Abstract

A modified bi-dimensional empirical mode decomposition (BEMD) method is proposed for sparsely decomposing a fringe pattern into two components, namely, a single intrinsic mode function (IMF) and a residue. The main idea of this method is a modified sifting process which employs morphological operations to detect ridges and troughs of the fringes, and uses weighted moving average algorithm to estimate envelopes of the IMF, replacing respective local extrema detection and envelope interpolation of conventional BEMDs. The background intensity of the fringe pattern is automatically removed by extracting the single IMF, thereby relieving the mode mixing problem of the BEMDs. A fast algorithm based on 2D convolution is also presented for reducing the calculation time to several seconds only. This approach is applied to process simulated and real fringe patterns, and the results obtained are compared with Fourier transform, discrete wavelet transform, and other EMD methods. The MATLAB code is downloadable at http://gr.xjtu.edu.cn/web/zhouxiang.

©2012 Optical Society of America

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    [Crossref]

2012 (1)

2011 (3)

2010 (1)

2009 (4)

S. Li, X. Su, W. Chen, and L. Xiang, “Eliminating the zero spectrum in Fourier transform profilometry using empirical mode decomposition,” J. Opt. Soc. Am. A 26(5), 1195–1201 (2009).
[Crossref] [PubMed]

X. Zhou, H. Zhao, and T. Jiang, “Adaptive analysis of optical fringe patterns using ensemble empirical mode decomposition algorithm,” Opt. Lett. 34(13), 2033–2035 (2009).
[Crossref] [PubMed]

X. Guanlei, W. Xiaotong, and X. Xiaogang, “Improved bi-dimensional EMD and Hilbert spectrum for the analysis of textures,” Pattern Recognit. 42(5), 718–734 (2009).
[Crossref]

S. M. A. Bhuiyan, N. Attoh-Okine, K. E. Barner, A. Y. Ayenu-Prah, and R. R. Adhami, “Bidimensional empirical mode decomposition using various interpolation techniques,” Adv. Adapt. Data Anal 01(02), 309–338 (2009).
[Crossref]

2008 (1)

2007 (5)

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16(8), 2080–2095 (2007).
[Crossref] [PubMed]

S. Ozder, O. Kocahan, E. Coşkun, and H. Göktaş, “Optical phase distribution evaluation by using an S-transform,” Opt. Lett. 32(6), 591–593 (2007).
[Crossref] [PubMed]

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45(2), 304–317 (2007).
[Crossref]

Q. Kemao, “On window size selection in the windowed Fourier ridges algorithm,” Opt. Lasers Eng. 45(12), 1186–1192 (2007).
[Crossref]

M. B. Bernini, G. E. Galizzi, A. Federico, and G. H. Kaufmann, “Evaluation of the 1D empirical mode decomposition method to smooth digital speckle pattern interferometry fringes,” Opt. Lasers Eng. 45(6), 723–729 (2007).
[Crossref]

2006 (1)

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Eliminating the zero spectrum in Fourier transform profilometry using a two-dimensional continuous wavelet transform,” Opt. Commun. 266(2), 482–489 (2006).
[Crossref]

2005 (4)

J. Nunes, S. Guyot, and E. Delechelle, “Texture analysis based on local analysis of the bidimensional empirical mode decomposition,” Mach. Vis. Appl. 16, 177–188 (2005).
[Crossref]

C. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE. Signal Proc. Lett. 12(10), 701–704 (2005).
[Crossref]

A. Linderhed, “Variable sampling of the empirical mode decomposition of two-dimensional signals,” Int. J. Wavelets Multiresolution Inf. Process. 03(03), 435–452 (2005).
[Crossref]

J. G. Zhong and J. W. Weng, “Phase retrieval of optical fringe patterns from the ridge of a wavelet transform,” Opt. Lett. 30(19), 2560–2562 (2005).
[Crossref] [PubMed]

2004 (4)

Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43(13), 2695–2702 (2004).
[Crossref] [PubMed]

J. G. Zhong and J. W. Weng, “Spatial carrier-fringe pattern analysis by means of wavelet transform: wavelet transform profilometry,” Appl. Opt. 43(26), 4993–4998 (2004).
[Crossref] [PubMed]

X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42(3), 245–261 (2004).
[Crossref]

J. Zhong and J. Weng, “Dilating Gabor transform for the fringe analysis of 3-D shape measurement,” Opt. Eng. 43(4), 895 (2004).
[Crossref]

2003 (2)

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and P. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput. 21(12), 1019–1026 (2003).
[Crossref]

A. Federico and G. H. Kaufmann, “Phase retrieval in digital speckle pattern interferometry by use of a smoothed space-frequency distribution,” Appl. Opt. 42(35), 7066–7071 (2003).
[Crossref] [PubMed]

2002 (1)

Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Proc. Lett. 9(3), 81–84 (2002).
[Crossref]

2001 (1)

J. A. Quiroga, J. Antonio Gómez-Pedrero, and Á. García-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197(1-3), 43–51 (2001).
[Crossref]

1999 (2)

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162(4-6), 205–210 (1999).
[Crossref]

L. R. Watkins, S. M. Tan, and T. H. Barnes, “Determination of interferometer phase distributions by use of wavelets,” Opt. Lett. 24(13), 905–907 (1999).
[Crossref] [PubMed]

1998 (1)

N. E. Huang, Z. Shen, S. R. Long, M. L. C. Wu, H. H. Shih, Q. N. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. London Ser. A-Math. Phys. Eng. Sci. 454(1971), 903–995 (1998).
[Crossref]

1997 (1)

1995 (1)

J. F. Lin and X. Y. Su, “Two-dimensional Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 34(11), 3297–3302 (1995).
[Crossref]

1983 (1)

Adhami, R. R.

S. M. A. Bhuiyan, N. Attoh-Okine, K. E. Barner, A. Y. Ayenu-Prah, and R. R. Adhami, “Bidimensional empirical mode decomposition using various interpolation techniques,” Adv. Adapt. Data Anal 01(02), 309–338 (2009).
[Crossref]

Aebischer, H. A.

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162(4-6), 205–210 (1999).
[Crossref]

Antonio Gómez-Pedrero, J.

J. A. Quiroga, J. Antonio Gómez-Pedrero, and Á. García-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197(1-3), 43–51 (2001).
[Crossref]

Attoh-Okine, N.

S. M. A. Bhuiyan, N. Attoh-Okine, K. E. Barner, A. Y. Ayenu-Prah, and R. R. Adhami, “Bidimensional empirical mode decomposition using various interpolation techniques,” Adv. Adapt. Data Anal 01(02), 309–338 (2009).
[Crossref]

Ayenu-Prah, A. Y.

S. M. A. Bhuiyan, N. Attoh-Okine, K. E. Barner, A. Y. Ayenu-Prah, and R. R. Adhami, “Bidimensional empirical mode decomposition using various interpolation techniques,” Adv. Adapt. Data Anal 01(02), 309–338 (2009).
[Crossref]

Barner, K. E.

S. M. A. Bhuiyan, N. Attoh-Okine, K. E. Barner, A. Y. Ayenu-Prah, and R. R. Adhami, “Bidimensional empirical mode decomposition using various interpolation techniques,” Adv. Adapt. Data Anal 01(02), 309–338 (2009).
[Crossref]

Barnes, T. H.

Bernini, M. B.

Bhuiyan, S. M. A.

S. M. A. Bhuiyan, N. Attoh-Okine, K. E. Barner, A. Y. Ayenu-Prah, and R. R. Adhami, “Bidimensional empirical mode decomposition using various interpolation techniques,” Adv. Adapt. Data Anal 01(02), 309–338 (2009).
[Crossref]

Bouaoune, Y.

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and P. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput. 21(12), 1019–1026 (2003).
[Crossref]

Bovik, A. C.

Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Proc. Lett. 9(3), 81–84 (2002).
[Crossref]

Bunel, P.

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and P. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput. 21(12), 1019–1026 (2003).
[Crossref]

Burton, D. R.

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Eliminating the zero spectrum in Fourier transform profilometry using a two-dimensional continuous wavelet transform,” Opt. Commun. 266(2), 482–489 (2006).
[Crossref]

Chen, W.

Coskun, E.

Cuevas, F. J.

Dabov, K.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16(8), 2080–2095 (2007).
[Crossref] [PubMed]

Damerval, C.

C. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE. Signal Proc. Lett. 12(10), 701–704 (2005).
[Crossref]

Delechelle, E.

J. Nunes, S. Guyot, and E. Delechelle, “Texture analysis based on local analysis of the bidimensional empirical mode decomposition,” Mach. Vis. Appl. 16, 177–188 (2005).
[Crossref]

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and P. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput. 21(12), 1019–1026 (2003).
[Crossref]

Egiazarian, K.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16(8), 2080–2095 (2007).
[Crossref] [PubMed]

Federico, A.

Foi, A.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16(8), 2080–2095 (2007).
[Crossref] [PubMed]

Galizzi, G. E.

M. B. Bernini, G. E. Galizzi, A. Federico, and G. H. Kaufmann, “Evaluation of the 1D empirical mode decomposition method to smooth digital speckle pattern interferometry fringes,” Opt. Lasers Eng. 45(6), 723–729 (2007).
[Crossref]

García-Botella, Á.

J. A. Quiroga, J. Antonio Gómez-Pedrero, and Á. García-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197(1-3), 43–51 (2001).
[Crossref]

Gdeisat, M. A.

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Eliminating the zero spectrum in Fourier transform profilometry using a two-dimensional continuous wavelet transform,” Opt. Commun. 266(2), 482–489 (2006).
[Crossref]

Göktas, H.

Guanlei, X.

X. Guanlei, W. Xiaotong, and X. Xiaogang, “Improved bi-dimensional EMD and Hilbert spectrum for the analysis of textures,” Pattern Recognit. 42(5), 718–734 (2009).
[Crossref]

Guyot, S.

J. Nunes, S. Guyot, and E. Delechelle, “Texture analysis based on local analysis of the bidimensional empirical mode decomposition,” Mach. Vis. Appl. 16, 177–188 (2005).
[Crossref]

Huang, N. E.

N. E. Huang, Z. Shen, S. R. Long, M. L. C. Wu, H. H. Shih, Q. N. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. London Ser. A-Math. Phys. Eng. Sci. 454(1971), 903–995 (1998).
[Crossref]

Jiang, T.

Katkovnik, V.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16(8), 2080–2095 (2007).
[Crossref] [PubMed]

Kaufmann, G. H.

Kemao, Q.

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45(2), 304–317 (2007).
[Crossref]

Q. Kemao, “On window size selection in the windowed Fourier ridges algorithm,” Opt. Lasers Eng. 45(12), 1186–1192 (2007).
[Crossref]

Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43(13), 2695–2702 (2004).
[Crossref] [PubMed]

Kocahan, O.

Lalor, M. J.

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Eliminating the zero spectrum in Fourier transform profilometry using a two-dimensional continuous wavelet transform,” Opt. Commun. 266(2), 482–489 (2006).
[Crossref]

Li, H. G.

Li, S.

Lin, J. F.

J. F. Lin and X. Y. Su, “Two-dimensional Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 34(11), 3297–3302 (1995).
[Crossref]

Linderhed, A.

A. Linderhed, “Variable sampling of the empirical mode decomposition of two-dimensional signals,” Int. J. Wavelets Multiresolution Inf. Process. 03(03), 435–452 (2005).
[Crossref]

Liu, H. H.

N. E. Huang, Z. Shen, S. R. Long, M. L. C. Wu, H. H. Shih, Q. N. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. London Ser. A-Math. Phys. Eng. Sci. 454(1971), 903–995 (1998).
[Crossref]

Long, S. R.

N. E. Huang, Z. Shen, S. R. Long, M. L. C. Wu, H. H. Shih, Q. N. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. London Ser. A-Math. Phys. Eng. Sci. 454(1971), 903–995 (1998).
[Crossref]

Marroquin, J. L.

Meignen, S.

C. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE. Signal Proc. Lett. 12(10), 701–704 (2005).
[Crossref]

Mutoh, K.

Niang, O.

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and P. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput. 21(12), 1019–1026 (2003).
[Crossref]

Nunes, J.

J. Nunes, S. Guyot, and E. Delechelle, “Texture analysis based on local analysis of the bidimensional empirical mode decomposition,” Mach. Vis. Appl. 16, 177–188 (2005).
[Crossref]

Nunes, J. C.

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and P. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput. 21(12), 1019–1026 (2003).
[Crossref]

Ozder, S.

Patorski, K.

Perrier, V.

C. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE. Signal Proc. Lett. 12(10), 701–704 (2005).
[Crossref]

Quiroga, J. A.

J. A. Quiroga, J. Antonio Gómez-Pedrero, and Á. García-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197(1-3), 43–51 (2001).
[Crossref]

Servin, M.

Shen, Z.

N. E. Huang, Z. Shen, S. R. Long, M. L. C. Wu, H. H. Shih, Q. N. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. London Ser. A-Math. Phys. Eng. Sci. 454(1971), 903–995 (1998).
[Crossref]

Shih, H. H.

N. E. Huang, Z. Shen, S. R. Long, M. L. C. Wu, H. H. Shih, Q. N. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. London Ser. A-Math. Phys. Eng. Sci. 454(1971), 903–995 (1998).
[Crossref]

Su, X.

Su, X. Y.

J. F. Lin and X. Y. Su, “Two-dimensional Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 34(11), 3297–3302 (1995).
[Crossref]

Takeda, M.

Tan, S. M.

Tung, C. C.

N. E. Huang, Z. Shen, S. R. Long, M. L. C. Wu, H. H. Shih, Q. N. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. London Ser. A-Math. Phys. Eng. Sci. 454(1971), 903–995 (1998).
[Crossref]

Waldner, S.

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162(4-6), 205–210 (1999).
[Crossref]

Wang, Z.

Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Proc. Lett. 9(3), 81–84 (2002).
[Crossref]

Watkins, L. R.

Weng, J.

J. Zhong and J. Weng, “Dilating Gabor transform for the fringe analysis of 3-D shape measurement,” Opt. Eng. 43(4), 895 (2004).
[Crossref]

Weng, J. W.

Wielgus, M.

Wu, M. L. C.

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Xiaotong, W.

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Yang, T.

Yen, N. C.

N. E. Huang, Z. Shen, S. R. Long, M. L. C. Wu, H. H. Shih, Q. N. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. London Ser. A-Math. Phys. Eng. Sci. 454(1971), 903–995 (1998).
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Zhao, H.

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N. E. Huang, Z. Shen, S. R. Long, M. L. C. Wu, H. H. Shih, Q. N. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. London Ser. A-Math. Phys. Eng. Sci. 454(1971), 903–995 (1998).
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[Crossref]

Proc. R. Soc. London Ser. A-Math. Phys. Eng. Sci. (1)

N. E. Huang, Z. Shen, S. R. Long, M. L. C. Wu, H. H. Shih, Q. N. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. London Ser. A-Math. Phys. Eng. Sci. 454(1971), 903–995 (1998).
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Figures (11)

Fig. 1
Fig. 1 Flowchart of the proposed algorithms (ridges are taken as example).

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Fig. 2
Fig. 2 Illustration of ridge detection and envelop estimation. (a) Simulated fringe pattern; (b) binary strips from the top-hat transformation; (c) the initial ridge map (red) and the refined ridge map (blue); (d) EDT of the refined ridge map; (e) the coarse and (f) smoothed upper envelopes; (g) window width distribution and (h) its histogram

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Fig. 3
Fig. 3 Decomposition of a mixed-carrier fringe pattern by MO-BEMD. (a) Simulated fringe pattern; (b) the IMF1; (c) the IMF2 and (d) the residue

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Fig. 4
Fig. 4 The 115th row signals and their spectra of the patterns shown in Fig. 3. (a) The original 115th row signal; (b) red curve: IMF1, blue curve: the ideal high carrier component; (c) red curve: IMF2, blue curve: the ideal low carrier component;; (d): red curve: the residue, blue curve: the ideal background component.(e)-(h) the FFT spectra of the left signals

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Fig. 5
Fig. 5 Decomposition of a noisy, closed fringe pattern. (a) The simulated fringe pattern and (b) its background; (c) the fringe pattern denoised by a low-pass filter; (d) the IMF and (e) the residue by MO-BEMD; (f) the reconstructed fringes from detailed coefficients at 1 to 4 levels and (g) the reconstructed background from approximation coefficients at 4th level by 2D DWT using a db9 wavelet; (h)–(l) the four IMFs and the residue by conventional BEMD

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Fig. 6
Fig. 6 Error evaluation of MO-BEMD on the patterns with different noise levels. (a) The improvements of IQIs after image preprocessing without denoising (red), with low-pass filter (green) and with BM3D (blue), respectively; (b) MSEs between the ideal background and the residues using corresponding preprocessing schemes.

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Fig. 7
Fig. 7 The decomposition results at four selected noise levels out of all levels in Fig. 6. The first column shows the noisy patterns at four IQIs. The next three columns contain pairs of IMFs and residues of MO-BEMD using different preprocessing schemes

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Fig. 8
Fig. 8 The phase error comparison of different methods. The phase error maps (a) by the frequency domain filtering technique and (b) by the MO-BEMD method. (c) The normalized pattern and (d) the phase error map by the normalized method. The analyzed fringe pattern is the same in Fig. 7 and its IQI is equal to 0.80.

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Fig. 9
Fig. 9 Decomposition of a projected fringe pattern on a plaster model. (a) Original fringe pattern; (b) the model under white light illumination; (c) the single IMF and (d) the residue by MO-BEMD; (e) the wrapped phase of the single IMF; (f) the reconstructed background by approximation coefficients at the 6th level of 2D DWT using a db9 wavelet; (g) the extracted background by Frequency domain filtering; (h)-(l) the results of the BEMD by subtracting from the pattern 1st, 1-2nd. 1-3rd, 1-4th, 1-5th IMFs, respectively.

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Fig. 10
Fig. 10 The relationship between the number of sifting iterations and MSE of the extracted IMF of the pattern in Fig. 2(a)

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Fig. 11
Fig. 11 Decomposition of a fingerprint image by MO-BEMD. (a) Original image; (b) the IMF; (c) the residue.

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Equations (14)

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I(x,y)=a(x,y)+b(x,y)cos[ φ(x,y) ]+n(x,y),
SD= ∑ x ∑ y [ S ′ (x,y)−S(x,y) ] 2 ∑ x ∑ y [ S(x,y) ] 2 .
I(x,y)= ∑ j=1 n c j (x,y) + r n (x,y).
T I (x,y)=I(x,y)−[ I(x,y)∘e(x,y) ],
D R (x,y)= min (s,t) { (x−s) 2 + (y−t) 2 ; R(s,t)=1 }.
D R R ′ = ∑ x ∑ y [R(x,y) ⊕ R ′ (x,y) ],
if ( x i , y i )∈ r ′ i , i=1,2,...N then E coarse up (x,y)=I( x i , y i ), ∀ (x,y)∈ r ′ i .
E smooth up (x,y)= 1 [ 2L(x,y)+1 ] 2 ∑ (s,t)∈ S xy E coarse up (s,t) ,
L(x,y)=round{ a⋅max (p,q)∈ r ′ i [ D R ′ (p,q) ] }; ∀(x,y)∈ r ′ i , i=1,2,...N,
C k up (x,y)= E coarse up (x,y)⊗ H k (x,y), k=1,2,...,K
H k (x,y)= 1 2π W k 2 e − x 2 + y 2 2 W k 2 .
E smooth up (x,y)= ∑ k=1 K C k up (x,y) ⋅ M k (x,y),
M k (x,y)={ 1, if L(x,y)= W k 0, otherwise , k=1,2,...,K.
I(x,y)=a⋅{ cos[ 1 8 πx+p(x,y) ]+cos[ 1 16 πx+ 1 2 p(x,y) ] }+b⋅ δp(x,y) δx +c,

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