Abstract

The problem of reconstructing digital data with nonideal analog interpolators is addressed. A theoretical viewpoint is taken which specifies optimal data reconstructors when a predisplay digital filter is employed. This theory allows for a weighted error definition so that solutions with different desired qualities may be obtained. Results for Gaussian-profiled interpolators show that a factor of ~100 in reconstruction-error reduction is achieved with a 13-tap finite-impulse-response predisplay digital filter. The problem of writing negative digital values with incoherent light spots is shown to be avoidable if the signal is reconstructed on photographic film.

© 1983 Optical Society of America

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References

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  1. A. A. Jamberdino, Opt. Eng. 20, 329 (1981).
  2. S. A. Tretter, Introduction to Discrete-Time Signal Processing (Wiley, New York, 1976).
  3. A. B. Carlson, Communication Systems (McGraw-Hill, New York, 1975).
  4. S. Zohar, J. Assoc. Comput. Mach. 16, 592 (1969).
    [CrossRef]

1981

A. A. Jamberdino, Opt. Eng. 20, 329 (1981).

1969

S. Zohar, J. Assoc. Comput. Mach. 16, 592 (1969).
[CrossRef]

Carlson, A. B.

A. B. Carlson, Communication Systems (McGraw-Hill, New York, 1975).

Jamberdino, A. A.

A. A. Jamberdino, Opt. Eng. 20, 329 (1981).

Tretter, S. A.

S. A. Tretter, Introduction to Discrete-Time Signal Processing (Wiley, New York, 1976).

Zohar, S.

S. Zohar, J. Assoc. Comput. Mach. 16, 592 (1969).
[CrossRef]

J. Assoc. Comput. Mach.

S. Zohar, J. Assoc. Comput. Mach. 16, 592 (1969).
[CrossRef]

Opt. Eng.

A. A. Jamberdino, Opt. Eng. 20, 329 (1981).

Other

S. A. Tretter, Introduction to Discrete-Time Signal Processing (Wiley, New York, 1976).

A. B. Carlson, Communication Systems (McGraw-Hill, New York, 1975).

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Figures (22)

Fig. 1
Fig. 1

Overall system.

Fig. 2
Fig. 2

Optimum reconstruction error. Unweighted frequency domain solution.

Fig. 3
Fig. 3

One-dimensional incoherent imaging system.

Fig. 4
Fig. 4

Optimum reconstruction error. Weighted frequency domain solution.

Fig. 5
Fig. 5

Reconstructed and ideal signals: weighted 1-tap solution.

Fig. 6
Fig. 6

Reconstructed and ideal signals: weighted 3-tap solution.

Fig. 7
Fig. 7

Reconstructed and ideal signals: weighted 7-tap solution.

Fig. 8
Fig. 8

Reconstructed and ideal signals: weighted 13-tap solution.

Fig. 9
Fig. 9

Error signal: weighted 1-tap solution.

Fig. 10
Fig. 10

Error signal: weighted 3-tap solution.

Fig. 11
Fig. 11

Error signal: weighted 7-tap solution.

Fig. 12
Fig. 12

Error signal: weighted 13-tap solution.

Fig. 13
Fig. 13

Display system with film model.

Fig. 14
Fig. 14

Display system nonlinearity.

Fig. 15
Fig. 15

Reconstructed and ideal signals as written onto film: weighted 1-tap solution.

Fig. 16
Fig. 16

Reconstructed and ideal signals as written onto film: weighted 3-tap solution.

Fig. 17
Fig. 17

Reconstructed and ideal signals as written onto film: weighted 7-tap solution.

Fig. 18
Fig. 18

Reconstructed and ideal signals as written onto film: weighted 13-tap solution.

Fig. 19
Fig. 19

Error signal as written onto film: weighted 1-tap solution.

Fig. 20
Fig. 20

Error signal as written onto film: weighted 3-tap solution.

Fig. 21
Fig. 21

Error signal as written onto film: weighted 7-tap solution.

Fig. 22
Fig. 22

Error signal as written onto film: weighted 13-tap solution.

Tables (2)

Tables Icon

Table I Optimum Display Filter Coefficients, W(ω) = 1

Tables Icon

Table II Optimum Display Filter Coefficients, W(ω) = tri2(ω/π)

Equations (57)

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f ¯ = [ f N f n + 1 f 0 f N 1 f N ] .
M = { g ( x ) : g ( x ) = i = N N a i u i ( x ) , u i V } ,
u i ( x ) = u ( x i Δ ) .
( f , g ) = X X w ( x ) f ( x ) g ( x ) d x ,
w ( x ) > 0
f = ( f , f ) 1 / 2 .
( f D g * , u k ) = 0 k ,
[ i f i h i ( x ) j g j u j ( x ) , u k ( x ) ] = 0 k .
i f i ( h i , u k ) = j g j ( u j , u k ) k ,
[ B ] f ¯ = [ G ] g ¯ .
g ¯ * = [ G ] 1 [ B ] f ¯ .
f i = { 1 i = 0 , 0 i 0 ,
a ¯ = [ G ] 1 b ¯ ,
b k = ( h o , u k ) ,
C ( z ) = i = N N a i z i .
e 2 = ( f D g * , f D g * ) .
e 2 = ( f D g * , f D ) ( f g * , g * ) .
e 2 = ( f D , f D ) ( g * , f D ) .
e 2 = w ( x ) [ sinc ( x Δ ) k = N N g * k u k ( x ) ] 2 d x .
e 2 = w ( x ) sinc 2 ( x Δ ) d x w ( x ) sinc ( x Δ ) k = N N g * k u k ( x ) d x .
G ( ω ) = S ( ω ) [ i = 0 N a i · cos ( i ω Δ ) + j i = 1 N a N + i sin ( i ω Δ ) ] ,
C ( z ) = a 0 + 1 2 { k = 1 N [ ( a k a N + k ) z k + ( a k + a N + k ) z k ] } .
( X , Y ) = W ( ω ) · X ¯ ( ω ) · Y ( ω ) d ω ,
( F G * , U k ) = 0 k ,
U k = { S ( ω ) cos ( k ω Δ ) , k = 0 , , N , S ( ω ) sin [ ( k N ) ω Δ ] , k = N + 1 , , 2 N + 1.
( F , U k ) = i = 0 2 N + 1 a i ( U i , U k ) k .
b ¯ = [ G ] a ¯ ,
b k = ( F , U k ) .
[ S ( ω ) sin ( k ω Δ ) , S ( ω ) cos ( l ω Δ ) ] = 0.
b ¯ E = [ G ] E a ¯ E ,
b ¯ 0 = [ G ] 0 a ¯ 0 .
e 2 = ( f G * , F G * ) .
e 2 = W ( ω ) | F ( ω ) G * ( ω ) | 2 d ω .
e 2 = | rect ( ω Δ 2 π ) G * ( ω ) | 2 d ω .
e 2 = π / Δ π / Δ | 1 G * ( ω ) | 2 d ω + 2 π / Δ | G * ( ω ) | 2 d ω .
F ( β ) G ¯ ( β ) d β = f ( α ) g ¯ ( α ) d α .
u i ( x ) = Gaus ( x i Δ σ ) .
Gaus ( x σ ) = 1 σ 2 π exp [ 1 2 ( x / σ ) 2 ] .
G i j = 1 2 π σ 2 w ( x ) × exp [ ( x i Δ ) 2 / 2 σ 2 ] exp [ ( x j Δ ) 2 / 2 σ 2 ] d x .
G i j = 1 2 σ π exp { [ ( Δ / 2 σ ) ( i j ) 2 ] } .
b i = 1 σ 2 π w ( x ) sinc ( x Δ ) exp [ ( x i Δ ) 2 / 2 σ 2 ] d x ,
sinc ( x ) sin ( π x ) π x .
F ( ω ) = rect ( ω Δ 2 π ) ,
S ( ω ) = exp [ ( σ ω ) 2 / 2 ] .
a ¯ E = [ G ] E 1 b ¯ E .
b k = π / Δ π / Δ W ( ω ) · exp [ ( σ ω ) 2 / 2 ] · cos ( k ω Δ ) d ω ,
G i j = W ( ω ) exp [ ( σ ω ) 2 ] cos ( i ω Δ ) cos ( j ω Δ ) d ω .
G i j = π 2 σ { exp [ ( 1 4 σ 2 ) ( i j ) 2 ] + exp [ ( 1 4 σ 2 ) ( i + j ) 2 ] } ,
W ( ω ) = | tri ˜ ( ω π ) | 2 ,
Film ( x ) = R ( 1 exp { [ a ( x x 0 ) ] b } ) + B ,
f ( x ) = 0.62 sinc 2 ( x ) + 0.21.
e 2 ( a o , , a l + Δ , , a N ) e 2 ( a ) + Δ 2 2 e 2 ( a ) a l 2 ,
2 e 2 a l = 2 W ( ω ) S 2 ( ω ) cos 2 ( ω ) d ω .
e 2 ( a + Δ ) e 2 ( a ) + l = 0 N Δ 2 2 2 e a l 2 .
e 2 ( a + Δ ) e 2 ( a ) + 3 ( N + 1 ) 2 2 b .
b = log 10 3 3 ( N + 1 ) / 2 · log 2 .
2 ( M , b ) = M 2 2 b 12 .

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