Abstract

We address performance modeling of superresolution (SR) techniques. Superresolution consists in combining several images of the same scene to produce an image with better resolution and contrast. We propose a discrete data-continuous reconstruction framework to conduct SR performance analysis and derive a theoretical expression of the reconstruction mean squared error (MSE) as a compact, computationally tractable function of signal-to-noise ratio (SNR), scene model, sensor transfer function, number of frames, interframe translation motion, and SR reconstruction filter. A formal expression for the MSE is obtained that allows a qualitative study of SR behavior. In particular we provide an original outlook on the balance between noise and aliasing reduction in linear SR. Explicit account for the SR reconstruction filter is an original feature of our model. It allows for the first time to study not only optimal filters but also suboptimal ones, which are often used in practice.

© 2009 Optical Society of America

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  46. M. Unser and J. Zerubia, “Generalized sampling: Stability and performance analysis,” IEEE Trans. Image Process. 12, 2941-2950 (1997).
  47. D. Seidner and M. Feder, “Noise amplification of periodic nonuniform sampling,” IEEE Trans. Image Process. 48, 275-277 (2000).
  48. B. Lucas and T. Kanade, “An iterative image registration technique with an application to stereo vision,” in Proceedings of 7th International Joint Conference Artificial Intelligence (IJCAI, 1981), pp. 674-679.
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    [CrossRef]
  50. G. Rochefort, F. Champagnat, G. Le Besnerais, and J.-F. Giovannelli, “An improved observation model for super-resolution under affine motion,” IEEE Trans. Image Process. 15, 3325-3337 (2006).
    [CrossRef] [PubMed]

2009 (1)

L. C. Pickup, D. P. Capel, S. J. Roberts, and A. Zisserman, “Bayesian methods for image super-resolution,” Comput. J. 52, 101-113 (2009).
[CrossRef]

2008 (2)

D. Robinson and D. G. Stork, “Joint digital-optical design of superresolution multiframe imaging systems,” Appl. Opt. 47, 11-20 (2008).
[CrossRef]

S. Ramani, D. Van De Ville, T. Blu, and M. Unser, “Nonideal sampling and regularization theory,” IEEE Trans. Signal Process. 56, 1055-1070 (2008).
[CrossRef]

2007 (5)

F. Sroubek, G. Cristobal, and J. Flusser, “A unified approach to superresolution and multichannel blind deconvolution,” IEEE Trans. Image Process. 16, 2322-2332 (2007).
[CrossRef] [PubMed]

D. Robinson, S. Farsiu, and P. Milanfar, “Optimal registration of aliased images using variable projection with applications to super-resolution,” Comput. J. bxm007v1-12 (2007).

A. W. M. van Eekeren, K. Schutte, O. R. Oudegeest, and L. van Vliet, “Performance evaluation of super-resolution reconstruction methods on real-world data,” EURASIP J. Advances Signal Process. 2007, 43953 (2007).

R. C. Hardie, “A fast image super-resolution algorithm using an adaptive Wiener filter,” IEEE Trans. Image Process. 16, 2953-2964 (2007).
[CrossRef] [PubMed]

S. Prasad, “Digital superresolution and the generalized sampling theorem,” J. Opt. Soc. Am. A 24, 311-325 (2007).
[CrossRef]

2006 (6)

D. Robinson and P. Milanfar, “Statistical performance analysis of super-resolution,” IEEE Trans. Image Process. 15, 1413-1428 (2006).
[CrossRef] [PubMed]

J. Shi and S. Reichenbach, “Image interpolation by two-dimensional parametric cubic convolution,” IEEE Trans. Image Process. 15, 1857-1870 (2006).
[CrossRef] [PubMed]

G. Rochefort, F. Champagnat, G. Le Besnerais, and J.-F. Giovannelli, “An improved observation model for super-resolution under affine motion,” IEEE Trans. Image Process. 15, 3325-3337 (2006).
[CrossRef] [PubMed]

D. J. Brady, “Micro-optics and megapixels,” Opt. Photonics News 17, 24-29 (2006).
[CrossRef]

J. Shi, S. E. Reichenbach, and J. D. Howe, “Small-kernel superresolution methods for microscanning imaging systems,” Appl. Opt. 6, 1203-1214 (2006).
[CrossRef]

Y. Eldar and M. Unser, “Nonideal sampling and interpolation from noisy observations in shift-invariant spaces,” IEEE Trans. Signal Process. 54, 2636-2651 (2006).
[CrossRef]

2005 (2)

Z. Wang and F. Qi, “Analysis of multiframe super-resolution reconstruction for image anti-aliasing and deblurring,” Image Vis. Comput. 23, 393-404 (2005).
[CrossRef]

M. Shimizu and M. Okutomi, “Sub-pixel estimation error cancellation on area-based matching,” Int. J. Comput. Vis. 63, 207-224 (2005).
[CrossRef]

2004 (4)

S. Baker and I. Matthews, “Lukas-Kanade 20 years on: A unifying framework,” Int. J. Comput. Vis. 56, 221-255 (2004).
[CrossRef]

Z. Lin and H.-Y. Shum, “Fundamental limits of reconstruction-based superresolution algorithms under local translation,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 83-97 (2004).
[CrossRef] [PubMed]

S. Farsiu, D. Robinson, M. Elad, and P. Milanfar, “Advances and challenges in super-resolution: special issue on high resolution image reconstruction,” Int. J. Imaging Syst. Technol. 14, 47-57 (2004).
[CrossRef]

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Fast and robust multiframe super-resolution,” IEEE Trans. Image Process. 13, 1327-1343 (2004).
[CrossRef] [PubMed]

2003 (2)

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: A technical overview,” IEEE Signal Process. Mag. 20, 21-36 (2003).
[CrossRef]

E. S. Lee and M. G. Kang, “Regularized adaptive high-resolution image reconstruction considering inaccurate subpixel registration,” IEEE Trans. Image Process. 12, 526-837 (2003).

2002 (1)

S. Baker and T. Kanade, “Limits on super-resolution and how to break them,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1167-1183 (2002).
[CrossRef]

2001 (4)

M. Elad and Y. Hel-Or, “A fast super-resolution reconstruction algorithm for pure translationnal motion and common space-invariant blur,” IEEE Trans. Image Process. 10, 1187-1193 (2001).
[CrossRef]

N. Nguyen, P. Milanfar, and G. Golub, “A computationally efficient superresolution image reconstruction algorithm,” IEEE Trans. Image Process. 10, 573-583 (2001).
[CrossRef]

A. J. Patti and Y. Altunbasak, “Artifact reduction for set theoretic super resolution image reconstruction with edge adaptative constraints and higher-order interpolants,” IEEE Trans. Image Process. 10, 179-186 (2001).
[CrossRef]

J. Tanida, T. Kumagai, K. Yamada, S. Miyatake, K. Ishida, T. Morimoto, N. Kondou, D. Miyazaki, and Y. Ichioka, “Thin observation module by bound optics (TOMBO): Concept and experimental verification,” Appl. Opt. 40, 1806-1813 (2001).
[CrossRef]

2000 (1)

D. Seidner and M. Feder, “Noise amplification of periodic nonuniform sampling,” IEEE Trans. Image Process. 48, 275-277 (2000).

1998 (1)

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, “High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system,” Opt. Eng. (Bellingham) 37, 247-260 (1998).
[CrossRef]

1997 (4)

M. Unser and J. Zerubia, “Generalized sampling: Stability and performance analysis,” IEEE Trans. Image Process. 12, 2941-2950 (1997).

A. J. Patti, M. I. Sezan, and A. M. Tekalp, “Superresolution video reconstruction with arbitrary sampling lattices and nonzero aperture time,” IEEE Trans. Image Process. 6, 1064-1076 (1997).
[CrossRef] [PubMed]

R. C. Hardie, K. J. Barnard, and E. E. Armstrong, “Joint MAP registration and high-resolution image estimation using a sequence of undersampled images,” IEEE Trans. Image Process. 6, 1621-1633 (1997).
[CrossRef] [PubMed]

M. Elad and A. Feuer, “Restoration of a single superresolution image from several blurred, noisy, and undersampled measured images,” IEEE Trans. Image Process. 6, 1646-1658 (1997).
[CrossRef] [PubMed]

1996 (1)

R. R. Schultz and R. L. Stevenson, “Extraction of high-resolution frames from video sequences,” IEEE Trans. Image Process. 5, 996-1011 (1996).
[CrossRef] [PubMed]

1993 (1)

S. P. Kim and W. Y. Su, “Recursive high-resolution reconstruction of blurred multiframe images,” IEEE Trans. Image Process. 2, 534-539 (1993).
[CrossRef] [PubMed]

1992 (1)

H. Ur and D. Gross, “Improved resolution from sub-pixel shifted pictures,” CVGIP: Graph. Models Image Process. 54, 181-186 (1992).
[CrossRef]

1991 (2)

S. E. Reichenbach, S. K. Park, and R. Narayanswamy, “Characterizing digital image acquisition devices,” Opt. Eng. (Bellingham) 30, 170-177 (1991).
[CrossRef]

M. Irani and S. Peleg, “Improving resolution by image registration,” Comput. Vis. Graph. Image Process. 52, 231-239 (1991).

1990 (1)

S. Kim, N. Bose, and H. Valenzuela, “Recursive reconstruction of high resolution image from noisy undersampled multiframes,” IEEE Transactions on Acoustics, IEEE Trans. Acoust., Speech, Signal Process. 38, 1013-1027 (1990).
[CrossRef]

1988 (1)

1987 (1)

H. R. Künsch, “Intrinsic autoregressions and related models on the two-dimensional lattice,” Biometrika 74, 517-524 (1987).

Altunbasak, Y.

A. J. Patti and Y. Altunbasak, “Artifact reduction for set theoretic super resolution image reconstruction with edge adaptative constraints and higher-order interpolants,” IEEE Trans. Image Process. 10, 179-186 (2001).
[CrossRef]

Armstrong, E. E.

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, “High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system,” Opt. Eng. (Bellingham) 37, 247-260 (1998).
[CrossRef]

R. C. Hardie, K. J. Barnard, and E. E. Armstrong, “Joint MAP registration and high-resolution image estimation using a sequence of undersampled images,” IEEE Trans. Image Process. 6, 1621-1633 (1997).
[CrossRef] [PubMed]

Baker, S.

S. Baker and I. Matthews, “Lukas-Kanade 20 years on: A unifying framework,” Int. J. Comput. Vis. 56, 221-255 (2004).
[CrossRef]

S. Baker and T. Kanade, “Limits on super-resolution and how to break them,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1167-1183 (2002).
[CrossRef]

Barnard, K. J.

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, “High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system,” Opt. Eng. (Bellingham) 37, 247-260 (1998).
[CrossRef]

R. C. Hardie, K. J. Barnard, and E. E. Armstrong, “Joint MAP registration and high-resolution image estimation using a sequence of undersampled images,” IEEE Trans. Image Process. 6, 1621-1633 (1997).
[CrossRef] [PubMed]

Blu, T.

S. Ramani, D. Van De Ville, T. Blu, and M. Unser, “Nonideal sampling and regularization theory,” IEEE Trans. Signal Process. 56, 1055-1070 (2008).
[CrossRef]

Bognar, J. G.

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, “High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system,” Opt. Eng. (Bellingham) 37, 247-260 (1998).
[CrossRef]

Bose, N.

S. Kim, N. Bose, and H. Valenzuela, “Recursive reconstruction of high resolution image from noisy undersampled multiframes,” IEEE Transactions on Acoustics, IEEE Trans. Acoust., Speech, Signal Process. 38, 1013-1027 (1990).
[CrossRef]

Brady, D. J.

D. J. Brady, “Micro-optics and megapixels,” Opt. Photonics News 17, 24-29 (2006).
[CrossRef]

Capel, D. P.

L. C. Pickup, D. P. Capel, S. J. Roberts, and A. Zisserman, “Bayesian methods for image super-resolution,” Comput. J. 52, 101-113 (2009).
[CrossRef]

D. P. Capel, “Image mosaicing and super-resolution,” Ph.D. thesis, (University of Oxford, 2001).

Champagnat, F.

G. Rochefort, F. Champagnat, G. Le Besnerais, and J.-F. Giovannelli, “An improved observation model for super-resolution under affine motion,” IEEE Trans. Image Process. 15, 3325-3337 (2006).
[CrossRef] [PubMed]

F. Champagnat, G. Le Besnerais, and C. Kulcsár, “Performance modeling of regularized, linear, spatially invariant super-resolution methods,” Tech. rep., RT 1/12265 DTIM (ONERA, 2008).

F. Champagnat and G. Le Besnerais, “A Fourier interpretation of super-resolution techniques,” in Proceedings of the IEEE International Conference on Image Processing 2005 (IEEE, 2005), Vol. 1, pp. 865-868.

F. Champagnat, C. Kulcsár, and G. Le Besnerais, “Continuous super-resolution for recovery of 1-D image features: Algorithm and performance modeling,” in 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 2006), Vol. 1, pp. 916-926.

Cristobal, G.

F. Sroubek, G. Cristobal, and J. Flusser, “A unified approach to superresolution and multichannel blind deconvolution,” IEEE Trans. Image Process. 16, 2322-2332 (2007).
[CrossRef] [PubMed]

Elad, M.

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Fast and robust multiframe super-resolution,” IEEE Trans. Image Process. 13, 1327-1343 (2004).
[CrossRef] [PubMed]

S. Farsiu, D. Robinson, M. Elad, and P. Milanfar, “Advances and challenges in super-resolution: special issue on high resolution image reconstruction,” Int. J. Imaging Syst. Technol. 14, 47-57 (2004).
[CrossRef]

M. Elad and Y. Hel-Or, “A fast super-resolution reconstruction algorithm for pure translationnal motion and common space-invariant blur,” IEEE Trans. Image Process. 10, 1187-1193 (2001).
[CrossRef]

M. Elad and A. Feuer, “Restoration of a single superresolution image from several blurred, noisy, and undersampled measured images,” IEEE Trans. Image Process. 6, 1646-1658 (1997).
[CrossRef] [PubMed]

Eldar, Y.

Y. Eldar and M. Unser, “Nonideal sampling and interpolation from noisy observations in shift-invariant spaces,” IEEE Trans. Signal Process. 54, 2636-2651 (2006).
[CrossRef]

Fales, C. L.

Farsiu, S.

D. Robinson, S. Farsiu, and P. Milanfar, “Optimal registration of aliased images using variable projection with applications to super-resolution,” Comput. J. bxm007v1-12 (2007).

S. Farsiu, D. Robinson, M. Elad, and P. Milanfar, “Advances and challenges in super-resolution: special issue on high resolution image reconstruction,” Int. J. Imaging Syst. Technol. 14, 47-57 (2004).
[CrossRef]

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Fast and robust multiframe super-resolution,” IEEE Trans. Image Process. 13, 1327-1343 (2004).
[CrossRef] [PubMed]

Feder, M.

D. Seidner and M. Feder, “Noise amplification of periodic nonuniform sampling,” IEEE Trans. Image Process. 48, 275-277 (2000).

Feuer, A.

M. Elad and A. Feuer, “Restoration of a single superresolution image from several blurred, noisy, and undersampled measured images,” IEEE Trans. Image Process. 6, 1646-1658 (1997).
[CrossRef] [PubMed]

Flusser, J.

F. Sroubek, G. Cristobal, and J. Flusser, “A unified approach to superresolution and multichannel blind deconvolution,” IEEE Trans. Image Process. 16, 2322-2332 (2007).
[CrossRef] [PubMed]

Giovannelli, J.-F.

G. Rochefort, F. Champagnat, G. Le Besnerais, and J.-F. Giovannelli, “An improved observation model for super-resolution under affine motion,” IEEE Trans. Image Process. 15, 3325-3337 (2006).
[CrossRef] [PubMed]

Golub, G.

N. Nguyen, P. Milanfar, and G. Golub, “A computationally efficient superresolution image reconstruction algorithm,” IEEE Trans. Image Process. 10, 573-583 (2001).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction à l'Optique de Fourier et à l'Holographie (Masson, Paris, 1972).

Gross, D.

H. Ur and D. Gross, “Improved resolution from sub-pixel shifted pictures,” CVGIP: Graph. Models Image Process. 54, 181-186 (1992).
[CrossRef]

Hardie, R. C.

R. C. Hardie, “A fast image super-resolution algorithm using an adaptive Wiener filter,” IEEE Trans. Image Process. 16, 2953-2964 (2007).
[CrossRef] [PubMed]

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, “High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system,” Opt. Eng. (Bellingham) 37, 247-260 (1998).
[CrossRef]

R. C. Hardie, K. J. Barnard, and E. E. Armstrong, “Joint MAP registration and high-resolution image estimation using a sequence of undersampled images,” IEEE Trans. Image Process. 6, 1621-1633 (1997).
[CrossRef] [PubMed]

Hel-Or, Y.

M. Elad and Y. Hel-Or, “A fast super-resolution reconstruction algorithm for pure translationnal motion and common space-invariant blur,” IEEE Trans. Image Process. 10, 1187-1193 (2001).
[CrossRef]

Howe, J. D.

J. Shi, S. E. Reichenbach, and J. D. Howe, “Small-kernel superresolution methods for microscanning imaging systems,” Appl. Opt. 6, 1203-1214 (2006).
[CrossRef]

Huang, T. S.

R. Y. Tsai and T. S. Huang, “Multiframe image restoration and registration,” in Advances in Computer Vision and Image Processing: Image Reconstruction from Incomplete ObservationsT.S.Huang, ed. (JAI Press, 1984), pp. 317-339.

Huck, F. O.

Ichioka, Y.

Irani, M.

M. Irani and S. Peleg, “Improving resolution by image registration,” Comput. Vis. Graph. Image Process. 52, 231-239 (1991).

Ishida, K.

Kanade, T.

S. Baker and T. Kanade, “Limits on super-resolution and how to break them,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1167-1183 (2002).
[CrossRef]

B. Lucas and T. Kanade, “An iterative image registration technique with an application to stereo vision,” in Proceedings of 7th International Joint Conference Artificial Intelligence (IJCAI, 1981), pp. 674-679.

Kang, M. G.

E. S. Lee and M. G. Kang, “Regularized adaptive high-resolution image reconstruction considering inaccurate subpixel registration,” IEEE Trans. Image Process. 12, 526-837 (2003).

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: A technical overview,” IEEE Signal Process. Mag. 20, 21-36 (2003).
[CrossRef]

Kent, J.

J. Kent and K. Mardia, “The link between kriging and thin plate splines,” in Probability, Statistics and Optimization: a Tribute to Peter Whittle (Wiley, 1994), pp. 325-339.

Kim, S.

S. Kim, N. Bose, and H. Valenzuela, “Recursive reconstruction of high resolution image from noisy undersampled multiframes,” IEEE Transactions on Acoustics, IEEE Trans. Acoust., Speech, Signal Process. 38, 1013-1027 (1990).
[CrossRef]

Kim, S. P.

S. P. Kim and W. Y. Su, “Recursive high-resolution reconstruction of blurred multiframe images,” IEEE Trans. Image Process. 2, 534-539 (1993).
[CrossRef] [PubMed]

Kondou, N.

Kulcsár, C.

F. Champagnat, C. Kulcsár, and G. Le Besnerais, “Continuous super-resolution for recovery of 1-D image features: Algorithm and performance modeling,” in 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 2006), Vol. 1, pp. 916-926.

F. Champagnat, G. Le Besnerais, and C. Kulcsár, “Performance modeling of regularized, linear, spatially invariant super-resolution methods,” Tech. rep., RT 1/12265 DTIM (ONERA, 2008).

Kumagai, T.

Künsch, H. R.

H. R. Künsch, “Intrinsic autoregressions and related models on the two-dimensional lattice,” Biometrika 74, 517-524 (1987).

Le Besnerais, G.

G. Rochefort, F. Champagnat, G. Le Besnerais, and J.-F. Giovannelli, “An improved observation model for super-resolution under affine motion,” IEEE Trans. Image Process. 15, 3325-3337 (2006).
[CrossRef] [PubMed]

F. Champagnat, G. Le Besnerais, and C. Kulcsár, “Performance modeling of regularized, linear, spatially invariant super-resolution methods,” Tech. rep., RT 1/12265 DTIM (ONERA, 2008).

F. Champagnat, C. Kulcsár, and G. Le Besnerais, “Continuous super-resolution for recovery of 1-D image features: Algorithm and performance modeling,” in 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 2006), Vol. 1, pp. 916-926.

F. Champagnat and G. Le Besnerais, “A Fourier interpretation of super-resolution techniques,” in Proceedings of the IEEE International Conference on Image Processing 2005 (IEEE, 2005), Vol. 1, pp. 865-868.

Lee, E. S.

E. S. Lee and M. G. Kang, “Regularized adaptive high-resolution image reconstruction considering inaccurate subpixel registration,” IEEE Trans. Image Process. 12, 526-837 (2003).

Lin, Z.

Z. Lin and H.-Y. Shum, “Fundamental limits of reconstruction-based superresolution algorithms under local translation,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 83-97 (2004).
[CrossRef] [PubMed]

Lucas, B.

B. Lucas and T. Kanade, “An iterative image registration technique with an application to stereo vision,” in Proceedings of 7th International Joint Conference Artificial Intelligence (IJCAI, 1981), pp. 674-679.

Mardia, K.

J. Kent and K. Mardia, “The link between kriging and thin plate splines,” in Probability, Statistics and Optimization: a Tribute to Peter Whittle (Wiley, 1994), pp. 325-339.

Matthews, I.

S. Baker and I. Matthews, “Lukas-Kanade 20 years on: A unifying framework,” Int. J. Comput. Vis. 56, 221-255 (2004).
[CrossRef]

McCormick, J. A.

Milanfar, P.

D. Robinson, S. Farsiu, and P. Milanfar, “Optimal registration of aliased images using variable projection with applications to super-resolution,” Comput. J. bxm007v1-12 (2007).

D. Robinson and P. Milanfar, “Statistical performance analysis of super-resolution,” IEEE Trans. Image Process. 15, 1413-1428 (2006).
[CrossRef] [PubMed]

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Fast and robust multiframe super-resolution,” IEEE Trans. Image Process. 13, 1327-1343 (2004).
[CrossRef] [PubMed]

S. Farsiu, D. Robinson, M. Elad, and P. Milanfar, “Advances and challenges in super-resolution: special issue on high resolution image reconstruction,” Int. J. Imaging Syst. Technol. 14, 47-57 (2004).
[CrossRef]

N. Nguyen, P. Milanfar, and G. Golub, “A computationally efficient superresolution image reconstruction algorithm,” IEEE Trans. Image Process. 10, 573-583 (2001).
[CrossRef]

Miyatake, S.

Miyazaki, D.

Morimoto, T.

Narayanswamy, R.

S. E. Reichenbach, S. K. Park, and R. Narayanswamy, “Characterizing digital image acquisition devices,” Opt. Eng. (Bellingham) 30, 170-177 (1991).
[CrossRef]

Nguyen, N.

N. Nguyen, P. Milanfar, and G. Golub, “A computationally efficient superresolution image reconstruction algorithm,” IEEE Trans. Image Process. 10, 573-583 (2001).
[CrossRef]

Okutomi, M.

M. Shimizu and M. Okutomi, “Sub-pixel estimation error cancellation on area-based matching,” Int. J. Comput. Vis. 63, 207-224 (2005).
[CrossRef]

Oudegeest, O. R.

A. W. M. van Eekeren, K. Schutte, O. R. Oudegeest, and L. van Vliet, “Performance evaluation of super-resolution reconstruction methods on real-world data,” EURASIP J. Advances Signal Process. 2007, 43953 (2007).

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes, 2nd ed. (McGraw-Hill, 1984).

Park, M. K.

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: A technical overview,” IEEE Signal Process. Mag. 20, 21-36 (2003).
[CrossRef]

Park, S. C.

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: A technical overview,” IEEE Signal Process. Mag. 20, 21-36 (2003).
[CrossRef]

Park, S. K.

S. E. Reichenbach, S. K. Park, and R. Narayanswamy, “Characterizing digital image acquisition devices,” Opt. Eng. (Bellingham) 30, 170-177 (1991).
[CrossRef]

C. L. Fales, F. O. Huck, J. A. McCormick, and S. K. Park, “Wiener restoration of sampled image data: end-to-end analysis,” J. Opt. Soc. Am. A 5, 300-314 (1988).
[CrossRef]

Patti, A. J.

A. J. Patti and Y. Altunbasak, “Artifact reduction for set theoretic super resolution image reconstruction with edge adaptative constraints and higher-order interpolants,” IEEE Trans. Image Process. 10, 179-186 (2001).
[CrossRef]

A. J. Patti, M. I. Sezan, and A. M. Tekalp, “Superresolution video reconstruction with arbitrary sampling lattices and nonzero aperture time,” IEEE Trans. Image Process. 6, 1064-1076 (1997).
[CrossRef] [PubMed]

Peleg, S.

M. Irani and S. Peleg, “Improving resolution by image registration,” Comput. Vis. Graph. Image Process. 52, 231-239 (1991).

Pickup, L. C.

L. C. Pickup, D. P. Capel, S. J. Roberts, and A. Zisserman, “Bayesian methods for image super-resolution,” Comput. J. 52, 101-113 (2009).
[CrossRef]

Prasad, S.

Qi, F.

Z. Wang and F. Qi, “Analysis of multiframe super-resolution reconstruction for image anti-aliasing and deblurring,” Image Vis. Comput. 23, 393-404 (2005).
[CrossRef]

Ramani, S.

S. Ramani, D. Van De Ville, T. Blu, and M. Unser, “Nonideal sampling and regularization theory,” IEEE Trans. Signal Process. 56, 1055-1070 (2008).
[CrossRef]

Reichenbach, S.

J. Shi and S. Reichenbach, “Image interpolation by two-dimensional parametric cubic convolution,” IEEE Trans. Image Process. 15, 1857-1870 (2006).
[CrossRef] [PubMed]

Reichenbach, S. E.

J. Shi, S. E. Reichenbach, and J. D. Howe, “Small-kernel superresolution methods for microscanning imaging systems,” Appl. Opt. 6, 1203-1214 (2006).
[CrossRef]

S. E. Reichenbach, S. K. Park, and R. Narayanswamy, “Characterizing digital image acquisition devices,” Opt. Eng. (Bellingham) 30, 170-177 (1991).
[CrossRef]

Roberts, S. J.

L. C. Pickup, D. P. Capel, S. J. Roberts, and A. Zisserman, “Bayesian methods for image super-resolution,” Comput. J. 52, 101-113 (2009).
[CrossRef]

Robinson, D.

D. Robinson and D. G. Stork, “Joint digital-optical design of superresolution multiframe imaging systems,” Appl. Opt. 47, 11-20 (2008).
[CrossRef]

D. Robinson, S. Farsiu, and P. Milanfar, “Optimal registration of aliased images using variable projection with applications to super-resolution,” Comput. J. bxm007v1-12 (2007).

D. Robinson and P. Milanfar, “Statistical performance analysis of super-resolution,” IEEE Trans. Image Process. 15, 1413-1428 (2006).
[CrossRef] [PubMed]

S. Farsiu, D. Robinson, M. Elad, and P. Milanfar, “Advances and challenges in super-resolution: special issue on high resolution image reconstruction,” Int. J. Imaging Syst. Technol. 14, 47-57 (2004).
[CrossRef]

Robinson, M. D.

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Fast and robust multiframe super-resolution,” IEEE Trans. Image Process. 13, 1327-1343 (2004).
[CrossRef] [PubMed]

Rochefort, G.

G. Rochefort, F. Champagnat, G. Le Besnerais, and J.-F. Giovannelli, “An improved observation model for super-resolution under affine motion,” IEEE Trans. Image Process. 15, 3325-3337 (2006).
[CrossRef] [PubMed]

Schultz, R. R.

R. R. Schultz and R. L. Stevenson, “Extraction of high-resolution frames from video sequences,” IEEE Trans. Image Process. 5, 996-1011 (1996).
[CrossRef] [PubMed]

Schutte, K.

A. W. M. van Eekeren, K. Schutte, O. R. Oudegeest, and L. van Vliet, “Performance evaluation of super-resolution reconstruction methods on real-world data,” EURASIP J. Advances Signal Process. 2007, 43953 (2007).

Seidner, D.

D. Seidner and M. Feder, “Noise amplification of periodic nonuniform sampling,” IEEE Trans. Image Process. 48, 275-277 (2000).

Sezan, M. I.

A. J. Patti, M. I. Sezan, and A. M. Tekalp, “Superresolution video reconstruction with arbitrary sampling lattices and nonzero aperture time,” IEEE Trans. Image Process. 6, 1064-1076 (1997).
[CrossRef] [PubMed]

Shi, J.

J. Shi, S. E. Reichenbach, and J. D. Howe, “Small-kernel superresolution methods for microscanning imaging systems,” Appl. Opt. 6, 1203-1214 (2006).
[CrossRef]

J. Shi and S. Reichenbach, “Image interpolation by two-dimensional parametric cubic convolution,” IEEE Trans. Image Process. 15, 1857-1870 (2006).
[CrossRef] [PubMed]

Shimizu, M.

M. Shimizu and M. Okutomi, “Sub-pixel estimation error cancellation on area-based matching,” Int. J. Comput. Vis. 63, 207-224 (2005).
[CrossRef]

Shum, H.-Y.

Z. Lin and H.-Y. Shum, “Fundamental limits of reconstruction-based superresolution algorithms under local translation,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 83-97 (2004).
[CrossRef] [PubMed]

Sroubek, F.

F. Sroubek, G. Cristobal, and J. Flusser, “A unified approach to superresolution and multichannel blind deconvolution,” IEEE Trans. Image Process. 16, 2322-2332 (2007).
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Stevenson, R. L.

R. R. Schultz and R. L. Stevenson, “Extraction of high-resolution frames from video sequences,” IEEE Trans. Image Process. 5, 996-1011 (1996).
[CrossRef] [PubMed]

Stork, D. G.

D. Robinson and D. G. Stork, “Joint digital-optical design of superresolution multiframe imaging systems,” Appl. Opt. 47, 11-20 (2008).
[CrossRef]

Su, W. Y.

S. P. Kim and W. Y. Su, “Recursive high-resolution reconstruction of blurred multiframe images,” IEEE Trans. Image Process. 2, 534-539 (1993).
[CrossRef] [PubMed]

Tanida, J.

Tekalp, A. M.

A. J. Patti, M. I. Sezan, and A. M. Tekalp, “Superresolution video reconstruction with arbitrary sampling lattices and nonzero aperture time,” IEEE Trans. Image Process. 6, 1064-1076 (1997).
[CrossRef] [PubMed]

Tsai, R. Y.

R. Y. Tsai and T. S. Huang, “Multiframe image restoration and registration,” in Advances in Computer Vision and Image Processing: Image Reconstruction from Incomplete ObservationsT.S.Huang, ed. (JAI Press, 1984), pp. 317-339.

Unser, M.

S. Ramani, D. Van De Ville, T. Blu, and M. Unser, “Nonideal sampling and regularization theory,” IEEE Trans. Signal Process. 56, 1055-1070 (2008).
[CrossRef]

Y. Eldar and M. Unser, “Nonideal sampling and interpolation from noisy observations in shift-invariant spaces,” IEEE Trans. Signal Process. 54, 2636-2651 (2006).
[CrossRef]

M. Unser and J. Zerubia, “Generalized sampling: Stability and performance analysis,” IEEE Trans. Image Process. 12, 2941-2950 (1997).

Ur, H.

H. Ur and D. Gross, “Improved resolution from sub-pixel shifted pictures,” CVGIP: Graph. Models Image Process. 54, 181-186 (1992).
[CrossRef]

Valenzuela, H.

S. Kim, N. Bose, and H. Valenzuela, “Recursive reconstruction of high resolution image from noisy undersampled multiframes,” IEEE Transactions on Acoustics, IEEE Trans. Acoust., Speech, Signal Process. 38, 1013-1027 (1990).
[CrossRef]

Van De Ville, D.

S. Ramani, D. Van De Ville, T. Blu, and M. Unser, “Nonideal sampling and regularization theory,” IEEE Trans. Signal Process. 56, 1055-1070 (2008).
[CrossRef]

van Eekeren, A. W. M.

A. W. M. van Eekeren, K. Schutte, O. R. Oudegeest, and L. van Vliet, “Performance evaluation of super-resolution reconstruction methods on real-world data,” EURASIP J. Advances Signal Process. 2007, 43953 (2007).

van Vliet, L.

A. W. M. van Eekeren, K. Schutte, O. R. Oudegeest, and L. van Vliet, “Performance evaluation of super-resolution reconstruction methods on real-world data,” EURASIP J. Advances Signal Process. 2007, 43953 (2007).

Wang, Z.

Z. Wang and F. Qi, “Analysis of multiframe super-resolution reconstruction for image anti-aliasing and deblurring,” Image Vis. Comput. 23, 393-404 (2005).
[CrossRef]

Watson, E. A.

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, “High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system,” Opt. Eng. (Bellingham) 37, 247-260 (1998).
[CrossRef]

Yamada, K.

Zerubia, J.

M. Unser and J. Zerubia, “Generalized sampling: Stability and performance analysis,” IEEE Trans. Image Process. 12, 2941-2950 (1997).

Zisserman, A.

L. C. Pickup, D. P. Capel, S. J. Roberts, and A. Zisserman, “Bayesian methods for image super-resolution,” Comput. J. 52, 101-113 (2009).
[CrossRef]

Appl. Opt. (3)

D. Robinson and D. G. Stork, “Joint digital-optical design of superresolution multiframe imaging systems,” Appl. Opt. 47, 11-20 (2008).
[CrossRef]

J. Tanida, T. Kumagai, K. Yamada, S. Miyatake, K. Ishida, T. Morimoto, N. Kondou, D. Miyazaki, and Y. Ichioka, “Thin observation module by bound optics (TOMBO): Concept and experimental verification,” Appl. Opt. 40, 1806-1813 (2001).
[CrossRef]

J. Shi, S. E. Reichenbach, and J. D. Howe, “Small-kernel superresolution methods for microscanning imaging systems,” Appl. Opt. 6, 1203-1214 (2006).
[CrossRef]

Biometrika (1)

H. R. Künsch, “Intrinsic autoregressions and related models on the two-dimensional lattice,” Biometrika 74, 517-524 (1987).

Comput. J. (2)

L. C. Pickup, D. P. Capel, S. J. Roberts, and A. Zisserman, “Bayesian methods for image super-resolution,” Comput. J. 52, 101-113 (2009).
[CrossRef]

D. Robinson, S. Farsiu, and P. Milanfar, “Optimal registration of aliased images using variable projection with applications to super-resolution,” Comput. J. bxm007v1-12 (2007).

Comput. Vis. Graph. Image Process. (1)

M. Irani and S. Peleg, “Improving resolution by image registration,” Comput. Vis. Graph. Image Process. 52, 231-239 (1991).

CVGIP: Graph. Models Image Process. (1)

H. Ur and D. Gross, “Improved resolution from sub-pixel shifted pictures,” CVGIP: Graph. Models Image Process. 54, 181-186 (1992).
[CrossRef]

EURASIP J. Advances Signal Process. (1)

A. W. M. van Eekeren, K. Schutte, O. R. Oudegeest, and L. van Vliet, “Performance evaluation of super-resolution reconstruction methods on real-world data,” EURASIP J. Advances Signal Process. 2007, 43953 (2007).

IEEE Signal Process. Mag. (1)

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: A technical overview,” IEEE Signal Process. Mag. 20, 21-36 (2003).
[CrossRef]

IEEE Trans. Acoust., Speech, Signal Process. (1)

S. Kim, N. Bose, and H. Valenzuela, “Recursive reconstruction of high resolution image from noisy undersampled multiframes,” IEEE Transactions on Acoustics, IEEE Trans. Acoust., Speech, Signal Process. 38, 1013-1027 (1990).
[CrossRef]

IEEE Trans. Image Process. (17)

S. P. Kim and W. Y. Su, “Recursive high-resolution reconstruction of blurred multiframe images,” IEEE Trans. Image Process. 2, 534-539 (1993).
[CrossRef] [PubMed]

M. Unser and J. Zerubia, “Generalized sampling: Stability and performance analysis,” IEEE Trans. Image Process. 12, 2941-2950 (1997).

D. Seidner and M. Feder, “Noise amplification of periodic nonuniform sampling,” IEEE Trans. Image Process. 48, 275-277 (2000).

G. Rochefort, F. Champagnat, G. Le Besnerais, and J.-F. Giovannelli, “An improved observation model for super-resolution under affine motion,” IEEE Trans. Image Process. 15, 3325-3337 (2006).
[CrossRef] [PubMed]

R. R. Schultz and R. L. Stevenson, “Extraction of high-resolution frames from video sequences,” IEEE Trans. Image Process. 5, 996-1011 (1996).
[CrossRef] [PubMed]

A. J. Patti, M. I. Sezan, and A. M. Tekalp, “Superresolution video reconstruction with arbitrary sampling lattices and nonzero aperture time,” IEEE Trans. Image Process. 6, 1064-1076 (1997).
[CrossRef] [PubMed]

R. C. Hardie, K. J. Barnard, and E. E. Armstrong, “Joint MAP registration and high-resolution image estimation using a sequence of undersampled images,” IEEE Trans. Image Process. 6, 1621-1633 (1997).
[CrossRef] [PubMed]

M. Elad and A. Feuer, “Restoration of a single superresolution image from several blurred, noisy, and undersampled measured images,” IEEE Trans. Image Process. 6, 1646-1658 (1997).
[CrossRef] [PubMed]

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Fast and robust multiframe super-resolution,” IEEE Trans. Image Process. 13, 1327-1343 (2004).
[CrossRef] [PubMed]

D. Robinson and P. Milanfar, “Statistical performance analysis of super-resolution,” IEEE Trans. Image Process. 15, 1413-1428 (2006).
[CrossRef] [PubMed]

F. Sroubek, G. Cristobal, and J. Flusser, “A unified approach to superresolution and multichannel blind deconvolution,” IEEE Trans. Image Process. 16, 2322-2332 (2007).
[CrossRef] [PubMed]

E. S. Lee and M. G. Kang, “Regularized adaptive high-resolution image reconstruction considering inaccurate subpixel registration,” IEEE Trans. Image Process. 12, 526-837 (2003).

M. Elad and Y. Hel-Or, “A fast super-resolution reconstruction algorithm for pure translationnal motion and common space-invariant blur,” IEEE Trans. Image Process. 10, 1187-1193 (2001).
[CrossRef]

N. Nguyen, P. Milanfar, and G. Golub, “A computationally efficient superresolution image reconstruction algorithm,” IEEE Trans. Image Process. 10, 573-583 (2001).
[CrossRef]

A. J. Patti and Y. Altunbasak, “Artifact reduction for set theoretic super resolution image reconstruction with edge adaptative constraints and higher-order interpolants,” IEEE Trans. Image Process. 10, 179-186 (2001).
[CrossRef]

J. Shi and S. Reichenbach, “Image interpolation by two-dimensional parametric cubic convolution,” IEEE Trans. Image Process. 15, 1857-1870 (2006).
[CrossRef] [PubMed]

R. C. Hardie, “A fast image super-resolution algorithm using an adaptive Wiener filter,” IEEE Trans. Image Process. 16, 2953-2964 (2007).
[CrossRef] [PubMed]

IEEE Trans. Pattern Anal. Mach. Intell. (2)

S. Baker and T. Kanade, “Limits on super-resolution and how to break them,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1167-1183 (2002).
[CrossRef]

Z. Lin and H.-Y. Shum, “Fundamental limits of reconstruction-based superresolution algorithms under local translation,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 83-97 (2004).
[CrossRef] [PubMed]

IEEE Trans. Signal Process. (2)

Y. Eldar and M. Unser, “Nonideal sampling and interpolation from noisy observations in shift-invariant spaces,” IEEE Trans. Signal Process. 54, 2636-2651 (2006).
[CrossRef]

S. Ramani, D. Van De Ville, T. Blu, and M. Unser, “Nonideal sampling and regularization theory,” IEEE Trans. Signal Process. 56, 1055-1070 (2008).
[CrossRef]

Image Vis. Comput. (1)

Z. Wang and F. Qi, “Analysis of multiframe super-resolution reconstruction for image anti-aliasing and deblurring,” Image Vis. Comput. 23, 393-404 (2005).
[CrossRef]

Int. J. Comput. Vis. (2)

M. Shimizu and M. Okutomi, “Sub-pixel estimation error cancellation on area-based matching,” Int. J. Comput. Vis. 63, 207-224 (2005).
[CrossRef]

S. Baker and I. Matthews, “Lukas-Kanade 20 years on: A unifying framework,” Int. J. Comput. Vis. 56, 221-255 (2004).
[CrossRef]

Int. J. Imaging Syst. Technol. (1)

S. Farsiu, D. Robinson, M. Elad, and P. Milanfar, “Advances and challenges in super-resolution: special issue on high resolution image reconstruction,” Int. J. Imaging Syst. Technol. 14, 47-57 (2004).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Eng. (Bellingham) (2)

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, “High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system,” Opt. Eng. (Bellingham) 37, 247-260 (1998).
[CrossRef]

S. E. Reichenbach, S. K. Park, and R. Narayanswamy, “Characterizing digital image acquisition devices,” Opt. Eng. (Bellingham) 30, 170-177 (1991).
[CrossRef]

Opt. Photonics News (1)

D. J. Brady, “Micro-optics and megapixels,” Opt. Photonics News 17, 24-29 (2006).
[CrossRef]

Other (9)

F. Champagnat, C. Kulcsár, and G. Le Besnerais, “Continuous super-resolution for recovery of 1-D image features: Algorithm and performance modeling,” in 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 2006), Vol. 1, pp. 916-926.

D. P. Capel, “Image mosaicing and super-resolution,” Ph.D. thesis, (University of Oxford, 2001).

J. Kent and K. Mardia, “The link between kriging and thin plate splines,” in Probability, Statistics and Optimization: a Tribute to Peter Whittle (Wiley, 1994), pp. 325-339.

J. W. Goodman, Introduction à l'Optique de Fourier et à l'Holographie (Masson, Paris, 1972).

R. Y. Tsai and T. S. Huang, “Multiframe image restoration and registration,” in Advances in Computer Vision and Image Processing: Image Reconstruction from Incomplete ObservationsT.S.Huang, ed. (JAI Press, 1984), pp. 317-339.

F. Champagnat, G. Le Besnerais, and C. Kulcsár, “Performance modeling of regularized, linear, spatially invariant super-resolution methods,” Tech. rep., RT 1/12265 DTIM (ONERA, 2008).

B. Lucas and T. Kanade, “An iterative image registration technique with an application to stereo vision,” in Proceedings of 7th International Joint Conference Artificial Intelligence (IJCAI, 1981), pp. 674-679.

F. Champagnat and G. Le Besnerais, “A Fourier interpretation of super-resolution techniques,” in Proceedings of the IEEE International Conference on Image Processing 2005 (IEEE, 2005), Vol. 1, pp. 865-868.

A. Papoulis, Probability, Random Variables and Stochastic Processes, 2nd ed. (McGraw-Hill, 1984).

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Figures (15)

Fig. 1
Fig. 1

SR experiment on resolution chart wedges. Left: ideal image of the chart. Bundle 4 is twice thinner than bundle 2, which is twice thinner than bundle 1. The dashed lines indicate the boundary of the high-frequency (HF) region selected for MSE computation. Image size is 1479 × 943   pixels . Right: interpolated LR image.

Fig. 2
Fig. 2

Left: observed response to a step-edge and response for a Gaussian PSF, σ h = 0.35 . Agreement is fairly correct. Right: modulation transfer function (MTF): Fourier transform of the Gaussian PSF, σ h = 0.35 . The Nyquist band boundaries are indicated by the two vertical dashed lines.

Fig. 3
Fig. 3

SR experiment on resolution chart wedges. Left: bicubic interpolation of the first LR frame. Right: SR using 40 images and PMF = 4 .

Fig. 4
Fig. 4

SR performance with 20 LR frames and a varying PMF. Top: SR reconstructions with PMF 2 (left), 3 (center), and 4 (right). Bottom: detail of bundle 2 from the upper part. Bundle 2 is not recovered with PMF = 2 as accurately as with PMF = 3 or PMF = 4 .

Fig. 5
Fig. 5

RMSE of SR reconstructions for different PMFs and an increasing number of input LR images.

Fig. 6
Fig. 6

PSDs of 1-D exponential correlation processes. Log-log graphs of Φ s ( ν ) = 100 ( α 2 + 4 π 2 ν 2 ) , for α = 0 , 0.01 , 0.1 , 1 . The Nyquist frequency ν s 2 is emphasized by a vertical dashed line. Except for α = 0 , Φ s has a typical lowpass behavior, with cutoff at α 2 π . The case α = 0 is a good approximation for ν > α 2 π .

Fig. 7
Fig. 7

Block diagram of data model with shifts τ 1 , , τ K .

Fig. 8
Fig. 8

Block diagram of CDC reconstruction with multichannel filter w 1 , , w K .

Fig. 9
Fig. 9

Histogram of the MMSE σ 0 2 ( τ ) for 10 4 samples τ drawn uniformly on [0,1). h is the box function, K = 5 , Φ s = 100 4 π 2 ν 2 , r b = 1 .

Fig. 10
Fig. 10

Statistics of the MMSE σ 0 2 ( τ ) for 10 4 samples τ drawn uniformly on [0,1). Curves σ 0 2 ( τ e ) and σ 0 , K 2 are also displayed. h is the box function, Φ s = 100 4 π 2 ν 2 , r b = 1 , K grows from 1 to 20.

Fig. 11
Fig. 11

Statistics of σ 0 2 ( τ ) for 10 4 samples τ drawn uniformly on [0,1). Curves σ 0 2 ( τ e ) and σ 0 , K 2 are also displayed. h is Gaussian ( σ h = 0.5 ) , Φ s = 100 4 π 2 ν 2 , r b = 1 , K grows from 1 to 20.

Fig. 12
Fig. 12

Curves depict σ 0 2 ( τ e ) , while crosses depict σ 0 , K 2 for different noise level: r b = 0.1 (solid curve), r b = 1 (dashed curve), r b = 10 (dotted curve). h is Gaussian, σ h = 0.25 , Φ s = 100 4 π 2 ν 2 .

Fig. 13
Fig. 13

Evolution of the MSE w.r.t.K for random τ, for the optimal Wiener filter (dashed curve) and the suboptimal one (dashed-dotted). The zero-shift case (dotted curve) and the equispaced case (open circles) appear as upper and lower bound, respectively. This is a 2-D case, Φ s = 100 ( ( 1 4 ) 2 + ν 2 ) ) 3 2 , h is Gaussian with σ h = 1 2 π , r b = 1 .

Fig. 14
Fig. 14

Evolution of the empirical RMSE w.r.t.K for estimated motion (open circles) and rounded motion, with different rounding accuracy: half pixel (open squares), 1 3   pixel (diamonds) and 1 4   pixel (plusses).

Fig. 15
Fig. 15

Evolution of the MSE w.r.t.K for random τ, for the optimal Wiener filter (solid curve) and suboptimal ones, with different rounding accuracy: 1 3   pixel (dashed curve), 1 4   pixel (dashed–dotted curve) and 1 10   pixel (open circles). This is a 2-D case, Φ s = 100 ( ( 0.25 ) 2 + ν 2 ) ) 3 2 , h is Gaussian with σ h = 1 2 π , r b = 1 .

Equations (67)

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h ̃ ( ν ) = R h ( x ) exp ( 2 i π ν x ) d x .
γ s ( x ) = E ( S ( x ) S ( 0 ) ) 2 = ρ s x .
F n = h S ( n ν s ) + B n , n Z ,
F n k = h S ( n τ k ) + B n k , { n Z k = 1 , , K } ,
S ̂ ( x ) = n F n T w ( x n ) ,
w ( x ) [ w 1 ( x ) , , w K ( x ) ] T ,
F n [ F n 1 , , F n K ] T .
S ̂ ( m M ) = n F n T w ( m M n ) , m Z ,
σ 2 ( x ) E ( S ( x ) S ̂ ( x ) ) 2 .
σ 2 ( x ) = r s ( 0 ) 2 n w T ( x n ) r f s ( x n ) + n , p w T ( x n ) R f ( p n ) w ( x p ) ,
r s ( y ) E ( S ( x ) S ( x + y ) ) ,
R f ( n ) E ( F n F n + m T ) ,
r f s ( y ) E ( F n S ( n + y ) ) .
σ ¯ 2 0 1 σ 2 ( x ) d x .
σ ¯ 2 = R ( Φ s ( ν ) 2 R e { w ̃ ( ν ) h ̃ ( ν ) Φ s ( ν ) v ν } + w ̃ ( ν ) Φ ν w ̃ ( ν ) ) d ν ,
Φ ν n Φ g ( ν + n ) v ν + n v ν + n + r b I .
Φ g Φ s h ̃ 2 ,
v ν [ exp ( 2 i π ν τ 1 ) , , exp ( 2 i π ν τ K ) ] T .
σ ¯ 2 = σ 0 2 + R ( w ̃ ( ν ) w ̃ 0 ( ν ) ) Φ ν ( w ̃ ( ν ) w ̃ 0 ( ν ) ) d ν ,
w ̃ 0 ( ν ) Φ s ( ν ) h ̃ ( ν ) Φ ν 1 v ν ,
σ 0 2 R Φ s ( ν ) ( 1 Φ s ( ν ) h ̃ ( ν ) 2 v ν Φ ν 1 v ν ) d ν .
w ̃ 0 = Φ s h ̃ K Φ s h ̃ 2 + r b v ν .
Φ ν = ( n Φ g ( ν + n ) ) v 0 v 0 + r b I .
σ 0 2 ( 0 ) = Φ s ( ν ) r b + K n 0 Φ g ( ν + n ) r b + K n Φ g ( ν + n ) d ν .
Φ ν = k ( p Φ g ( ν + k + K p ) ) v ν + k v ν + k + r b I .
Φ ν v ν = ( r b + K p Φ g ( ν + K p ) ) v ν .
σ 0 2 ( τ e ) = Φ s ( ν ) r b + K n 0 Φ g ( ν + K n ) r b + K n Φ g ( ν + K n ) d ν .
σ 0 , K 2 σ 0 2 ( τ ) σ 0 2 ( 0 ) ,
σ 0 , K 2 R r b Φ s K Φ s h ̃ 2 + r b .
σ 0 2 ( τ e ) σ 0 2 ( τ ) ,
Φ a ( ν , K ) n 0 Φ g ( ν + K n )
σ 0 2 ( τ e ) = Φ s ( ν ) r b + K Φ a r b + K Φ s h ̃ 2 + K Φ a d ν .
σ 0 2 ( τ e ) = 2 ν ¯ c + Φ s + [ ν ¯ c , ν ¯ c ] Φ s ( 1 Φ g Φ g + r b K + Φ a ( , K ) ) .
σ ¯ 2 = σ 0 2 + R ( w ̃ w ̃ 0 ) Φ ( w ̃ w ̃ 0 ) .
w ̃ ( ν ) = w ̃ ( ν ) v ν ,
w ̃ 0 ( ν ) = Φ s h ̃ n Φ g ( ν + n ) v 0 v n 2 K + r b .
σ ¯ 2 ( w ̃ 0 , τ ) Φ s ( 1 Φ g Φ g + r b K + n 0 Φ g ( + n ) v 0 v n 2 K 2 ) .
v 0 v n 2 K 2 = k exp ( 2 i π n τ k ) K 2 ,
2 ν ¯ c + Φ s σ 0 2 ( τ ) σ ¯ 2 ( w ̃ 0 , τ ) ;
J ( s ) = f A s 2 + λ R ( s ) .
n R f ( n ) exp ( 2 i π ν n ) = n Φ g ( ν + n ) v ν + n v ν + n + r b I
= Φ ν .
n r h r s ( n + τ k τ l ) exp ( 2 i π ν n ) = n Φ g ( ν + n ) exp ( 2 i π ( ν + n ) ( τ k τ l ) ) .
σ ¯ 2 = r s ( 0 ) 2 R w T ( x ) r f s ( x ) d x + 0 1 n , p w T ( x n ) R f ( p n ) w ( x p ) .
Q = 0 1 n , p w T ( x n ) R f ( p n ) w ( x p ) d x
= 0 1 n , p w T ( x n ) R f ( p ) w ( x n p ) d x
= p ( 0 1 n w T ( x n ) R f ( p ) w ( x n p ) ) d x
= p R w T ( x ) R f ( p ) w ( x p ) d x .
r f s ( x ) = [ h r s ( x + τ 1 ) , , h r s ( x + τ k ) ] T ,
r ̃ f s ( ν ) h ̃ ( ν ) Φ s ( ν ) v ν .
2 R w ̃ ( ν ) r ̃ f s ( ν ) d ν = 2 R w ̃ ( ν ) h ̃ ( ν ) Φ s ( ν ) v ν d ν ,
2 R w T ( x ) r f s ( x ) d x = 2 R R e { w ̃ ( ν ) h ̃ ( ν ) Φ s ( ν ) v ν } d ν .
Q = p R w ̃ ( ν ) R f ( p ) w ̃ ( ν ) exp ( 2 i π ν p ) d ν
= R w ̃ ( ν ) Φ ν w ̃ ( ν ) d ν .
σ 0 , K 2 R r b Φ s K Φ s h ̃ 2 + r b σ 0 2 ( τ ) σ 0 2 ( 0 ) .
a Φ ν a = r b a 2 + n Φ g ( ν + n ) v ν + n a 2 .
a Φ ν a a 2 ( r b + K n Φ g ( ν + n ) ) .
σ 0 2 ( τ ) Φ s ( ν ) r b + K n 0 Φ g ( ν + n ) r b + K n Φ g ( ν + n ) d ν .
ϕ s f ( ν ) Φ s ( ν ) Φ s ( ν ) h ̃ ( ν ) ( v ν Φ ν 1 v ν ) h ̃ ( ν ) Φ s ( ν ) .
ϕ s f ( ν ) = Φ s ( ν ) 1 + Φ g ( ν ) v ν Φ ¯ ν 1 v ν .
a T Φ ¯ ν a = r b a 2 + n 0 Φ g ( ν + n ) v ν + n a 2 ,
ϕ s f ( ν ) r b Φ s ( ν ) r b + K Φ s ( ν ) h ̃ ( ν ) 2 ,
w ̃ 0 ( ν ) = Φ s h ̃ n Φ g ( ν + n ) v 0 v n 2 K + r b ,
ψ s f w ̃ ( ν ) Φ ν w ̃ ( ν ) 2 R e { w ̃ ( ν ) h ̃ ( ν ) Φ s ( ν ) v ν } + Φ s ( ν ) .
2 R e { w ̃ ( ν ) h ̃ ( ν ) Φ s ( ν ) v ν } = 2 R e { K h ̃ ( ν ) Φ s ( ν ) w ̃ ( ν ) } .
w ̃ ( ν ) Φ ν w ̃ ( ν ) = r b w ̃ 2 + n Φ g ( ν + n ) w ̃ v ν + n 2 .
w ̃ ( ν ) Φ ν w ̃ ( ν ) = w ̃ ( ν ) 2 ( K r b + n Φ g ( ν + n ) v 0 v n 2 ) .

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