Abstract

Spatial filtering of two-dimensional pictorial data as an extension of one-dimensional filter theory is applied to the problem of enhancing the detection of localized objects which are superimposed upon a noisy background.

Four types of filters are derived. These are the linear, quadratic, general statistical, and decision filters. Each filter is of the “matched” type, the different designs being associated with various degrees of knowledge about the noise statistics.

A computer simulation of the linear and general statistical filters was done and examples are shown.

© 1962 Optical Society of America

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References

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  1. P. M. Duffieux, “L’integrale de Fourier et ses applications en optique,” thesis, Universite de Besancon, 1946.
  2. A. L. Cauchy, “Memoire sur Diverses Formulaes dé Analyse,” Compt. rend. 12, 283 (1841).
  3. P. Elias, D. S. Grey, and Z. Robinson, J. Opt. Soc. Am. 42, 127 (1952).
    [Crossref]
  4. U. Grenander and M. Rosenblatt, Statistical Analysis of Stationary Time Series (John Wiley Sons, Inc., New York, 1957), pp. 29, 30.
  5. D. O. North, “Analysis of Factors which Determine Signal–Noise Discrimination in Pulsed Carrier Systems,” R.C.A. Tech. Rept. PTR 6C (June1943).
  6. N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series (Technology Press MIT, Cambridge, Massachusetts, and John Wiley Sons, Inc., New York, 1949).
  7. D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill Book Company, Inc., New York, 1960), pp. 773–810.
  8. A. Wald, Statistical Decision Functions (John Wiley Sons, Inc., New York, 1950).
  9. This may very well not be the best criterion to use if certain costs can be assigned to the two kinds of errors, but it will be sufficient to illustrate the general form of the result for decision-theory problems of this type.

1952 (1)

1841 (1)

A. L. Cauchy, “Memoire sur Diverses Formulaes dé Analyse,” Compt. rend. 12, 283 (1841).

Cauchy, A. L.

A. L. Cauchy, “Memoire sur Diverses Formulaes dé Analyse,” Compt. rend. 12, 283 (1841).

Duffieux, P. M.

P. M. Duffieux, “L’integrale de Fourier et ses applications en optique,” thesis, Universite de Besancon, 1946.

Elias, P.

Grenander, U.

U. Grenander and M. Rosenblatt, Statistical Analysis of Stationary Time Series (John Wiley Sons, Inc., New York, 1957), pp. 29, 30.

Grey, D. S.

Middleton, D.

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill Book Company, Inc., New York, 1960), pp. 773–810.

North, D. O.

D. O. North, “Analysis of Factors which Determine Signal–Noise Discrimination in Pulsed Carrier Systems,” R.C.A. Tech. Rept. PTR 6C (June1943).

Robinson, Z.

Rosenblatt, M.

U. Grenander and M. Rosenblatt, Statistical Analysis of Stationary Time Series (John Wiley Sons, Inc., New York, 1957), pp. 29, 30.

Wald, A.

A. Wald, Statistical Decision Functions (John Wiley Sons, Inc., New York, 1950).

Wiener, N.

N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series (Technology Press MIT, Cambridge, Massachusetts, and John Wiley Sons, Inc., New York, 1949).

Compt. rend. (1)

A. L. Cauchy, “Memoire sur Diverses Formulaes dé Analyse,” Compt. rend. 12, 283 (1841).

J. Opt. Soc. Am. (1)

Other (7)

U. Grenander and M. Rosenblatt, Statistical Analysis of Stationary Time Series (John Wiley Sons, Inc., New York, 1957), pp. 29, 30.

D. O. North, “Analysis of Factors which Determine Signal–Noise Discrimination in Pulsed Carrier Systems,” R.C.A. Tech. Rept. PTR 6C (June1943).

N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series (Technology Press MIT, Cambridge, Massachusetts, and John Wiley Sons, Inc., New York, 1949).

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill Book Company, Inc., New York, 1960), pp. 773–810.

A. Wald, Statistical Decision Functions (John Wiley Sons, Inc., New York, 1950).

This may very well not be the best criterion to use if certain costs can be assigned to the two kinds of errors, but it will be sufficient to illustrate the general form of the result for decision-theory problems of this type.

P. M. Duffieux, “L’integrale de Fourier et ses applications en optique,” thesis, Universite de Besancon, 1946.

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Figures (14)

Fig. 1
Fig. 1

Neighborhood modification process for a 3×3 size.

Fig. 2
Fig. 2

Electronic linear filter. (a) Continuous weighting. (b) Discrete weighting.

Fig. 3
Fig. 3

Spatial linear filter.

Fig. 4
Fig. 4

Optimization criteria depiction of the effect of filter optimization. Criteria for a one-dimensional filter.

Fig. 5
Fig. 5

Computer flow chart for optimum filter using rms ratio criterion.

Fig. 6
Fig. 6

Computer flow chart for optimum filter using variance ratio criterion. Processing by subtracting average intensity from the output is also shown.

Fig. 7
Fig. 7

Computer flow chart for general statistical filter.

Equations (55)

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i c i ( s i + n i ) av / [ ( i c i n i ) 2 av ] 1 2 .
R = [ i c i ( s i + n i av ) ] 2 / ( i c i n i ) 2 av ,
R / c n = 0             n = 1 , 2 , .
Φ k = a σ .
k = a Φ - 1 σ .
R v = ( i c i s i ) 2 / i c i ( n i - φ ) 2 av
c i ( n i - φ ) 2 av = i j c i c j v i j ,
ν i j = ( n i - φ ) ( n j - φ ) av = n i n j av - φ 2 = φ i j - φ 2 .
( i j a i j σ i σ j ) 2 / ( i j a i j n i n j ) 2 av .
i j a i j φ i j k l = b σ k σ l             for             k , l = 1 , , N .
a 1 b 1 a 2 b 2 a N b N ν ( ξ ) d ξ 1 d ξ 2 d ξ N ,
a i < n i < b i             for             i = 1 , , N ,
n = ( n 1 n N ) .
a b ν ( ξ ) d ξ
ν ( ξ ) d ξ = 1
a 1 b 1 a 2 b 2 a N b N σ ( η ) d η 1 d η 2 d η N ,
a i < s i < b i             for             i = 1 , , N ,
s = ( s 1 s N ) .
f ( η ) σ ( η ) d η             and             f 2 ( ξ ) ν ( ξ ) d ξ .
R = [ f ( η ) σ ( η ) d η ] 2 / f 2 ( ξ ) ν ( ξ ) d ξ ,
φ ( ξ ) = A [ σ ( ξ ) / ν ( ξ ) ] ,
δ ( d 0 : ξ ) + δ ( d 1 : ξ ) = 1.
β = σ ( η ) δ ( d 0 : η ) d η , α = ν ( η ) δ ( d 1 : η ) d η .
T = p σ ( η ) δ ( d 0 : η ) d η + q ν ( η ) δ ( d 1 : η ) d η .
T = p + [ q ν ( η ) - p σ ( η ) ] δ ( d 1 : η ) d η .
δ ( d 1 : η ) = { 1 if q ν ( η ) - p σ ( η ) 0 , 0 if q ν ( η ) - p σ ( η ) > 0 ,
δ ( d 1 : η ) = 1 if p σ ( η ) / q ν ( η ) 1 , 0 if p σ ( η ) / q ν ( η ) < 1.
ϕ ( R ) = 1 S r I ( r ) I ( r + R ) ,
i c i φ i n = a σ n             n = 1 , 2 , ,
4 4 4 4 9 4 4 4 4
4 44 4
0 + 0 = 0 , 0 + 1 1 + 0 1 + 1 } = 1.
010 010 000 010 010 011 and 011 111 111 110 010 010 010 000 010.
x = ( x 1 x 2 x 9 ) ,
( Output at P ) = i = 1 25 c i σ i ,
( i c i n i ) 2 av = c i c j φ i j ,
[ i c i ( s i + n ¯ i ) ] 2 = ( i c i σ i ) 2 ,
( i j c i c j φ i j ) 2 σ n ( i c i σ i ) - ( i c i σ i ) 2 2 ( i c i φ i n ) = 0 ,
i c i φ i n = a σ n ,
Φ k = a σ
R ( k ) = ( a σ Φ - 1 σ ) 2 / a 2 ( σ Φ - 1 σ ) = σ Φ - 1 σ .
R ( k + h ) = ( a α + h σ ) 2 / ( a 2 α + 2 a h σ + h Φ h ) ,
R ( k + h ) = ( a α + γ ) 2 / ( a 2 α + 2 a γ + β ) .
F ( h ) = ( γ 2 - α β ) / ( α a 2 + 2 γ a + β ) .
α β - γ 2 = ( σ Φ - 1 σ ) ( h Φ h ) - ( h σ ) 2 = h ( α Φ - σ σ ) h ,
R = [ f ( ξ ) σ ( ξ ) d ξ ] 2 f 2 ( η ) ν ( η ) d η ,
R ( , δ ) = { [ φ ( ξ ) + δ ( ξ ) ] σ ( ξ ) d ξ } 2 [ φ ( η ) + δ ( η ) ] 2 ν ( η ) d η ,
( φ 2 ν ) ( φ σ ) ( δ σ ) = ( φ σ ) 2 ( φ δ ν ) .
s ( t σ - s φ ν ) δ = 0.
s ( t σ - s φ ν ) = 0 for all ξ .
φ ( ξ ) = A [ σ ( ξ ) / ν ( ξ ) ] .
ϕ ( R ) = { 20.25 , R 0 28.50 , R = 0.
010 111 010.
010 011 010.
010 000 010 010 111 111 011 and 110 000 010 010 010.