Abstract

We consider the problem of reconstructing remotely obtained images from image-plane detector arrays. Although the individual detectors may be larger than the blur spot of the imaging optics, high-resolution reconstructions can be obtained by scanning or rotating the image with respect to the detector. As an alternative to matrix inversion or least-squares estimation [ Appl. Opt. 6, 3615 ( 1987)], the method of convex projections is proposed. We show that readily obtained prior knowledge can be used to obtain good-quality imagery with reduced data. The effect of noise on the reconstruction process is considered.

© 1989 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. A. Richards, Remote Sensing Digital Image Analysis (Springer-Verlag, New York, 1986).
    [Crossref]
  2. A detector of area A, with quantum efficiency μ exposed to light of uniform intensity I at frequency ν will have a signal-to-noise ratio SNR = μIAΔt/hν if the integration time is Δt and a Poisson process is assumed.
  3. B. R. Frieden, H. H. G. Aumann, “Image reconstruction from multiple 1-D scans using filtered localized projections,” Appl. Opt. 26, 3615–3621 (1987).
    [Crossref] [PubMed]
  4. P. Oskoui-Fard, H. Stark, “Tomographic image reconstruction using the theory of convex projections,”IEEE Trans. Med. Imag. MI-3, 45–58 (1988).
    [Crossref]
  5. M. I. Sezan, H. Stark, “Tomographic image reconstruction from incomplete data by convex projections and direct Fourier method,”IEEE Trans. Med. Imag. MI-3, 91–98 (1984).
    [Crossref]
  6. A. Lent, H. Tuy, “An iterative method for the extrapolation of band-limited functions,”J. Math. Anal. Appl. 83, 554–565 (1981).
    [Crossref]
  7. A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,”IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
    [Crossref]
  8. R. W. Gerchberg, “Super resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
    [Crossref]
  9. B. P. Medoff, W. R. Broady, M. Nassi, A. Macovski, “Iterative convolution backprojection algorithms for image reconstruction from limited data,”J. Opt. Soc. Am. 72, 1493–1500 (1983).
    [Crossref]
  10. Weak convergence, also known as inner product convergence, is identical to the more familiar strong convergence, i.e., ||f− fk|| → 0, in the finite-dimensional case.
  11. One should make a distinction between the original function, which generated the measurements and is assumed to lie in the set of functions that are square integrable over some domain, and the set of reconstructed images composed of square pixels or of linear combinations of some other basis functions. From a finite set of measurements, one obtains a reconstructed image f(x, y) that is an element of the finite dimension set of possible reconstructed images. This reconstructed image is taken to be an approximation of the function that generated the measurements. The original function can be recovered exactly only if it happens to lie in the set of possible reconstructed images.
  12. D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1–theory,”IEEE Trans. Med. Imag. MI-1, 81–94 (1982).
    [Crossref]
  13. L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,”IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).

1988 (1)

P. Oskoui-Fard, H. Stark, “Tomographic image reconstruction using the theory of convex projections,”IEEE Trans. Med. Imag. MI-3, 45–58 (1988).
[Crossref]

1987 (1)

1984 (1)

M. I. Sezan, H. Stark, “Tomographic image reconstruction from incomplete data by convex projections and direct Fourier method,”IEEE Trans. Med. Imag. MI-3, 91–98 (1984).
[Crossref]

1983 (1)

1982 (1)

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1–theory,”IEEE Trans. Med. Imag. MI-1, 81–94 (1982).
[Crossref]

1981 (1)

A. Lent, H. Tuy, “An iterative method for the extrapolation of band-limited functions,”J. Math. Anal. Appl. 83, 554–565 (1981).
[Crossref]

1975 (1)

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,”IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[Crossref]

1974 (2)

R. W. Gerchberg, “Super resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[Crossref]

L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,”IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).

Aumann, H. H. G.

Broady, W. R.

Frieden, B. R.

Gerchberg, R. W.

R. W. Gerchberg, “Super resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[Crossref]

Lent, A.

A. Lent, H. Tuy, “An iterative method for the extrapolation of band-limited functions,”J. Math. Anal. Appl. 83, 554–565 (1981).
[Crossref]

Logan, B. F.

L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,”IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).

Macovski, A.

Medoff, B. P.

Nassi, M.

Oskoui-Fard, P.

P. Oskoui-Fard, H. Stark, “Tomographic image reconstruction using the theory of convex projections,”IEEE Trans. Med. Imag. MI-3, 45–58 (1988).
[Crossref]

Papoulis, A.

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,”IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[Crossref]

Richards, J. A.

J. A. Richards, Remote Sensing Digital Image Analysis (Springer-Verlag, New York, 1986).
[Crossref]

Sezan, M. I.

M. I. Sezan, H. Stark, “Tomographic image reconstruction from incomplete data by convex projections and direct Fourier method,”IEEE Trans. Med. Imag. MI-3, 91–98 (1984).
[Crossref]

Shepp, L. A.

L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,”IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).

Stark, H.

P. Oskoui-Fard, H. Stark, “Tomographic image reconstruction using the theory of convex projections,”IEEE Trans. Med. Imag. MI-3, 45–58 (1988).
[Crossref]

M. I. Sezan, H. Stark, “Tomographic image reconstruction from incomplete data by convex projections and direct Fourier method,”IEEE Trans. Med. Imag. MI-3, 91–98 (1984).
[Crossref]

Tuy, H.

A. Lent, H. Tuy, “An iterative method for the extrapolation of band-limited functions,”J. Math. Anal. Appl. 83, 554–565 (1981).
[Crossref]

Webb, H.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1–theory,”IEEE Trans. Med. Imag. MI-1, 81–94 (1982).
[Crossref]

Youla, D. C.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1–theory,”IEEE Trans. Med. Imag. MI-1, 81–94 (1982).
[Crossref]

Appl. Opt. (1)

IEEE Trans. Circuits Syst. (1)

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,”IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[Crossref]

IEEE Trans. Med. Imag. (3)

P. Oskoui-Fard, H. Stark, “Tomographic image reconstruction using the theory of convex projections,”IEEE Trans. Med. Imag. MI-3, 45–58 (1988).
[Crossref]

M. I. Sezan, H. Stark, “Tomographic image reconstruction from incomplete data by convex projections and direct Fourier method,”IEEE Trans. Med. Imag. MI-3, 91–98 (1984).
[Crossref]

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1–theory,”IEEE Trans. Med. Imag. MI-1, 81–94 (1982).
[Crossref]

IEEE Trans. Nucl. Sci. (1)

L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,”IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).

J. Math. Anal. Appl. (1)

A. Lent, H. Tuy, “An iterative method for the extrapolation of band-limited functions,”J. Math. Anal. Appl. 83, 554–565 (1981).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Acta (1)

R. W. Gerchberg, “Super resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[Crossref]

Other (4)

J. A. Richards, Remote Sensing Digital Image Analysis (Springer-Verlag, New York, 1986).
[Crossref]

A detector of area A, with quantum efficiency μ exposed to light of uniform intensity I at frequency ν will have a signal-to-noise ratio SNR = μIAΔt/hν if the integration time is Δt and a Poisson process is assumed.

Weak convergence, also known as inner product convergence, is identical to the more familiar strong convergence, i.e., ||f− fk|| → 0, in the finite-dimensional case.

One should make a distinction between the original function, which generated the measurements and is assumed to lie in the set of functions that are square integrable over some domain, and the set of reconstructed images composed of square pixels or of linear combinations of some other basis functions. From a finite set of measurements, one obtains a reconstructed image f(x, y) that is an element of the finite dimension set of possible reconstructed images. This reconstructed image is taken to be an approximation of the function that generated the measurements. The original function can be recovered exactly only if it happens to lie in the set of possible reconstructed images.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (13)

Fig. 1
Fig. 1

(a) Rectangular detector of size ab; (b) detector over image f(x, y), displaced by xj and yj and rotated through an angle θj.

Fig. 2
Fig. 2

Rotation between detector array and image field to get additional information.

Fig. 3
Fig. 3

Scanning configuration of detectors. Information is obtained as detectors scan the image field in a vertical direction.

Fig. 4
Fig. 4

Reconstruction of high-resolution images from full data sets, using convex projections without reference-image constraint. The initial image is zero image, and 128 detectors of size 8 × 4 were used in 32 views. (a) Original image, (b) reconstruction after 10 iterations, (c) reconstruction after 50 iterations, (d) reconstruction after 100 iterations.

Fig. 5
Fig. 5

Image sometimes used as the initial image and/or the reference image. It consists of a uniform ellipse.

Fig. 6
Fig. 6

Reconstruction using convex projections after 100 iterations: (a) the initial image is that of Fig. 5, (b) the initial image is zero image. The reference-image constraint is not in use.

Fig. 7
Fig. 7

Error history of reconstructions by convex projections: solid curve shows the reconstruction error when f0 = 0, dashed curve shows the error for the value of f0 shown in Fig. 5.

Fig. 8
Fig. 8

Limited-view reconstructions using POCS after 100 iterations. The initial image f0 is that of Fig. 5. Reconstructions are shown with (a) 20, (b) 40, (c) 60, and (d) 90 deg of missing data.

Fig. 9
Fig. 9

Same as Fig. 8 but with the reference-image constraint and the amplitude level constraint omitted. Also, the initial image is f0 = 0. This image is reconstructed from data only, whereas that of Fig. 8 uses prior knowledge.

Fig. 10
Fig. 10

Effects of noise on POCS reconstructions after 100 iterations. (a) Original image, (b) reconstruction from noiseless data, (c) reconstruction from 50-dB SNR data, (d) reconstruction from 30-dB SNR data.

Fig. 11
Fig. 11

Comparison of POCS and FLP reconstruction algorithms for the two-point blur. (a) Unprocessed detector image, (b) LP image, (c) FLP image, (d) POCS images after 100 iterations with f0 = 0 and all constraints in effect.

Fig. 12
Fig. 12

Same as Fig. 11 except for more severe four-point blur. (a) Unprocessed detector image, (b) LP image, (c) FLP image, (d) POCS image after 100 iterations.

Fig. 13
Fig. 13

Effect of dead detectors on POCS and FLP reconstruction algorithms. Two-point horizontal blur is in effect. (a) Unprocessed detector image, (b) LP image, (c) FLP image using correct inverse filter known a priori,(d) POCS image after 30 iterations.

Equations (37)

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

f ( k + 1 ) = P m P m - 1 P 2 P 1 f ( k ) ,             f ( 0 ) f 0 ,
d j = - f ( x , y ) σ j ( x , y ) d x d y .
σ j ( x , y ) = rect [ ( x - x j ) cos θ j + ( y - y j ) sin θ j a ] × rect [ ( y - y j ) cos θ j + ( x - x j ) sin θ j b ]
d j = l k f ( l , k ) σ j ( l , k ) ,
σ j ( l , k ) = { 0 when pixel centered at ( l , k ) lies wholly outside detector footprint 1 when pixel centered at ( l , k ) lies wholly within detector footprint r j ( 0 < r j < 1 ) when pixel centered at ( l , k ) lies partially within the detector footprint .
C j = { f ( x , y ) : ( f , σ j ) = d j } .
g = P j h = { h ( x , y ) if ( h , σ j ) = d j h ( x , y ) + d j - ( h , σ j ) ( σ j , σ j ) σ j ( x , y ) otherwise .
f n + 1 ( x , y ) = { f n ( x , y ) if ( f n , σ j ) = d j f n ( x , y ) + d j - ( f n , σ j ) ( σ j , σ j ) otherwise .
C A = { g ( x , y ) : α g ( x , y ) β , β > α } .
g = P A h = { α if h ( x , y ) < α h ( x , y ) if α h ( x , y ) β β if h ( x , y ) > β .
C E = { g ( x , y ) : g 2 E } ,
g = P E h = { h ( x , y ) if h 2 E ( E / E h ) 1 / 2 h ( x , y ) if h 2 > E .
C R = { g ( x , y ) : g - f R R } .
g = P R h = { h if h - f R R f R + R h - f R h - f R if h - f R > R .
C s = { g ( x , y ) : g ( x , y ) = 0 for ( x , y ) A } .
g = P s h = { h ( x , y ) if ( x , y ) A 0 otherwise .
f n + 1 ( x , y ) = P K . P 1 f n ,             n = 0 , 1 , ,
μ = α I ,
σ 2 = α I ,
α = A D Δ t h ν .
log α = 0.1 ( SNR - 10 log I ) .
log α n = 0.1 ( SNR - 10 log I n ) ,             n = 1 , , K ,
I = HO ,
O p = H T I = H T HO .
H + [ H T H ] - 1 ,
O ^ = [ H T H ] - 1 H T I O ^ LS .
( μ h 1 + ( 1 - μ ) h 2 , σ j ) = μ ( h 1 , σ j ) + ( 1 - μ ) ( h 2 , σ j ) = μ d j + ( 1 - μ ) d j = d j
( h n - h , σ j ) h n - h σ j 0 ,
( h n - h , σ j ) 0 ,
( h n , σ j ) ( h , σ j ) = d j .
min g C j - ( h - g ) 2 d x d y
min g C j l , k [ h ( l , k ) - g ( l , k ) ] 2
min - ( h - g ) 2 d x d y ,
min - ( h - g ) 2 d x d y + λ [ ( g , σ j ) - d j ] ,
g ( x , y ) = h ( x , y ) - λ 2 σ j .
λ 2 = ( h , σ j ) - d j ( σ j , σ j ) .
g ( x , y ) = h ( x , y ) + d j - ( h , σ j ) ( σ j , σ j ) σ j ( x , y ) ,

Metrics