for BEMD especially in the image filtering application. We propose to apply the BEMD approach with the adequate interpolation function in the image denoising domain. After that, we combine the BEMD with the DWT to improve the BEMD denoising method. The analysis is done using real images. Simulations are made to focus mainly on the effect of interpolation methods by providing less or negligible control on the other parameters of the BEMD process. The study is believed to work as a guideline in the area of BEMD based real image in the filtering application.

Index Terms- Bi-dimensional Empirical mode Decomposition, Intrinsic Mode Function, scattered data interpolation, Radial Basis Function, image filtering.

I. INTRODUCTION The Empirical Mode Decomposition (EMD) technique has

been developed to analyze time-frequency distribution of nonlinear and non-stationary signals. It is an adaptive decomposition which can decompose any stationary or non-stationary signal into its intrinsic mode functions (IMFs), providing well-defined instantaneous and/or local frequency information about a signal [1].

The Empirical Mode Decomposition used to analyze the two-dimensional signals is called Bi-dimensional Empirical Mode Decomposition (BEMD). BEMD approach is better than Fourier, wavelet and other decomposition algorithms in image decomposition[2]. The BEMD method has been successively used for texture analysis, edge detection, texture classification [3], segmentation, compression and content based medical image retrieval [4][5]. The performance of this

method depends on detection of extrema points and the interpolation of the scattered extrema points [6].

The interpolation is considered as a delicate stage of this method and which has a remarkable influence on the result of the decomposition. Here, we search to experimentally understand in the choice in the majority of research carried out until now (to improve the BEMD) of the Thin-Plate as interpolation function compared to the other existing functions.

In this paper, a real image has been composed used by the BEMD method. Then, many interpolation functions are used for further verification of the results. Various metrics are considered to evaluate the effectiveness or performance, or both, of each of the interpolation techniques in the case of the BEMD process.

The results may be used in the image filtering with the BEMD approach. The choice of the interpolation function influences the efficiency of the BEMD filtering approach.

This paper shows the influence of interpolation function in image filtering on the visual quality of denoised image in term of Peak Signal to Noise ratio (PSNR) and Mean Squared Error (MSE).

The rest of the paper is organized as follows. Section 2 gives a brief overview of the BEMD process. Section 3 outlines the interpolation techniques used in this study. Then in section 4, we describe the proposed noise reduction approach based in the BEMD combining with the DWT technique . Simulation results and some analysis and discussion of the interpolation function results are shown in Sec. 5. The section 6 shows the experimental results and the comparison of BEMD in image filtering with the method used in this application. Finally, concluding remarks are given in Sec. 7.

1st International Conference on Advanced Technologies for Signal and Image Processing - ATSIP'2014 March 17-19, 2014, Sousse, Tunisia MIA-151

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II. BIDIMENSIONAL EMPIRICAL MODALE DECOMPOSITION (BEMD)

The EMD method is an adaptive decomposition and suitable for analysis of non-linear and non-stationary processes. This decomposition technique extracts a finite number of oscillatory components or AM-FM functions (Amplitude Modulation, Frequency Modulation), called IMF (Intrinsic Mode Function), directly from the data. IMFs are monocomponent functions that have well defined instantaneous frequencies. The EMD does not use any pre-determined filter or basis functions, which is quite different from Gabor analysis and Wavelet analysis [6].

This technique extended to analyze two-dimensional data called the Bidimensional EMD (BEMD) [7]. This novel approach is a highly adaptive decomposition. The image characterization is based on its decomposition in intrinsic mode function (IMF). The image can be decomposed into a redundant set of signals called IMF and a residue. The reconstruction process consists of adding all the IMF’s and the residue. The original image is reconstructed without distortion or loss of information. An IMF is characterized by two specific properties [8]:

• The number of zero crossing and the number of extrema points is equal or differs only by one. • It has a zero local mean.

Given an image I, the sifting process of BEMD can be defined as follows [9]:

• Step 1) Fixed. ξ, i←1 • Step 2) r i-1← I (residue) • Step 3) Extract the ith IMF: a. hi,j← ri-1, j←1 (j, iteration of the loop of sifting). b. Extract local maxima and minima of hi,j-1 c. Compute upper envelope and lower envelope functions Ui,j-1 and Li,j-1 by interpolating, respectively, local minima and local maxima of hi,j-1 d. Compute local mean surface e. Update: hi,j ←hi,j-1-mi, j←j+1

2)( 1,1,

1,−−

−

+← jiji

ji

LUm (1)

f. Calculate stopping criterion: (Standard Deviation). g. Repeat steps (b)-(f) until SD(j) < ξ, let. IMFi ←hi,j (i eme IMF).

• Step 4) Update residue:

ri←ri-1 - IMFi • Step 5) Repeat steps (3) with i ←i +1 until the

number of extrema in ri is less than 2 i.e., the residue does not contain any extrema points.

The Standard Deviation (SD) is the criteria to stop sifting process, computed from two consecutive sifting (2).

∑= −

−−=

T

i ji

jiji

h

hhjSD

12

1,

1,,)( (2)

T: the number of iterations. When the decomposition is complete, the original image

may be reconstructed by adding all the IMFs and the last residue (3)

∑+

=

=1

1treconstruc

K

iiCI (3)

Where, Ci is the ith IMF or residue and k is the number of

IMFs.

Fig. 1 Bidimensional Empirical mode decomposition flow chart.

III. INTERPOLATION TECHNIQUES IN BEMD APPROACH

The surface interpolation from 2D scattered data (local maxima and minima points is a crucial and an important part of BEMD method [10]. The Scattered data interpolation refers to compute the suitable surface model that approximates arbitrarily distributed discrete data samples. The process of reconstructing smooth surfaces from discrete data can be achieved either by interpolation or by approximation [11].

In order to extract the various IMFs in the decomposition process, the interpolation is a one of the problems for the estimation of surfaces formed by the extrema [12]. In fact, the construction of two maximum and minimum envelopes is performed by interpolating the respective maxima and minima [13].

The dispersion of these extrema drives a problem of choice of the interpolation function. To do so, we are conducted to study and compare different interpolation functions and especially Radial Basis Functions (RBF).

The RBF based interpolation methods are examples of global interpolation methods for scattered data points. RBF methods impose fewer restrictions on the geometry of the interpolation centers and are suited to problems

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where the interpolation centers do not form a regular grid as in the case of local maxima or minima maps of images or textures. In addition, RBF methods are some of the most elegant schemes from a mathematic point of view. RBFs are well known to provide powerful tools for high-fidelity reconstruction of surfaces from a selected set of sparse and irregular samples [14].

There are a multitude of functions that can be used as a basic function for radial basis functions. In this context, we propose to examine the best basic function for RBFs through a comparative study.

In the sifting process and after extraction the extrema, one is faced with a serious problem of interpolating the extrema points indeed to create the envelopes. Contrary to the signals, the points to be interpolated are not disposed on a regular grid, but they are called dispersed scattered data [14]. This form requires a more elaborate and suited method to this type of problem. In general, the following radial basis functions are used [12]:

• xx =)(ϕ Linear function

• 1)( 2 += xxϕ Multi- Quadrique function

• xex −=)(ϕ Gaussian function

• )log()( xxx =ϕ Logarithmic function

• )log()( 2 xxx =ϕ Thin-plate spline function

To do the right choice of the RBF function in the interpolation phase, we applied all these functions RBF to a gray level image “Lena.bmp”. The decomposition of this image is presented in the figure 2.

(a) original Image

(b) IMF1

(c) residue1

(d) IMF2

(e) residue 2

(f) IMF3

(g) residue 3

(h) Reconstructed image

Fig. 2 The decomposition of the lena.bmp in three modes with the BEMD

approach

IV. BEMD APPROCH IN IMAGE FILTERING APPLICATION For a Gaussian noisy image, BEMD approach is very

effective for decomposition and denoising images, but during decomposition can be see that the first IMF and probably the second can contains the largest amount of noise.

The wavelet transform is applied to filter the first and second IMF to improve the visual quality of the denoised image in term of the PSNR criterion. Several research projects have used the BEMD as filtering approach but it does not give good results in terms of visual quality. In this approach, they assumed that the noise is low amplitude that the extrema points.

Therefore, the first IMF contains almost all the noise. So, to reduce the noise in a noisy image, they just reject all values lower than a specified threshold in the extrema points of the first IMF. They leave only the extrema points which are considered of significant points [12].

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Our approach consists in using the Wavelet Transform for filtering the first and the second IMF because the first IMF does not contain the amount of noise. In fact, this noise can also exist in the second IMF. This approach can improve visual quality and better performances than the traditional methods of image denoising. The description of the proposed approach is illustrated in the figure 3.

Fig. 3. Denoising image approach based in BEMD combined with DWT.

V. THE CHOICE OF INTERPOLATION FUNCTION IN

BEMD APPROACH

A. Experimental Result

The interpolation phase is the most important phase in the decomposition in the approach BEMD. So to improve the efficiency of this approach we propose the best interpolation technique among the RBF techniques used in the BEMD approach.

Scattered data interpolation is used to interpolate the maxima and the minima points to produce 2D continuous surfaces, called the upper and the lower envelopes, respectively. The average of these two envelopes results in the mean envelope, which indicates the local mean of the data. Since the maxima and the minima maps change in each iteration, the corresponding interpolated surfaces should also change at each iteration.

In this section we will present a comparative study in the objective to choose the best basis function interpolation method for the radial basis function used in BEMD approach.

In this study we will show the results of the decomposition of the image “lena.bmp” by the BEMD approach with the use of existing techniques, indicating in every step the number of maxima and minima of interpolation functions.

Figure 4 and figure 5 shows the number of maxima and minima respectively, with different technique of RBF interpolation.

Fig. 4 Number of minima for different IMFs in the BEMD decomposition approach

Fig. 5 Number of maxima for different IMFs in the BEMD decomposition approach

B. Discussion

According to the existing curves in figure 4 and 5 we note that the use of a Gaussian function as interpolation function causes strong amplifications of extrema and the number of extrema is not reduced and remains more or less constant during decomposition. This leads to decomposition in several IMFs and consequently execution time during the decomposition big enough and the reconstruction is perfect independent from the interpolation function. We notice that using a Gaussian function as the interpolation function causes strong amplifications extrema and the number of extrema is not reduced and remains roughly constant during the decomposition. This leads to decomposition in several IMFs and consequently execution time during the decomposition big

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enough and the reconstruction of the image loses good visual quality.

The curves also show the interest of the use of the logarithmic function as the interpolation RBF function. This function shows the speed of the decomposition and which does not result in amplification of the extrema values.

The application of the linear and the Multi-Quadrique function during interpolation, we noticed that the decomposition is fast seen many MFIs extracted which leads to a rapid passage of high frequency to low frequency. At residue, the number of maxima and minima is not the same that will influence the stopping criterion is to say, will be controlled by the minima or maxima.

In the case of the decomposition using the Thin-plate as a basic function, we notice during implementation, the characteristics of the IMF and the stop criterion of the screening process are respected. As the decomposition is faster and with a residue which contains more extrema.

From this comparative study between the different interpolation functions, we used with the Thin-plate function in the image filtering because of its speed and best results.

VI. BEMD IMAGE FILTERING WITH THIN PLATE INTERPOLATION FUNCTION

In this section, we applied the BEMD approach with the DWT technique for image denoising. In this application we used the thin plate function interpolation already chosen. The proposed image approach has been applied on several natural grayscale images (512×512) that are contaminated by additive Gaussian noise with different noise levels σ = 5, 10, 15, 20 and 30.

The figure 6 shows the performance of our approach based in BEMD with DWT compared with the DWT and the median filter like methods of denoising.

In comparison with traditional methods, table 1 and figure 6 illustrate the performance of our approach based in BEMD with DWT compared with the DWT and the median filter as nonlinear methods of denoising.

TABLE I. COMPARISON RESULTS FOR DENOISING METHODS IN TERM OF PSNR (DB)

Fig. 6. PSNR performance of the proposed method compared with the DWT and the Median filter

As illustration, the proposed approach gives better results to

the levels of visual quality compared with the BEMD with the histogram approach (figure 7).

σ noisy Median DWT BEMD

5 34,16 34,02 35,04 37,70

10 28,12 29,55 30,80 32,97

15 24,58 25,78 28,40 30,30

20 22,09 24,03 26,01 28,50

30 18,57 20,37 23,03 25,90

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Fig. 7. (a) The noisy image with Gaussian noise σ=20, (b) denoised image

with BEMD with DWT (2 IMF filtred), (c) BEMD with histogram.

VII. CONCLUSION

The BEMD is adaptive image decomposition. The obtained IMFs can be seen as results of filter bank. The results of a simple use of the BEMD for image processing prompted us to improve it to have a very powerful tool in this field.

In the decomposition process, local maxima and minima of the image are extracted at each iteration and then interpolated to form the upper and the lower envelopes, respectively. The number of two-dimensional intrinsic mode functions resulting from the decomposition and their properties are highly dependent on the method of interpolation. Though a few methods of interpolation have been tested and applied to the BEMD process.

In this paper, comprehensive experiments on BEMD have been made using RBF based interpolation methods. The purpose of the analyses is to find the effect of interpolation methods on the performance of BEMD and to determine some preferable methods for the BEMD process. In the analysis, the emphasis is mainly given on the interpolation methods to govern the decomposition by providing less control over the other parameters. It has been observed that thin-plate function interpolation interpolators provide better results than the others, which is clearly supported by the simulation using real grayscale images in this paper.

The choice of the interpolation function improves the efficiency of the BEMD approach in the field image filtering.

The proposed approach has been tested on several natural grayscale images (512×512) that are contaminated by additive Gaussian noise with different noise levels. The results presented in comparison with DWT show the efficiency of our proposed approach which can improve effectively image visual quality and performs better than traditional methods of image denoising.

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