BIDIMENSIONAL EMPIRICAL MODE DECOMPOSITION USING VARIOUS INTERPOLATION TECHNIQUES

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1 Advances in Adaptive Data Analysis Vol. 1, No. 2 (2009) c World Scientific Publishing Company BIDIMENSIONAL EMPIRICAL MODE DECOMPOSITION USING VARIOUS INTERPOLATION TECHNIQUES SHARIF M. A. BHUIYAN, NII O. ATTOH-OKINE, KENNETH E. BARNER, ALBERT Y. AYENU-PRAH and REZA R. ADHAMI, Department of Electrical and Computer Engineering University of Alabama in Huntsville Huntsville, AL 35899, USA, Department of Civil and Environmental Engineering University of Delaware Newark, DE 19716, USA Department of Electrical and Computer Engineering University of Delaware Newark, DE 19716, USA bhuiyas@ece.uah.edu okine@ce.udel.edu barner@ece.udel.edu albert@udel.edu adhami@ece.uah.edu Scattered 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 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/or applied to the BEMD process, many others remain to be tested. This paper evaluates the performance of some of the widely used surface interpolation techniques to identify one or more good choices of such methods for envelope estimation in BEMD. The interpolation techniques studied in this paper include various radial basis function interpolators and Delaunay triangulation based interpolators. The analysis is done first using a synthetic texture image and then using two different real texture images. Simulations are made to focus mainly on the effect of interpolation methods by providing less or negligible control on the other parameters or factors of the BEMD process. Keywords: Bidimensional empirical mode decomposition; intrinsic mode function; scattered data interpolation; radial basis function; Delaunay triangulation; nonlinear and non-stationary data analysis. 1. Introduction The empirical mode decomposition (EMD) technique has been developed with a view to analyze time-frequency distribution of nonlinear and nonstationary data. 1,2 309

2 310 S. M. A. Bhuiyan et al. It is an adaptive decomposition with which any complicated signal can be decomposed into its intrinsic mode functions (IMFs), providing well-defined instantaneous and/or local frequency information about a signal. This decomposition technique has also been extended for analyzing images or 2D data, which is known as bidimensional EMD (BEMD), image EMD (IEMD) 2D EMD, etc BEMD is a promising image processing algorithm that can be applied in various real-world problems, e.g., medical image analysis, pattern analysis, and texture analysis, Both EMD and BEMD require finding local maxima and minima points and subsequent interpolation of those points at each iteration of the process. One dimensional (1D) extrema points are obtained using either a sliding window or a local derivative; 2D extrema points are obtained using a sliding window or various morphologic operations. 1 5 Cubic spline interpolation is preferred for 1D interpolation, while thin-plate spline, B-spline, Delaunay triangulation with cubic interpolation, finiteelement method, etc., have been used as 2D scattered data interpolation for envelope surface estimation in BEMD, 1 7 where Delaunay triangulation and finite-element method based surface estimation result in relatively faster decomposition than the others. Beside true 2D implementation of the BEMD process, 1D implementation of the BEMD process is also studied in the literature, where each row and/or each column of the 2D data is actually processed by 1D EMD, which makes it a faster process However, it has been found that this 1D implementation results in poorer IMF components compared to the 2D implementation due to the fact that the former ignores the correlation among the rows and/or columns of a 2D image. 11 BEMD decomposition and the resulting 2D IMFs are governed by the method of extrema detection, criteria for stopping the iterations for each IMF and interpolation techniques. Though all of these factors are important for successful decomposition, the interpolation method may be considered the most crucial. Most of the scattered data interpolation techniques to produce 2D surfaces are themselves iterative processes. In the case of BEMD, it is very likely that the maxima or minima map does not contain any interpolation centers at the boundary region, which may be more severe for the later modes of decomposition. Hence, some kind of boundary processing to introduce additional interpolation centers at the boundary may also be required for successful decomposition. 11,12 Interpolation of the local maxima points is needed to form the upper envelope, and interpolation of the local minima points is needed to form the lower envelope of the data/image. The average of the upper and the lower envelopes of the image gives the mean envelope. One of the purposes of the BEMD decomposition is to get 2D IMFs with zero local mean defined by the mean envelope, which further plays a significant role for orthogonal decomposition. Hence, the accuracy of the envelopes in terms of shape and smoothness is very important, which calls for the need to identify an appropriate 2D scattered data interpolation technique for BEMD.

3 Bidimensional Empirical Mode Decomposition 311 There are a few 2D scattered data interpolation techniques that are employed for the BEMD process so far, and there are few works that compares the BEMD performance for few interpolation techniques. 13,14 But there is no work in the literature that compares BEMD performance for many interpolation techniques with detailed simulation/analyses. In this paper, BEMD simulation is performed using radial basis function (RBF) based interpolations and Delaunay triangulation (DT) based interpolations. To determine the true effect of interpolation techniques, boundary processing is not employed in this analysis. In fact, boundary processing is computationally extensive, and there may not be a universally standard technique to accomplish this. Hence, an interpolation technique that results in a good decomposition even without boundary processing of the data may be preferable. In this paper, a synthetic texture image with three component textures has been used for the initial simulation. Then, two real texture images 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 for the case of the BEMD process. The results may be used as a guideline for selecting an appropriate interpolation for BEMD based on specific goals and system constraints. 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. Simulation results are given in Sec. 4, while analysis and discussion of the results are given in Sec. 5. Finally, concluding remarks are given in Sec BEMD Overview EMD as well as BEMD decomposes a signal into its IMFs based basically on the local frequency or oscillation information. The first IMF contains the highest local frequencies of oscillation, the final IMF contains the lowest local frequencies of oscillation and the residue contains the trend of the data. Application of Hilbert transform to the 1D IMFs produces analytic signals that are used to analyze the time frequency distribution known as the Hilbert spectral analysis (HSA); EMD plus HSA together form what is known as the Hilbert Huang transform (HHT). 1 It is claimed and experimentally shown that the HHT performs better than the other existing techniques of analyzing the time frequency distribution of nonstationary and nonlinear data. 1 Like time frequency distribution with EMD, acquiring the space spatial frequency distribution of 2D data/image may be possible with BEMD. Although direct estimation of the horizontal and vertical frequencies of 2D IMFs has been studied, 15 2D HHT has not yet been reported in the literature. The IMFs obtained from EMD are expected to have the following properties 1 5 : (i) in the whole data set, the number of extrema and the number of zero crossings must be equal or differ by at most one; (ii) there should be only one mode of oscillation between two successive zero crossings; (iii) at any point, the mean value of the envelope defined by the local maxima and the envelope defined by the local

4 312 S. M. A. Bhuiyan et al. minima is zero; and (iv) the IMFs are locally orthogonal. In fact property (i) ensures property (ii), and vice versa. The definition and properties of the BIMFs are slightly different from the IMFs. It is sufficient for BIMFs to follow only the final two (iii and iv) properties given above. 3,4 In fact, due to the properties of an image and the BEMD process, it is not possible to satisfy the first two properties (i and ii) given above in the case of BIMFs, since the maxima and minima points are defined in a 2D scenario for an image. Let the original image be denoted as I, a2dimfasf, and the final residue as R. In the process ith IMF F i is obtained from its source image S i (alternately known in the literature as intermediate residue), where S i = S i 1 F i 1 and S 1 = I. It requires one or more iterations to obtain F i, where the temporary state of an IMF (TS-IMF) in jth iteration can be denoted as F Tj. Using the above-mentioned definitions, the steps of the BEMD process can be summarized as below 1 5 : (i) Set i =1andS i = I. (ii) Set j =1andF Tj = S i. (iii) Obtain the local maxima and minima maps of F Tj, denoted as P j and Q j, respectively. (iv) Form the upper envelope (UE) and lower envelope (LE) of F Tj, denoted as U Ej and L Ej, from the maxima and minima points in P j and Q j, respectively. (v) Find the mean envelope (ME) as M Ej =(U Ej L Ej )/2. (vi) Calculate F Tj+1 as F Tj+1 = F Tj M Ej. (vii) Check if F Tj+1 follows the 2D IMF properties. This can be ensured by finding the standard deviation (SD), denoted as D, between F Tj+1 and F Tj. 1 5,16,17 D may be defined in the following different ways 1 5 : D = D = M x=1 y=1 M x=1 y=1 N F Tj+1 (x, y) F Tj (x, y) 2 F Tj (x, y) 2, (1a) N F Tj+1 (x, y) F Tj (x, y) 2 F Tj+1 (x, y) 2, (1b) D = M x=1 N y=1 F Tj+1(x, y) F Tj (x, y) 2 M x=1 N y=1 F Tj(x, y) 2, (1c) where (x, y) is the coordinate and M (N) is the total number of rows (columns) in the 2D data. Although all three SD measures in Eqs. (1a), (1b), and (1c) provide a global measure of SD, the third one causes the fastest convergence while the second one causes the slowest convergence. It should be noted that the definition of SD affects the number of required iterations to achieve a given threshold and thus the amount of sifting per IMF; it does not have any contribution to the calculation of UE, LE, or ME in a particular iteration. To satisfy the nearly zero local envelope mean for each IMF, the

5 Bidimensional Empirical Mode Decomposition 313 SD threshold can be set below 0.5 for Eq. (1b), and below 0.05 for Eqs. (1a) and (1c) for practical purposes. (viii) If F Tj+1 meetsthecriteriagiveninstep(vii),thentakef i = F Tj+1 ;set S i+1 = S i F i and i = i+1; go to step (ix). Otherwise, set j = j+1, go to step (iii) and continue up to step (viii). (ix) If S i has less than three extrema points then the final residue R = S i ;and the decomposition is complete. Otherwise, go to step (ii) and continue up to step (ix). Except for the truncation error of the digital computer, the summation of all the IMFs and the final residue, denoted as C, returns the original data/image back as given by K C = F i + R = I, (2) i=1 where K is the total number of IMFs excluding the residue. Following Ref. 1 an orthogonality index (OI), denoted as O, may be defined for 2D IMFs and residue as follows: O = M x=1 y=1 N K+1 i=1 K+1 j=1 F i (x, y)f j (x, y) 2, (3) C (x, y) where F K+1 denotes the final residue R. A low value of OI indicates a good decomposition in terms of local orthogonality among the components. The whole process of decomposition of an image into its 2D IMFs in order of local spatial scales is known as sifting. The decomposition of any image into 2D IMFs is not a unique process. The number of IMFs and their characteristics depends on the extrema detection method, interpolation technique, and stopping criteria of the iterations for each IMF. In that sense, there are an infinite number of IMF sets for each image. 16 Although the SD threshold criterion is used to satisfy the zero envelope mean for each IMF, this criterion sometimes causes over sifting of the data and results in a poor decomposition 16,17 causing the IMF to be a constant-amplitude frequency modulated function lacking the intrinsic amplitude variations. To overcome this problem, various additional stopping criteria may be employed. 16,17 3. Interpolation Techniques Surface interpolation from 2D scattered data (local maxima and minima points) is a crucial and important part of BEMD method. Scattered data interpolation refers to the computation of a 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.

6 314 S. M. A. Bhuiyan et al. Since the solution of the interpolation problem coincides with the data points, one might assume it to be the more accurate reconstruction. But if the samples carry some noise, an approximating surface will be more adequate. Therefore, the choice of the appropriate method depends on the specific structure of the problem In this section several RBF based surface interpolation and several DT based surface interpolation techniques are reviewed in brief. Some of these interpolation methods are later tested in terms of BEMD performance in this paper RBF based interpolation 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 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. 18,21 RBFs are well known to provide powerful tools for high-fidelity reconstruction of surfaces from a selected set of sparse and irregular samples, in which context they have intensively been studied for more than twenty years. 22,23 However, a maximum number of interpolation centers with which an RBF interpolator can work may pose some limitations on its usefulness. Classical examples for these functions are given by Duchon s thin plate splines 24,25 and Hardy s multiquadrics, 21 where developments during this decade have provided compactly supported RBF (CSRBF) and RBFs with multizone decomposition. Consider f : R d R a real-valued function of d variables that is to be approximated by s : R d R, given the values {f(x i ) : i = 1, 2, 3,...,N}, where {x i : i =1, 2, 3,...,N} is a set of distinct points in R d called the nodes of operation. The RBF approximation s is given by where s(x) =p m (x)+ N λ i φ( x x i ), x R d, λ i R, (4) i=1 s is the radial basis function (RBF); p m is a low-degree polynomial, typically linear or quadratic, a member of the mth degree polynomials in d variables; denotes the Euclidean norm; the λ i s are the RBF coefficients; φ is a real-valued function called the basis function; and the x i s are the RBF centers. Some examples of popular choices of φ and the names of the corresponding RBFs are given in Table 1. In Table 1 the constant c for the cases of Hardy s multiquadrics, inverse multiquadrics, exponential splines and Gaussian splines may be selected based on specific need and performance. 26

7 Bidimensional Empirical Mode Decomposition 315 Table 1. Choices of φ for various RBFs. RBFs Choices of φ Linear φ(r) =r Cubic splines φ(r) =r 3 Thin-plate splines φ(r) =r 2 log(r) Hardy s multiquadrics φ(r) = r 2 + c 2a Inverse multiquadrics φ(r) = 1 r 2 +c 2 Exponential splines φ(r) =e cr Gaussian splines φ(r) =e cr2 Compactly supported splines φ(r) =(1 r) mb a c is a constant that governs the shape/spread of the basis function b m is a function of spatial dimension 3.2. DT based interpolation Given a set of data points, the DT is a set of lines connecting each point to its natural neighbors. The DT is related to the Voronoi diagram the circle circumscribed about a DT has its center at the vertex of a Voronoi polygon. Voronoi diagram refers to the partitioning of a plane with n points into convex polygons such that each polygon contains exactly one generating point, and every point in a given polygon is closer to its generating point than to any other. 27 A Voronoi diagram is sometimes also known as a Dirichlet tessellation. The cells are called Dirichlet regions, Thiessen polytopes, or Voronoi polygons. The DT is a triangulation that is equivalent to the nerve of the cells in a Voronoi diagram, i.e., that triangulation of the convex hull of the points in the diagram in which every circumcircle of a triangle is an empty circle. 28 The DT can be used to create a triangular grid for scattered data points. Subsequent piecewise interpolation on triangles can be used to create an approximated surface. 6,29 The interpolation methods for this purpose may be linear, cubic or nearest neighbor. 4. Simulation Results RBF based and DT based 2D interpolation methods are suitable for the type of scattered data appearing in the maxima or minima map in the BEMD process Hence, these two types of interpolation techniques are considered in this paper. A comprehensive study of BEMD process has been made for these two types of major interpolation techniques using synthetic as well as real texture images. Extrema (maxima and minima points) detection is required prior to the interpolation. In this simulation extrema points are found by using neighboring window method employinga3 3 window. Generally, a 3 3 window results in an optimum extrema map for a given 2D data. In this process a 3 3 window is moved all over the image/2d data points. If the pixel corresponding to the center of the window has a value that is strictly higher than all of its neighbors (i.e., eight neighboring pixels) within the window, then it is considered a local maximum; similarly, if the same

8 316 S. M. A. Bhuiyan et al. pixel has a value that is lower than all its neighbors, then it is considered a local minimum. In this work, a slightly different neighboring window is considered for the corner and boundary pixels rather than ignoring them for finding local maxima or minima, where each corner pixel has only three neighbors and each boundary pixel has only five neighbors to be considered. This method may result in one or more maxima/minima points at the boundary region without any other heuristic processing and thus may reduce the boundary effects of the interpolation in the BEMD process. In the local maxima (minima) image, all but the local maxima (minima) points are set to zero. An SD threshold is the fundamental criterion for stopping the iterations for a particular IMF. To avoid over-sifting, various additional stopping criteria may be employed as for the case of 1D EMD. 16,17,30 In this simulation, besides the original SD threshold criterion, a maximum number of iterations for each IMF is set as an additional criterion. However, this additional criterion is set in such a way that the decomposition is mostly governed by the interpolation technique, not by the stopping criteria BEMD simulation using synthetic texture Synthetic texture The synthetic texture is composed of three components. Each component is synthesized from horizontal and vertical approximate sinusoidal waveforms having different but closely spaced frequencies. The first texture component consists of higher frequencies, the second component consists of medium frequencies, and the last component consists of very low frequencies. The amplitudes and frequencies of various components are given in Table 2. The OI among the original synthetic texture components and the mean value of each component are given in Table 3. These values are also useful for comparing the resulting IMFs with the actual components. For the simulation, the synthetic texture is chosen to be pixels. Hence, the sampling frequency, or the number of data points, is taken accordingly. Onedimensional horizontal and vertical sinusoidal components are shown in Figs. 1 and 2, respectively. 2D plots of each texture component along with their horizontal and vertical elements are shown in Figs The final composite synthetic texture image is given in Fig. 6. Diagonal 1D slices of all three texture components are Table 2. Amplitude and frequencies of 1D sinusoidal waves used to generate the synthetic texture. Horizontal Vertical Component 1 Amplitude Frequency Component 2 Amplitude Frequency Component 3 Amplitude Frequency

9 Bidimensional Empirical Mode Decomposition 317 Table 3. Global mean of synthetic texture components and their orthogonality index. Global mean Orthogonality index (OI) Component Component Component Fig. 1. 1D sine wave components in the horizontal direction of the synthetic texture. Fig. 2. 1D sine wave components in the vertical direction of the synthetic texture. shown in Fig. 7. Although formation of the synthetic texture components using horizontal and vertical approximate sinusoidal waveforms gives regular texture grids, only the BEMD can directly generate similar texture components back from the composite synthetic texture, whereas the Fourier decomposition fails to do so. For example, the pixel composite synthetic texture image has 256 frequency components in the horizontal direction and another 256 frequency components in (a) (b) (c) Fig. 3. For highest frequency synthetic texture component: (a) 2D texture from horizontal sine waves, (b) 2D texture from vertical sine waves, and (c) 2D texture from addition of (a) and (b).

10 318 S. M. A. Bhuiyan et al. (a) (b) (c) Fig. 4. For medium frequency synthetic texture component: (a) 2D texture from horizontal sine waves, (b) 2D texture from vertical sine waves, and (c) 2D texture from addition of (a) and (b). (a) (b) (c) Fig. 5. For lowest frequency synthetic texture component: (a) 2D texture from horizontal sine waves, (b) 2D texture from vertical sine waves, and (c) 2D texture from addition of (a) and (b). the vertical direction. Thus it can be decomposed into 65,536 component images by applying Fourier decomposition, where each component image corresponds to a combination of one horizontal frequency element and one vertical frequency element. The reconstruction/approximation of the original texture image requires addition of hundreds of dominating image components obtained by Fourier analysis Extrema detection Detection of maxima and minima points is required for each iteration of the process. As the source image changes from one iteration to the next, the extrema points also change in number and sometimes in position at each iteration. However, for a clear understanding of the extrema detection, the minima and maxima images and their 3D mesh plots for the original image are given in Figs. 8 and 9, respectively.

11 Bidimensional Empirical Mode Decomposition 319 Fig. 6. The pixel synthetic texture image Interpolation 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. However, for illustration purposes, the interpolated envelopes along with the original data for various interpolation methods are given in Fig. 10 for the original synthetic texture image. In Fig. 10, the left column shows 3D mesh plots of a32 32-pixel region of the original data and corresponding envelopes, while the right column shows the diagonal slices (intensity profiles) of the original data and corresponding envelopes. From Fig. 10, it is found that the RBF interpolators produce complete 2D surfaces that might be useful for the BEMD process. On the other hand, most of the DT based interpolators can produce a surface within a rectangle region that bounds the largest x- andy-coordinates of the maxima/minima data. Hence, in its original form, DT cannot be used for BEMD for all kinds of data. In cases where it may be possible to have some maxima or minima points at the extreme coordinates as in Ref. 6, DT may be suitable over the other techniques, still with the addition of some data points to meet the requirement of convexhull. But in the iterative process it is not certain that there will be some extrema points at the boundaries, and it may also be impractical to put in additional data points for this interpolation. Although DT followed by the nearest neighbor interpolation can produce a surface up to the boundary, the interpolation is very poor and fails to produce meaningful IMF components. Based on these facts, DT may not be suitable, in general, for the BEMD process and is not considered for the subsequent analysis of this paper Results For the simulation using synthetic texture, an SD threshold of 0.5 and a maximum of 1000 iterations are used as the stopping criteria. For RBF interpolators where

12 320 S. M. A. Bhuiyan et al. (a) (b) (c) (d) Fig. 7. 1D diagonal slices (intensity profiles) of the synthetic texture image and its components: (a) original texture, (b) component-1, (c) component-2, and (d) component-3. Fig. 8. Minima image and corresponding mesh plot of the original synthetic texture image. needed, the inverse of the approximate average distance among the interpolation centers is used as the value of the constant c. The IMFs and residue obtained from the synthetic texture for RBF interpolation techniques are shown in Fig. 11, where it is observed that some techniques result in more than three components. Figure 12

13 Bidimensional Empirical Mode Decomposition 321 Fig. 9. Maxima image and corresponding mesh plot of the original synthetic texture image. (a) RBF-linear (b) RBF-cubic Fig. 10. Envelopes and original image: left, 3D mesh plots of a pixel region and right, diagonal slices (intensity profiles) of the original image and corresponding envelopes.

14 322 S. M. A. Bhuiyan et al. (c) RBF-thin-plate spline (d) RBF-multiquadric (e) RBF-inverse multiquadric Fig. 10. (Continued )

15 Bidimensional Empirical Mode Decomposition 323 (f) RBF-exponential (g) RBF-Gaussian (h) DT-linear Fig. 10. (Continued )

16 324 S. M. A. Bhuiyan et al. (i) DT-cubic (j) DT-nearest neighbor Fig. 10. (Continued ) depicts the diagonal slice (intensity profile) of each component for the interpolation methods that result in only three components so that they can be compared with the original intensity profiles. The total number of IMFs including the final residue, total number of iterations needed, and OI for various RBF interpolation techniques are summarized in Table 4. To compare the resulting IMFs with the corresponding original components, several metrics are considered, which are peak signal-to-noise ratio (PSNR), mean square error (MSE), mean absolute error (MAE), and correlation coefficient (CC). For the interpolation methods, which generate more than three components, even though the first IMF bears some resemblance to the actual first component of the synthetic texture, the other two original components cannot be compared with more than two remaining IMF components. Consequently, IMF-wise iterations, achieved SD, global mean, MSE, MAE, CC, and PSNR are displayed in Tables 5 11 only for the interpolation techniques that result in three components. As mentioned previously, these parameters are used to compare the IMF components with the actual components. Although this comparison is possible only for the synthetic case, it gives a picture of the performance of BEMD process for various interpolation techniques.

17 Bidimensional Empirical Mode Decomposition 325 (a) RBF-linear (b) RBF-cubic (c) RBF-thin-plate spline Fig D IMFs and final residue obtained from the synthetic texture for different RBF interpolators.

18 326 S. M. A. Bhuiyan et al. (d) RBF-multiquadric (e) RBF-inverse multiquadric (f) RBF-exponential (g) RBF-Gaussian Fig. 11. (Continued )

19 Bidimensional Empirical Mode Decomposition 327 (a) RBF-cubic (b) RBF-thin-plate spline (c) RBF-inverse multiquadric (d) RBF-Gaussian Fig D diagonal slices (intensity profiles) of the 2D IMFs, final residue, and their sum for different RBF interpolators. Table 4. Comparison among different RBF interpolators with respect to the synthetic texture image of Fig. 6: total IMFs, total iterations, and OI. RBF interpolators Total number of IMFs/components Total iterations needed OI Linear Cubic Thin-plate spline Multiquadric Inverse multiquadric Exponential Gaussian BEMD simulation using real images Different interpolation techniques are next applied to several real texture images. In this case, the simulation parameters for various steps of the BEMD process are kept the same as for the case of synthetic texture except the SD threshold which

20 328 S. M. A. Bhuiyan et al. is set to 10. A higher value of the SD threshold is chosen to limit the number of iterations. Since the SD value after the first iteration is very high, the SD threshold of 10 may be considered acceptable. It may not ensure a perfect local zero mean for the IMFs, but it may be better than applying additional stopping criteria in the sense that any additional stopping criteria can cause an even higher stopping point SD value. 16 Hence, the SD threshold value of 10 can adequately serve the purpose of finding the effect of interpolation methods on the BEMD process. For the true images it is not possible to compare the decomposed IMFs with the actual or true components, since there are no ground-truth IMFs. Hence the total number of Table 5. Comparison among different RBF interpolators with respect to the synthetic texture image of Fig. 6: iterations needed. RBFinterpolators IMF1 IMF2 Cubic Thin-plate spline Inverse multiquadric Gaussian Table 6. Comparison among different RBF interpolators with respect to the synthetic texture image of Fig. 6: achieved SD. RBF interpolators IMF 1 IMF 2 Cubic Thin-plate spline Inverse multiquadric Gaussian Table 7. Comparison among different RBF interpolators with respect to the synthetic texture image of Fig. 6: global mean. RBFinterpolators IMF1 IMF2 Residue Cubic Thin-plate spline Inverse multiquadric Gaussian Table 8. Comparison among different RBF interpolators with respect to the synthetic texture image of Fig. 6: MSE. RBF interpolators IMF 1 IMF 2 Residue Sum of all components Cubic e 028 Thin-plate spline e 028 Inverse multiquadric e 028 Gaussian e 028

21 Bidimensional Empirical Mode Decomposition 329 Table 9. Comparison among different RBF interpolators with respect to the synthetic texture image of Fig. 6: MAE. RBF interpolators IMF 1 IMF 2 Residue Sum of all components Cubic e 015 Thin-plate spline e 015 Inverse multiquadric e 015 Gaussian e 015 Table 10. Comparison among different RBF interpolators with respect to the synthetic texture image of Fig. 6: CC. RBF interpolators IMF 1 IMF 2 Residue Sum of all components Cubic Thin-plate spline Inverse multiquadric Gaussian Table 11. Comparison among different RBF interpolators with respect to the synthetic texture image of Fig. 6: PSNR. RBF interpolators IMF 1 IMF 2 Residue Sum of all components Cubic Thin-plate spline Inverse multiquadric Gaussian obtained IMFs/components, the OI, the total number of iterations, and the number of iterations needed for the first five IMFs are considered as the performance indices. The images used are two texture images from the Brodatz texture set, 31 namely D18 and D27. Each of the images used in the simulation is pixels, as shown in Fig. 13; the images are taken as cropped regions from the original pixel images. As sample representation, the generated IMFs for D27 are shown in Figs. 14(a) 14(g) for various RBF interpolators. The comparison among all the (a) (b) Fig. 13. The pixel real texture images taken from Brodatz texture set [28]: (a) D18 and (b) D27.

22 330 S. M. A. Bhuiyan et al. (a) RBF-linear Fig D IMFs and final residue obtained from real texture image D27 for different RBF interpolators.

23 Bidimensional Empirical Mode Decomposition 331 (b) RBF-cubic (c) RBF-thin-plate spline Fig. 14. (Continued ) different RBF interpolators for BEMD of D18 is given in Table 12 and for BEMD of D27 is given in Table Analyses and Discussion For the synthetic texture, each of the RBF-based interpolation techniques results in a different set of IMFs or components. This decomposition is definitely based on the local spatial scales of the texture. However, RBF-linear, RBF-multiquadric,

24 332 S. M. A. Bhuiyan et al. (d) RBF-multiquadric (e) RBF-inverse multiquadric Fig. 14. (Continued ) and RBF-exponential interpolation methods result in more than three components as seen in Fig. 11 and Table 4, making them different from the actual synthetic texture components. From Fig. 11 it is further observed that the RBF-linear fails to produce meaningful components. These components are not enough orthogonal as well, which can be understood from the corresponding OIs in Table 4. Although the RBF-multiquadric interpolator produces some kind of meaningful components, its performance is not as good as RBF-cubic, RBF-thin-plate spline, RBF-inverse multiquadric, or RBF-Gaussian. Indeed, from the simulation with synthetic texture

25 Bidimensional Empirical Mode Decomposition 333 (f) RBF-exponential Fig. 14. (Continued ) image, RBF-cubic, RBF-thin-plate spline, RBF-inverse multiquadric, and RBF- Gaussian interpolators appear to be effective and/or the preferred methods for BEMD. Although the total number of iterations needed for the entire decomposition process is a factor indicating the computation involved, the number of iterations for the first few IMFs is more significant in this regard. Generally, the first iteration for the first IMF requires more computation and time than the first iteration for the second IMF, and so on. On the other hand, in general, the first IMF requires more iterations than the subsequent IMFs. This can be attributed to the fact that the number of extrema in the corresponding source images (alternately known in the

26 334 S. M. A. Bhuiyan et al. (g) RBF-Gaussian Fig. 14. (Continued ) Table 12. Comparison among different RBF interpolators with respect to D18 image of Fig. 13: total IMFs, iterations, and OI. RBF interpolators Total Iterations needed OI number of components Total IMF 1 IMF 2 IMF 3 IMF 4 IMF 5 Linear Cubic Thin-plate spline Multiquadric Inverse multiquadric Exponential Gaussian literature as intermediate residues) of subsequent modes decreases, which further affects the interpolation algorithm. Table 5 shows that IMF-1 needs the lowest number of iterations for RBF-Gaussian, while IMF-2 needs the lowest number of iterations for RBF-cubic; and IMF-1 needs the highest number of iterations for RBF-cubic, while IMF-2 needs the highest number of iterations for RBF-thin-plate spline. Achieved SD is an indication of local mean of the IMFs, where a lower SD indicates that the local mean is closer to zero. Since, for the synthetic texture, the desired SD threshold is achieved with all the RBF interpolation methods, as can be seen in Table 6, no conclusion can be drawn from the achieved SD based only on the synthetic texture decomposition. Although zero global mean does not necessarily indicate zero local mean, the global means of the decomposed components are given in Table 7 to compare them with the original components.

27 Bidimensional Empirical Mode Decomposition 335 Table 13. Comparison among different RBF interpolators with respect to D27 image of Fig. 13: total IMFs, iterations, and OI. RBF interpolators Total Iterations needed OI number of components Total IMF 1 IMF 2 IMF 3 IMF 4 IMF 5 Linear Cubic Thin-plate spline Multiquadric Inverse multiquadric Exponential Gaussian MSE and MAE between the decomposed components and the corresponding original components, as well as between their sum and the synthetic texture for the four preferred RBF interpolation methods are given in Tables 8 and 9. These tables show that for IMF-1 RBF-Gaussian causes the lowest MSE or MAE, while RBF-thin-plate spline causes the highest MSE or MAE. For IMF-2 and residue, RBF-cubic causes the lowest MSE or MAE. RBF-inverse multiquadric causes the highest MSE or MAE for IMF-2, while RBF-Gaussian causes the highest MSE or MAE for the residue. MSE or MAE is almost zero for the sum of decomposed components with all the RBF interpolators. Correlation coefficient (CC) may be a good measure to evaluate the match between the original components and the decomposed components. All the CC values as observed in Table 10 are good enough for the four preferred interpolation methods indicating successful decomposition into nearly original components. However, the performance of RBF-cubic may be better than the others. PSNR provides similar information about the effect of interpolation methods, as do the MSE and MAE. From Tables 4 to 11 and from Figs. 11 and 12, it appears that the RBF-cubic performs better than all other studied interpolators for this particular synthetic texture. Like the synthetic case, the total number of decomposed components, total iterations, and OIs are used for performance evaluation of BEMD with real texture images for different interpolation methods. As sample cases, the number of iterations for the first five IMFs are also observed for comparison based on the fact that the first few IMFs generally require more iterations and computation than the later IMFs. It has already been mentioned that the first 2D IMF provides the highest local spatial scales (finest local variation) of the image, while the final 2D IMF provides the lowest local spatial scales (coarsest local variation) of the image; and the final residue gives the trend of the image. Thus IMFs should always have some resemblance to the image at different scales of local spatial variation. Hence, visually evaluating the IMFs and matching them with the actual image provides additional performance measures. The simulation results in terms of the various performance indices given in Tables 12 and 13, and the characteristics of the IMFs obtained for the real texture images indicate that the effects of different interpolation methods

28 336 S. M. A. Bhuiyan et al. are quite similar for the synthetic texture. For example, RBF-linear and RBFexponential interpolators result in a large number of IMFs with a high value of OI (lower local orthogonality). Beside these, the IMFs do not resemble the original textures. Hence, RBF-linear and RBF-exponential interpolators do not appear to be suitable for the BEMD process. The simulation using real texture images indicates that RBF-cubic, RBF-thin-plate spline, RBF-multiquadric, RBF-inverse multiquadric, and RBF-Gaussian interpolators can be good candidates for BEMD. However, simulation results from Tables 12 and 13 as well as visual evaluation of the 2D IMFs emphasize that RBF-cubic, and RBF-thin-plate spline interpolators may be preferable for all general-purpose decompositions using BEMD. Based on the characteristics of obtained interpolated surface with DT based interpolation methods, it has already been claimed in Sec that DT based interpolation methods are not suitable for BEMD unless additional manipulation is done. Careful observation of the envelope shapes shown in Fig. 10 and the envelope shapes for other images and/or source images at different iterations (not shown in this paper), and corresponding 2D IMFs indicates that the mean envelope having locally convex or concave like surface bounded by zero crossings may provide more meaningful decomposition along with higher local orthogonality (lower OI) and zero local mean of the components. Hence, the 2D surface interpolation methods producing locally convex or concave like smoother surfaces may be preferable. For this reason RBF-cubic, RBF-thin-plate spline, RBF-inverse multiquadric, and RBF- Gaussian interpolators provide better results than the others. This fact may be attributed to the required IMF characteristic as defined for 1D IMFs. 1 However, further investigation may be required to find the true reason for the success of a particular interpolation method in BEMD. 6. Conclusion In this paper, comprehensive experiments on BEMD have been made using DT and 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 RBF-cubic, RBF-thinplate spline, RBF-inverse multiquadric, and RBF-Gaussian interpolators provide better results than the others, which is clearly supported by the simulation using both synthetic and real images in this paper. However, it may be difficult to reach a conclusion about the superiority of one particular interpolation method for BEMD, based on the experiments on only a few images, but it certainly sheds light on a few methods performing better than the others. On the other hand, it is believed that the results in this paper may be helpful to select a particular interpolation method depending on the specific needs, for example, local orthogonality, local mean, and total number of IMFs.

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