Abstract

A unified two-step approach for evaluation of deterministic and statistical modulation transfer functions (MTF’s) is applied to a time-delay-and-integration charge-coupled imager. The deterministic MTF’s include the well-known spatial and temporal aperture MTF’s, as well as the charge-coupled imager interpixel and intrapixel synchronism MTF’s, which are derived here. These latter MTF’s originate from nonsynchronous motion (velocity mismatch) between the image on the focal plane and the charge packets. The statistical evaluation results in phase and jitter MTF’s.

© 1993 Optical Society of America

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References

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  1. D. F. Barbe, “Time delay and integration image sensors,” in Solid-State Imaging, P. G. Jespers, F. Van der Wiele, M. H. White, eds. (Noordhoff, Gröningen, The Netherlands, 1976), pp. 659–671.
    [CrossRef]
  2. R. J. Arguello, “Image chain analysis of high resolution, high speed CCD film reader system,” presented at the Tenth Annual Modeling and Simulation Conference, School of Engineering, University of Pittsburgh, Pittsburgh, Pa., 25–27 April 1979.
  3. D. F. Barbe, “Imaging devices using the charge-coupled concept,” Proc. IEEE 63, 38–67 (1975).
    [CrossRef]
  4. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, London, 1975), p. 480.
  5. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chaps. 5 and 6, pp. 77–140.
  6. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), Chap. 4, p. 83.
  7. T. S. Lomheim, L. W. Schumann, R. M. Shima, J. S. Thompson, W. F. Woodward, “Electro-optical hardware considerations in measuring the imaging capability of scanned time-delay-and-integrate charge-coupled imagers,” Opt. Eng. 29, 911–927 (1990).
    [CrossRef]
  8. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products, 4th ed. (Academic, New York, 1965), p. 29.

1990 (1)

T. S. Lomheim, L. W. Schumann, R. M. Shima, J. S. Thompson, W. F. Woodward, “Electro-optical hardware considerations in measuring the imaging capability of scanned time-delay-and-integrate charge-coupled imagers,” Opt. Eng. 29, 911–927 (1990).
[CrossRef]

1975 (1)

D. F. Barbe, “Imaging devices using the charge-coupled concept,” Proc. IEEE 63, 38–67 (1975).
[CrossRef]

Arguello, R. J.

R. J. Arguello, “Image chain analysis of high resolution, high speed CCD film reader system,” presented at the Tenth Annual Modeling and Simulation Conference, School of Engineering, University of Pittsburgh, Pittsburgh, Pa., 25–27 April 1979.

Barbe, D. F.

D. F. Barbe, “Imaging devices using the charge-coupled concept,” Proc. IEEE 63, 38–67 (1975).
[CrossRef]

D. F. Barbe, “Time delay and integration image sensors,” in Solid-State Imaging, P. G. Jespers, F. Van der Wiele, M. H. White, eds. (Noordhoff, Gröningen, The Netherlands, 1976), pp. 659–671.
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, London, 1975), p. 480.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chaps. 5 and 6, pp. 77–140.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products, 4th ed. (Academic, New York, 1965), p. 29.

Lomheim, T. S.

T. S. Lomheim, L. W. Schumann, R. M. Shima, J. S. Thompson, W. F. Woodward, “Electro-optical hardware considerations in measuring the imaging capability of scanned time-delay-and-integrate charge-coupled imagers,” Opt. Eng. 29, 911–927 (1990).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), Chap. 4, p. 83.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products, 4th ed. (Academic, New York, 1965), p. 29.

Schumann, L. W.

T. S. Lomheim, L. W. Schumann, R. M. Shima, J. S. Thompson, W. F. Woodward, “Electro-optical hardware considerations in measuring the imaging capability of scanned time-delay-and-integrate charge-coupled imagers,” Opt. Eng. 29, 911–927 (1990).
[CrossRef]

Shima, R. M.

T. S. Lomheim, L. W. Schumann, R. M. Shima, J. S. Thompson, W. F. Woodward, “Electro-optical hardware considerations in measuring the imaging capability of scanned time-delay-and-integrate charge-coupled imagers,” Opt. Eng. 29, 911–927 (1990).
[CrossRef]

Thompson, J. S.

T. S. Lomheim, L. W. Schumann, R. M. Shima, J. S. Thompson, W. F. Woodward, “Electro-optical hardware considerations in measuring the imaging capability of scanned time-delay-and-integrate charge-coupled imagers,” Opt. Eng. 29, 911–927 (1990).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, London, 1975), p. 480.

Woodward, W. F.

T. S. Lomheim, L. W. Schumann, R. M. Shima, J. S. Thompson, W. F. Woodward, “Electro-optical hardware considerations in measuring the imaging capability of scanned time-delay-and-integrate charge-coupled imagers,” Opt. Eng. 29, 911–927 (1990).
[CrossRef]

Opt. Eng. (1)

T. S. Lomheim, L. W. Schumann, R. M. Shima, J. S. Thompson, W. F. Woodward, “Electro-optical hardware considerations in measuring the imaging capability of scanned time-delay-and-integrate charge-coupled imagers,” Opt. Eng. 29, 911–927 (1990).
[CrossRef]

Proc. IEEE (1)

D. F. Barbe, “Imaging devices using the charge-coupled concept,” Proc. IEEE 63, 38–67 (1975).
[CrossRef]

Other (6)

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, London, 1975), p. 480.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chaps. 5 and 6, pp. 77–140.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), Chap. 4, p. 83.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products, 4th ed. (Academic, New York, 1965), p. 29.

D. F. Barbe, “Time delay and integration image sensors,” in Solid-State Imaging, P. G. Jespers, F. Van der Wiele, M. H. White, eds. (Noordhoff, Gröningen, The Netherlands, 1976), pp. 659–671.
[CrossRef]

R. J. Arguello, “Image chain analysis of high resolution, high speed CCD film reader system,” presented at the Tenth Annual Modeling and Simulation Conference, School of Engineering, University of Pittsburgh, Pittsburgh, Pa., 25–27 April 1979.

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Figures (4)

Fig. 1
Fig. 1

Portion of a sinusoidal wave of infinite extent (within dashed lines; the solid lines represent lines of constant phase) with spatial frequency vector k0 traveling in the x direction with velocity v across a single pixel located at (x1, y1).

Fig. 2
Fig. 2

Ideal TDI implementation.

Fig. 3
Fig. 3

Portion of sinusoidal wave of infinite extent (within dashed lines; the solid lines represent lines of constant phase) with spatial frequency vector k0, traveling with velocity v + Δv across four TDI column pixels. The first pixel (n = 1) is located at the focal-plane coordinates (x1, y1).

Fig. 4
Fig. 4

Portion of sinusoidal focal-plane irradiance scanning across multiphase pixels of length lx and M = 4 phase gates per pixel: (a) Irradiance at t = −T/8 is shown as a solid curve and irradiance at t = T/8 is shown as a dashed curve. (b), (c), (d) Successive portions of the scanning irradiance and the moving charge packet within a charge-coupled imager. During each sub-pixel integration time (of duration T/4) the photocharge is integrated over the pixel length lx.

Equations (54)

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Q ( ρ 1 , t 1 , λ ) = R 0 ( λ ) - d t A t ( t 1 - t ) - d x A x ( x 1 - x ) × - d y A y ( y 1 - y ) h ( ρ , t , λ ) .
Q ( ρ 1 , t 1 ) = Δ λ d λ Q ( ρ 1 , t 1 , λ ) .
H ( k , t , λ ) = T OP ( k , λ ) H g ( k , t , λ ) .
T OP ( k , λ ) = MTF OP ( k , λ ) exp [ i Φ ( k , λ ) ] ,
h g ( ρ , t , λ , k 0 , v , φ ) = τ ( λ ) H 0 ( λ ) [ 1 + m cos ( k 0 · ρ - k 0 · v t + φ ) ] ,
h ( ρ , t , λ , k 0 , v , φ ) = τ ( λ ) H 0 ( λ ) { 1 + m MTF OP ( k 0 , λ ) × cos [ k 0 · ρ - k 0 · v t + φ + Φ ( k 0 , λ ) ] } .
Q ( ρ 1 , t 1 , λ , k 0 , v , φ ) = l x l y TR 0 ( λ ) τ ( λ ) H 0 ( λ ) { 1 + m MTF OP ( k 0 , λ ) l x l y T × t 1 t 1 + T d t x 1 - ½ l x x 1 + ½ l x d x y 1 - ½ l y y 1 + ½ l y d y × cos [ k 0 · ρ - k 0 · v t + φ + Φ ( k 0 , λ ) ] } ,
Q N ( ρ 1 , t 1 , λ , k 0 , v + Δ v , φ ) = R 0 ( λ ) n = 1 N - d t A t , n ( t 1 - t ) - d x A x , n ( x 1 - x ) × - d y A y ( y 1 - y ) h ( ρ , t , λ , k 0 , v + Δ v , φ ) ,
Q N ( ρ 1 , t 1 , λ , k 0 , v + Δ v , φ ) = R 0 ( λ ) n = 1 N m = 1 M t 1 + ( n - 1 ) T + ( m - 1 ) T / M - ½ T / M t 1 + ( n - 1 ) T + m T / M - ½ T / M d t × x 1 + ( n - 1 ) l x + ( m - 1 ) l x / M - ½ l x x 1 + ( n - 1 ) l x + ( m - 1 ) l x / M + ½ l x d x × y 1 - ½ l y y 1 + ½ l y d y h ( ρ , t , λ , k 0 , v + Δ v , φ ) ,
Q N ( ρ 1 , t 1 , λ , k 0 , v + Δ v , φ ) = N l x l y TR 0 ( λ ) τ ( λ ) H 0 ( λ ) { 1 + m MTF OP ( k 0 , λ ) × MTF A ( k 0 ) MTF SYNC ( k 0 ) cos [ k 0 · ρ 1 - k 0 · ( v + Δ v ) t 1 + φ + Φ ( k 0 , λ ) + δ ] } ,
MTF A ( k 0 ) = sinc ( ½ k 0 l x cos θ ) sinc ( ½ k 0 l y sin θ ) × sinc [ ½ k 0 · ( v + Δ v ) T / M ] .
MTF SYNC ( k 0 ) = sinc ( ½ N k 0 · Δ v T ) sinc ( ½ k 0 · Δ v T ) sinc ( ½ k 0 · Δ v T ) sinc ( ½ k 0 · Δ v T / M ) ,
δ = - 1 2 ( N - 1 ) k 0 · Δ v T - 1 2 M - 1 M k 0 · Δ v T ,
μ F ˜ = F ( u ˜ ) u ˜ = a b d u f u ˜ ( u ) F ( u ) ,
σ F ˜ 2 = [ F ( u ˜ ) - μ F ˜ ] 2 u ˜ = a b d u f u ˜ ( u ) [ F ( u ) - μ F ˜ ] 2 .
μ Q ˜ = Q N ( ρ ˜ 1 , t 1 , λ , k 0 , v + Δ v , φ ˜ ) x ˜ 1 , y ˜ 1 , φ ˜ .
μ Q ˜ = - d x 1 - d y 1 f x ˜ 1 , y ˜ 1 ( x 1 , y 1 ) × - π π d φ f φ ˜ ( φ ) Q N ( ρ 1 , t 1 , λ , k 0 , v + Δ v , φ ) ,
f x ˜ , y ˜ ( x , y ) = exp { - ( ξ x 2 - 2 r ξ x ξ y + ξ y 2 ) / [ 2 ( 1 - r 2 ) ] } 2 π σ x ˜ σ y ˜ ( 1 - r 2 ) 1 / 2 ,
ξ x = ( x - μ x ˜ ) / σ x ˜ ,
μ x ˜ = x ˜ ,
σ x ˜ 2 = ( x ˜ - μ x ˜ ) 2 ,
f x ˜ , y ˜ ( x , y ) = f x ˜ ( x ) f y ˜ ( y ) ,
Q N ( ρ ˜ 1 , t 1 , λ , k 0 , v + Δ v , φ ˜ ) x ˜ 1 , y ˜ 1 = N l x l y TR 0 ( λ ) τ ( λ ) H 0 ( λ ) { 1 + m MTF OP ( k 0 , λ ) × MTF A ( k 0 ) MTF SYNC ( k 0 ) MTF JIT ( k 0 ) × cos [ k 0 · μ ρ ˜ 1 - k 0 · ( v + Δ v ) t 1 + φ ˜ + Φ ( k 0 , λ ) + δ ] } ,
MTF JIT ( k 0 ) = exp ( - ½ k 0 2 σ x ˜ 2 cos 2 θ ) exp ( - ½ k 0 2 σ y ˜ 2 sin 2 θ ) .
σ x ˜ 2 = 0 d f S x ˜ ( f ) .
MTF JIT ( k 0 ) = exp ( - k 0 2 σ 2 / 2 ) ,
f φ ˜ ( φ ) = { 1 / ( Δ φ ) - ½ Δ φ φ ½ Δ φ 0 otherwise .
μ Q ˜ = Q N ( ρ ˜ 1 , t 1 , λ , k 0 , v + Δ v , φ ˜ ) x ˜ 1 , y ˜ 1 , φ ˜ = N l x l y TR 0 ( λ ) τ ( λ ) H 0 ( λ ) { 1 + m MTF OP ( k 0 , λ ) × MTF A ( k 0 ) MTF SYNC ( k 0 ) MTF JIT ( k 0 ) MTF PH ( k 0 ) × cos [ k 0 · μ ρ ˜ 1 - k 0 · ( v + Δ v ) t 1 + Φ ( k 0 , λ ) + δ ] } .
MTF PH ( k 0 ) = sinc ( ½ Δ φ ) ,
σ Q ˜ 2 = ( Q ˜ - μ Q ˜ ) 2 x ˜ 1 , y ˜ 1 , φ ˜ ,
σ Q ˜ 2 = [ N l x l y TR 0 ( λ ) τ ( λ ) H 0 ( λ ) ] 2 × [ m MTF OP ( k 0 , λ ) MTF A ( k 0 ) MTF SYNC ( k 0 ) ] 2 × [ ½ + ½ sinc ( Δ φ ) exp ( - 2 k 0 2 σ 2 ) × cos ( 2 k 0 · μ ρ ˜ 1 + 2 C ) - MTF PH 2 ( k 0 ) × MTF JIT 2 ( k 0 ) cos 2 ( k 0 · μ ρ ˜ 1 + C ) ] ,
Q N ( ρ 1 , t 1 , λ , k 0 , v + Δ v , φ ) = τ ( λ ) H 0 ( λ ) R 0 ( λ ) n = 1 N m = 1 M t 1 + ( n - 1 ) T + ( m - 1 ) T / M - ½ T / M t 1 + ( n - 1 ) T + m T / M - ½ T / M d t × x 1 + ( n - 1 ) l x + ( m - 1 ) l x / M - ½ l x x 1 + ( n - 1 ) l x + ( m - 1 ) l x / M + ½ l x d x × y 1 - ½ l y y 1 + ½ l y d y { 1 + m MTF OP ( k 0 , λ ) × cos [ k 0 · ρ - k 0 · ( v + Δ v ) t + φ + Φ ( k 0 , λ ) ] } .
I = y 1 - ½ l y y 1 + ½ l y d y [ 1 + A cos ( k 0 y y + B ) ] ,
I = l y [ 1 + A sinc ( ½ k 0 y l y ) cos ( k 0 y y 1 + B ) ] ,
sin ( c + d ) - sin ( c - d ) = 2 cos ( c ) sin ( d )
Q N ( ρ 1 , t 1 , λ , k 0 , v + Δ v , φ ) = τ ( λ ) H 0 ( λ ) R 0 ( λ ) N l x l y T { 1 + m MTF OP ( k 0 , λ ) × MTF A ( k 0 ) 1 N n = 1 N m = 1 M cos [ k 0 · ρ 1 - k 0 · ( v + Δ v ) t 1 + φ + Φ ( k 0 , λ ) - ( n - 1 + m - 1 M ) k 0 · Δ v T ] } .
k = 0 n - 1 cos ( x + k y ) = cos [ x + 1 2 ( n - 1 ) y ] sin ( n y / 2 ) cosec ( y / 2 )
Q N ( ρ ˜ 1 ) ρ ˜ 1 = A [ 1 + B cos ( k 0 · ρ ˜ 1 + C ) ] ρ ˜ 1 = A [ 1 + B cos ( k 0 x x ˜ 1 + k 0 y y ˜ 1 + C ) x ˜ 1 , y ˜ 1 ] ,
A = N l x l y TR 0 ( λ ) τ ( λ ) H 0 ( λ ) ,
B = m MTF OP ( k 0 , λ ) MTF A ( k 0 ) MTF SYNC ( k 0 ) ,
C = - k 0 · ( v + Δ v ) t 1 + φ + Φ ( k 0 , λ ) + δ .
cos ( k 0 x x ˜ 1 + D ) x ˜ 1 = - d x 1 exp [ - ½ ( x 1 - μ x ˜ 1 ) 2 / σ x ˜ 1 2 ] ( 2 π ) 1 / 2 σ x ˜ 1 cos ( k 0 x x 1 + D ) = exp ( - k 0 x 2 σ x ˜ 1 2 / 2 ) cos ( k 0 x μ x ˜ 1 + D ) .
Q ˜ N x ˜ 1 , y ˜ 1 = A [ 1 + B MTF JIT ( k 0 ) × cos ( k 0 x μ x 1 + k 0 y μ y 1 + C ) ] ,
Q N ( φ ˜ ) x ˜ 1 , y ˜ 1 , φ ˜ = - π π d φ f φ ( φ ) Q N ( φ ) x ˜ 1 , y ˜ 1 ,
Q N ( φ ) x ˜ 1 , y ˜ 1 = A [ 1 + B cos ( C + φ ) ] ,
Q N ( φ ˜ ) x ˜ 1 , y ˜ 1 , φ ˜ = - Δ φ / 2 Δ φ / 2 d φ 1 Δ φ A [ 1 + B cos ( C + φ ) ] ,
Q N ( φ ˜ ) x ˜ 1 , y ˜ 1 , φ ˜ = A [ 1 + B 2 sin ( ½ Δ φ ) Δ φ cos ( C ) ] = A [ 1 + B MTF PH ( k 0 ) cos ( C ) ] .
σ Q ˜ 2 = - d x 1 - d y 1 f x ˜ 1 , y ˜ 1 ( x 1 , y 1 ) - π π d φ f φ ˜ ( φ ) × [ Q N ( ρ 1 , t 1 , λ , k 0 , v + Δ v , φ ) - μ Q ˜ ] 2 .
Q ˜ N = A [ 1 + B cos ( k 0 · ρ ˜ 1 + φ ˜ + C ) ] .
A = N l x l y TR 0 ( λ ) τ ( λ ) H 0 ( λ ) ,
B = m MTF OP ( k 0 , λ ) MTF A ( k 0 ) MTF SYNC ( k 0 ) .
μ Q ˜ = Q ˜ N φ ˜ , ρ ˜ 1 = A [ 1 + B MTF JIT ( k 0 ) MTF PH ( k 0 ) cos ( k 0 · μ ρ ˜ 1 + C ) ] ,
σ Q ˜ 2 = Q ˜ N 2 φ ˜ , ρ ˜ 1 - μ Q ˜ 2 .
σ Q ˜ 2 = A 2 B 2 [ ½ + ½ sinc ( Δ φ ) × exp ( - 2 k 0 2 σ 2 ) cos ( 2 k 0 · μ ρ ˜ 1 + 2 C ) - MTF PH 2 ( k 0 ) MTF JIT 2 ( k 0 ) cos 2 ( k 0 · μ ρ ˜ 1 + C ) ] ,

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