Abstract

This paper presents a new technique for measuring the location of point images in an image-sampling system. Primary attention is given to the situation in which the image is under sampled. Such under sampling could be the result of data-storage or bandwidth limitations, and would introduce errors into standard methods of image location. The new technique is valid for under-sampled as well as adequately sampled images, and consists of a linear-filtering procedure of the same complexity required by standard location-measurement techniques. The problem is formulated as one of statistical-estimation theory, the optimum-location procedures being derived for an additive-gaussian-noise environment, and for conditions of known and unknown image irradiance.

© 1973 Optical Society of America

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References

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  1. C. W. Helstrom, Statistical Theory of Signal Detection (Pergamon, New York, 1960).
  2. D. Middleton, An Introduction to Statistical Communication Theory (McGraw–Hill, New York, 1960).
  3. C. E. Shannon, Proc. IRE 37, 10 (1949).
    [Crossref]
  4. M. Schwartz, W. Bennett, and S. Stein, Communication Systems and Techniques (McGraw–Hill, New York, 1966).

1949 (1)

C. E. Shannon, Proc. IRE 37, 10 (1949).
[Crossref]

Bennett, W.

M. Schwartz, W. Bennett, and S. Stein, Communication Systems and Techniques (McGraw–Hill, New York, 1966).

Helstrom, C. W.

C. W. Helstrom, Statistical Theory of Signal Detection (Pergamon, New York, 1960).

Middleton, D.

D. Middleton, An Introduction to Statistical Communication Theory (McGraw–Hill, New York, 1960).

Schwartz, M.

M. Schwartz, W. Bennett, and S. Stein, Communication Systems and Techniques (McGraw–Hill, New York, 1966).

Shannon, C. E.

C. E. Shannon, Proc. IRE 37, 10 (1949).
[Crossref]

Stein, S.

M. Schwartz, W. Bennett, and S. Stein, Communication Systems and Techniques (McGraw–Hill, New York, 1966).

Proc. IRE (1)

C. E. Shannon, Proc. IRE 37, 10 (1949).
[Crossref]

Other (3)

M. Schwartz, W. Bennett, and S. Stein, Communication Systems and Techniques (McGraw–Hill, New York, 1966).

C. W. Helstrom, Statistical Theory of Signal Detection (Pergamon, New York, 1960).

D. Middleton, An Introduction to Statistical Communication Theory (McGraw–Hill, New York, 1960).

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Figures (2)

Fig. 1
Fig. 1

One-dimensional transfer function and impulse response.

Fig. 2
Fig. 2

Examples of under sampling.

Equations (35)

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t ( x , y ) = T ( k x , k y ) e j 2 π ( x k x + y k y ) d k x d k y ,
f = k υ .
R ( x , y ) = υ ( x + x , y + y ) t ( x , y ) d x d y .
R ( x , y ) = υ ( x , y ) t ( x x , y y ) d x d y .
t ( x x , y y ) = i = j = t ( i Δ x , j Δ y ) × sin c [ 2 π Δ x ( x x i Δ x ) ] × sin c [ 2 π Δ y ( y y j Δ y ) ] ,
υ ( x , y ) = i = j = υ ( i Δ x , j Δ y ) × sin c [ 2 π Δ x ( x i Δ x ) ] × sinc [ 2 π Δ y ( y j Δ y ) ] ,
R ( x , y ) = i = j = i = j = υ ( i Δ x , j Δ y ) t ( i Δ x , j Δ y ) × sin c [ 2 π Δ x ( x ( i i ) Δ x ) ] × sin c [ 2 π Δ y ( y ( j j ) Δ y ) ] .
R ( x , y ) = i = j = υ ( i Δ x , j Δ y ) × k = l = t [ ( i k ) Δ x , ( j l ) Δ y ] × sin c [ 2 π Δ x ( x k Δ x ) ] sin c [ 2 π Δ y ( y l Δ y ) ] .
R ( x , y ) = k = l = R ( k , l ) sin c [ 2 π Δ x ( x k Δ x ) ] × sin c [ 2 π Δ y ( y l Δ y ) ] ,
R ( k , l ) = i = j = υ ( i Δ x , j Δ y ) t [ ( i k ) Δ x , ( j l ) Δ y ]
R ( x , y ) = i = j = υ ( i Δ x , j Δ y ) G ( i , j , x , y ) ,
G ( i , j , x , y ) = k = l = t [ ( i k ) Δ x , ( j l ) Δ y ) ] × sin c [ 2 π Δ x ( x k Δ x ) ] sinc [ 2 π Δ y ( y l Δ y ) ] = t ( i Δ x x , j Δ y y ) .
R ( x , y ) = i = j = υ ( i Δ x , j Δ y ) t ( i Δ x x , j Δ y y ) .
F ( k ) = 1 + cos α k , | k | π / α .
f ( x ) = F ( k ) e j 2 π x k d k = 2 π α [ sin c 2 π 2 x α + 1 2 sin c ( 2 π x + α ) π α + 1 2 sin c ( 2 π x α ) π α ] .
k s 2 π α .
L ( x y ) = ( 2 π σ 2 ) ( N 1 + 1 2 ) ( N 2 + 1 2 ) exp ( { i = N 1 N 1 j = N 2 N 2 × [ υ ( i Δ x , j Δ y ) t ( i Δ x x , j Δ y y ) ] 2 2 σ 2 } ) ,
i = N 1 N 1 j = N 2 N 2 [ υ ( i Δ x , j Δ y ) t ( i Δ x x , j Δ y y ) ] 2 2 = i = N 1 N 1 j = N 2 N 2 { υ ( i Δ x , j Δ y ) t ( i Δ x x , j Δ y y ) 1 2 t 2 ( i Δ x x , j Δ y y ) 1 2 υ 2 ( i Δ x , j Δ y ) } .
μ ( x , y ) = i = N 1 N 1 j = N 2 N 2 { υ ( i Δ x , j Δ y ) t ( i Δ x x , j Δ y y ) 1 2 t 2 ( i Δ x x , j Δ y y ) } .
I ˆ = i = N 1 N 1 j = N 2 N 2 υ ( i Δ x , j Δ y ) t ( i Δ x x , j Δ y y ) / i = N 1 N 1 j = N 2 N 2 t 2 ( i Δ x x , j Δ y y ) .
μ ( x , y ) = 1 2 { i = N 1 N 1 j = N 2 N 2 υ ( i Δ x , j Δ y ) t ( i Δ x x , j Δ y y ) } 2 / i = N 1 N 1 j = N 2 N 2 t 2 ( i Δ x x , j Δ y y ) .
ρ ( x , y ) = i = N 1 N 1 j = N 2 N 2 υ ( i Δ x , j Δ y ) t ( i Δ x x , j Δ y y ) / [ i = N 1 N 1 j = N 2 N 2 υ 2 ( i Δ x , j Δ y ) i = N 1 N 1 j = N 2 N 2 t 2 ( i Δ x x , j Δ y y ) ] 1 2 .
| ρ ( x y ) | 1.
[ i = N 1 N 1 j = N 2 N 2 υ 2 ( i Δ x , j Δ y ) ] 1 2
ρ 1 ( x y ) = i = N 1 N 1 j = N 2 N 2 υ ( i Δ x , j Δ y ) t ( i Δ x x , j Δ y y ) / [ i = N 1 N 1 j = N 2 N 2 t 2 ( i Δ x x , j Δ y y ) ] 1 2 .
μ ( x , y ) = 1 2 ρ 1 2 ( x , y ) .
i = N 1 N 1 j = N 2 N 2 { t 2 ( i Δ x x , j Δ y y ) + n ( i Δ x , j Δ y ) t ( i Δ x x , j Δ y y ) } .
σ [ i = N 1 N 1 j = N 2 N 2 t 2 ( i Δ x x , j Δ y y ) ] 1 2 .
SNR = i = N 1 N 1 j = N 2 N 2 t 2 ( i Δ x x , j Δ y y ) / σ [ i = N 1 N 1 j = N 2 N 2 t 2 ( i Δ x x , j Δ y y ) ] 1 2 = ( 1 / σ ) [ i = N 1 N 1 j = N 2 N 2 t 2 ( i Δ x x , j Δ y y ) ] 1 2 .
t ( x ) = i = t ( i Δ x ) sin c [ 2 π Δ x ( x i Δ x ) ] .
t ( x ) = i = t ( i Δ x x ) sin c [ 2 π Δ x ( x + x i Δ x ) ] .
= t 2 ( x ) d x .
= i = j = t ( i Δ x ) t ( j Δ x ) × sin c [ 2 π Δ x ( x i Δ x ) ] sin c [ 2 π Δ x ( x j Δ x ) ] d x = i = j = t ( i Δ x ) t ( j Δ x ) δ i j = i = t 2 ( i Δ x ) .
= i = j = t ( i Δ x x ) t ( j Δ x x ) δ i j = i = t 2 ( i Δ x x ) .
i = t 2 ( i Δ x ) = i = t 2 ( i Δ x x ) .