Abstract

An iterative technique is proposed for improving the quality of reconstructions from projections when the number of projections is small or the angular range of projections is limited. The technique consists of transforming repeatedly between image and transform spaces and applying a priori object information at each iteration. The approach is a generalization of the Gerchberg-Papoulis algorithm, a technique for extrapolating in the Fourier domain by imposing a space-limiting constraint on the object in the spatial domain. A priori object data that may be applied, in addition to truncating the image beyond the known boundaries of the object, include limiting the maximum range of variation of the physical parameter being imaged. The results of computer simulations show clearly how the process of forcing the image to conform to a priori object data reduces artifacts arising from limited data available in the Fourier domain.

© 1981 Optical Society of America

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References

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  1. J. L. Harris, J. Opt. Soc. Am. 54, 931 (1964).
    [CrossRef]
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 133–136.
  3. G. Toraldo di Francia, Nuovo Cimento Suppl. 9, 426 (1952).
    [CrossRef]
  4. R. N. Bracewell, “Image Reconstruction in Radio Astronomy,” in Image Reconstruction from Projections, G. T. Herman, Ed. (Springer, Berlin, 1979).
    [CrossRef]
  5. R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).
  6. W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).
  7. R. W. Gerchberg, Opt. Acta 21, 709 (1974).
    [CrossRef]
  8. A. Papoulis, IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
    [CrossRef]
  9. S. J. Norton, M. Linzer, Ultrason. Imaging 1, 154 (1979).
    [CrossRef] [PubMed]
  10. S. J. Norton, M. Linzer, Ultrason. Imaging 1, 210 (1979).
    [CrossRef] [PubMed]
  11. Strictly speaking, setting data to zero beyond an upper cutoff frequency in step (b) contradicts the assumption of a space-limited object, since such an object has a spectrum of infinite extent. (To put it another way, a spectrum of finite extent cannot be analytic.) Thus, in principle, the method will fail to converge if space-limiting and band-limiting constraints are imposed simultaneously. However, if one has reason to suspect that an overwhelming fraction of the object energy is contained within a certain finite radius in the frequency plane and that energy beyond this radius consists essentially of noise, setting that portion of the spectrum to zero should not drastically affect convergence of the technique to an acceptable image. (In practice, the region outside the cutoff radius would not be set to zero abruptly; rather a smoothly decreasing apodizing function would be used.) Moreover, in practice, one is primarily interested in frequency-plane interpolation, i.e., filling in missing gaps in the object spectrum within some reasonable cutoff radius rather than extrapolating the spectrum beyond that radius.

1979 (2)

S. J. Norton, M. Linzer, Ultrason. Imaging 1, 154 (1979).
[CrossRef] [PubMed]

S. J. Norton, M. Linzer, Ultrason. Imaging 1, 210 (1979).
[CrossRef] [PubMed]

1975 (1)

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

1974 (1)

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

1972 (1)

R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).

1964 (1)

1952 (1)

G. Toraldo di Francia, Nuovo Cimento Suppl. 9, 426 (1952).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, “Image Reconstruction in Radio Astronomy,” in Image Reconstruction from Projections, G. T. Herman, Ed. (Springer, Berlin, 1979).
[CrossRef]

Gerchberg, R. W.

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

R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 133–136.

Harris, J. L.

Linzer, M.

S. J. Norton, M. Linzer, Ultrason. Imaging 1, 154 (1979).
[CrossRef] [PubMed]

S. J. Norton, M. Linzer, Ultrason. Imaging 1, 210 (1979).
[CrossRef] [PubMed]

Norton, S. J.

S. J. Norton, M. Linzer, Ultrason. Imaging 1, 154 (1979).
[CrossRef] [PubMed]

S. J. Norton, M. Linzer, Ultrason. Imaging 1, 210 (1979).
[CrossRef] [PubMed]

Papoulis, A.

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

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).

W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).

Toraldo di Francia, G.

G. Toraldo di Francia, Nuovo Cimento Suppl. 9, 426 (1952).
[CrossRef]

IEEE Trans. Circuits Syst. (1)

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

J. Opt. Soc. Am. (1)

Nuovo Cimento Suppl. (1)

G. Toraldo di Francia, Nuovo Cimento Suppl. 9, 426 (1952).
[CrossRef]

Opt. Acta (1)

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

Optik (1)

R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).

Ultrason. Imaging (2)

S. J. Norton, M. Linzer, Ultrason. Imaging 1, 154 (1979).
[CrossRef] [PubMed]

S. J. Norton, M. Linzer, Ultrason. Imaging 1, 210 (1979).
[CrossRef] [PubMed]

Other (4)

Strictly speaking, setting data to zero beyond an upper cutoff frequency in step (b) contradicts the assumption of a space-limited object, since such an object has a spectrum of infinite extent. (To put it another way, a spectrum of finite extent cannot be analytic.) Thus, in principle, the method will fail to converge if space-limiting and band-limiting constraints are imposed simultaneously. However, if one has reason to suspect that an overwhelming fraction of the object energy is contained within a certain finite radius in the frequency plane and that energy beyond this radius consists essentially of noise, setting that portion of the spectrum to zero should not drastically affect convergence of the technique to an acceptable image. (In practice, the region outside the cutoff radius would not be set to zero abruptly; rather a smoothly decreasing apodizing function would be used.) Moreover, in practice, one is primarily interested in frequency-plane interpolation, i.e., filling in missing gaps in the object spectrum within some reasonable cutoff radius rather than extrapolating the spectrum beyond that radius.

W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).

R. N. Bracewell, “Image Reconstruction in Radio Astronomy,” in Image Reconstruction from Projections, G. T. Herman, Ed. (Springer, Berlin, 1979).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 133–136.

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Figures (7)

Fig. 1
Fig. 1

(a) Angular range of the transmission projections is limited to θmax by obstacles; (b) angular range of the reflectivity projections is limited to θmax by an enclosing obstacle; (c) 2-D Fourier transform of the range has a missing sector due to the limited-angle projection data.

Fig. 2
Fig. 2

Schematic diagram of iterative procedure. Measured projection data and a priori object information are applied in each iteration in steps (b) and (d).

Fig. 3
Fig. 3

Simulated reconstruction in absence of noise: (a) object; (b) limited-angle projection data; (c) image obtained by inverse transformation of (b); (d) image obtained by inverse transformation of (b) after application of a single-step interpolation operation to the limited data to fill in the missing sector; (e) image after two iterations; (f) image after 100 iterations.

Fig. 4
Fig. 4

Simulated reconstruction with additive Gaussian noise (SNR = 2): (a) object; (b) limited-angle projection data; (c) image obtained by inverse transformation of (b); (d) image obtained by inverse transformation of (b) after application of a single-step interpolation operation to the limited data to fill in the missing sector; (e) image after two iterations; (f) image after 100 iterations.

Fig. 5
Fig. 5

Simulated reconstruction in absence of noise.

Fig. 6
Fig. 6

Simulated reconstruction with additive Gaussian noise (SNR = 2).

Fig. 7
Fig. 7

En is the mean square difference between the object f and the nth iteration of the image fn normalized to the mean square difference between the object and the image f ^ computed using the measured projection data without a priori object information, i.e., En = ∫∫|f(x,y) − fn(x,y)|2dxdy/E, where E = ∫∫|f(x,y) − f ^ (x,y)|2dxdy. Dn is the mean square difference between successive iterations and is also normalized to E, i.e., Dn = ∫∫|fn(x,y) − fn−1(x,y)|2dxdy/E.

Equations (8)

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F n ( u , v ) = I f n ( x , y ) exp [ i ( u x + v y ) ] d x d y ,
g n + 1 ( x , y ) = 1 ( 2 π ) 2 - - G n ( u , v ) exp [ - i ( u x + v y ) ] d u d v ,
f n + 1 ( x , y ) = g n + 1 ( x , y ) P I ( x , y ) ,
P I ( x , y ) = { 1 for ( x , y ) I 0 for ( x , y ) I .
f n + 1 ( x , y ) = { f n + 1 ( x , y ) when t l ( x , y ) f n + 1 ( x , y ) t u ( x , y ) , t u ( x , y ) when t u ( x , y ) < f n + 1 ( x , y ) , t l ( x , y ) when f n + 1 ( x , y ) < t l ( x , y ) ,
G n ( u , v ) = F n ( u , v ) + [ F ( u , v ) - F n ( u , v ) ] P D ( u , v ) = { F ( u , v ) for ( u , v ) D F n ( u , v ) for ( u , v ) D ,
P D ( u , v ) = { 1 for ( u , v ) D 0 for ( u , v ) D ,
E n = I | f ( x , y ) - f n ( x , y ) | 2 d x d y = 1 ( 2 π ) 2 - - | F ( u , v ) - F n ( u , v ) | 2 d u d v > 1 ( 2 π ) 2 - - | F ( u , v ) - G n ( u , v ) | 2 d u d v = - - | f ( x , y ) - g n + 1 ( x , y ) | 2 d x d y > I | f ( x , y ) - f n + 1 ( x , y ) | 2 d x d y = E n + 1 .

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