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.

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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).

Other (4)

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).

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

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

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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 ) .