Indonesian J our nal of Electrical Engineering and Computer Science V ol. 38, No. 1, April 2025, pp. 172 ∼ 181 ISSN: 2502-4752, DOI: 10.11591/ijeecs.v38.i1.pp172-181 ❒ 172 Enhancing BEMD decomposition using adapti v e support size f or CSRBF functions Mohammed Arrazaki 1 , Othman El Ouahabi 2 , Mohamed Zohry 1 , Adel Bab bah 3 1 Department of Mathematics, F aculty of Sciences, Uni v ersity AbdelMalek Essaadi, T etouan, Morocco 2 National School of Applied Sciences, Uni v ersity AbdelMalek Essaadi, T angier , Morocco 3 Department of Mathematics, F aculty of Polydisciplinary , Uni v ersity of AbdelMalek Essaadi, Larache, Morocco Article Inf o Article history: Recei v ed Jul 27, 2024 Re vised Oct 16, 2024 Accepted Oct 30, 2024 K eyw ords: BEMD decomposition CSRBF functions Intrinsic mode functions Orthogonality inde x Synthetic te xture image T ime-frequenc y analysis W endland functions ABSTRA CT Despite their widespread de v elopment, the F ourier transform and w a v elet trans- form are still unsuita ble for analyzing non-stationary and non-linear signals. T o address this limitation, bidi mensional empirical mode decomposition (BEMD) has emer ged as a promising technique. BEMD ef fecti v ely e xtracts structures at v arious scales and frequencies b ut f aces signicant computational comple xity , primarily during the e xtremum interpolation phase. T o mitig ate this, dif ferent interpolation functions were presented and suggested, with BEMD using com- pactly supported radial basis functions (BEMD-CSRBF) sho wing promising re- sults in reducing computational cost while maintaining decomposition quality . Ho we v er , the choice of support size for CSRBF functions signicantly impacts the quality of BEMD. This article presents an enhancement to the BEMD- CSRBF algorithm by adjusting the CSRBF support size based on the e xtrema distrib ution of the image. Our method’ s results sho w a signicant impro v ement in the BEMD-CSR BF algorithm’ s quality . Furthermore, when compare d to the other tw o approaches to BEMD, it sho ws higher accurac y in terms of both in- trinsic mode function (IMF) quality and computational ef cienc y . This is an open access article under the CC BY -SA license . Corresponding A uthor: Mohammed Arrazaki Department of Mathematics, F aculty of Sciences, Uni v ersity AbdelMalek Essaadi BP . 2121 M’Hannech II, 93030 T etouan, Morocco Email: rezaki mohamed@hotmail.com 1. INTR ODUCTION Huang et al. [1] introduced empirical mode decomposition (EMD) a s an ef fecti v e method for analyz- ing non-stationary and non-linear signals in one-dimensional (1D). It has been sho wn to be ef cient for signal denoising [2]. The EMD approach has prompt ed researchers to de v elop the technique for bidimensional sig- nals. The 2D e xtension of EMD w as created by Nunes et al. [3], it is kno wn as bidimensional empirical mode decomposition (BEMD), and i t k eeps t he same concept as EMD via decomposition of an image to a set of intrinsic modal functions (IMFs) using an iterati v e process. This technique has made it possible to de v elop ne w methods in the analysis and processing of images that can be applied to an y image, especially te xtured images, the results of which sho w better performance compared to e xisting decomposition techniques [3]. BEMD has been applied in dif ferent imaging areas such as te xture analysis [4], [5], image inde xing [6], image classica- tion [7]–[10], image w atermarking [11], [12], image se gmentation [13], and fractal analysis [14]. The quality or performance of an IMF depends on the quality of preceding IMFs. The choice of the stopping criterion for the sifting process is therefore v ery important and is based on the follo wing tw o conditions [3]: J ournal homepage: http://ijeecs.iaescor e .com Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian J Elec Eng & Comp Sci ISSN: 2502-4752 ❒ 173 - F or each IMF , there are an equal number of zero crossings and e xtrema. - Each IMF is symmet rical with respect to the local mean. In addition, the signal is assumed to ha v e at least tw o e xtrema. These conditions are by denition the properties of an IMF . The principle of the BEMD requires the follo wing phases [3]: 1. Initialize r 0 = m (the residual) and k = 1 (the inde x number of IMF). 2. Extract the k th IMF (sifting process): a. Initialize h 0 = r k − 1 and j = 1 . b . j=1 Extract the local minima and maxima of h j − 1 . c. Compute the upper en v elope and lo wer en v elope functions x j − 1 and y j − 1 by interpolating, respec- ti v ely , the local minima and local maxima of h j − 1 . d. Compute the mean en v elope: m j − 1 = ( x j − 1 + y j − 1 ) / 2 e. Update h j = h j − 1 − m j − 1 and j = j + 1 . f. Calculate the stopping criterion: S D ( j ) = 1 M × N P M m =1 P T t =0 ( h j − 1 − h j ) 2 h 2 j − 1 + ϵ where ϵ is a (weak) term eliminating an y di visions by zero. g. Decision: repeat steps (b) through (f) until S D j < S D max and then put d k = h j ( k th IMF). 3. Update the residue r k = r k − 1 − d k . 4. Repeat steps 1–3 with k = k + 1 until the number of e xtrema in r k is less than 2. When the decomposition is achie v ed, we can write the signal in the follo wing form: I = P P k =1 I M F k + r P +1 The stopping criterion is v alid if S D does not e xceed S D max (certain predened threshold), we use S D max between 0 . 2 and 0 . 5 because this v alue gi v es satisf actory results in practice. Figure 1 presents an e xample of the BEMD decomposition of the original te xture image as Figure 1(a), this image is decomposed into three IMFs in Figures 1(b) to 1(d) and the residue in Figure 1(e), which illustrate a multiscale decomposition from high frequencies to lo w frequencies of the original image. Ho we v er , a real obstacle to the implementation of this method is the computational comple xity , most of which is consumed in creating the upper and lo wer en v elopes by interpolated functions, lik e the radial basis functions (RBF) [2]. T o solv e this problem, some w orks ha v e been proposed with a less e xpensi v e technique, such as using Delaunay triangulation [15], nite elements [16] or by utilizing a lter to obtain the upper and lo wer en v elopes [17]. In the same conte xt, Bhuiyan et al. [18] suggested using the statistical lters Max and Min accompanied by a smoothing operator repeated se v eral ti mes when generating the upper and lo wer en v elopes, b ut there are se v eral limitations such as: determining the correct lter size and the number of iterations of the smoothing operator . (a) (b) (c) (d) (e) Figure 1. Ex emple of BEMD decomposition of te xture image: (a) original image, (b) IMF 1, (c) IMF 2, (d) IMF 3, and (e) residue The BEMD using compactly supported radial basis functions (BEMD-CSRBF) [19] produces good results in terms of computational comple xity and BEMD decomposition quality , particularly for the rst IMFs. Ho we v er , using a x ed support size for CSRBF functions in BEMD mak es e xtracting lo w frequencies (last Enhancing BEMD decomposition using adaptive support size for CSRBF functions (Mohammed Arr azaki) Evaluation Warning : The document was created with Spire.PDF for Python.
174 ❒ ISSN: 2502-4752 IMFs) dif cult. Especially since this decomposition is iterati v e, and each iteration produces a dif ferent number of e xtrema and a dif ferent distrib ution in space. Also, the number of e xtrema decreases after each IMF is e xtracted. Which mak es using a x ed support size to e xtract all IMFs inef cient. In this paper , we suggested an approach to adjust the support size during the BEMD algorithm. T o e xtract the rst IMF , we determine the CSRBF function support size (initial support) based on one of the distances in [18], because it is deri v ed based on the distrib ution of the e xtrema. Considering that the e xtrema continuously reduce during the decomposition process, we just double the size of the initial support after e xtracting each IMF without the need to recalculate pre vious distances, thus a v oiding increasing the comple xity of the computation. This approach demonstrates enhanced quality in the B EMD-CSRBF decomposition and pro v es its ef cac y when compared to other BEMD decomposition methods, both in terms of the quality of IMFs and the comple xity of computations. 2. METHOD 2.1. Compactly supported radial basis functions As our research is founded on the emplo yment of the CSRBFs in the BEMD algorithm, the present study aims to e xplicate the characteristics of CSRBF functions belonging to this specic cate gory . Notably , a f amily of radial basis functions with compact support w as rst introduced in the mid-1990s, as e videnced by the w orks of W u in 1995 [20], and W endland in 1999 [21]. It should be noted that there are other types of CSRBF functions [22], [23]. Generally , a basis radial function with compact support is gi v en by the e xpression [24]: ϕ l ,k ( r ) = (1 − r ) n + p ( r ) k ≥ 1 (1) with (1 − r ) n + = ( (1 − r ) n r ∈ [0 , 1] 0 r > 1 (2) where p ( r ) is one of the polynomials prescribed by W u or W endland, the indices l and 2 k represent respecti v ely the space dimension and smoothness of the function. The T able 1 contains some functions of W u and W endland. The dif culty wi th using CSRBFs is the size of the support. In the follo wing, we look at the inuence of the support size on the BEMD decomposition, especially since the e xtrema to be interpolated are reduced during the BEMD. W e used the W endland function φ 3 , 1 used in [19] as the CSRBF function. T able 1. CSRBF functions of W u and W endland Smoothness SPD W u functions ψ 1 , 3 ( r ) = (1 − r ) 6 + (5 r 5 + 30 r 4 + 72 r 3 + 82 r 2 + 36 r + 6) C 4 R 3 ψ 2 , 3 ( r ) = (1 − r ) 5 + (5 r 4 + 25 r 3 + 48 r 2 + 40 r + 8) C 2 R 3 ψ 3 , 3 ( r ) = (1 − r ) 4 + (5 r 3 + 20 r 2 + 29 r + 16) C 0 R 3 W endland functions φ 3 , 1 ( r ) = (1 − r ) 4 + (4 r + 1) C 2 R 3 φ 3 , 2 ( r ) = (1 − r ) 6 + (35 r 2 + 18 r + 3) C 4 R 3 φ 3 , 3 ( r ) = (1 − r ) 8 + (32 r 3 + 25 r 2 + 8 r + 1) C 6 R 3 2.2. BEMD-CSRBF with adjusting the support size T o determine the size of support for the CSRBF function for the rst IMF , we chose one of the distances used in [18] as the support size to ens ure that each e xtrema center of the image’ s support contains other e xtrema points. As mentioned in [18], in order to determine the 4 distances, we must rst e xtract the local m aximal and the local minimal from the image, we calculate the Euclidean distance between each local maximal element and its nearest maximal element. Subsequently , we generate a table of distances called adjacent maximal distance array (TdmaxA). Lik e wise, we calculat e the table of adjacent minimal distance array (TdminA). nally , the size of the support is chosen from these distances. w = d 1 = min { min { T dmaxA } , min { T dminA }} (3) Indonesian J Elec Eng & Comp Sci, V ol. 38, No. 1, April 2025: 172–181 Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian J Elec Eng & Comp Sci ISSN: 2502-4752 ❒ 175 w = d 2 = max { min { T dmaxA } , min { T dminA }} (4) w = d 3 = min { max { T dmaxA } , max { T dminA }} (5) w = d 4 = max { max { T dmaxA } , max { T dminA }} (6) Kno wing that the e xtrema decrease progressi v ely during the decomposition, we just multiply the initial support size by 2 after e xtracting each IMF . Then, from the second IMF , the support size is estimated without recalculating the prior distances or using a data partitioning approach, thus a v oiding increasi ng the comple xity of the computation. Then our methode follo ws the follo wing steps: i) Identify the e xtrema (both the maximal a n d the minimal) of our image I, using the neighborhood windo w approach to e xtract e xtrema points from 2D. ii) Determine the support size initial θ using one of distances in [18]. iii) Generate the rst IMF with a CSRBF function using the initial support size θ . i v) Generate the k-ieme IMF ( k ≥ 2 ) with support size equal to 2 k − 1 ∗ θ . Our research focuses on the interpolation procedure, specically on adjusting the support size of the CSRBF function. T o do this, we k ept the same CSRBF function (W endland function φ 3 , 1 ) that w as used in [19]. In our BEMD approach, the distance d 4 w as used as the initial support size for the CSRBF function. This decision w as based on this distance’ s impro v ed performance in comparison to other distances mentioned in [18], and limiting the maximal number of allo wed iterations (MN AI) to 5 for each IMF in order to pre v ent o v ertting. Finally , the standard de viation (SD) is used as the basic stopping criterion, with a limit of 0.5. 2.3. Ev aluation T o e v aluate the ef cac y of our method, we rst decomposed cameraman image using BEMD-CSRBF with an adjustable support size and compared it with traditional BEMD-CSRBF decomposition. This w as done to see the impact of the adjusted support size on the number of e xtrema during the proposed BEMD and the po wer of e xtraction of lo w frequencies. In addition, to thoroughly e v aluate our approach, we conducted a comparati v e analysis between the IMFs generated by the ne w BEMD method and those produced by alternati v e methods such as : F ABEMD [18] and BEMD-VNW [25]. The selection of F ABEMD and BEMD-VNW w as based on their ef fecti v eness and rapidity compared to other EBMD approaches. On the other hand, we chose the synthetic te xture image (STI) because it enables us to e v aluate the ef cac y of the BEMD approach using the orthogonality inde x (OI). The OI w as de v eloped in order to e v aluate the quality of IMFs [18]. This inde x’ s denition is as: O I = M X x =1 N X y =1 K +1 X i =1 K +1 X j =1 BEMC i − BEMC j P 2 B E M C ( x, y ) (7) W e refer to the IMFs and the residue as bidimensional empirical multimodal components (B EMCs). A smaller OI v alue indicates an optimal decomposition with respect to local orthogonality . In general, OI v alues of 0.1 or less are often re g arded as suf cient. Ho we v er , the F ABEMD is a BEMD approach that does a w ay with the interpolation phase. The upper and lo wer en v elopes are obtained from the image’ s e xtrema using order -stat istics lter , with just one iteration for each IMF (MN AI=1). Ho we v er , in order to pre v ent signicant discont inuities, both en v elopes need to ha v e a smoothing operator applied multiple times, which increases the computation time. While f ast BEMD based on v ariable neighborhood windo w method (BEMD- VNW) suggests replacing the square windo w used in the F ABEMD approach with a disc windo w , which has an isotropic structure element windo w , considering that the isotropic structural element windo w is more compatible with the image’ s properties, adjacent maximal and minimal v alues are a v eraged to determine the appropriate size for the windo w . 3. RESUL TS AND DISCUSSION 3.1. BEMD-CSRBF with adjusting the support size In this section, we used the cameraman image of size 128 × 128 as a simulation image in Figure 2. T able 2 sho ws the de v elopment of e xtrema during the IMFs with a support size of 10 for the CSRBF function. Enhancing BEMD decomposition using adaptive support size for CSRBF functions (Mohammed Arr azaki) Evaluation Warning : The document was created with Spire.PDF for Python.
176 ❒ ISSN: 2502-4752 The results corres ponding to a support size of 20 are represented belo w in T able 3. T able 2 sho ws that the de v elopment of the number of e xtrema (eit her the maximal or the minimal) remains stable, especially from the third IMF bet ween 136 and 156 for maximal and between 119 and 121 for minimal, with an e x ecution time of 10.30 seconds. At the size of support 20 as sho wn in T able 3, we nd that the de v elopment of the number of e xtrema is f aster , b ut in the last tw o IMFs, it remains stable with fe wer e xtremes. with an increase in e x ecution time (19.34 seconds), which is logical when we use a lar ger support size of the CSRBF functions in the inter - polation phase. Figure 2. Original image T able 2. Number of e xtrema and e x ecution time during BEMD decomposition with support size equal to 10 Example in Arabic Max Min IMF 1 656 637 IMF 2 214 199 IMF 3 156 121 IMF 4 136 119 T ime 10.30 s T able 3. Number of e xtrema and e x ecution time during BEMD decomposition with support size equal to 20 Number of e xtrema Max Min IMF1 708 694 IMF2 105 96 IMF3 53 53 IMF4 44 45 T ime 19.34 s In both cases, the number of e xtrema i s stable in the last IMFs due to a discontinuity of the signal using a x ed support size. Because the support no longer contains e xtrema for estimating the best en v elopes (upper and lo wer). This inuenced the e xtraction of lo w frequencies, as demonstrated in Figures 3 and 4. Figures 3(a) to 3(d) depict the IMFs e xtracted from the cameraman image with a x ed support size of 10, while Figures 4(a) to 4(d) display the IMFs e xtracted with a x ed support size of 20. T able 4 displays the e v olution of the number of e xtrema points, including both maximal and minimal v alues, as well as the time tak en for the BEMD decomposition of the cameraman image using a no v el method. This table sho ws ho w the number of e xtrema v aries with each IMF , of fering important insight into the perfor - mance and ef cienc y of the algorithm. Additionally , Figure 5 illustrates this image’ s BEMD decomposition, and Figures 5(a) to 5(d) sho w all of these IMFs. Indonesian J Elec Eng & Comp Sci, V ol. 38, No. 1, April 2025: 172–181 Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian J Elec Eng & Comp Sci ISSN: 2502-4752 ❒ 177 According to T able 4, we observ e that the number of e xtrema during BEMD is gradually decre asing from the rst IMF (708 for maxima and 694 for minima) to the fourth IMF (9 for maxima and 10 for minima). This type of e v olution allo ws us to e xtract the lo w frequencies, as we see in Figure 5. W e also noted that the calculation time (12.19 seconds) remains acceptable, especially since we ha v e ensured the e xtraction of lo w frequencies. (a) (b) (c) (d) Figure 3. BEMD-CSRBF with a support size of 10 on (a) IMF 1, (b) IMF 2, (c) IMF 3, and (d) IMF 4 (a) (b) (c) (d) Figure 4. BEMD-CSRBF with a support size of 20 on (a) IMF 1, (b) IMF 2, (c) IMF 3, and (d) IMF 4 T able 4. Number of e xtrema and e x ecution time during BEMD decomposition with adjusting support size Number of e xtrema Max Min IMF1 708 694 IMF2 116 117 IMF3 27 29 IMF4 9 10 T ime 12.19 s (a) (b) (c) (d) Figure 5. BEMD-CSRBF with adjusting support size: (a) IMF 1, (b) IMF 2, (c) IMF 3, and (d) IMF 4 3.2. Analyses and discussion using a synthetic textur e image In this section, we utilized a STI of size 256 × 256 pix els. The STI w as generated by combining three synthetic component images (SCIs). Both the STI and SCIs are sho wn in Figure 6. W e create each SCI by emplo ying sinusoidal w a v es with slight v ariation in f requenc y in both the horizontal and v ertical directions. Enhancing BEMD decomposition using adaptive support size for CSRBF functions (Mohammed Arr azaki) Evaluation Warning : The document was created with Spire.PDF for Python.
178 ❒ ISSN: 2502-4752 Figure 6(a) illustrates the higher frequencies (SCI 1), Figure 6(b) illustrates the middle frequencies (SCI 2), Figure 6(c) illustrates the lo w frequencies (SCI 3), and F igure 6(d) presents the STI. T able 5 presents the global mean of each component SCI and the OI v alues of the STI. The STI in Figure 6(c) w as used to apply a no v el algorithm. Figures 7 to 9 demonstrate that IMFs resembled the original SCIs. Figure 7 sho ws the IMFs as sho wn in Figures 7(a) to 7(c) and their summation in Figure 7(d) resulting from the F ABEMD. The IMFs of t he STI as sho wn in Figures 8(a) to 8(c) and their summation in Figures 8(d) corresponding to BEMD-VNW are presented in Figure 8. And Figure 9 present the IMFs in Figures 9(a) to 9(c) and their summation in Figure 9(d) resulting from the pr o pos ed BEMD. The proposed BEMD generates 3 IMFs (SCI), which is the same number as the original STI. This indicates that the decomposition of the method is appropriate. (a) (b) (c) (d) Figure 6. STI and their SCIs; (a) SCI-1, (b) SCI-2, (c) SCI-3, and (d) STI obtained by adding (a) to (c) T able 5. Inde x of orthogonality and global mean of SCIs SCI 1 SCI 2 SCI 3 Global mean 0.0046 0.0622 -0.287 OI 0.0488 (a) (b) (c) (d) Figure 7. STI decomposition using F ABEMD: (a) IMF 1, (b) IMF 2, (c) IMF 3, (d) STI obtained by adding (a) to (c) (a) (b) (c) (d) Figure 8. STI decomposition using BEMD based on v ariable neighborhood: (a) IMF 1, (b) IMF 2, (c) IMF 3, (d) STI obtained by adding (a) to (c) Indonesian J Elec Eng & Comp Sci, V ol. 38, No. 1, April 2025: 172–181 Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian J Elec Eng & Comp Sci ISSN: 2502-4752 ❒ 179 (a) (b) (c) (d) Figure 9. STI decomposition using proposed BEMD: (a) IMF 1, (b) IMF 2, (c) IMF 3, (d) STI obtained by adding (a) to (c) T o conduct a more thorough e v aluation of the no v el BEMD and tw o other BEMD methods, T able 6 presents the number of acquired IMFs, the time required, and the OI v alues of the decomposition of the STI for each algorithm. W e observ ed that the number of IMFs remains 3 in the proposed BEMD and other methods of BEMD. This ef fecti v ely supports the number of components (SCIs) of STI, indicating that the y e xhibit good decompositions. Compared to the e x ecution time and OI v alues, we observ e that the suggested approach has an impro v ed e x ecution time of 3.23 seconds and a more ef fecti v ely OI v alue of 0.0975. In general, t h e se characteristics indicate that the proposed is an appropriate choice for the decompo- sition of the STI in Figure 1(d) in comparison with the F ABEMD and BEMD-VNW algorithms, kno wing that the OI v alues and time required for the corresponding no v el method are less than those corresponding to other BEMD methods. which mak es this method a f a v ourable choice for v arious imagery applications, especially those focused on feature e xtraction, including image inde xing and f acial recognition. T able 6. Comparing the proposed BEMD, F ABEMD and BEMD-VNW for the STI, including the total number of BEMCs, total time needed, and OI Proposed BEMD F ABEMD BEMD-VNW T otal no. of IMFs 3 3 3 T otal time (seconds) 3.23 4,09 3.78 OI 0.0975 0.1102 0.1049 4. CONCLUSION This article presents an impro v ement to the BEMD-CSRBF algorithm for ef cient lo w frequenc y e xtraction (the last IMFs). This method is based on adjusting the support size of the CSRBF function in the BEMD algorithm. T o e xtract the rst IMF , we determine the support size of the CSRBF function based on the e xtrem a distrib ution, and then the other IMFs are e xtracted using a dynamic support size based on the initial support size. This approach demonstrates impro v ed qu a lity in the BEMD-CSRBF decomposition and pro v es its ef fecti v eness compared to other BEMD decomposition methods, both in terms of quality of IMFs and comput ational comple xity . Future research will focus on the CSRBF function’ s adapti v e choice depending on the image used and its frequenc y content. REFERENCES [1] N. E. Huang et al. , “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis, ” Pr oceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences , v ol. 454, no. 1971, pp. 903–995, Mar . 1998, doi: 10.1098/rspa.1998.0193. [2] P . Flandrin, G. Rill ing, and P . Gonca lv es, “Empirical mode decomposition as a lter bank, ” IEEE Signal Pr ocessing Letter s , v ol. 11, no. 2, pp. 112–114, Feb . 2004, doi: 10.1109/LSP .2003.821662. [3] J. -C. Nunes, Y . Bouaoune, E. Delechelle, O. Niang, and P . Bunel, “Image analysis by bidimensional empirical mode decomposi- tion, ” Ima g e and V ision Computing , v ol. 21, no. 12, pp. 1019–1026, No v . 2003, doi: 10.1016/S0262-8856(03)00094-5. [4] Sumanto, A. Buono, K. Priandana, B. P . Silalahi, and E. S. Hendrastuti, “T e xture analysis of citrus leaf images using BEMD for Huanglongbing disease diagnosis, ” J urnal Online Informatika , v ol. 8, no. 1, pp. 115–121, Jun. 2023, doi: 10.15575/join.v8i1.1075. [5] Z. Y ang, D. Qi, and L. Y ang, “ Signal period analysis based on Hilbert-Huang transform and its application to te xture analysis, ” in Pr oceedings - Thir d International Confer ence on Ima g e and Gr aphics , 2004, pp. 430–433, doi: 10.1109/icig.2004.129. [6] A. Sabri, M. Karoud, and H. T . A. Aarab, “ An ef cient image retrie v al approach based on spati al correlation of the e xtrema points of the IMEs, ” Re vie w Liter atur e And Arts Of The Americas , v ol. 3, no. No v ember , pp. 597–605, 2008. Enhancing BEMD decomposition using adaptive support size for CSRBF functions (Mohammed Arr azaki) Evaluation Warning : The document was created with Spire.PDF for Python.
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An, “F ast bi-dimensiona l empirical mode decomposition (BEMD) based on v ariable neighborhood windo w method, ” Multimedia T ools and Applications , v ol. 78, no. 7, pp. 8889–8910, Apr . 2019, doi: 10.1007/s11042-018-6629-6. BIOGRAPHIES OF A UTHORS Mohammed Arrazaki recei v ed his Ph.D. de gree in 2021 for his research w ork in applied mathematics from F aculty of Sciences T etouan, Morocco. His elds of interest are applied mathe- matics, image analysis, image processing, machine learning, and deep learning. He can be contacted at email: rezaki mohamed@hotmail.com. Othman El Ouahabi is a Ph.D. student in National School of Applied Sciences of T angier , Morocco. His elds of interest are image processing, machine learning, and deep neural netw orks. He can be contacted at email: othman.elouahabi1@etu.uae.ac.ma. Indonesian J Elec Eng & Comp Sci, V ol. 38, No. 1, April 2025: 172–181 Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian J Elec Eng & Comp Sci ISSN: 2502-4752 ❒ 181 Mohamed Zohry is a professor at the F aculty of Sciences T etouan, Morocco. He recei v ed his Ph.D. de gree in 1990 from F aculty of Sciences, Rabat, Morocco. And Ph.D. in 1996 at Granada Uni v ersity (Spain). His research areas include analysis, geometry and topology , number theory , ap- plied mathematics, statistics, and probability theory . He can be contacted at email: zohry@hotmail.fr . Adel Bab bah is a professor at the Polydisciplinary F aculty of Larache, Morocco. He recei v ed his Ph.D. de gree in 2016 from F aculty of Sciences, T etouan, Morocco. His research areas include functional analysis, operators theory , applied mathematics, and Comple x analysis. He can be contacted at email: adel.groupe@gmail.com. Enhancing BEMD decomposition using adaptive support size for CSRBF functions (Mohammed Arr azaki) Evaluation Warning : The document was created with Spire.PDF for Python.