Abstract

In this paper, we propose the use of geometric moments to the field of nonblind image deblurring. Using the developed relationship of geometric moments for original and blurred images, a mathematical formulation based on the Euler–Lagrange identity and variational techniques is proposed. It uses an iterative procedure to deblur the image in moment domain. The theoretical framework is validated by a set of experiments. A comparative analysis of the results obtained using the spatial and moment domains are evaluated using a quality assessment method known as the Blind/Reference-less Image Spatial Quality Evaluator (BRISQUE). The results show that the proposed method yields a higher quality score when compared with the spatial domain method for the same number of iterations.

© 2014 Optical Society of America

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References

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  1. J. Chen, W. Dong, H. Feng, Z. Xu, and Q. Li, “High quality non-blind image deconvolution using the fields of experts prior,” Optik 124, 3601–3606 (2013).
    [Crossref]
  2. M. Almeida and M. Figueiredo, “Parameter estimation for blind and non-blind deblurring using residual whiteness measures,” IEEE Trans. Image Process. 22, 2751–2763 (2013).
    [Crossref]
  3. S. Tao, W. Dong, H. Feng, Z. Xu, and Q. Li, “Non-blind image deconvolution using natural image gradient prior,” Optik 124, 6599–6605 (2013).
    [Crossref]
  4. S. Tang, W. Gong, W. Li, and W. Wang, “Non-blind image deblurring method by local and nonlocal total variation models,” Signal Process. 94, 339–349 (2014).
    [Crossref]
  5. E. Vera, M. Vega, R. Molina, and A. K. Katsaggelos, “Iterative image restoration using nonstationary priors,” Appl. Opt. 52, D-102–D-110 (2013).
    [Crossref]
  6. D. S. Stoker, J. Wedd, E. Lavelle, and J. van der Laan, “Restoration and recognition of distant, blurry irises,” Appl. Opt. 52, 1864–1875 (2013).
    [Crossref]
  7. R. Fergus, B. Singh, A. Hertzmann, S. T. Roweis, and W. T. Freeman, “Removing camera shake from a single photograph,” ACM Trans. Graph. 25, 787–794 (2006).
    [Crossref]
  8. M. Ben-Ezra and S. Nayar, “Motion deblurring using hybrid imaging,” in Computer Vision and Pattern Recognition (IEEE, 2003), Vol. 1, pp. I-657–I-664.
  9. L. Yuan, J. Sun, L. Quan, and H.-Y. Shum, “Image deblurring with blurred/noisy image pairs,” ACM Trans. Graph. 26, 1–10 (2007).
    [Crossref]
  10. R. Raskar, A. Agrawal, and J. Tumblin, “Coded exposure photography: motion deblurring using fluttered shutter,” ACM Trans. Graph. 25, 795–804 (2006).
    [Crossref]
  11. A. Levin, “Blind motion deblurring using image statistics,” in Advances in Neural Information Processing Systems, B. Schölkopf, J. Platt, and T. Hofmann, eds. (MIT, 2006), pp. 841–848.
  12. J. Jia, “Single image motion deblurring using transparency,” in Computer Vision and Pattern Recognition Conference (IEEE, 2007), pp. 1–8.
  13. T. Chan and J. Shen, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods (Society for Industrial and Applied Mathematics, 2005).
  14. J. Flusser, B. Zitova, and T. Suk, Moments and Moment Invariants in Pattern Recognition (Wiley, 2009).
  15. R. Mukundan, S. Ong, and A. Lee, “Image analysis by Tchebichef moments,” IEEE Trans. Image Process. 10, 1357–1364 (2001).
    [Crossref]
  16. P.-T. Yap, R. Paramesran, and S.-H. Ong, “Image analysis by Krawtchouk moments,” IEEE Trans. Image Process. 12, 1367–1377 (2003).
    [Crossref]
  17. A. Mittal, A. Moorthy, and A. Bovik, “No-reference image quality assessment in the spatial domain,” IEEE Trans. Image Process. 21, 4695–4708 (2012).
    [Crossref]
  18. http://classes.soe.ucsc.edu/ee264/Fall11/LecturePDF/5-LocalOperations.pdf .

2014 (1)

S. Tang, W. Gong, W. Li, and W. Wang, “Non-blind image deblurring method by local and nonlocal total variation models,” Signal Process. 94, 339–349 (2014).
[Crossref]

2013 (5)

E. Vera, M. Vega, R. Molina, and A. K. Katsaggelos, “Iterative image restoration using nonstationary priors,” Appl. Opt. 52, D-102–D-110 (2013).
[Crossref]

J. Chen, W. Dong, H. Feng, Z. Xu, and Q. Li, “High quality non-blind image deconvolution using the fields of experts prior,” Optik 124, 3601–3606 (2013).
[Crossref]

M. Almeida and M. Figueiredo, “Parameter estimation for blind and non-blind deblurring using residual whiteness measures,” IEEE Trans. Image Process. 22, 2751–2763 (2013).
[Crossref]

S. Tao, W. Dong, H. Feng, Z. Xu, and Q. Li, “Non-blind image deconvolution using natural image gradient prior,” Optik 124, 6599–6605 (2013).
[Crossref]

D. S. Stoker, J. Wedd, E. Lavelle, and J. van der Laan, “Restoration and recognition of distant, blurry irises,” Appl. Opt. 52, 1864–1875 (2013).
[Crossref]

2012 (1)

A. Mittal, A. Moorthy, and A. Bovik, “No-reference image quality assessment in the spatial domain,” IEEE Trans. Image Process. 21, 4695–4708 (2012).
[Crossref]

2007 (1)

L. Yuan, J. Sun, L. Quan, and H.-Y. Shum, “Image deblurring with blurred/noisy image pairs,” ACM Trans. Graph. 26, 1–10 (2007).
[Crossref]

2006 (2)

R. Raskar, A. Agrawal, and J. Tumblin, “Coded exposure photography: motion deblurring using fluttered shutter,” ACM Trans. Graph. 25, 795–804 (2006).
[Crossref]

R. Fergus, B. Singh, A. Hertzmann, S. T. Roweis, and W. T. Freeman, “Removing camera shake from a single photograph,” ACM Trans. Graph. 25, 787–794 (2006).
[Crossref]

2003 (1)

P.-T. Yap, R. Paramesran, and S.-H. Ong, “Image analysis by Krawtchouk moments,” IEEE Trans. Image Process. 12, 1367–1377 (2003).
[Crossref]

2001 (1)

R. Mukundan, S. Ong, and A. Lee, “Image analysis by Tchebichef moments,” IEEE Trans. Image Process. 10, 1357–1364 (2001).
[Crossref]

Agrawal, A.

R. Raskar, A. Agrawal, and J. Tumblin, “Coded exposure photography: motion deblurring using fluttered shutter,” ACM Trans. Graph. 25, 795–804 (2006).
[Crossref]

Almeida, M.

M. Almeida and M. Figueiredo, “Parameter estimation for blind and non-blind deblurring using residual whiteness measures,” IEEE Trans. Image Process. 22, 2751–2763 (2013).
[Crossref]

Ben-Ezra, M.

M. Ben-Ezra and S. Nayar, “Motion deblurring using hybrid imaging,” in Computer Vision and Pattern Recognition (IEEE, 2003), Vol. 1, pp. I-657–I-664.

Bovik, A.

A. Mittal, A. Moorthy, and A. Bovik, “No-reference image quality assessment in the spatial domain,” IEEE Trans. Image Process. 21, 4695–4708 (2012).
[Crossref]

Chan, T.

T. Chan and J. Shen, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods (Society for Industrial and Applied Mathematics, 2005).

Chen, J.

J. Chen, W. Dong, H. Feng, Z. Xu, and Q. Li, “High quality non-blind image deconvolution using the fields of experts prior,” Optik 124, 3601–3606 (2013).
[Crossref]

Dong, W.

J. Chen, W. Dong, H. Feng, Z. Xu, and Q. Li, “High quality non-blind image deconvolution using the fields of experts prior,” Optik 124, 3601–3606 (2013).
[Crossref]

S. Tao, W. Dong, H. Feng, Z. Xu, and Q. Li, “Non-blind image deconvolution using natural image gradient prior,” Optik 124, 6599–6605 (2013).
[Crossref]

Feng, H.

S. Tao, W. Dong, H. Feng, Z. Xu, and Q. Li, “Non-blind image deconvolution using natural image gradient prior,” Optik 124, 6599–6605 (2013).
[Crossref]

J. Chen, W. Dong, H. Feng, Z. Xu, and Q. Li, “High quality non-blind image deconvolution using the fields of experts prior,” Optik 124, 3601–3606 (2013).
[Crossref]

Fergus, R.

R. Fergus, B. Singh, A. Hertzmann, S. T. Roweis, and W. T. Freeman, “Removing camera shake from a single photograph,” ACM Trans. Graph. 25, 787–794 (2006).
[Crossref]

Figueiredo, M.

M. Almeida and M. Figueiredo, “Parameter estimation for blind and non-blind deblurring using residual whiteness measures,” IEEE Trans. Image Process. 22, 2751–2763 (2013).
[Crossref]

Flusser, J.

J. Flusser, B. Zitova, and T. Suk, Moments and Moment Invariants in Pattern Recognition (Wiley, 2009).

Freeman, W. T.

R. Fergus, B. Singh, A. Hertzmann, S. T. Roweis, and W. T. Freeman, “Removing camera shake from a single photograph,” ACM Trans. Graph. 25, 787–794 (2006).
[Crossref]

Gong, W.

S. Tang, W. Gong, W. Li, and W. Wang, “Non-blind image deblurring method by local and nonlocal total variation models,” Signal Process. 94, 339–349 (2014).
[Crossref]

Hertzmann, A.

R. Fergus, B. Singh, A. Hertzmann, S. T. Roweis, and W. T. Freeman, “Removing camera shake from a single photograph,” ACM Trans. Graph. 25, 787–794 (2006).
[Crossref]

Jia, J.

J. Jia, “Single image motion deblurring using transparency,” in Computer Vision and Pattern Recognition Conference (IEEE, 2007), pp. 1–8.

Katsaggelos, A. K.

E. Vera, M. Vega, R. Molina, and A. K. Katsaggelos, “Iterative image restoration using nonstationary priors,” Appl. Opt. 52, D-102–D-110 (2013).
[Crossref]

Lavelle, E.

Lee, A.

R. Mukundan, S. Ong, and A. Lee, “Image analysis by Tchebichef moments,” IEEE Trans. Image Process. 10, 1357–1364 (2001).
[Crossref]

Levin, A.

A. Levin, “Blind motion deblurring using image statistics,” in Advances in Neural Information Processing Systems, B. Schölkopf, J. Platt, and T. Hofmann, eds. (MIT, 2006), pp. 841–848.

Li, Q.

S. Tao, W. Dong, H. Feng, Z. Xu, and Q. Li, “Non-blind image deconvolution using natural image gradient prior,” Optik 124, 6599–6605 (2013).
[Crossref]

J. Chen, W. Dong, H. Feng, Z. Xu, and Q. Li, “High quality non-blind image deconvolution using the fields of experts prior,” Optik 124, 3601–3606 (2013).
[Crossref]

Li, W.

S. Tang, W. Gong, W. Li, and W. Wang, “Non-blind image deblurring method by local and nonlocal total variation models,” Signal Process. 94, 339–349 (2014).
[Crossref]

Mittal, A.

A. Mittal, A. Moorthy, and A. Bovik, “No-reference image quality assessment in the spatial domain,” IEEE Trans. Image Process. 21, 4695–4708 (2012).
[Crossref]

Molina, R.

E. Vera, M. Vega, R. Molina, and A. K. Katsaggelos, “Iterative image restoration using nonstationary priors,” Appl. Opt. 52, D-102–D-110 (2013).
[Crossref]

Moorthy, A.

A. Mittal, A. Moorthy, and A. Bovik, “No-reference image quality assessment in the spatial domain,” IEEE Trans. Image Process. 21, 4695–4708 (2012).
[Crossref]

Mukundan, R.

R. Mukundan, S. Ong, and A. Lee, “Image analysis by Tchebichef moments,” IEEE Trans. Image Process. 10, 1357–1364 (2001).
[Crossref]

Nayar, S.

M. Ben-Ezra and S. Nayar, “Motion deblurring using hybrid imaging,” in Computer Vision and Pattern Recognition (IEEE, 2003), Vol. 1, pp. I-657–I-664.

Ong, S.

R. Mukundan, S. Ong, and A. Lee, “Image analysis by Tchebichef moments,” IEEE Trans. Image Process. 10, 1357–1364 (2001).
[Crossref]

Ong, S.-H.

P.-T. Yap, R. Paramesran, and S.-H. Ong, “Image analysis by Krawtchouk moments,” IEEE Trans. Image Process. 12, 1367–1377 (2003).
[Crossref]

Paramesran, R.

P.-T. Yap, R. Paramesran, and S.-H. Ong, “Image analysis by Krawtchouk moments,” IEEE Trans. Image Process. 12, 1367–1377 (2003).
[Crossref]

Quan, L.

L. Yuan, J. Sun, L. Quan, and H.-Y. Shum, “Image deblurring with blurred/noisy image pairs,” ACM Trans. Graph. 26, 1–10 (2007).
[Crossref]

Raskar, R.

R. Raskar, A. Agrawal, and J. Tumblin, “Coded exposure photography: motion deblurring using fluttered shutter,” ACM Trans. Graph. 25, 795–804 (2006).
[Crossref]

Roweis, S. T.

R. Fergus, B. Singh, A. Hertzmann, S. T. Roweis, and W. T. Freeman, “Removing camera shake from a single photograph,” ACM Trans. Graph. 25, 787–794 (2006).
[Crossref]

Shen, J.

T. Chan and J. Shen, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods (Society for Industrial and Applied Mathematics, 2005).

Shum, H.-Y.

L. Yuan, J. Sun, L. Quan, and H.-Y. Shum, “Image deblurring with blurred/noisy image pairs,” ACM Trans. Graph. 26, 1–10 (2007).
[Crossref]

Singh, B.

R. Fergus, B. Singh, A. Hertzmann, S. T. Roweis, and W. T. Freeman, “Removing camera shake from a single photograph,” ACM Trans. Graph. 25, 787–794 (2006).
[Crossref]

Stoker, D. S.

Suk, T.

J. Flusser, B. Zitova, and T. Suk, Moments and Moment Invariants in Pattern Recognition (Wiley, 2009).

Sun, J.

L. Yuan, J. Sun, L. Quan, and H.-Y. Shum, “Image deblurring with blurred/noisy image pairs,” ACM Trans. Graph. 26, 1–10 (2007).
[Crossref]

Tang, S.

S. Tang, W. Gong, W. Li, and W. Wang, “Non-blind image deblurring method by local and nonlocal total variation models,” Signal Process. 94, 339–349 (2014).
[Crossref]

Tao, S.

S. Tao, W. Dong, H. Feng, Z. Xu, and Q. Li, “Non-blind image deconvolution using natural image gradient prior,” Optik 124, 6599–6605 (2013).
[Crossref]

Tumblin, J.

R. Raskar, A. Agrawal, and J. Tumblin, “Coded exposure photography: motion deblurring using fluttered shutter,” ACM Trans. Graph. 25, 795–804 (2006).
[Crossref]

van der Laan, J.

Vega, M.

E. Vera, M. Vega, R. Molina, and A. K. Katsaggelos, “Iterative image restoration using nonstationary priors,” Appl. Opt. 52, D-102–D-110 (2013).
[Crossref]

Vera, E.

E. Vera, M. Vega, R. Molina, and A. K. Katsaggelos, “Iterative image restoration using nonstationary priors,” Appl. Opt. 52, D-102–D-110 (2013).
[Crossref]

Wang, W.

S. Tang, W. Gong, W. Li, and W. Wang, “Non-blind image deblurring method by local and nonlocal total variation models,” Signal Process. 94, 339–349 (2014).
[Crossref]

Wedd, J.

Xu, Z.

S. Tao, W. Dong, H. Feng, Z. Xu, and Q. Li, “Non-blind image deconvolution using natural image gradient prior,” Optik 124, 6599–6605 (2013).
[Crossref]

J. Chen, W. Dong, H. Feng, Z. Xu, and Q. Li, “High quality non-blind image deconvolution using the fields of experts prior,” Optik 124, 3601–3606 (2013).
[Crossref]

Yap, P.-T.

P.-T. Yap, R. Paramesran, and S.-H. Ong, “Image analysis by Krawtchouk moments,” IEEE Trans. Image Process. 12, 1367–1377 (2003).
[Crossref]

Yuan, L.

L. Yuan, J. Sun, L. Quan, and H.-Y. Shum, “Image deblurring with blurred/noisy image pairs,” ACM Trans. Graph. 26, 1–10 (2007).
[Crossref]

Zitova, B.

J. Flusser, B. Zitova, and T. Suk, Moments and Moment Invariants in Pattern Recognition (Wiley, 2009).

ACM Trans. Graph. (3)

L. Yuan, J. Sun, L. Quan, and H.-Y. Shum, “Image deblurring with blurred/noisy image pairs,” ACM Trans. Graph. 26, 1–10 (2007).
[Crossref]

R. Raskar, A. Agrawal, and J. Tumblin, “Coded exposure photography: motion deblurring using fluttered shutter,” ACM Trans. Graph. 25, 795–804 (2006).
[Crossref]

R. Fergus, B. Singh, A. Hertzmann, S. T. Roweis, and W. T. Freeman, “Removing camera shake from a single photograph,” ACM Trans. Graph. 25, 787–794 (2006).
[Crossref]

Appl. Opt. (2)

E. Vera, M. Vega, R. Molina, and A. K. Katsaggelos, “Iterative image restoration using nonstationary priors,” Appl. Opt. 52, D-102–D-110 (2013).
[Crossref]

D. S. Stoker, J. Wedd, E. Lavelle, and J. van der Laan, “Restoration and recognition of distant, blurry irises,” Appl. Opt. 52, 1864–1875 (2013).
[Crossref]

IEEE Trans. Image Process. (4)

M. Almeida and M. Figueiredo, “Parameter estimation for blind and non-blind deblurring using residual whiteness measures,” IEEE Trans. Image Process. 22, 2751–2763 (2013).
[Crossref]

R. Mukundan, S. Ong, and A. Lee, “Image analysis by Tchebichef moments,” IEEE Trans. Image Process. 10, 1357–1364 (2001).
[Crossref]

P.-T. Yap, R. Paramesran, and S.-H. Ong, “Image analysis by Krawtchouk moments,” IEEE Trans. Image Process. 12, 1367–1377 (2003).
[Crossref]

A. Mittal, A. Moorthy, and A. Bovik, “No-reference image quality assessment in the spatial domain,” IEEE Trans. Image Process. 21, 4695–4708 (2012).
[Crossref]

Optik (2)

S. Tao, W. Dong, H. Feng, Z. Xu, and Q. Li, “Non-blind image deconvolution using natural image gradient prior,” Optik 124, 6599–6605 (2013).
[Crossref]

J. Chen, W. Dong, H. Feng, Z. Xu, and Q. Li, “High quality non-blind image deconvolution using the fields of experts prior,” Optik 124, 3601–3606 (2013).
[Crossref]

Signal Process. (1)

S. Tang, W. Gong, W. Li, and W. Wang, “Non-blind image deblurring method by local and nonlocal total variation models,” Signal Process. 94, 339–349 (2014).
[Crossref]

Other (6)

M. Ben-Ezra and S. Nayar, “Motion deblurring using hybrid imaging,” in Computer Vision and Pattern Recognition (IEEE, 2003), Vol. 1, pp. I-657–I-664.

http://classes.soe.ucsc.edu/ee264/Fall11/LecturePDF/5-LocalOperations.pdf .

A. Levin, “Blind motion deblurring using image statistics,” in Advances in Neural Information Processing Systems, B. Schölkopf, J. Platt, and T. Hofmann, eds. (MIT, 2006), pp. 841–848.

J. Jia, “Single image motion deblurring using transparency,” in Computer Vision and Pattern Recognition Conference (IEEE, 2007), pp. 1–8.

T. Chan and J. Shen, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods (Society for Industrial and Applied Mathematics, 2005).

J. Flusser, B. Zitova, and T. Suk, Moments and Moment Invariants in Pattern Recognition (Wiley, 2009).

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Figures (3)

Fig. 1.
Fig. 1.

(a) Estimated m 1 ( f ^ ) and (b) estimated m 2 ( f ^ ) .

Fig. 2.
Fig. 2.

Error versus iterations.

Fig. 3.
Fig. 3.

Error variation with estimated σ .

Tables (2)

Tables Icon

Table 1. Deblurring Using Spatial and Moment Domain for Binary Images with Different Gaussian Kernel and Mask Size with Corresponding BRISQUE

Tables Icon

Table 2. Deblurring Using Spatial and Moment Domain Approaches for Test Images of Different Size, Gaussian Kernel, Mask Size, with Corresponding BRISQUE

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

m p ( f ) = x = 1 N x p f ( x ) .
m p ( g ) = i = 0 p ( p i ) m i ( f ) m p i ( h ) ,
E p = ( i = 0 p ( p i ) m i ( f ^ ) m p i ( h ) m p ( g ) ) 2 ,
E p m p ( f ^ ) = 2 m 0 ( h ) ( i = 0 p ( p i ) m i ( f ^ ) m p i ( h ) m p ( g ) ) .
m p ( f ^ ) t = E p m p ( f ^ ) .
m p ( f ^ ) t = 2 m 0 ( h ) ( i = 0 p ( p i ) m i ( f ^ ) m p i ( h ) m p ( g ) ) .
m p ( f ^ ) ( n + 1 ) = m p ( f ^ ) ( n ) 2 m 0 ( h ) Δ t × ( i = 0 p ( p i ) m i ( f ^ ) m p i ( h ) m p ( g ) ) ,
m p , q ( f ^ ) ( n + 1 ) = m p , q ( f ^ ) ( n ) 2 m 0 , 0 ( h ) Δ t × ( k = 0 p j = 0 q ( p k ) ( q j ) m k , j ( f ^ ) m p k , q j ( h ) m p , q ( g ) ) ,
m p ( f ) = { 16,43,149,559,2213,9103 } ,
m p ( h ) = { 1 , 1 , 1 } ,
m p ( g ) = { 3,4,6,10,18,34,66,130 } ,

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