## Tensor factorization for model-free space-variant blind deconvolution of the single- and multi-frame multi-spectral image |

Optics Express, Vol. 18, Issue 17, pp. 17819-17833 (2010)

http://dx.doi.org/10.1364/OE.18.017819

Acrobat PDF (5423 KB)

### Abstract

The higher order orthogonal iteration (HOOI) is used for a single-frame and multi-frame space-variant blind deconvolution (BD) performed by factorization of the tensor of blurred multi-spectral image (MSI). This is achieved by conversion of BD into blind source separation (BSS), whereupon sources represent the original image and its spatial derivatives. The HOOI-based factorization enables an essentially unique solution of the related BSS problem with orthogonality constraints imposed on factors and the core tensor of the Tucker3 model of the image tensor. In contrast, the matrix factorization-based unique solution of the same BSS problem demands sources to be statistically independent or sparse which is not true. The consequence of such an approach to BD is that it virtually does not require *a priori* information about the possibly space-variant point spread function (PSF): neither its model nor size of its support. For the space-variant BD problem, MSI is divided into blocks whereupon the PSF is assumed to be a space-invariant within the blocks. The success of proposed concept is demonstrated in experimentally degraded images: defocused single-frame gray scale and red-green-blue (RGB) images, single-frame gray scale and RGB images blurred by atmospheric turbulence, and a single-frame RGB image blurred by a grating (photon sieve). A comparable or better performance is demonstrated in relation to the blind Richardson-Lucy algorithm which, however, requires *a priori* information about parametric model of the blur.

© 2010 OSA

## 1. Introduction

5. I. Kopriva, “Single-frame multichannel blind deconvolution by nonnegative matrix factorization with sparseness constraints,” Opt. Lett. **30**(23), 3135–3137 (2005). [CrossRef] [PubMed]

6. I. Kopriva, “3D tensor factorization approach to single-frame model-free blind-image deconvolution,” Opt. Lett. **34**(18), 2835–2837 (2009). [CrossRef] [PubMed]

17. I. Kopriva, “Approach to blind image deconvolution by multiscale subband decomposition and independent component analysis,” J. Opt. Soc. Am. A **24**(4), 973–983 (2007). [CrossRef]

21. J. G. Daugman, ““Complete Discrete 2-D Gabor Transforms by Neural Networks for Image Analysis and Compression,” IEEE Trans. Acoust. Speech Signal Process. **36**(7), 1169–1179 (1988). [CrossRef]

22. J. Lin and A. Zhang, “Fault feature separation using wavelet-ICA filter,” NDT Int. **38**(6), 421–427 (2005). [CrossRef]

23. M. E. Davies and C. J. James, “Source separation using single channel ICA,” Signal Process. **87**(8), 1819–1832 (2007). [CrossRef]

24. H. G. Ma, Q. B. Jiang, Z. Q. Liu, G. Liu, and Z. Y. Ma, “A novel blind source separation method for single-channel signal,” Signal Process. **90**(12), 3232–3241 (2010). [CrossRef]

*n*

^{th}order expansion requires the source image to be

*n*-times differentiable at the origin. This smoothness requirement is expected, most likely, to limit performance of proposed approach to blind image deconvolution when the degradation process is strong. Then, additional terms in the implicit Taylor expansion are necessary requiring a higher level of smoothness of the source image and this might not be fulfilled. Likewise, images with sharp boundaries are also expected to be deconvolved less effectively by the proposed approach. Nevertheless, an important contribution of the proposed approach to blind image deconvolution remains: no information about model or size of support of the PSF is required. Due to the reasons discussed, the proposed method should be considered complementary rather than competing to physically constrained iterative blind image deconvolution algorithms for situations when physically based constraints are difficult or impossible to be defined. Due to the same reasons, a comparison of the proposed method against applications of specific blind deconvolution methods (for example, against methods developed for deconvolution of turbulence degraded image in astronomy) is not fair because these application-specific methods will almost surely perform better if all required

*a priori*information are provided to them. Therefore, it is my intention in the experimental section of this paper to demonstrate success of the proposed approach by deconvolving experimental images degraded by various types of blurs. In the last example, the blur caused by a grating does not even have a physical analogy in some practical situation. Thus, it is impossible to find a proper model for it. Blind Richardson-Lucy (R-L) algorithm [25

25. D. A. Fish, A. M. Brinicombe, E. R. Pike, and J. G. Walker, “Blind deconvolution by means of the Richardson-Lucy algorithm,” J. Opt. Soc. Am. A **12**(1), 58–65 (1995). [CrossRef]

26. D. S. C. Biggs and M. Andrews, “Acceleration of iterative image restoration algorithms,” Appl. Opt. **36**(8), 1766–1775 (1997). [CrossRef] [PubMed]

*i*) experimental de-focused single-frame gray scale and red-green-blue (RGB) images, (

*ii*) experimental multi-frame gray scale and RGB images degraded by atmospheric turbulence and, (

*iii*) experimental single-frame RGB image degraded by a grating. Conclusions are given in Section 5.

## 2. Basics of tensor notation, Tucker 3 model and HOOI decomposition

*N*denotes the number of modes or order of the tensor

*i*

_{1}= 1,...,

*I*

_{1},

*i*

_{2}= 1,...,

*I*

_{2}, and

*i*= 1, ...,

_{N}*I*. This is the standard notation adopted for use in multi-way analysis [29

_{N}29. H. A. L. Kiers, “Towards a standardized notation and terminology in multiway analysis,” J. Chemometr. **14**(3), 105–122 (2000). [CrossRef]

*n*-mode product between the core tensor

30. L. R. Tucker, “Some mathematical notes on three-mode factor analysis,” Psychometrika **31**(3), 279–311 (1966). [CrossRef] [PubMed]

32. B. W. Bader, and T. G. Kolda, *MATLAB Tensor Toolbox version 2.2**.*http://csmr.ca.sandia.gov/~tkolda/TensorToolbox.

## 3. Multi-dimensional linear mixture model of degraded image

**H**with an original source image

**F**:where

*M*denotes half of the PSF support size. In Eq. (9) presence of the additive noise is ignored in order to focus on an essential issue: model-free blind image deconvolution. It is however clear that some form of image de-noising is performed, if necessary, prior to executing deconvolution. It is also assumed that the unknown original image

**F**is

*n*

^{th}order smooth implying that it is

*n*-times differentiable at the origin (

*i*

_{1},

*i*

_{2}), whereupon

*n*represents the order of spatial derivatives (terms) in the Taylor expansion of

*i*

_{1},

*i*

_{2}). An implicit Taylor expansion of the original image

**F**(

*i*

_{1}

*-s,i*

_{2}

*-t*) around (

*i*

_{1},

*i*

_{2}), is used to convert blind image deconvolution into blind source separation [20,5

5. I. Kopriva, “Single-frame multichannel blind deconvolution by nonnegative matrix factorization with sparseness constraints,” Opt. Lett. **30**(23), 3135–3137 (2005). [CrossRef] [PubMed]

6. I. Kopriva, “3D tensor factorization approach to single-frame model-free blind-image deconvolution,” Opt. Lett. **34**(18), 2835–2837 (2009). [CrossRef] [PubMed]

17. I. Kopriva, “Approach to blind image deconvolution by multiscale subband decomposition and independent component analysis,” J. Opt. Soc. Am. A **24**(4), 973–983 (2007). [CrossRef]

*i*

_{1}and

*i*

_{2}directions respectively. By using Eq. (10), Eq. (9) can be re-written as:

5. I. Kopriva, “Single-frame multichannel blind deconvolution by nonnegative matrix factorization with sparseness constraints,” Opt. Lett. **30**(23), 3135–3137 (2005). [CrossRef] [PubMed]

6. I. Kopriva, “3D tensor factorization approach to single-frame model-free blind-image deconvolution,” Opt. Lett. **34**(18), 2835–2837 (2009). [CrossRef] [PubMed]

17. I. Kopriva, “Approach to blind image deconvolution by multiscale subband decomposition and independent component analysis,” J. Opt. Soc. Am. A **24**(4), 973–983 (2007). [CrossRef]

*a*

_{1}to

*a*

_{6}absorbed into themselves the coefficients of the PSF:

*M*. Hence, blind image deconvolution could be converted to blind source separation provided that a multi-channel version of the blurred image [Eq. (11)] is available. As in [20,5

**30**(23), 3135–3137 (2005). [CrossRef] [PubMed]

**34**(18), 2835–2837 (2009). [CrossRef] [PubMed]

**24**(4), 973–983 (2007). [CrossRef]

21. J. G. Daugman, ““Complete Discrete 2-D Gabor Transforms by Neural Networks for Image Analysis and Compression,” IEEE Trans. Acoust. Speech Signal Process. **36**(7), 1169–1179 (1988). [CrossRef]

**24**(4), 973–983 (2007). [CrossRef]

**30**(23), 3135–3137 (2005). [CrossRef] [PubMed]

**34**(18), 2835–2837 (2009). [CrossRef] [PubMed]

**24**(4), 973–983 (2007). [CrossRef]

*I*

_{3}= 17 images. Thus, by together merging a blurred image [Eq. (11)] with filtered images [Eq. (12)], the multichannel image forms a 3D tensor

*N*-order Tucker3 tensor model [Eq. (1)], this scenario is characterized by

*N*= 3. Based on Eq. (6), it follows that an estimate of the tensor of source images is obtained by means of the HOOI-based tensor factorization as

*j =*1 i.e. the estimate of the original image corresponds to the first source image.

*I*

_{3}blocks, where PSF is assumed to be a space-invariant within each block. Then, each of the

*I*

_{3}image blocks is filtered by means of a 2D Gabor filter bank yielding in overall a 4D multi-channel image tensor

*j*= 1. The original image itself is then obtained by a proper rearranging of the blocks into a matrix.

*I*

_{3}represents the number of spectral bands. Filtering each of the

*I*

_{3}images by means of a 2D Gabor filter bank yields a 4D multi-channel image tensor

*j*= 1. The physical interpretation of tensor modes associated with blind deconvolution scenarios considered in this paper is provided in the Table 1.

## 4. Experimental results for images blurred by de-focus, atmospheric turbulence and grating

*a priori*information is provided to them.

*i*) de-focused RGB image shown in Fig. 1 , as well as its gray scale version, with dimensions of 384 × 512 pixels. Solutions of blind deconvolution problems classified in Table 1 as 2, 3 and 4 have been demonstrated on this image. The image has been recorded by digital camera in a manually de-focused mode and has been used previously in [5

**30**(23), 3135–3137 (2005). [CrossRef] [PubMed]

**34**(18), 2835–2837 (2009). [CrossRef] [PubMed]

**24**(4), 973–983 (2007). [CrossRef]

*ii*) RGB and gray scale multi-frame images of the Washington monument blurred by atmospheric turbulence. Four randomly chosen frames with dimensions of 160 × 80 pixels are shown in Fig. 6a . The gray scale version of this image sequence has been used previously in [33

33. I. Kopriva, Q. Du, H. Szu, and W. Wasylkiwskyj, “Independent Component Analysis Approach to Image Sharpening in the Presence of Atmospheric Turbulence,” Opt. Commun. **233**(1-3), 7–14 (2004). [CrossRef]

*iii*) single-frame RGB image blurred by a grating. The image with dimensions of 301 × 351 pixels is shown in the leftmost picture of Fig. 10 . The grating-caused blur does not have a real physical equivalent such as turbulence, de-focus or motion. Therefore, it is virtually impossible to define a proper model or other type of

*a priori*information required by physically constrained iterative blind deconvolution algorithms such as exemplified by [7

7. F. Li, X. Jia, and D. Fraser, “Superresolution Reconstruction of Multispectral Data for Improved Image Classification,” IEEE Geosci. Remote Sens. Lett. **6**(4), 689–693 (2009). [CrossRef]

14. M. Šorel and J. Flusser, “Space-variant restoration of images degraded by camera motion blur,” IEEE Trans. Image Process. **17**(2), 105–116 (2008). [CrossRef] [PubMed]

25. D. A. Fish, A. M. Brinicombe, E. R. Pike, and J. G. Walker, “Blind deconvolution by means of the Richardson-Lucy algorithm,” J. Opt. Soc. Am. A **12**(1), 58–65 (1995). [CrossRef]

26. D. S. C. Biggs and M. Andrews, “Acceleration of iterative image restoration algorithms,” Appl. Opt. **36**(8), 1766–1775 (1997). [CrossRef] [PubMed]

*a priori*information about the blur in the form of a parametric model. As other model-based deconvolution methods, it is sensitive to model misspecifications and a miss-estimation of the values of model parameters [1]. As can be seen in Figs. 4 , 9 and 10, the blind R-L algorithm yielded results of equal or lower quality than those obtained by the proposed tensor factorization model-free approach to blind deconvolution, despite the fact that the optimal choice of a blur model and model parameters were supplied to the algorithm. Figures 4 and 9 also illustrate sensitivity of the blind R-L method to a slight over-estimation of model parameters. In Fig. 10 the best result for blind R-L is shown after many attempts to choose the blur model and model parameters.

### 4.1 Space-variant blind deconvolution of de-focused single-frame gray scale image

*I*

_{1}= 48,

*I*

_{2}= 64,

*I*

_{3}= 64, and

*I*

_{4}= 17. The image has been divided into 64 blocks with the size of 48 × 64 pixels and PSF is assumed to be the space-invariant within each block. Figure 2 -left shows a result of the 4D tensor factorization approach. Figure 2-right shows a corresponding result, with poorer resolution, obtained when each block has been deconvolved separately through 3D tensor factorization. This result is explained by the fact that there is only one scaling indeterminacy associated with the solution of the 4D tensor factorization problem, while there are

*I*

_{3}= 64 scaling indeterminacies associated with the solutions of 3D tensor factorization problems. White artifact lines and a sharp difference between the few blocks are visible in restored image and are the consequence of an adopted block-wise approach to space-variant blind deconvolution. However, the super-resolution effect of the deconvolution method is demonstrated. I consider the block-wise induced artifacts as an open problem that can possibly be reduced through some continuity (smoothness) constraints either directly through factorization or in a form of post-processing. Choosing the optimal size of the image blocks is also an open issue to be explored in the future.

### 4.2 Space-invariant blind deconvolution of de-focused single-frame multi-spectral image

*I*

_{1}= 384,

*I*

_{2}= 512,

*I*

_{3}= 3 and

*I*

_{4}= 17. Figure 3 -left shows the result of a 4D tensor factorization approach to model-free blind deconvolution. A corresponding result, with poorer resolution, is presented in Fig. 3-right, and also in [7

7. F. Li, X. Jia, and D. Fraser, “Superresolution Reconstruction of Multispectral Data for Improved Image Classification,” IEEE Geosci. Remote Sens. Lett. **6**(4), 689–693 (2009). [CrossRef]

*I*

_{3}= 3 scaling indeterminacies associated with the solutions of 3D tensor factorization problems. The result obtained by model-free blind deconvolution has served as a reference to optimize parameters for a blind R-L algorithm, the results of which are shown in Fig. 4. Knowing that the RGB image was de-focused, a disk with the proper radius was supplied to the algorithm as a blur model. The optimal result is shown in Fig. 4-left. As shown in Fig. 4-right, over-estimating the disk radius for 1 pixel only deteriorated the performance of the blind R-L method significantly.

### 4.3 Space-variant blind deconvolutin of a de-focused single-frame multi-spectral image

### 4.4 Space-invariant blind deconvolution of a multi-frame multi-spectral image blurred by atmospheric turbulence

*I*

_{1}= 160,

*I*

_{2}= 80,

*I*

_{3}= 3,

*I*

_{4}= 4 and

*I*

_{5}= 17. Figure 6b-left shows an average of the four frames shown in Fig. 6a, while Fig. 6b-right shows an edge map extracted from Fig. 6b-left by Canny's algorithm with the threshold set to 0.21. Figure 7a shows four source image frames restored with a proposed tensor factorization approach, while Fig. 7b-left shows the average of these four source image frames. Figure 7b-right shows an edge map extracted from Fig. 7b-left by Canny's algorithm with the threshold set again to 0.21. Details like windows and the top of the monument are reconstructed. Some of these details (windows) are missed on edges extracted from turbulence degraded image. They can be recovered by reducing the threshold (increasing the sensitivity) of Canny's algorithm. However, in this case, turbulence-induced artifacts would be picked up as well.

### 4.5 Space-invariant blind deconvolution of multi-frame gray scale image blurred by atmospheric turbulence

*I*

_{1}= 160,

*I*

_{2}= 80,

*I*

_{3}= 4 and

*I*

_{4}= 17. Figure 8a shows four source image frames obtained with the proposed tensor factorization approach, while Fig. 8b-left shows an average of the four source image frames shown in Fig. 8a. Figure 8b-right shows an edge map extracted from Fig. 8b-left by Canny's algorithm with the threshold set again to 0.21. Again, details like windows and the top of the monument are reconstructed. Figure 9a-left shows the image restored from a gray scale version of the average of the four blurred frames, shown in Fig. 6a, by means of a blind R-L algorithm. The corresponding edge map extracted by Canny's method with a threshold set again to 0.21 is shown in Fig. 9a-right. Knowledge of the type of degradation has been used to specify the correct model of the blur that is required by blind R-L algorithm: 2D Gaussian with a support size of 18 pixels and a standard deviation of 1.3 pixels has been used to obtain the shown result. The result obtained by the model-free approach, shown in Fig. 8b, has been used as a reference to obtain optimal values of the parameters of the 2D Gaussian. Yet, the blind R-L algorithm with parameters optimized for a blur model yielded a result that is not better than the one obtained by a model-free blind deconvolution approach. Figure 9b shows the result obtained by a blind R-L algorithm when the standard deviation is over-estimated to 1.9 pixels. This, again, illustrates sensitivity of the blind R-L algorithm to miss-estimation of model parameters. No such problems arise with the proposed model-free approach to blind deconvolution.

### 4.6 Space-(in)variant blind deconvolutin of single-frame multi-spectral image blurred by a grating

*I*

_{1}= 150,

*I*

_{2}= 117,

*I*

_{3}= 6,

*I*

_{4}= 3 and

*I*

_{5}= 17. A grating blurred RGB image is shown in the leftmost picture in Fig. 10. The true image is composed of a palette of color pens, and a painting on the white board. Since there is no “real life” physical analogy to this grating-caused blur, it is virtually impossible to select a specialized method to perform blind deconvolution of this grating-blurred image. The picture second from the left in Fig. 10 shows the result obtained by a solution of space-variant blind deconvolution problem 4. The picture second from the right in Fig. 10 shows the result obtained by a space-invariant blind deconvolution problem 3 that results in 4D tensor factorization. The tensor (1) of the blurred image is characterized with:

*I*

_{1}= 301,

*I*

_{2}= 351,

*I*

_{3}= 3 and

*I*

_{4}= 17. The best result obtained by blind R-L algorithm for a space-invariant problem is shown in the rightmost picture of Fig. 10. It has been obtained after many attempts to find both the proper model and parameters of the blur. Still, obtained results is not better than the one show in the picture second from right and obtained by a solution of space-invariant blind deconvolution problem 3. It is, however, visible that the space-variant version of blind deconvolution really improved the spatial resolution of the restored image significantly in relation to images restored by space-invariant blind deconvolution methods. This result has been obtained truly without any

*a priori*information provided to the blind deconvolution algorithm.

## 5. Conclusion and future work

*i*) the HOOI-based factorization of the tensor of the blurred image is essentially unique with no hard constraints imposed on source images compared to matrix factorization based methods; (

*ii*) neither model nor size of the support of the point spread function is required to be

*a priori*known or estimated. However, use of the implicit Taylor expansion implies a certain level of smoothness of the original image. This might limit the performance of the proposed approach to blind image deconvolution when the blurring process is strong or the original image contains sharp boundaries. Nevertheless, the proposed method is expected to be useful in scenarios when

*a priori*information required by physically constrained iterative blind deconvolution methods are difficult or impossible to define. The two fundamental issues are considered to be important exploring in the future work: optimal selection of the size of the image blocks and neutralization of block-wise induced artifacts associated with space-variant deconvolution; sequence partitioning-based methods as a possible replacement of Gabor filter bank approach to single-channel blind source separation.

## Appendix: Mode-*n* unfolding

*n*unfolding of a tensor

*n*fibers of the tensor

*i*

_{1},

*i*

_{2},...,

*i*

_{N}) maps onto a matrix element (

*i*

_{n},

*j*), where

*i*

_{1},

*i*

_{2},

*i*

_{3}) has a corresponding position (

*i*

_{n},

*j*) in the mode-

*n*unfolded matrix

*n*=1,2,3) as follows: mode-

*1*:

*j*=

*i*

_{2}+ (

*i*

_{3}- 1)

*I*

_{2}; mode-

*2*:

*j*=

*i*

_{1}+ (

*i*

_{3}- 1)

*I*

_{1}; mode-

*3*:

*j*=

*i*

_{1}+ (

*i*

_{2}- 1)

*I*

_{1}.

## Acknowledgments

## References and links

1. | R. L. Lagendijk, and J. Biemond, |

2. | M. R. Banham and A. K. Katsaggelos, “Digital Image Restoration,” IEEE Signal Process. Mag. |

3. | D. Kundur and D. Hatzinakos, “Blind Image Deconvolution,” IEEE Signal Process. Mag. |

4. | P. Campisi, and K. Egiazarian, eds., |

5. | I. Kopriva, “Single-frame multichannel blind deconvolution by nonnegative matrix factorization with sparseness constraints,” Opt. Lett. |

6. | I. Kopriva, “3D tensor factorization approach to single-frame model-free blind-image deconvolution,” Opt. Lett. |

7. | F. Li, X. Jia, and D. Fraser, “Superresolution Reconstruction of Multispectral Data for Improved Image Classification,” IEEE Geosci. Remote Sens. Lett. |

8. | H. Ji and C. Fermüller, “Robust wavelet-based super-resolution reconstruction: theory and algorithm,” IEEE Trans. Pattern Anal. Mach. Intell. |

9. | T. J. Schulz, “Multiframe blind deconvolution of astronomical images,” J. Opt. Soc. Am. A |

10. | R. Fergus, B. Singh, A. Hertzmann, S. T. Roweis, and W. T. Freeman, “Removing Camera Shake from a Single Photograph,” ACM Trans. Graph. |

11. | Q. Shan, J. Jia, and A. Agarwala, “High-quality Motion Deblurring from a Single Image,” ACM Trans. Graph. |

12. | S. Cho and S. Lee, “Fast Motion Deblurring,” ACM Trans. Graph. |

13. | J. Miskin, and D. J. C. MacKay, “Ensemble Learning for Blind Image Separation and Deconvolution,” in: |

14. | M. Šorel and J. Flusser, “Space-variant restoration of images degraded by camera motion blur,” IEEE Trans. Image Process. |

15. | J. Bardsley, S. Jefferies, J. Nagy, and R. Plemmons, “A computational method for the restoration of images with an unknown, spatially-varying blur,” Opt. Express |

16. | E. F. Y. Hom, F. Marchis, T. K. Lee, S. Haase, D. A. Agard, and J. W. Sedat, “AIDA: an adaptive image deconvolution algorithm with application to multi-frame and three-dimensional data,” J. Opt. Soc. Am. A |

17. | I. Kopriva, “Approach to blind image deconvolution by multiscale subband decomposition and independent component analysis,” J. Opt. Soc. Am. A |

18. | L. De Lathauwer, B. De Moor, and J. Vandewalle, “A multilinear singular value decomposition,” SIAM J. Matrix Anal. Appl. |

19. | L. De Lathauwer, B. De Moor, and J. Vandewalle, “On the best rank-1 and rank-(R1,R2,…,RN) approximation of higher-order tensors,” SIAM J. Matrix Anal. Appl. |

20. | S. Umeyama, “Blind deconvolution of blurred images by use of ICA,” Electron Commun. Jpn |

21. | J. G. Daugman, ““Complete Discrete 2-D Gabor Transforms by Neural Networks for Image Analysis and Compression,” IEEE Trans. Acoust. Speech Signal Process. |

22. | J. Lin and A. Zhang, “Fault feature separation using wavelet-ICA filter,” NDT Int. |

23. | M. E. Davies and C. J. James, “Source separation using single channel ICA,” Signal Process. |

24. | H. G. Ma, Q. B. Jiang, Z. Q. Liu, G. Liu, and Z. Y. Ma, “A novel blind source separation method for single-channel signal,” Signal Process. |

25. | D. A. Fish, A. M. Brinicombe, E. R. Pike, and J. G. Walker, “Blind deconvolution by means of the Richardson-Lucy algorithm,” J. Opt. Soc. Am. A |

26. | D. S. C. Biggs and M. Andrews, “Acceleration of iterative image restoration algorithms,” Appl. Opt. |

27. | A. Cichocki, R. Zdunek, A. H. Phan, and S. I. Amari, |

28. | A. Cichocki, and S. Amari, |

29. | H. A. L. Kiers, “Towards a standardized notation and terminology in multiway analysis,” J. Chemometr. |

30. | L. R. Tucker, “Some mathematical notes on three-mode factor analysis,” Psychometrika |

31. | A. Cichocki, and A. H. Phan. Fast Local Algorithms for Large Scale Nonnegative Matrix and Tensor Factorizations. IEICE Trans Fundamentals 2009; E92-A(3): 708–721. |

32. | B. W. Bader, and T. G. Kolda, |

33. | I. Kopriva, Q. Du, H. Szu, and W. Wasylkiwskyj, “Independent Component Analysis Approach to Image Sharpening in the Presence of Atmospheric Turbulence,” Opt. Commun. |

**OCIS Codes**

(100.1830) Image processing : Deconvolution

(100.3010) Image processing : Image reconstruction techniques

(100.3190) Image processing : Inverse problems

(100.6640) Image processing : Superresolution

(100.6890) Image processing : Three-dimensional image processing

**ToC Category:**

Image Processing

**History**

Original Manuscript: June 21, 2010

Revised Manuscript: July 23, 2010

Manuscript Accepted: July 24, 2010

Published: August 3, 2010

**Citation**

Ivica Kopriva, "Tensor factorization for model-free space-variant blind deconvolution of the single- and multi-frame multi-spectral image," Opt. Express **18**, 17819-17833 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-17-17819

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### References

- R. L. Lagendijk, and J. Biemond, Iterative Identification and Restoration of Images (KAP, 1991).
- M. R. Banham and A. K. Katsaggelos, “Digital Image Restoration,” IEEE Signal Process. Mag. 14(2), 24–41 (1997). [CrossRef]
- D. Kundur and D. Hatzinakos, “Blind Image Deconvolution,” IEEE Signal Process. Mag. 13(3), 43–64 (1996). [CrossRef]
- P. Campisi and K. Egiazarian, eds., Blind Image Deconvolution (CRC Press, Boca Raton, 2007).
- I. Kopriva, “Single-frame multichannel blind deconvolution by nonnegative matrix factorization with sparseness constraints,” Opt. Lett. 30(23), 3135–3137 (2005). [CrossRef] [PubMed]
- I. Kopriva, “3D tensor factorization approach to single-frame model-free blind-image deconvolution,” Opt. Lett. 34(18), 2835–2837 (2009). [CrossRef] [PubMed]
- F. Li, X. Jia, and D. Fraser, “Superresolution Reconstruction of Multispectral Data for Improved Image Classification,” IEEE Geosci. Remote Sens. Lett. 6(4), 689–693 (2009). [CrossRef]
- H. Ji and C. Fermüller, “Robust wavelet-based super-resolution reconstruction: theory and algorithm,” IEEE Trans. Pattern Anal. Mach. Intell. 31(4), 649–660 (2009). [CrossRef] [PubMed]
- T. J. Schulz, “Multiframe blind deconvolution of astronomical images,” J. Opt. Soc. Am. A 10(5), 1064–1073 (1993). [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(3), 787–794 (2006). [CrossRef]
- Q. Shan, J. Jia, and A. Agarwala, “High-quality Motion Deblurring from a Single Image,” ACM Trans. Graph. 27(3), 1 (2008). [CrossRef]
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