Abstract

The mathematical generalization of image restoration by recursive methods furnished by D. C. Youla [ IEEE Trans. Circuits Syst. CAS 25, 695– 702 ( 1978)] is used to show that arbitrary L2 (i.e., square-integrable) images can be reconstructed from two projections without any a priori assumption regarding the mathematical properties of the object, such as space-limitedness or band-limitedness. Recursive algorithms are given to restore images from (1) extended segments and low-pass spectra and (2) short segments and high-pass spectra. Using the alternating projection theorem, we prove monotonic convergence (in the norm) to the original image.

© 1981 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS 25, 695–702 (1978).
  2. A. Papoulis, “A new algorithm in spectral analysis and bandlimited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
    [Crossref]
  3. R. W. Gershberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
    [Crossref]
  4. J. von Neumann, The Geometry of Orthogonal Spaces, Vol. II (Princeton U. Press, Princeton, N.J., 1950).
  5. H. Stark and et al., “Direct Fourier reconstruction in computer tomography,” IEEE Trans. Acoust. Speech Signal Process.ASSP-29 (1981) (to be published).
  6. H. Stark, “Some film-noise measurements by diffraction of coherent light,” Appl. Opt. 10, 333–337 (1971).
    [Crossref] [PubMed]
  7. A. Lent and H. Tuy, “An iterative method for the extrapolation of bandlimited functions,” (State University of New York at Buffalo, Amherst, N. Y., 1979).
  8. M. Davison and F. A. Grunbaum, “Convolution algorithm for arbitary projection angles,” IEEE Trans. Nucl. Sci. NS-26, 2670–2673 (1979).
    [Crossref]

1979 (1)

M. Davison and F. A. Grunbaum, “Convolution algorithm for arbitary projection angles,” IEEE Trans. Nucl. Sci. NS-26, 2670–2673 (1979).
[Crossref]

1978 (1)

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS 25, 695–702 (1978).

1975 (1)

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

1974 (1)

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

1971 (1)

Davison, M.

M. Davison and F. A. Grunbaum, “Convolution algorithm for arbitary projection angles,” IEEE Trans. Nucl. Sci. NS-26, 2670–2673 (1979).
[Crossref]

Gershberg, R. W.

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

Grunbaum, F. A.

M. Davison and F. A. Grunbaum, “Convolution algorithm for arbitary projection angles,” IEEE Trans. Nucl. Sci. NS-26, 2670–2673 (1979).
[Crossref]

Lent, A.

A. Lent and H. Tuy, “An iterative method for the extrapolation of bandlimited functions,” (State University of New York at Buffalo, Amherst, N. Y., 1979).

Papoulis, A.

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

Stark, H.

H. Stark, “Some film-noise measurements by diffraction of coherent light,” Appl. Opt. 10, 333–337 (1971).
[Crossref] [PubMed]

H. Stark and et al., “Direct Fourier reconstruction in computer tomography,” IEEE Trans. Acoust. Speech Signal Process.ASSP-29 (1981) (to be published).

Tuy, H.

A. Lent and H. Tuy, “An iterative method for the extrapolation of bandlimited functions,” (State University of New York at Buffalo, Amherst, N. Y., 1979).

von Neumann, J.

J. von Neumann, The Geometry of Orthogonal Spaces, Vol. II (Princeton U. Press, Princeton, N.J., 1950).

Youla, D. C.

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS 25, 695–702 (1978).

Appl. Opt. (1)

IEEE Trans. Circuits Syst. (2)

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS 25, 695–702 (1978).

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

IEEE Trans. Nucl. Sci. (1)

M. Davison and F. A. Grunbaum, “Convolution algorithm for arbitary projection angles,” IEEE Trans. Nucl. Sci. NS-26, 2670–2673 (1979).
[Crossref]

Opt. Acta (1)

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

Other (3)

J. von Neumann, The Geometry of Orthogonal Spaces, Vol. II (Princeton U. Press, Princeton, N.J., 1950).

H. Stark and et al., “Direct Fourier reconstruction in computer tomography,” IEEE Trans. Acoust. Speech Signal Process.ASSP-29 (1981) (to be published).

A. Lent and H. Tuy, “An iterative method for the extrapolation of bandlimited functions,” (State University of New York at Buffalo, Amherst, N. Y., 1979).

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 (17)

Fig. 1
Fig. 1

The image function f(x) = 5[1 + (x/5)2]−1.

Fig. 2
Fig. 2

The fast Fourier transform of f(x) in Fig. 1.

Fig. 3
Fig. 3

Initial estimate, m1, of f(x) for α = 53%, β = 26%.

Fig. 4
Fig. 4

Restored image from estimate shown in Fig. 3 after 100 iterations of Eq. (23).

Fig. 5
Fig. 5

Initial estimate, m1, of f(x) for α = 53%, β = 53%.

Fig. 6
Fig. 6

Restored image from estimate shown in Fig. 5 after 100 iterations of Eq. (23).

Fig. 7
Fig. 7

Initial estimate, m2, of f(x) for α′ = 46%, β′ = 28%.

Fig. 8
Fig. 8

Restored image from estimate shown in Fig. 7 after 100 iterations of Eq. (35).

Fig. 9
Fig. 9

Initial estimate, m2, of f(x) for α′ = 46%, β′ = 46%.

Fig. 10
Fig. 10

Restored image from estimate shown in Fig. 9 after 100 iterations of Eq. (35).

Fig. 11
Fig. 11

A small digitized circle on a 128 × 128 pixel grid.

Fig. 12
Fig. 12

The fast Fourier transform of Fig. 11.

Fig. 13
Fig. 13

The spectrum obtained from the given data g. Equivalently one could be given this spectral segment and generate g, as shown in Fig. 14.

Fig. 14
Fig. 14

The given data g. The spectrum in Fig. 13 and g form a Fourier-transform pair.

Fig. 15
Fig. 15

The corrected spectrum after the first correction. It consists of the spectrum of g (Fig. 13) to which has been added the spectrum of PbPaf.

Fig. 16
Fig. 16

Reconstruction after one pass: the signal f2(x).

Fig. 17
Fig. 17

The restored signal f10(x) obtained after nine iterations of Eq. 53.

Tables (2)

Tables Icon

Table 1 Results of Restoration from an Extended Segment and the Low-Pass Spectrum

Tables Icon

Table 2 Results of Restoration from a Short Segment and the High-Pass Spectrum

Equations (56)

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

( f , g ) - f ( x ) g * ( x ) d x .
f = g + h ,
- g ( x ) h * ( x ) d x = 0.
g = P f ,
h = Q f .
g ( x ) = rect ( x 2 a ) f ( x ) ,
h ( x ) = [ 1 - rect ( x 2 a ) ] f ( x ) .
g ( x ) = f ( x ) * sin x b π x ,
P b ( P a ) = ϕ ,
A = 1 - Q a P b ,
P a f = rect ( x 2 X a ) f ( x ) ,
Q a f = [ 1 - rect ( x 2 X a ) ] f ( x ) ,
P b f = rect ( x 2 X b ) f ( x ) ,
Q b f = [ 1 - rect ( x 2 X b ) ] f ( x ) .
P b ( P a ) ϕ ,
g ( x ) = g 1 ( x ) rect ( x - X M X b - X a ) + g 2 ( x ) rect ( x + X M X b - X a )
P b ( P a ) = ϕ .
f = [ - f ( x ) 2 d x ] 1 / 2 .
g - f = 0.
g ( x ) = [ 1 - rect ( x 2 X a ) ] f ( x )             ( extended segment ) ,
h ( x ) = - 1 [ rect ( ω 2 b ) F ( ω ) ]             ( low - pass signal ) ,
m 1 = P b f + Q b Q a f
= ( 1 - Q b ) f + Q b Q a f = ( 1 - Q b P a ) f = A 1 f ,
f k + 1 = m 1 + Q b P a f k ,             k = 1 ,             f 1 = m 1
f k = r = 0 k - 1 ( Q b P a ) r ( 1 - Q b P a ) f = f - ( Q b P a ) k f .
lim k f k = f - lim k ( Q b P a ) k f .
lim k ( Q b P a ) k = f c ,
lim k f k = f .
k + 1 = ( Q b P a ) k ,
k + 1 Q b P a k ,
P a f = rect ( x 2 X a ) f ( x ) = g ( x )             ( segment ) ,
Q b f = - 1 { [ 1 - rect ( ω 2 b ) ] F ( ω ) } h ( x )             ( high - pass signal ) ,
m 2 = g + Q a h
= P a f + Q a Q b f = ( 1 - Q a ) f + Q a Q b f = ( 1 - Q a P b ) f
= A 2 f ,
f k + 1 = m 2 + Q a P b f k ,             k = 1 ,             f 1 = m 2
f k = f - ( Q a P b ) k f .
g = Q a f = Q a P b f = ( 1 - P a ) P b f = ( 1 - P a P b ) f = A 3 f ,
f k + 1 = g + P a P b f k ,             k = 1 ,             f 1 = g
f k + 1 ( x ) = g ( x ) + rect ( x 2 X a ) ( f k ( x ) * sin b x π x ) .
- X a X a ϕ i ( s ) sin b ( x - s ) π ( x - s ) d s = λ i ϕ i ( x ) ,
1 > λ 0 > λ 1 , > > λ n , > 0 as n .
- X a X a ϕ k ( x ) ϕ n ( x ) d x = { λ n , k = n 0 , k n .
f ( x ) = k = 0 α k ϕ k ( x ) ,
α k = ( g , ϕ k ) λ k .
α k + Δ α k = ( g , ϕ k ) λ k + ( n , ϕ k ) λ k ,
h ( x ) f ( x ) - g ( x ) ,
f ( x ) = k = 0 β k ϕ k ( x ) ,
β k = ( h , ϕ k ) 1 - λ k .
Δ β k = ( n , ϕ k ) 1 - λ k ,
Δ f ( x ) = k = 0 ( n , ϕ k ) 1 - λ k .
Δ f ( x ) = k = 0 k 0 ( n , ϕ k ) 1 - λ k + k = k 0 + 1 ( n , ϕ k ) 1 - λ k .
k = k 0 + 1 ( n , ϕ k ) 2 ( 1 - λ k ) 2 < 1 M 2 k = k 0 + 1 ( n , ϕ k ) 2 1 M 2 n 2
f ( x ) = 5 1 + ( x 5 ) 2 ,
g = Q b f = ( 1 - P b ) P a f = ( 1 - P b P a ) f ,
f k + 1 = g + P b P a f k ,             k = 1 ,             f 1 = g ,