Abstract

Spline functions, because of their highly desirable interpolating and approximating characteristics, are used as a potential alternative to the conventional pulse approximation method in digital image processing. In space-invariant imaging systems, the object and point-spread function are represented by a class of spline functions called B-splines. Exploiting the convolutional property of B-splines, the deterministic part of the degraded image is another B-spline of higher degree. A minimum norm principle leading to pseudoinversion is used for the restoration of space-invariant degradations with underdetermined and overdetermined models. The singular-value-decomposition technique is used to determine the pseudoinverse.

© 1978 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  2. T. N. E. Greville, Ed., Theory and Applications of Spline Functions (Academic, New York, 1969).
  3. J. C. Holladay, Math. Tables Aids. Comput. 11, 233 (1957).
    [CrossRef]
  4. C. de Boor, J. Math. Mech. 12, 747 (1963).
  5. I. J. Schoenberg, Proc. Nat. Acad. Sci. U.S.A. 52, 947 (1964).
    [CrossRef]
  6. L. L. Schumaker, in Ref. 2, p. 87.
  7. H. B. Curry, I. J. Schoenberg, J. Anal. Math. 17, 71 (1966).
    [CrossRef]
  8. W. K. Pratt, Digital Image Processing (Wiley, New York, 1978), Chap. 9.
  9. H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977), Chap. 7.
  10. M. J. Peyrovian, A. A. Sawchuk, University of Southern California Image Processing Institute Technical Report 620 (1975), p. 86.
  11. A. Albert, Regression and the Moore-Penrose Pseudoinverse (Academic, New York, 1972).
  12. E. D. Nering, Linear Algebra and Matrix Theory (Wiley, New York, 1970).
  13. G. H. Golub, C. Reinsch, Numer. Math. 14, 403 (1970).
    [CrossRef]
  14. A. A. Sawchuk, M. J. Peyrovian, J. Opt. Soc. Am. 65, 712 (1975).
    [CrossRef]

1975 (1)

1970 (1)

G. H. Golub, C. Reinsch, Numer. Math. 14, 403 (1970).
[CrossRef]

1966 (1)

H. B. Curry, I. J. Schoenberg, J. Anal. Math. 17, 71 (1966).
[CrossRef]

1964 (1)

I. J. Schoenberg, Proc. Nat. Acad. Sci. U.S.A. 52, 947 (1964).
[CrossRef]

1963 (1)

C. de Boor, J. Math. Mech. 12, 747 (1963).

1957 (1)

J. C. Holladay, Math. Tables Aids. Comput. 11, 233 (1957).
[CrossRef]

Albert, A.

A. Albert, Regression and the Moore-Penrose Pseudoinverse (Academic, New York, 1972).

Andrews, H. C.

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977), Chap. 7.

Curry, H. B.

H. B. Curry, I. J. Schoenberg, J. Anal. Math. 17, 71 (1966).
[CrossRef]

de Boor, C.

C. de Boor, J. Math. Mech. 12, 747 (1963).

Golub, G. H.

G. H. Golub, C. Reinsch, Numer. Math. 14, 403 (1970).
[CrossRef]

Goodman, J. W.

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

Holladay, J. C.

J. C. Holladay, Math. Tables Aids. Comput. 11, 233 (1957).
[CrossRef]

Hunt, B. R.

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977), Chap. 7.

Nering, E. D.

E. D. Nering, Linear Algebra and Matrix Theory (Wiley, New York, 1970).

Peyrovian, M. J.

A. A. Sawchuk, M. J. Peyrovian, J. Opt. Soc. Am. 65, 712 (1975).
[CrossRef]

M. J. Peyrovian, A. A. Sawchuk, University of Southern California Image Processing Institute Technical Report 620 (1975), p. 86.

Pratt, W. K.

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978), Chap. 9.

Reinsch, C.

G. H. Golub, C. Reinsch, Numer. Math. 14, 403 (1970).
[CrossRef]

Sawchuk, A. A.

A. A. Sawchuk, M. J. Peyrovian, J. Opt. Soc. Am. 65, 712 (1975).
[CrossRef]

M. J. Peyrovian, A. A. Sawchuk, University of Southern California Image Processing Institute Technical Report 620 (1975), p. 86.

Schoenberg, I. J.

H. B. Curry, I. J. Schoenberg, J. Anal. Math. 17, 71 (1966).
[CrossRef]

I. J. Schoenberg, Proc. Nat. Acad. Sci. U.S.A. 52, 947 (1964).
[CrossRef]

J. Anal. Math. (1)

H. B. Curry, I. J. Schoenberg, J. Anal. Math. 17, 71 (1966).
[CrossRef]

J. Math. Mech. (1)

C. de Boor, J. Math. Mech. 12, 747 (1963).

J. Opt. Soc. Am. (1)

Math. Tables Aids. Comput. (1)

J. C. Holladay, Math. Tables Aids. Comput. 11, 233 (1957).
[CrossRef]

Numer. Math. (1)

G. H. Golub, C. Reinsch, Numer. Math. 14, 403 (1970).
[CrossRef]

Proc. Nat. Acad. Sci. U.S.A. (1)

I. J. Schoenberg, Proc. Nat. Acad. Sci. U.S.A. 52, 947 (1964).
[CrossRef]

Other (8)

L. L. Schumaker, in Ref. 2, p. 87.

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

T. N. E. Greville, Ed., Theory and Applications of Spline Functions (Academic, New York, 1969).

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978), Chap. 9.

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977), Chap. 7.

M. J. Peyrovian, A. A. Sawchuk, University of Southern California Image Processing Institute Technical Report 620 (1975), p. 86.

A. Albert, Regression and the Moore-Penrose Pseudoinverse (Academic, New York, 1972).

E. D. Nering, Linear Algebra and Matrix Theory (Wiley, New York, 1970).

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Figures (10)

Fig. 1
Fig. 1

B-spline of degree 0.

Fig. 2
Fig. 2

B-spline of degree 1.

Fig. 3
Fig. 3

B-spline of degree of 2.

Fig. 4
Fig. 4

B-spline of degree 3 (cubic B-spline).

Fig. 5
Fig. 5

Original and blurred image line.

Fig. 6
Fig. 6

Restored image line with and without using splines.

Fig. 7
Fig. 7

Original image.

Fig. 8
Fig. 8

Image blurred by motion.

Fig. 9
Fig. 9

Image restored using an overdetermined model.

Fig. 10
Fig. 10

Singular values for motion blur.

Equations (48)

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f s ( x ) = i = f ( i Δ x ) δ ( x i Δ x ) .
F s ( u ) = i = F ( u i Δ x ) ,
Δ x [ 1 / ( 2 B x ) ] .
f ( x ) = i = f ( i 2 B x ) sinc [ 2 B x ( x i 2 B x ) ]
[ 1 , x , . . . , x m , ( x x 1 ) + m , ( x x 2 ) + m , . . . , ( x x n ) + m ] ,
( x x i ) + m = { 0 if x x i ( x x i ) m if x > x i .
M i ( x ) = ( m + 1 ) l = i i + m + 1 ( x x l ) + m ω ( x l ) ,
ω ( x l ) = ( 1 ) m + 1 j = i j l i + m + 1 ( x l x j ) .
B 2 ( x ) = 3 [ ( x + 1.5 ) + 2 6 ( x + 0.5 ) + 2 2 + ( x 0.5 ) + 2 2 ( x 1.5 ) + 2 6 ] ,
B 3 ( x ) = 4 [ ( x + 2 ) + 3 24 ( x + 1 ) + 3 6 + ( x ) + 3 4 ( x 1 ) + 3 6 + ( x 2 ) + 3 24 ] .
g ( x ) = h ( x ξ ) f ( ξ ) d ξ ,
f ( x ) = i f i B m ( x x i ) ,
h ( x ) = j h j B n ( x x j ) ,
g ( x ) = i j f i h j B m ( x x i ) * B n ( x x j ) .
B m ( x x i ) * B n ( x x j ) = B m + n + 1 ( x x i x j )
k g k B m + n + 1 ( x k Δ x ) = i j f i h j B m + n + 1 [ x ( i + j ) Δ x ] ,
g = Hf ,
h ( x ) = 15 56 [ 1 ( x 3.5 ) 2 ] 2 , 3.5 x 3.5 = 0 , elsewhere
g = Hf ,
f ̂ = H 1 g ,
minimize f 2 ,
g Hf 2 ,
W ( f ) = g Hf 2 + δ 2 f
W f = 2 H t ( g Hf ) + 2 δ 2 f = 0 ,
f ̂ = lim δ 0 ( H t H + δ 2 I ) 1 H t g ,
H + = lim δ 0 ( H t H + δ 2 I ) 1 H t
f ̂ = H + g
( H t H H t + δ 2 H t ) = H t ( H H t + δ 2 I ) = ( H t H + δ 2 I ) H t ,
( H t H + δ 2 I ) 1 H t = H t ( H H t + δ 2 I ) 1 ,
H + = lim δ 0 H t ( H H t + δ 2 I ) 1 .
f ( ξ ) = i = 1 M f i B m ( ξ ξ i ) ,
g ( x ) = i = 1 N g i B ( x x i ) ,
W ( f ) = [ g ( x ) h ( x , ξ ) f ( ξ ) d ξ ] 2 d x + δ 2 [ f ( ξ ) ] 2 d ξ ,
f ̂ = ( P + δ 2 B ) 1 Qg ,
f ̂ = [ f ̂ 1 , f ̂ 2 , . . . , f ̂ m ] t , g = [ g 1 , g 2 , . . . , g N ] t , B m ( ξ ) = [ B m ( ξ ξ 1 ) , B m ( ξ ξ 2 ) , . . . , B m ( ξ ξ M ) ] t , B n ( x ) = [ B n ( x x 1 ) , B n ( x x 2 ) , . . . , B n ( x x N ) ] t , p ( x ) = h ( x , ξ ) B m ( ξ ) d ξ , P = p ( x ) p t ( x ) d x , B = B m ( ξ ) B m ( ξ ) t d ξ , Q = p ( x ) B n ( x ) t d x .
q t Pq = q t p ( x ) p t ( x ) q d x = [ q t p ( x ) ] 2 d x 0 .
a = max ( x l , x 1 ) ,
b = min ( x + l , x N ) ,
g ( x ) = a b h ( x ξ ) f ( ξ ) d ξ .
g ( i ) = j = K 1 K 2 c i j h ( i j ) f ( j ) ,
K 1 = max ( i L , 1 ) ,
K 2 = min ( i + L , N ) ,
M = N + L ( L + 1 ) + 1 = N + 2 L .
g = Hf ,
h = ( h L , . . . , h 1 , h 0 , h 1 , . . . , h L )
H f ̂ 1 H f ̂ 2 = Hg Hg = 0
H ( f ̂ 1 f ̂ 2 ) = 0 .
g = Hf ,

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