Abstract

Recently there has been a significant interest in reconstructing a high-resolution (HR) image based on a set of low-resolution (LR) images with relative displacement. These images are typically undersampled with respect to the image spectrum of a HR image. I show that, although ideally a resolution increase of N times is possible with N LR images, in a practical system noise is a limiting factor that increases substantially as we approach this theoretical superresolution limit. For one dimension and a special case with two LR images, I present an analytical result of the noise amplification as a function of their displacement. This is defined as a condition number of the superresolution system, with the associated definitions of a well-conditioned and ill-conditioned superresolution reconstruction system.

© 2003 Optical Society of America

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References

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  2. R. W. Gerchberg, Opt. Acta 21, 709 (1974).
    [CrossRef]
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    [CrossRef]
  4. N. Nguyen, P. Milanfar, and G. Golub, IEEE Trans. Image Process. 10, 573 (2001).
    [CrossRef]
  5. H. Shekarforoush and R. Chellappa, J. Opt. Soc. Am. A 16, 481 (1999).
    [CrossRef]
  6. S. C. Park, M. K. Park, and M. G. Kang, IEEE Signal Process. Mag. 20, 21 (2003).
    [CrossRef]
  7. R. N. Bracewell, Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986).
  8. R. J. Marks II, Introduction to Shannon Sampling and Interpolation Theory (Springer-Verlag, New York, 1991).
    [CrossRef]
  9. D. A. Linden, Proc. IRE 47, 1219 (1959).
    [CrossRef]
  10. K. R. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1996).

2003

S. C. Park, M. K. Park, and M. G. Kang, IEEE Signal Process. Mag. 20, 21 (2003).
[CrossRef]

2001

N. Nguyen, P. Milanfar, and G. Golub, IEEE Trans. Image Process. 10, 573 (2001).
[CrossRef]

1999

1996

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

K. R. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1996).

1991

R. J. Marks II, Introduction to Shannon Sampling and Interpolation Theory (Springer-Verlag, New York, 1991).
[CrossRef]

1986

R. N. Bracewell, Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986).

1975

A. Papoulis, IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
[CrossRef]

1974

R. W. Gerchberg, Opt. Acta 21, 709 (1974).
[CrossRef]

1959

D. A. Linden, Proc. IRE 47, 1219 (1959).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986).

Castleman, K. R.

K. R. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1996).

Chellappa, R.

Gerchberg, R. W.

R. W. Gerchberg, Opt. Acta 21, 709 (1974).
[CrossRef]

Golub, G.

N. Nguyen, P. Milanfar, and G. Golub, IEEE Trans. Image Process. 10, 573 (2001).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Kang, M. G.

S. C. Park, M. K. Park, and M. G. Kang, IEEE Signal Process. Mag. 20, 21 (2003).
[CrossRef]

Linden, D. A.

D. A. Linden, Proc. IRE 47, 1219 (1959).
[CrossRef]

Marks II, R. J.

R. J. Marks II, Introduction to Shannon Sampling and Interpolation Theory (Springer-Verlag, New York, 1991).
[CrossRef]

Milanfar, P.

N. Nguyen, P. Milanfar, and G. Golub, IEEE Trans. Image Process. 10, 573 (2001).
[CrossRef]

Nguyen, N.

N. Nguyen, P. Milanfar, and G. Golub, IEEE Trans. Image Process. 10, 573 (2001).
[CrossRef]

Papoulis, A.

A. Papoulis, IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
[CrossRef]

Park, M. K.

S. C. Park, M. K. Park, and M. G. Kang, IEEE Signal Process. Mag. 20, 21 (2003).
[CrossRef]

Park, S. C.

S. C. Park, M. K. Park, and M. G. Kang, IEEE Signal Process. Mag. 20, 21 (2003).
[CrossRef]

Shekarforoush, H.

IEEE Signal Process. Mag.

S. C. Park, M. K. Park, and M. G. Kang, IEEE Signal Process. Mag. 20, 21 (2003).
[CrossRef]

IEEE Trans. Circuits Syst.

A. Papoulis, IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
[CrossRef]

IEEE Trans. Image Process.

N. Nguyen, P. Milanfar, and G. Golub, IEEE Trans. Image Process. 10, 573 (2001).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Acta

R. W. Gerchberg, Opt. Acta 21, 709 (1974).
[CrossRef]

Proc. IRE

D. A. Linden, Proc. IRE 47, 1219 (1959).
[CrossRef]

Other

K. R. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1996).

R. N. Bracewell, Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1986).

R. J. Marks II, Introduction to Shannon Sampling and Interpolation Theory (Springer-Verlag, New York, 1991).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

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Figures (3)

Fig. 1
Fig. 1

Interlaced sampling.

Fig. 2
Fig. 2

Noise magnification in interlaced sampling. Reconstruction with (a) a=0.3 and (b) a=0.15.

Fig. 3
Fig. 3

Condition number of the superresolution reconstruction system.

Equations (16)

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

δ=0.61λ/NA,
1XcombxX1Xn=-δxX-n=n=-δx-nX,
gˆx=sincxX*1XcombxXgx.
AxXsinc2xX-π cot aπxXsinc2xX,
gˆx=AxX*1XcombxXgx+A-xX*1Xcombx-aXgx.
gx=gx0+x-x0gx0+x-x022!gx0+.
σa,b2=n=-A2b+n+A2a-b+nσ2.
σa2=12σa,b2+σa,b+0.52.
Q1n=-A20.5+n,
Q2n=-A2a-0.5+n,
Q1=-A2xcombx-0.5dx=A˜s*A˜s*exp-jπscombss=0,
A˜s=-m1-m=c0<s<1-m¯1-m¯=d-1<s<0,0otherwise
Q1=c2 expjπ+2cd+d2 exp-jπ=-c2+2cd-d2.
Q2=-c2m¯+2cd-d2m.
σa2=121-1+m¯c2+4cd-1+md2σ2=121-m2+6m+11-m2σ2=4σ21-m1-m¯.
K41-exp-j2πa1-expj2πa

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