Abstract

We present a new method to measure the modulation transfer function (MTF) beyond the Nyquist frequency of a multichannel imaging system for which all the channels have parallel optical axes. Such a multichannel optical system produces a set of undersampled subimages. If the subimages contain nonredundant information, high spatial frequencies are folded between low spatial frequencies, leading to the possible extraction of frequencies higher than the Nyquist frequency. The measurement of the MTF of the multichannel system leads to the estimation of the resolution enhancement of the final image that can be obtained by applying a postprocessing algorithm to the collection of undersampled subimages. Experimental images are presented to validate this method.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]

2009 (2)

2008 (5)

2007 (4)

L. C. Laycock and V. A. Handerek, “Multi-aperture imaging device for airborne platforms,” Proc. SPIE 6737, 673709(2007).
[CrossRef]

J. Fanning, J. Miller, J. Park, G. Tener, J. Reynolds, P. O’Shea, C. Halford, and R. Driggers, “IR system field performance with superresolution,” Proc. SPIE 6543, 65430Z (2007).
[CrossRef]

K. Krapels, R. G. Driggers, E. Jacobs, S. Burks, and S. Young, “Characteristics of infrared imaging systems that benefit from superresolution reconstruction,” Appl. Opt. 46, 4594–4603(2007).
[CrossRef] [PubMed]

A. V. Kanaev, J. R. Ackerman, E. F. Fleet, and D. A. Scribner, “TOMBO sensor with scene-independent superresolution processing,” Opt. Lett. 32, 2855–2857 (2007).
[CrossRef] [PubMed]

2006 (4)

2005 (2)

2004 (3)

2003 (1)

R. Völkel, M. Eisner, and K. J. Weible, “Miniaturized imaging systems,” Microelectron. Eng. 67–68, 461–472 (2003).
[CrossRef]

2001 (2)

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]

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]

1999 (1)

1996 (1)

M. Chambon, J. Primot, and M. Girard, “Modulation transfer function assessment for sampled imaging system: application of the generalized line spread function to a standard infrared camera,” Infrared Phys. Technol. 37, 619–626 (1996).
[CrossRef]

1995 (1)

M. A. Chambliss, J. A. Dawson, and E. J. Borg, “Measuring the MTF of undersampled staring IRFPA sensors using 2D discrete Fourier transform,” Proc. SPIE 2470, 312–324 (1995).
[CrossRef]

1991 (1)

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

1989 (1)

1977 (1)

A. Papoulis, “Generalized sampling expansion,” IEEE Trans. Circuits Syst. 24, 652–654 (1977).
[CrossRef]

Ackerman, J. R.

Barnard, R.

R. Barnard, V. P. Pauca, T. C. Torgersen, R. J. Plemmons, S. Prasad, J. van der Gracht, J. Nagy, J. Chung, G. Behrmann, S. Mathews, and M. Mirotznik, “High-resolution iris image reconstruction from low-resolution imagery,” Proc. SPIE 6313, D1–D13 (2006).

Behrmann, G.

R. Barnard, V. P. Pauca, T. C. Torgersen, R. J. Plemmons, S. Prasad, J. van der Gracht, J. Nagy, J. Chung, G. Behrmann, S. Mathews, and M. Mirotznik, “High-resolution iris image reconstruction from low-resolution imagery,” Proc. SPIE 6313, D1–D13 (2006).

Borg, E. J.

M. A. Chambliss, J. A. Dawson, and E. J. Borg, “Measuring the MTF of undersampled staring IRFPA sensors using 2D discrete Fourier transform,” Proc. SPIE 2470, 312–324 (1995).
[CrossRef]

Brady, D.

Bräuer, A.

Burks, S.

Caes, M.

Cao, F.

C.-L. Tisse, F. Guichard, and F. Cao, “Does resolution really increase image quality?” Proc. SPIE 6817, 68170Q (2008).
[CrossRef]

Carriere, J.

Chambliss, M. A.

M. A. Chambliss, J. A. Dawson, and E. J. Borg, “Measuring the MTF of undersampled staring IRFPA sensors using 2D discrete Fourier transform,” Proc. SPIE 2470, 312–324 (1995).
[CrossRef]

Chambon, M.

M. Chambon, J. Primot, and M. Girard, “Modulation transfer function assessment for sampled imaging system: application of the generalized line spread function to a standard infrared camera,” Infrared Phys. Technol. 37, 619–626 (1996).
[CrossRef]

Champagnat, F.

Chen, C.

Choi, K.

Chung, J.

R. Barnard, V. P. Pauca, T. C. Torgersen, R. J. Plemmons, S. Prasad, J. van der Gracht, J. Nagy, J. Chung, G. Behrmann, S. Mathews, and M. Mirotznik, “High-resolution iris image reconstruction from low-resolution imagery,” Proc. SPIE 6313, D1–D13 (2006).

Dannberg, P.

Dawson, J. A.

M. A. Chambliss, J. A. Dawson, and E. J. Borg, “Measuring the MTF of undersampled staring IRFPA sensors using 2D discrete Fourier transform,” Proc. SPIE 2470, 312–324 (1995).
[CrossRef]

Deschamps, J.

G. Druart, N. Guérineau, R. Haïdar, M. Tauvy, S. Thétas, S. Rommeluère, J. Primot, J. Deschamps, and E. Lambert, “MULTICAM: a miniature cryogenic camera for infrared detection,” Proc. SPIE 6992, 699215 (2008).

Driggers, R.

J. Fanning, J. Miller, J. Park, G. Tener, J. Reynolds, P. O’Shea, C. Halford, and R. Driggers, “IR system field performance with superresolution,” Proc. SPIE 6543, 65430Z (2007).
[CrossRef]

Driggers, R. G.

Druart, G.

G. Druart, N. Guérineau, R. Haïdar, S. Thétas, J. Taboury, S. Rommeluère, J. Primot, and M. Fendler, “Demonstration of an infrared microcamera inspired by Xenos peckii vision,” Appl. Opt. 48, 3368–3374 (2009).
[CrossRef] [PubMed]

G. Druart, N. Guérineau, R. Haïdar, M. Tauvy, S. Thétas, S. Rommeluère, J. Primot, J. Deschamps, and E. Lambert, “MULTICAM: a miniature cryogenic camera for infrared detection,” Proc. SPIE 6992, 699215 (2008).

Duparré, J.

Eisner, M.

R. Völkel, M. Eisner, and K. J. Weible, “Miniaturized imaging systems,” Microelectron. Eng. 67–68, 461–472 (2003).
[CrossRef]

Elad, M.

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]

Fanning, J.

J. Fanning, J. Miller, J. Park, G. Tener, J. Reynolds, P. O’Shea, C. Halford, and R. Driggers, “IR system field performance with superresolution,” Proc. SPIE 6543, 65430Z (2007).
[CrossRef]

Fendler, M.

Fleet, E. F.

Gibbons, R.

Girard, M.

M. Chambon, J. Primot, and M. Girard, “Modulation transfer function assessment for sampled imaging system: application of the generalized line spread function to a standard infrared camera,” Infrared Phys. Technol. 37, 619–626 (1996).
[CrossRef]

Guérineau, N.

Guichard, F.

C.-L. Tisse, F. Guichard, and F. Cao, “Does resolution really increase image quality?” Proc. SPIE 6817, 68170Q (2008).
[CrossRef]

Haïdar, R.

G. Druart, N. Guérineau, R. Haïdar, S. Thétas, J. Taboury, S. Rommeluère, J. Primot, and M. Fendler, “Demonstration of an infrared microcamera inspired by Xenos peckii vision,” Appl. Opt. 48, 3368–3374 (2009).
[CrossRef] [PubMed]

G. Druart, N. Guérineau, R. Haïdar, M. Tauvy, S. Thétas, S. Rommeluère, J. Primot, J. Deschamps, and E. Lambert, “MULTICAM: a miniature cryogenic camera for infrared detection,” Proc. SPIE 6992, 699215 (2008).

Halford, C.

J. Fanning, J. Miller, J. Park, G. Tener, J. Reynolds, P. O’Shea, C. Halford, and R. Driggers, “IR system field performance with superresolution,” Proc. SPIE 6543, 65430Z (2007).
[CrossRef]

Handerek, V. A.

L. C. Laycock and V. A. Handerek, “Multi-aperture imaging device for airborne platforms,” Proc. SPIE 6737, 673709(2007).
[CrossRef]

Haney, M. W.

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]

Holst, G. C.

G. C. Holst, “Infrared imaging system testing,” in Infrared and Electro-optical Systems Handbook, M.C.Dudzik, ed. (SPIE, 1993), Vol. 4, pp. 223–232.

Howe, J. D.

Ichioka, Y.

Ishida, K.

Jacobs, E.

Kanaev, A. V.

Kitamura, Y.

Kondou, N.

Krapels, K.

Kulcsàr, C.

Kumagai, T.

Lambert, E.

G. Druart, N. Guérineau, R. Haïdar, M. Tauvy, S. Thétas, S. Rommeluère, J. Primot, J. Deschamps, and E. Lambert, “MULTICAM: a miniature cryogenic camera for infrared detection,” Proc. SPIE 6992, 699215 (2008).

Laycock, L. C.

L. C. Laycock and V. A. Handerek, “Multi-aperture imaging device for airborne platforms,” Proc. SPIE 6737, 673709(2007).
[CrossRef]

Le Besnerais, G.

Lohmann, A. W.

Masaki, Y.

Mathews, S.

R. Barnard, V. P. Pauca, T. C. Torgersen, R. J. Plemmons, S. Prasad, J. van der Gracht, J. Nagy, J. Chung, G. Behrmann, S. Mathews, and M. Mirotznik, “High-resolution iris image reconstruction from low-resolution imagery,” Proc. SPIE 6313, D1–D13 (2006).

Matthes, A.

Miller, J.

J. Fanning, J. Miller, J. Park, G. Tener, J. Reynolds, P. O’Shea, C. Halford, and R. Driggers, “IR system field performance with superresolution,” Proc. SPIE 6543, 65430Z (2007).
[CrossRef]

Mirotznik, M.

R. Barnard, V. P. Pauca, T. C. Torgersen, R. J. Plemmons, S. Prasad, J. van der Gracht, J. Nagy, J. Chung, G. Behrmann, S. Mathews, and M. Mirotznik, “High-resolution iris image reconstruction from low-resolution imagery,” Proc. SPIE 6313, D1–D13 (2006).

Miyamoto, M.

Miyatake, S.

Miyazaki, D.

Morimoto, T.

Nagy, J.

R. Barnard, V. P. Pauca, T. C. Torgersen, R. J. Plemmons, S. Prasad, J. van der Gracht, J. Nagy, J. Chung, G. Behrmann, S. Mathews, and M. Mirotznik, “High-resolution iris image reconstruction from low-resolution imagery,” Proc. SPIE 6313, D1–D13 (2006).

Nitta, K.

O’Shea, P.

J. Fanning, J. Miller, J. Park, G. Tener, J. Reynolds, P. O’Shea, C. Halford, and R. Driggers, “IR system field performance with superresolution,” Proc. SPIE 6543, 65430Z (2007).
[CrossRef]

Papoulis, A.

A. Papoulis, “Generalized sampling expansion,” IEEE Trans. Circuits Syst. 24, 652–654 (1977).
[CrossRef]

Park, J.

J. Fanning, J. Miller, J. Park, G. Tener, J. Reynolds, P. O’Shea, C. Halford, and R. Driggers, “IR system field performance with superresolution,” Proc. SPIE 6543, 65430Z (2007).
[CrossRef]

Park, S. K.

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

Pauca, V. P.

R. Barnard, V. P. Pauca, T. C. Torgersen, R. J. Plemmons, S. Prasad, J. van der Gracht, J. Nagy, J. Chung, G. Behrmann, S. Mathews, and M. Mirotznik, “High-resolution iris image reconstruction from low-resolution imagery,” Proc. SPIE 6313, D1–D13 (2006).

Pitsianis, N.

Plemmons, R. J.

R. Barnard, V. P. Pauca, T. C. Torgersen, R. J. Plemmons, S. Prasad, J. van der Gracht, J. Nagy, J. Chung, G. Behrmann, S. Mathews, and M. Mirotznik, “High-resolution iris image reconstruction from low-resolution imagery,” Proc. SPIE 6313, D1–D13 (2006).

Prasad, S.

R. Barnard, V. P. Pauca, T. C. Torgersen, R. J. Plemmons, S. Prasad, J. van der Gracht, J. Nagy, J. Chung, G. Behrmann, S. Mathews, and M. Mirotznik, “High-resolution iris image reconstruction from low-resolution imagery,” Proc. SPIE 6313, D1–D13 (2006).

Prather, D.

Primot, J.

G. Druart, N. Guérineau, R. Haïdar, S. Thétas, J. Taboury, S. Rommeluère, J. Primot, and M. Fendler, “Demonstration of an infrared microcamera inspired by Xenos peckii vision,” Appl. Opt. 48, 3368–3374 (2009).
[CrossRef] [PubMed]

G. Druart, N. Guérineau, R. Haïdar, M. Tauvy, S. Thétas, S. Rommeluère, J. Primot, J. Deschamps, and E. Lambert, “MULTICAM: a miniature cryogenic camera for infrared detection,” Proc. SPIE 6992, 699215 (2008).

N. Guérineau, J. Primot, M. Tauvy, and M. Caes, “Modulation transfer function measurement of an infrared focal plane array by use of the self-imaging property of a canted periodic target,” Appl. Opt. 38, 631–637 (1999).
[CrossRef]

M. Chambon, J. Primot, and M. Girard, “Modulation transfer function assessment for sampled imaging system: application of the generalized line spread function to a standard infrared camera,” Infrared Phys. Technol. 37, 619–626 (1996).
[CrossRef]

Pshenay-Severin, E.

Rarayanswamy, R.

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

Reichenbach, S. E.

J. Shi, S. E. Reichenbach, and J. D. Howe, “Small-kernel superresolution methods for microscanning imaging systems,” Appl. Opt. 45, 1203–1214 (2006).
[CrossRef] [PubMed]

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

Reynolds, J.

J. Fanning, J. Miller, J. Park, G. Tener, J. Reynolds, P. O’Shea, C. Halford, and R. Driggers, “IR system field performance with superresolution,” Proc. SPIE 6543, 65430Z (2007).
[CrossRef]

Robinson, M. D.

Rommeluère, S.

G. Druart, N. Guérineau, R. Haïdar, S. Thétas, J. Taboury, S. Rommeluère, J. Primot, and M. Fendler, “Demonstration of an infrared microcamera inspired by Xenos peckii vision,” Appl. Opt. 48, 3368–3374 (2009).
[CrossRef] [PubMed]

G. Druart, N. Guérineau, R. Haïdar, M. Tauvy, S. Thétas, S. Rommeluère, J. Primot, J. Deschamps, and E. Lambert, “MULTICAM: a miniature cryogenic camera for infrared detection,” Proc. SPIE 6992, 699215 (2008).

Schreiber, P.

Schulz, T.

Schulz, T. J.

Scribner, D. A.

Shankar, M.

Shi, J.

Shogenji, R.

Stork, D. G.

Taboury, J.

Tanida, J.

Tauvy, M.

G. Druart, N. Guérineau, R. Haïdar, M. Tauvy, S. Thétas, S. Rommeluère, J. Primot, J. Deschamps, and E. Lambert, “MULTICAM: a miniature cryogenic camera for infrared detection,” Proc. SPIE 6992, 699215 (2008).

N. Guérineau, J. Primot, M. Tauvy, and M. Caes, “Modulation transfer function measurement of an infrared focal plane array by use of the self-imaging property of a canted periodic target,” Appl. Opt. 38, 631–637 (1999).
[CrossRef]

Te Kolste, R.

Tener, G.

J. Fanning, J. Miller, J. Park, G. Tener, J. Reynolds, P. O’Shea, C. Halford, and R. Driggers, “IR system field performance with superresolution,” Proc. SPIE 6543, 65430Z (2007).
[CrossRef]

Thétas, S.

G. Druart, N. Guérineau, R. Haïdar, S. Thétas, J. Taboury, S. Rommeluère, J. Primot, and M. Fendler, “Demonstration of an infrared microcamera inspired by Xenos peckii vision,” Appl. Opt. 48, 3368–3374 (2009).
[CrossRef] [PubMed]

G. Druart, N. Guérineau, R. Haïdar, M. Tauvy, S. Thétas, S. Rommeluère, J. Primot, J. Deschamps, and E. Lambert, “MULTICAM: a miniature cryogenic camera for infrared detection,” Proc. SPIE 6992, 699215 (2008).

Tisse, C.-L.

C.-L. Tisse, F. Guichard, and F. Cao, “Does resolution really increase image quality?” Proc. SPIE 6817, 68170Q (2008).
[CrossRef]

Torgersen, T. C.

R. Barnard, V. P. Pauca, T. C. Torgersen, R. J. Plemmons, S. Prasad, J. van der Gracht, J. Nagy, J. Chung, G. Behrmann, S. Mathews, and M. Mirotznik, “High-resolution iris image reconstruction from low-resolution imagery,” Proc. SPIE 6313, D1–D13 (2006).

Tünnermann, A.

van der Gracht, J.

R. Barnard, V. P. Pauca, T. C. Torgersen, R. J. Plemmons, S. Prasad, J. van der Gracht, J. Nagy, J. Chung, G. Behrmann, S. Mathews, and M. Mirotznik, “High-resolution iris image reconstruction from low-resolution imagery,” Proc. SPIE 6313, D1–D13 (2006).

Völkel, R.

R. Völkel, M. Eisner, and K. J. Weible, “Miniaturized imaging systems,” Microelectron. Eng. 67–68, 461–472 (2003).
[CrossRef]

Weible, K. J.

R. Völkel, M. Eisner, and K. J. Weible, “Miniaturized imaging systems,” Microelectron. Eng. 67–68, 461–472 (2003).
[CrossRef]

Willett, R.

Yamada, K.

Young, S.

Appl. Opt. (14)

A. W. Lohmann, “Scaling laws for lens systems,” Appl. Opt. 28, 4996–4998 (1989).
[CrossRef] [PubMed]

N. Guérineau, J. Primot, M. Tauvy, and M. Caes, “Modulation transfer function measurement of an infrared focal plane array by use of the self-imaging property of a canted periodic target,” Appl. Opt. 38, 631–637 (1999).
[CrossRef]

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Figures (17)

Fig. 1
Fig. 1

Moduli of the pixel transfer function ( | TF pixel | ) and of the optical transfer function ( | OTF | ).

Fig. 2
Fig. 2

Aliasing on the MTF of an optical system.

Fig. 3
Fig. 3

(a) Miniaturization of an optical system by downscaling its focal length f and its diameter Φ by the same factor M. (b) Reduction of the number of resolved points in the image while maintaining the same pixel pitch on the detector.

Fig. 4
Fig. 4

(a) Replication of a miniaturized imaging system, resulting in a multichannel optical system based on the TOMBO principle. (b) Image obtained on the detector surface.

Fig. 5
Fig. 5

GLSF method: (a) object target frequency spectrum, (b) effects of filtering by the pixel transfer function and sampling on the input spatial-frequency spectrum.

Fig. 6
Fig. 6

CPTT: (a) object target frequency spectrum, (b) effects of filtering by the pixel transfer function and sampling on the input spatial-frequency spectrum.

Fig. 7
Fig. 7

Two replications of the MTF of the multichannel system in the Fourier domain. The distances that enable us to find a condition on the value of the tilt angle are indicated.

Fig. 8
Fig. 8

Integer pairs that can be deduced from an integer pair ( n , m ) corresponding to a tilt angle φ between 0 ° and 45 ° .

Fig. 9
Fig. 9

Shifts between three adjacent optical channels.

Fig. 10
Fig. 10

(a) Optimal value of the tilt angle, for which a bright spot corresponding to a high spatial frequency is at the center of the square formed by four low-frequency bright spots. (b) Situation in which a bright spot corresponding to a high spatial frequency is at half the distance between two low- frequency bright spots.

Fig. 11
Fig. 11

(a) Mechanical scheme of MULTICAM. (b) Optical scheme of one optical channel.

Fig. 12
Fig. 12

Experimental setup for the measurement of the PSF of MULTICAM.

Fig. 13
Fig. 13

Image of a nonresolved point source acquired by MULTICAM. The image is presented in negative contrast.

Fig. 14
Fig. 14

MTF of MULTICAM. Inset, spatial frequencies greater than the Nyquist frequency; the arrow pointing towards the bottom of the inset shows frequencies lower than the Nyquist frequency; the arrow pointing towards the top of the inset shows the folded frequencies greater than the Nyquist frequency. The image is presented in negative contrast.

Fig. 15
Fig. 15

Experimental global MTF of MULTICAM. The green (upper) part of the plot is a representation of the MTF for frequencies lower than the Nyquist frequency. The blue (lower) part of the plot is a representation of the MTF for frequencies greater than the Nyquist frequency. Some level planes are indicated by the solid black curves.

Fig. 16
Fig. 16

Section of the MTF measured experimentally and compared with the theoretical MTF.

Fig. 17
Fig. 17

(a) Metallic target used as the object. (b) Image of the target acquired by MULTICAM. (c) One of the frames acquired by MULTICAM. Inset, Fourier transform of the image, illustrated below the Nyquist frequency. (d) Superresolved image. Inset, Fourier transform of the image, illustrated below the pixel cutoff frequency.

Equations (13)

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

MTF = | OTF × TF pixel | .
ν max = min ( 1 / ( 2 p s ) , ν c , 1 / t pix ) ,
IFOV = 1 f ν max ,
IFOV = 1 f × f Ny = 2 p s f .
p s t pix / 2 .
PSF m ( x , y ) = [ PSF e ( x , y ) * PSF pixel ( x , y ) ] * [ Ш d , d ( x , y ) × rect ( x a ) × rect ( y b ) ] ,
MTF m ( ν x , ν y ) = | [ OTF e ( ν x , ν y ) × TF pixel ( ν x , ν y ) ] × [ Ш 1 / d , 1 / d ( ν x , ν y ) * ( sinc ( a ν x ) × sinc ( b ν y ) ) ] | ,
cos φ = n p s d , sin φ = m p s d ,
n 2 + m 2 = k 2 = ( d p s ) 2 .
cos φ = n k , sin φ = m k .
tan φ = m n .
Δ x 1 = Δ y 2 = d cos φ ,
Δ y 1 = Δ x 2 = d sin φ .

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