Abstract

A high-precision and fast algorithm for computation of Jacobi–Fourier moments (JFMs) is presented. A fast recursive method is developed for the radial polynomials that occur in the kernel function of the JFMs. The proposed method is numerically stable and very fast in comparison with the conventional direct method. Moreover, the algorithm is suitable for computation of the JFMs of the highest orders. The JFMs are generic expressions to generate orthogonal moments changing the parameters α and β of Jacobi polynomials. The quality of the description of the proposed method with α and β parameters known is studied. Also, a search is performed of the best parameters, α and β, which significantly improves the quality of the reconstructed image and recognition. Experiments are performed on standard test images with various sets of JFMs to prove the superiority of the proposed method in comparison with the direct method. Furthermore, the proposed method is compared with other existing methods in terms of speed and accuracy.

© 2013 Optical Society of America

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2013

T. Hoang and S. Tabbone, “Errata and comments on ‘generic orthogonal moments: Jacobi–Fourier moments for invariant image description,” Pattern Recogn. 46, 3148–3155 (2013).
[CrossRef]

C. Singh, E. Walia, and R. Upneja, “Accurate calculation of Zernike moments,” Inf. Sci. 233, 255–275 (2013).
[CrossRef]

2012

C. Singh and R. Upneja, “Accurate computation of orthogonal Fourier–Mellin moments,” J. Math. Imaging Vision 44, 411–431 (2012).
[CrossRef]

E. Walia, C. Singh, and A. Goyal, “On the fast computation of orthogonal Fourier–Mellin moments with improved numerical stability,” J. Real-Time Image Process. 7, 247–256 (2012).
[CrossRef]

R. Biswas and S. Biswas, “Polar Zernike moments and rotational invariance,” Opt. Eng. 51, 087204 (2012).
[CrossRef]

2011

N. V. S. Sree Rathna Lakshmi and C. Manoharan, “An automated system for classification of micro calcification in mammogram based on Jacobi moments,” IJCTE 3, 431–434 (2011).
[CrossRef]

K. M. Hosny, M. A. Shouman, and H. M. Abdel Salam, “Fast computation of orthogonal Fourier–Mellin moments in polar coordinates,” J. Real-Time Image Process. 6, 73–80 (2011).
[CrossRef]

C. Toxqui-Quitl, L. Gutierrez-Lazcano, A. Padilla-Vivanco, and C. Camacho-Bello, “Gray-level image reconstruction using Bessel–Fourier moments,” Proc. SPIE 8011, 80112T (2011).
[CrossRef]

2010

B. Xiao, J. F. Ma, and X. Wang, “Image analysis by Bessel–Fourier moments,” Pattern Recogn. 43, 2620–2629 (2010).
[CrossRef]

2009

C. Toxqui-Quitl, A. Padilla-Vivanco, and J. Baez-Rojas, “Classification of mechanical parts using an optical-digital system and the Jacobi–Fourier moments,” Proc. SPIE 7389, 738934 (2009).
[CrossRef]

H. Hu and P. Zi-liang, “Computation of orthogonal Fourier–Mellin moments in two coordinate systems,” J. Opt. Soc. Am. A 26, 1080–1084 (2009).
[CrossRef]

2007

A. Padilla-Vivanco, G. Urcid-Serrano, F. Granados-Agustín, and A. Cornejo-Rodríguez, “Comparative analysis of pattern reconstruction using orthogonal moments,” Opt. Eng. 46, 017002 (2007).
[CrossRef]

Y. Xin, M. Pawlak, and S. Liao, “Accurate computation of Zernike moments in polar coordinates,” IEEE Trans. Image Process. 16, 581–587 (2007).
[CrossRef]

C. Y. Wee and R. Paramesran, “On the computational aspects of Zernike moments,” Image Vis. Comput. 25, 967–980 (2007).
[CrossRef]

Z. Ping, H. Ren, J. Zou, Y. Sheng, and W. Bo, “Generic orthogonal moments: Jacobi–Fourier moments for invariant image description,” Pattern Recogn. 40, 1245–1254 (2007).
[CrossRef]

G. A. Papakostas, Y. S. Boutalis, D. A. Karras, and B. G. Mertzios, “Fast numerically stable computation of orthogonal Fourier–Mellin moments,” IET Comput. Vis. 1, 11–16 (2007).
[CrossRef]

2004

2003

2002

1998

S. X. Liao and M. Pawlak, “On the accuracy of Zernike moments for image analysis,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 1358–1364 (1998).
[CrossRef]

1995

R. Mukundan and K. R. Ramakrishnan, “Fast computation of Legendre and Zernike moments,” Pattern Recogn. 28, 1433–1442 (1995).
[CrossRef]

1994

1981

R. G. Keys, “Cubic convolution interpolation for digital image processing,” IEEE Trans. Acoust. Speech Signal Process. 29, 1153–1160 (1981).
[CrossRef]

1980

1954

A. B. Bhatia and E. Wolf, “On the circular polynomials of Zernike and related orthogonal sets,” Proc. Cambridge Philos. Soc. 50, 40–48 (1954).
[CrossRef]

Abdel Salam, H. M.

K. M. Hosny, M. A. Shouman, and H. M. Abdel Salam, “Fast computation of orthogonal Fourier–Mellin moments in polar coordinates,” J. Real-Time Image Process. 6, 73–80 (2011).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Functions with Formulas, Graphs and Mathematical Tables (Dover, 1964).

Amu, G.

Baez-Rojas, J.

C. Toxqui-Quitl, A. Padilla-Vivanco, and J. Baez-Rojas, “Classification of mechanical parts using an optical-digital system and the Jacobi–Fourier moments,” Proc. SPIE 7389, 738934 (2009).
[CrossRef]

Bhatia, A. B.

A. B. Bhatia and E. Wolf, “On the circular polynomials of Zernike and related orthogonal sets,” Proc. Cambridge Philos. Soc. 50, 40–48 (1954).
[CrossRef]

Biswas, R.

R. Biswas and S. Biswas, “Polar Zernike moments and rotational invariance,” Opt. Eng. 51, 087204 (2012).
[CrossRef]

Biswas, S.

R. Biswas and S. Biswas, “Polar Zernike moments and rotational invariance,” Opt. Eng. 51, 087204 (2012).
[CrossRef]

Bo, W.

Z. Ping, H. Ren, J. Zou, Y. Sheng, and W. Bo, “Generic orthogonal moments: Jacobi–Fourier moments for invariant image description,” Pattern Recogn. 40, 1245–1254 (2007).
[CrossRef]

H. Ren, Z. Ping, W. Bo, W. Wu, and Y. Sheng, “Multi-distorted invariant image recognition with radial-harmonic-Fourier moments,” J. Opt. Soc. Am. A 20, 631–637 (2003).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Boutalis, Y. S.

G. A. Papakostas, Y. S. Boutalis, D. A. Karras, and B. G. Mertzios, “Fast numerically stable computation of orthogonal Fourier–Mellin moments,” IET Comput. Vis. 1, 11–16 (2007).
[CrossRef]

Camacho-Bello, C.

C. Toxqui-Quitl, L. Gutierrez-Lazcano, A. Padilla-Vivanco, and C. Camacho-Bello, “Gray-level image reconstruction using Bessel–Fourier moments,” Proc. SPIE 8011, 80112T (2011).
[CrossRef]

C. Camacho-Bello, C. Toxqui-Quitl, and A. Padilla-Vivanco, “Gait recognition by Jacobi–Fourier moments,” in Frontiers in Optics/Laser Science XXVII, OSA Technical Digest (Optical Society of America, 2011), paper JTuA19.

Cornejo-Rodríguez, A.

A. Padilla-Vivanco, G. Urcid-Serrano, F. Granados-Agustín, and A. Cornejo-Rodríguez, “Comparative analysis of pattern reconstruction using orthogonal moments,” Opt. Eng. 46, 017002 (2007).
[CrossRef]

Goyal, A.

E. Walia, C. Singh, and A. Goyal, “On the fast computation of orthogonal Fourier–Mellin moments with improved numerical stability,” J. Real-Time Image Process. 7, 247–256 (2012).
[CrossRef]

Granados-Agustín, F.

A. Padilla-Vivanco, G. Urcid-Serrano, F. Granados-Agustín, and A. Cornejo-Rodríguez, “Comparative analysis of pattern reconstruction using orthogonal moments,” Opt. Eng. 46, 017002 (2007).
[CrossRef]

Gutierrez-Lazcano, L.

C. Toxqui-Quitl, L. Gutierrez-Lazcano, A. Padilla-Vivanco, and C. Camacho-Bello, “Gray-level image reconstruction using Bessel–Fourier moments,” Proc. SPIE 8011, 80112T (2011).
[CrossRef]

Hasi, S.

Hoang, T.

T. Hoang and S. Tabbone, “Errata and comments on ‘generic orthogonal moments: Jacobi–Fourier moments for invariant image description,” Pattern Recogn. 46, 3148–3155 (2013).
[CrossRef]

Hosny, K. M.

K. M. Hosny, M. A. Shouman, and H. M. Abdel Salam, “Fast computation of orthogonal Fourier–Mellin moments in polar coordinates,” J. Real-Time Image Process. 6, 73–80 (2011).
[CrossRef]

Hu, H.

Karras, D. A.

G. A. Papakostas, Y. S. Boutalis, D. A. Karras, and B. G. Mertzios, “Fast numerically stable computation of orthogonal Fourier–Mellin moments,” IET Comput. Vis. 1, 11–16 (2007).
[CrossRef]

Keys, R. G.

R. G. Keys, “Cubic convolution interpolation for digital image processing,” IEEE Trans. Acoust. Speech Signal Process. 29, 1153–1160 (1981).
[CrossRef]

Liao, S.

Y. Xin, M. Pawlak, and S. Liao, “Accurate computation of Zernike moments in polar coordinates,” IEEE Trans. Image Process. 16, 581–587 (2007).
[CrossRef]

Liao, S. X.

S. X. Liao and M. Pawlak, “On the accuracy of Zernike moments for image analysis,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 1358–1364 (1998).
[CrossRef]

Ma, J. F.

B. Xiao, J. F. Ma, and X. Wang, “Image analysis by Bessel–Fourier moments,” Pattern Recogn. 43, 2620–2629 (2010).
[CrossRef]

Manoharan, C.

N. V. S. Sree Rathna Lakshmi and C. Manoharan, “An automated system for classification of micro calcification in mammogram based on Jacobi moments,” IJCTE 3, 431–434 (2011).
[CrossRef]

Mertzios, B. G.

G. A. Papakostas, Y. S. Boutalis, D. A. Karras, and B. G. Mertzios, “Fast numerically stable computation of orthogonal Fourier–Mellin moments,” IET Comput. Vis. 1, 11–16 (2007).
[CrossRef]

Mukundan, R.

R. Mukundan and K. R. Ramakrishnan, “Fast computation of Legendre and Zernike moments,” Pattern Recogn. 28, 1433–1442 (1995).
[CrossRef]

Padilla-Vivanco, A.

C. Toxqui-Quitl, L. Gutierrez-Lazcano, A. Padilla-Vivanco, and C. Camacho-Bello, “Gray-level image reconstruction using Bessel–Fourier moments,” Proc. SPIE 8011, 80112T (2011).
[CrossRef]

C. Toxqui-Quitl, A. Padilla-Vivanco, and J. Baez-Rojas, “Classification of mechanical parts using an optical-digital system and the Jacobi–Fourier moments,” Proc. SPIE 7389, 738934 (2009).
[CrossRef]

A. Padilla-Vivanco, G. Urcid-Serrano, F. Granados-Agustín, and A. Cornejo-Rodríguez, “Comparative analysis of pattern reconstruction using orthogonal moments,” Opt. Eng. 46, 017002 (2007).
[CrossRef]

C. Camacho-Bello, C. Toxqui-Quitl, and A. Padilla-Vivanco, “Gait recognition by Jacobi–Fourier moments,” in Frontiers in Optics/Laser Science XXVII, OSA Technical Digest (Optical Society of America, 2011), paper JTuA19.

Papakostas, G. A.

G. A. Papakostas, Y. S. Boutalis, D. A. Karras, and B. G. Mertzios, “Fast numerically stable computation of orthogonal Fourier–Mellin moments,” IET Comput. Vis. 1, 11–16 (2007).
[CrossRef]

Paramesran, R.

C. Y. Wee and R. Paramesran, “On the computational aspects of Zernike moments,” Image Vis. Comput. 25, 967–980 (2007).
[CrossRef]

Pawlak, M.

Y. Xin, M. Pawlak, and S. Liao, “Accurate computation of Zernike moments in polar coordinates,” IEEE Trans. Image Process. 16, 581–587 (2007).
[CrossRef]

S. X. Liao and M. Pawlak, “On the accuracy of Zernike moments for image analysis,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 1358–1364 (1998).
[CrossRef]

Ping, Z.

Ping, Z. L.

Ramakrishnan, K. R.

R. Mukundan and K. R. Ramakrishnan, “Fast computation of Legendre and Zernike moments,” Pattern Recogn. 28, 1433–1442 (1995).
[CrossRef]

Ren, H.

Z. Ping, H. Ren, J. Zou, Y. Sheng, and W. Bo, “Generic orthogonal moments: Jacobi–Fourier moments for invariant image description,” Pattern Recogn. 40, 1245–1254 (2007).
[CrossRef]

H. Ren, Z. Ping, W. Bo, W. Wu, and Y. Sheng, “Multi-distorted invariant image recognition with radial-harmonic-Fourier moments,” J. Opt. Soc. Am. A 20, 631–637 (2003).
[CrossRef]

Shen, L. X.

Sheng, Y.

Z. Ping, H. Ren, J. Zou, Y. Sheng, and W. Bo, “Generic orthogonal moments: Jacobi–Fourier moments for invariant image description,” Pattern Recogn. 40, 1245–1254 (2007).
[CrossRef]

H. Ren, Z. Ping, W. Bo, W. Wu, and Y. Sheng, “Multi-distorted invariant image recognition with radial-harmonic-Fourier moments,” J. Opt. Soc. Am. A 20, 631–637 (2003).
[CrossRef]

Sheng, Y. L.

Shouman, M. A.

K. M. Hosny, M. A. Shouman, and H. M. Abdel Salam, “Fast computation of orthogonal Fourier–Mellin moments in polar coordinates,” J. Real-Time Image Process. 6, 73–80 (2011).
[CrossRef]

Singh, C.

C. Singh, E. Walia, and R. Upneja, “Accurate calculation of Zernike moments,” Inf. Sci. 233, 255–275 (2013).
[CrossRef]

E. Walia, C. Singh, and A. Goyal, “On the fast computation of orthogonal Fourier–Mellin moments with improved numerical stability,” J. Real-Time Image Process. 7, 247–256 (2012).
[CrossRef]

C. Singh and R. Upneja, “Accurate computation of orthogonal Fourier–Mellin moments,” J. Math. Imaging Vision 44, 411–431 (2012).
[CrossRef]

Sree Rathna Lakshmi, N. V. S.

N. V. S. Sree Rathna Lakshmi and C. Manoharan, “An automated system for classification of micro calcification in mammogram based on Jacobi moments,” IJCTE 3, 431–434 (2011).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Functions with Formulas, Graphs and Mathematical Tables (Dover, 1964).

Tabbone, S.

T. Hoang and S. Tabbone, “Errata and comments on ‘generic orthogonal moments: Jacobi–Fourier moments for invariant image description,” Pattern Recogn. 46, 3148–3155 (2013).
[CrossRef]

Teague, M. R.

Toxqui-Quitl, C.

C. Toxqui-Quitl, L. Gutierrez-Lazcano, A. Padilla-Vivanco, and C. Camacho-Bello, “Gray-level image reconstruction using Bessel–Fourier moments,” Proc. SPIE 8011, 80112T (2011).
[CrossRef]

C. Toxqui-Quitl, A. Padilla-Vivanco, and J. Baez-Rojas, “Classification of mechanical parts using an optical-digital system and the Jacobi–Fourier moments,” Proc. SPIE 7389, 738934 (2009).
[CrossRef]

C. Camacho-Bello, C. Toxqui-Quitl, and A. Padilla-Vivanco, “Gait recognition by Jacobi–Fourier moments,” in Frontiers in Optics/Laser Science XXVII, OSA Technical Digest (Optical Society of America, 2011), paper JTuA19.

Upneja, R.

C. Singh, E. Walia, and R. Upneja, “Accurate calculation of Zernike moments,” Inf. Sci. 233, 255–275 (2013).
[CrossRef]

C. Singh and R. Upneja, “Accurate computation of orthogonal Fourier–Mellin moments,” J. Math. Imaging Vision 44, 411–431 (2012).
[CrossRef]

Urcid-Serrano, G.

A. Padilla-Vivanco, G. Urcid-Serrano, F. Granados-Agustín, and A. Cornejo-Rodríguez, “Comparative analysis of pattern reconstruction using orthogonal moments,” Opt. Eng. 46, 017002 (2007).
[CrossRef]

Walia, E.

C. Singh, E. Walia, and R. Upneja, “Accurate calculation of Zernike moments,” Inf. Sci. 233, 255–275 (2013).
[CrossRef]

E. Walia, C. Singh, and A. Goyal, “On the fast computation of orthogonal Fourier–Mellin moments with improved numerical stability,” J. Real-Time Image Process. 7, 247–256 (2012).
[CrossRef]

Wang, X.

B. Xiao, J. F. Ma, and X. Wang, “Image analysis by Bessel–Fourier moments,” Pattern Recogn. 43, 2620–2629 (2010).
[CrossRef]

Wee, C. Y.

C. Y. Wee and R. Paramesran, “On the computational aspects of Zernike moments,” Image Vis. Comput. 25, 967–980 (2007).
[CrossRef]

Wolf, E.

A. B. Bhatia and E. Wolf, “On the circular polynomials of Zernike and related orthogonal sets,” Proc. Cambridge Philos. Soc. 50, 40–48 (1954).
[CrossRef]

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Wu, R. G.

Wu, W.

Xiao, B.

B. Xiao, J. F. Ma, and X. Wang, “Image analysis by Bessel–Fourier moments,” Pattern Recogn. 43, 2620–2629 (2010).
[CrossRef]

Xin, Y.

Y. Xin, M. Pawlak, and S. Liao, “Accurate computation of Zernike moments in polar coordinates,” IEEE Trans. Image Process. 16, 581–587 (2007).
[CrossRef]

Yang, X.

Zi-liang, P.

Zou, J.

Z. Ping, H. Ren, J. Zou, Y. Sheng, and W. Bo, “Generic orthogonal moments: Jacobi–Fourier moments for invariant image description,” Pattern Recogn. 40, 1245–1254 (2007).
[CrossRef]

Appl. Opt.

IEEE Trans. Acoust. Speech Signal Process.

R. G. Keys, “Cubic convolution interpolation for digital image processing,” IEEE Trans. Acoust. Speech Signal Process. 29, 1153–1160 (1981).
[CrossRef]

IEEE Trans. Image Process.

Y. Xin, M. Pawlak, and S. Liao, “Accurate computation of Zernike moments in polar coordinates,” IEEE Trans. Image Process. 16, 581–587 (2007).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell.

S. X. Liao and M. Pawlak, “On the accuracy of Zernike moments for image analysis,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 1358–1364 (1998).
[CrossRef]

IET Comput. Vis.

G. A. Papakostas, Y. S. Boutalis, D. A. Karras, and B. G. Mertzios, “Fast numerically stable computation of orthogonal Fourier–Mellin moments,” IET Comput. Vis. 1, 11–16 (2007).
[CrossRef]

IJCTE

N. V. S. Sree Rathna Lakshmi and C. Manoharan, “An automated system for classification of micro calcification in mammogram based on Jacobi moments,” IJCTE 3, 431–434 (2011).
[CrossRef]

Image Vis. Comput.

C. Y. Wee and R. Paramesran, “On the computational aspects of Zernike moments,” Image Vis. Comput. 25, 967–980 (2007).
[CrossRef]

Inf. Sci.

C. Singh, E. Walia, and R. Upneja, “Accurate calculation of Zernike moments,” Inf. Sci. 233, 255–275 (2013).
[CrossRef]

J. Math. Imaging Vision

C. Singh and R. Upneja, “Accurate computation of orthogonal Fourier–Mellin moments,” J. Math. Imaging Vision 44, 411–431 (2012).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Real-Time Image Process.

K. M. Hosny, M. A. Shouman, and H. M. Abdel Salam, “Fast computation of orthogonal Fourier–Mellin moments in polar coordinates,” J. Real-Time Image Process. 6, 73–80 (2011).
[CrossRef]

E. Walia, C. Singh, and A. Goyal, “On the fast computation of orthogonal Fourier–Mellin moments with improved numerical stability,” J. Real-Time Image Process. 7, 247–256 (2012).
[CrossRef]

Opt. Eng.

R. Biswas and S. Biswas, “Polar Zernike moments and rotational invariance,” Opt. Eng. 51, 087204 (2012).
[CrossRef]

A. Padilla-Vivanco, G. Urcid-Serrano, F. Granados-Agustín, and A. Cornejo-Rodríguez, “Comparative analysis of pattern reconstruction using orthogonal moments,” Opt. Eng. 46, 017002 (2007).
[CrossRef]

Pattern Recogn.

B. Xiao, J. F. Ma, and X. Wang, “Image analysis by Bessel–Fourier moments,” Pattern Recogn. 43, 2620–2629 (2010).
[CrossRef]

R. Mukundan and K. R. Ramakrishnan, “Fast computation of Legendre and Zernike moments,” Pattern Recogn. 28, 1433–1442 (1995).
[CrossRef]

T. Hoang and S. Tabbone, “Errata and comments on ‘generic orthogonal moments: Jacobi–Fourier moments for invariant image description,” Pattern Recogn. 46, 3148–3155 (2013).
[CrossRef]

Z. Ping, H. Ren, J. Zou, Y. Sheng, and W. Bo, “Generic orthogonal moments: Jacobi–Fourier moments for invariant image description,” Pattern Recogn. 40, 1245–1254 (2007).
[CrossRef]

Proc. Cambridge Philos. Soc.

A. B. Bhatia and E. Wolf, “On the circular polynomials of Zernike and related orthogonal sets,” Proc. Cambridge Philos. Soc. 50, 40–48 (1954).
[CrossRef]

Proc. SPIE

C. Toxqui-Quitl, L. Gutierrez-Lazcano, A. Padilla-Vivanco, and C. Camacho-Bello, “Gray-level image reconstruction using Bessel–Fourier moments,” Proc. SPIE 8011, 80112T (2011).
[CrossRef]

C. Toxqui-Quitl, A. Padilla-Vivanco, and J. Baez-Rojas, “Classification of mechanical parts using an optical-digital system and the Jacobi–Fourier moments,” Proc. SPIE 7389, 738934 (2009).
[CrossRef]

Other

C. Camacho-Bello, C. Toxqui-Quitl, and A. Padilla-Vivanco, “Gait recognition by Jacobi–Fourier moments,” in Frontiers in Optics/Laser Science XXVII, OSA Technical Digest (Optical Society of America, 2011), paper JTuA19.

M. Abramowitz and I. A. Stegun, Handbook of Functions with Formulas, Graphs and Mathematical Tables (Dover, 1964).

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

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Figures (16)

Fig. 1.
Fig. 1.

Cartesian pixel region for computing JFMs.

Fig. 2.
Fig. 2.

Graphs of Jacobi polynomials of order n=23 through the conventional method and the recurrence relation. (a) J23(1,1,r2), (b) J23(2,2,r), and (c) J23(2,3/2,r).

Fig. 3.
Fig. 3.

Polar pixel representation of an image.

Fig. 4.
Fig. 4.

Concentric sector Ωuv or polar pixel.

Fig. 5.
Fig. 5.

Boat image in a polar pixel scheme.

Fig. 6.
Fig. 6.

Boat image for different scales and rotations. (a) Original 512×512 pixel test image. (b) The image in (a) is scaled by k=0.75. (c) The image in (a) is scaled by k=0.75 and rotated by g=15°. (d) The image in (a) is scaled by k=0.50. (e) The image in (a) is scaled by k=0.50 and rotated by g=15°.

Fig. 7.
Fig. 7.

256×256 pixel reconstructions of the image in Fig. 4(d) using pseudo-Jacobi–Fourier moments, Legendre–Fourier moments, and Chebyshev–Fourier moments. The maximum orders of reconstruction are 20, 100, and 200.

Fig. 8.
Fig. 8.

256×256 pixel reconstructions of the image in Fig. 4(d) using orthogonal Fourier–Mellin moments, ZMs, and JFMs with α=8 and β=2. The maximum orders of reconstruction are 20, 60, 100, and 200.

Fig. 9.
Fig. 9.

NIRE of the 256×256 pixel boat image for different orthogonal moments.

Fig. 10.
Fig. 10.

Search space of the optimal parameters. (a) Low-order moments, L=20. (b) High-order moments, L=200.

Fig. 11.
Fig. 11.

Relative error between the first 100 JFMs of Figs. 6(a) and 6(c). (a) Results from the proposed method. (b) Results from the direct method.

Fig. 12.
Fig. 12.

NMSE for different scales. (a) Error calculated between Figs. 6(a) and 6(b). (b) Error computed between Figs. 6(a) and 6(d).

Fig. 13.
Fig. 13.

PSNR of the 256×256 pixel reconstruction of the boat image, corrupted by an additive Gaussian noise of zero mean and σ2=0, 0.02, 0.05, 0.01.

Fig. 14.
Fig. 14.

Boat image corrupted by an additive Gaussian noise of zero mean and σ2=0.02, 0.05, 0.01 and its respective reconstruction, with L=78, α=8, and β=2.

Fig. 15.
Fig. 15.

Computing times for different methods with a 256×256 pixel image.

Fig. 16.
Fig. 16.

Computation of NIRE for different methods with a 256×256 pixel image.

Tables (2)

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Table 1. Weights (ηk) and Location of Sampling Points (zk) for 10-Point Gaussian Quadrature

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Table 2. Comparison of Computations by Method

Equations (42)

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An,m=02π01f(r,θ)Pnm(r,θ)rdrdθ,
Pnm(r,θ)=Jn(α,β,r)exp(jmθ).
Jn(α,β,r)=w(α,β,r)bn(α,β)Gn(α,β,r),
Gn(α,β,r)=n!Γ(β)Γ(α+n)×s=0n(1)sΓ(α+n+s)(ns)!s!Γ(β+s)rs,
bn(α,β)=n!Γ2(β)Γ(αβ+n+1)Γ(β+n)Γ(α+n)(α+2n),
w(α,β,r)=(1r)αβrβ1,
A˜n,m=i=0M1j=0N1f(ri,j,θi,j)P˜nm(ri,j,θi,j),
ri,j=xi2+yj2ri,j1,θi,j=arctan(yjxi),
xi=1+2iN1,yj=1+2jM1,
DnJn(α,β,r)=(Cn1+12r)Jn1(α,β,r)Dn1Jn2(α,β,r),
Cn=(α1)(2βα1)(2n+α1)(2(n1)+α1),
Dn=4n(n+αβ)(n+β1)(n+α1)(2n+α1)2(2n+α)(2n+α2).
J0(α,β,r)=w(α,β,r)b0(α,β),
J1(α,β,r)=J0(α,β,r)(α+2)βαβ+1(1α+1βr).
A^nm=u=1Uv=1(2u1)Vf^(ruv,θuv)ωnm(ruv,θuv),
ωnm(ruv,θuv)=ΩuvJn(α,β,r)exp(jmθ)rdrdθ,=ruv(s)ruv(e)Jn(α,β,r)rdrθuv(s)θuv(e)exp(jmθ)dθ,=I1×I2,
I2={jm[exp(jmθuv(e))exp(jmθuv(s))],m0θuv(e)θuv(s),m=0.
I1=ruv(e)ruv(s)2k=110ηkJn(α,β,ruv(e)ruv(s)2zk+ruv(e)+ruv(s)2).
u(x)={32|x|352|x|2+10<|x|<112|x|3+52|x|24|x|+21<|x|<202<|x|.
f^(ruv,θuv)=i=k1k+2j=l1l+2f(i,j)u(ki)u(lj),
f˜(i,j)=n=0Lm=0L|A^nm|Jn(α,β,rij)exp(jmθij),
NIRE=i=0N1j=0M1[f˜(i,j)f(i,j)]2i=0N1j=0M1f2(i,j).
Ψ(α,β)=1pL=1pNIRE(L,α,β),
A^nm(γ,k)=A^nmeimγ.
|A^nm(γ,k)|=|A^nm|,
1k2|A^nm(γ,k)|=|A^nm|,
Mnm=02π01rnf(r/k,θγ)exp(imθ)rdrdθ,
M^nm=02π01rnf(r,θ)exp(imθ)rdrdθ,
k=[M10/M00][M^10/M^00].
δAnm=1k2|A^nm(γ,k)||A^nm||A^nm|,
NMSE(|A^nm|,|A^nm(γ,k)|)=1L2n=1Lm=1L(1k2|A^nm(γ,k)||A^nm|)2|A^nm|2,
PSNR=10log10(2552MSE).
MSE=1N×Mi=0N1j=0M1(f(i,j)f*(i,j))2,
Rm+2sm(r)=rmJs(m+1,m+1,r2),
Cs=m2(2s+m)(2s+m2),
Ds=2s(s+m)(2s+m)(2s+m+1)(2s+m1).
J0(m+1,m+1,r2)=1,
J1(m+1,m+1,r2)=m+3m+1r2.
JFMn,mα,β=02π01f(r,θ)Jn(α,β,r)eiθmdrdθ,
ZMn,m=02π01f(r,θ)Rnm(r)eiθmdrdθ,
ZMm+2s,m=02π01f(r,θ)rmJs(m+1,m+1,r2)eiθmdrdθ.
FMs,mm+1,m+1=02π01f(r,θ)Js(m+1,m+1,r)eiθmdrdθ,

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