Abstract

We investigate the problem of obtaining minimum-bias spectral estimates under the constraint of finite spectral-window bandwidth. The problem is meaningful for spectral estimation with a coherent optical system in which the power-spectrum estimate, i.e., the Fourier irradiance, is physically available for smoothing. An example that deals with the smoothing of statistically unstable optical data is furnished in connection with the problem of particle-size estimation with an optical-digital computer. We show that the smoothing of the irradiance spectrum prior to particle-size estimation furnishes more satisfactory results than operating with unsmoothed data.

© 1975 Optical Society of America

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References

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  1. H. Stark and B. Dimitriadis, J. Opt. Soc. Am. 65, 425/973E (1975).
  2. G. M. Jenkins and D. B. Watts, Spectral Analysis and Its Applications (Holden–Day, San Fransisco, 1969), p. 251.
  3. A. Papoulis, IEEE Trans. IT-19, 1 (1973).
  4. W. L. Anderson and R. E. Beissner, Appl. Opt. 10, 1503 (1971).
    [Crossref] [PubMed]
  5. Ref. 2, p. 211.

1975 (1)

H. Stark and B. Dimitriadis, J. Opt. Soc. Am. 65, 425/973E (1975).

1973 (1)

A. Papoulis, IEEE Trans. IT-19, 1 (1973).

1971 (1)

Anderson, W. L.

Beissner, R. E.

Dimitriadis, B.

H. Stark and B. Dimitriadis, J. Opt. Soc. Am. 65, 425/973E (1975).

Jenkins, G. M.

G. M. Jenkins and D. B. Watts, Spectral Analysis and Its Applications (Holden–Day, San Fransisco, 1969), p. 251.

Papoulis, A.

A. Papoulis, IEEE Trans. IT-19, 1 (1973).

Stark, H.

H. Stark and B. Dimitriadis, J. Opt. Soc. Am. 65, 425/973E (1975).

Watts, D. B.

G. M. Jenkins and D. B. Watts, Spectral Analysis and Its Applications (Holden–Day, San Fransisco, 1969), p. 251.

Appl. Opt. (1)

IEEE Trans. (1)

A. Papoulis, IEEE Trans. IT-19, 1 (1973).

J. Opt. Soc. Am. (1)

H. Stark and B. Dimitriadis, J. Opt. Soc. Am. 65, 425/973E (1975).

Other (2)

G. M. Jenkins and D. B. Watts, Spectral Analysis and Its Applications (Holden–Day, San Fransisco, 1969), p. 251.

Ref. 2, p. 211.

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Figures (8)

FIG. 1
FIG. 1

Optical-digital computer (ODC). Coherent light is filtered and expanded by a spatial filter (a) and collimated by the lens (b). The sample is held in a phase-matching liquid (c) and placed against the Fourier-transforming lens (d). The spectrum is observed in the back-focal plane xfyf. A TV camera (e) scans the spectrum and transmits the information to the computer, (g) which furnishes the least-squares estimate N ˆ.

FIG. 2
FIG. 2

Unsmoothed irradiance spectrum of a large number of circular particles of uniform radii.

FIG. 3
FIG. 3

Radial scan through the spectrum in Fig. 2.

FIG. 4
FIG. 4

Scan in Fig. 3 smoothed by the optimum window.

FIG. 5
FIG. 5

Input scene consisting of particles from three classes.

FIG. 6
FIG. 6

Unsmoothed irradiance spectrum of the input scene (Fig. 5).

FIG. 7
FIG. 7

The function (3/8K)f(x).

FIG. 8
FIG. 8

Comparison of several smoothing windows: (a) cosine, (b) cone, (c) sine, (d) uniform, (e) optimum.

Tables (1)

Tables Icon

TABLE I Summary of results.

Equations (63)

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r ( x ) = y ( x ) * i = 1 N δ ( x - x i ) ,
S ( u ) R ( u ) 2 N = Y ( u ) 2 [ 1 + n ( u ) ] ,
n ( u ) 1 N i k N N e - j 2 π u ( x i - x k ) ,
f ( z ) = 1 L ( 1 - z L ) rect z 2 L
e - j 2 π u z = sinc 2 u L .
n ( u ) = N - 1 ( N - 1 ) N sinc 2 u L N sinc 2 u L
S ( u ) = S ( u ) [ 1 + N sinc 2 u L ] .
N μ L ,
S ( u ) = S ( u ) [ 1 + μ L sinc 2 u L ] S ( u )
var [ S ( u ) ] = S ( u ) 2 n 2 ( u ) ,
n 2 ( u ) = ( 1 / N 2 ) [ N ( N - 1 ) ( N - 2 ) ( N - 3 ) sinc 4 u L + 2 N ( N - 1 ) ( N - 2 ) sinc 2 u L + 2 N ( N - 1 ) ( N - 2 ) sinc 2 u L sinc 2 2 u L + N ( N - 1 ) sinc 2 2 u L + N ( N - 1 ) ] .
n 2 ( u ) 1
var [ S ( u ) ] S ( u ) 2 1
( i ) W ( u ) = 0 ,             u > c / 2 ( ii ) - W ( u ) 2 d u = K , ( iii ) - W ( u ) d u = 1.
W ( u ) = 1 / c , u < c / 2 = 0 , otherwise .
S W ( u ) = - S ( u ) W ( u - u ) d u + - S ( u ) n ( u ) W ( u - u ) d u .
S W ( u ) = S ( u ) [ 1 + - n ( u ) W ( u - u ) d u ] .
S W ( u ) = S ( u ) [ 1 + - n ( u ) W ( u - u ) d u ] S ( u ) for L large and u 1 / L .
var [ S W ( u ) ] = S ( u ) 2 | 1 + - n ( u ) W ( u - u ) d u | 2 - S ( u ) 2 S ( u ) 2 - - n ( u ) n * ( u ) W ( u - u ) W ( u - u ) d u d u S ( u ) 2 { - - sinc 2 ( u - u ) L W ( u - u ) W ( u - u ) d u d u + - - sinc 2 ( u + u ) L W ( u - u ) W ( u - u ) d u d u } .
var [ S W ( u ) ] = S ( u ) 2 K - 1 / K 1 / K ( 1 - K x ) sinc 2 x L d x K L S ( u ) 2 .
var [ S W ( u ) ] S W ( u ) 2 var [ S W ( u 1 ) ] S ( u ) 2 K L ,
B ( u , v ) = S W ( u , v ) - S ( u , v ) ,
B ( u , v ) = B ( ω ) π 2 1 ω ω [ ω S ( ω ) ω ] 0 ρ 3 W ( ρ ) d ρ
0 ρ 3 W ( ρ ) d ρ
( i ) W ( ρ ) = 0 ,             ρ c ( ii ) 0 ρ W 2 ( ρ ) d ρ = K , ( iii ) 2 π 0 ρ W ( ρ ) d ρ = 1.
W 0 ( ρ ) = 3 K π ( 1 - 3 2 π 2 K ρ 2 ) ,             ρ < c 0 = 0 ,             ρ c 0 ,
c 0 = 1 π ( 2 3 K ) 1 / 2 .
D 2 π 0 ρ 3 W ( ρ ) d ρ ,
H ( ω ) = i = 1 L N i G i ( ω ) + n ( ω ) ,
H ( ω ) = H ( ω ) = i = 1 L N i G i ( ω ) ,
H ( ω 1 ) = i = 1 L N i G i ( ω 1 ) , H ( ω m ) = i = 1 L N i G i ( ω m ) ,
H = G N .
N LS = [ G T G ] - 1 G T H .
H W ( l , n ) = i k H ( i , k ) W 0 ( l - i , n - k ) ,
H W = W 0 H ,
H = G N + n ,
H W = W 0 G N + W 0 n .
N ˆ LS = [ G T G ] - 1 G T [ W 0 G N + W 0 n ] .
N ˆ LS [ G T G ] - 1 G T W 0 G N .
G ( ω ) = G ( ω ) + n G ( ω ) ,
G W = W 0 G = W 0 G + W 0 n G W 0 G ,
N ˆ LS [ ( W 0 G ) T W 0 G ] - 1 [ W 0 G ] T [ W 0 G N + W 0 n ] .
N ˆ LS N
W 0 ( ρ ) = 3 K π ( 1 - 3 2 π 2 K ρ 2 ) ,             ρ < c 0 = 0 ,             ρ c 0 ,
c 0 = 1 π ( 2 3 K ) 1 / 2 .
h W - h ρ ( h W ) = 0 ,
W 0 ( ρ ) = 3 K π ( 1 - 3 2 π 2 K ρ 2 ) ,             ρ < c 0 = 0 ,             ρ c 0 .
D ( W ) = 2 π 0 ρ 3 W ( ρ ) d ρ .
W ( ρ ) = W 0 ( ρ ) + δ W ,
Δ D ( W ) - D ( W 0 ) .
Δ = 2 π 0 c ρ 3 δ W ( ρ ) d ρ .
2 π 0 c ρ W ( ρ ) d ρ = 1
0 c ρ δ W ( ρ ) d ρ = 0.
0 c ρ W 2 ( ρ ) d ρ = K ,
0 c ρ 3 δ W ( ρ ) d ρ = ( 9 K 2 π 3 ) - 1 0 c ρ ( δ W ) 2 d ρ 0.
( 9 K π 3 ) - 1 + 3 4 K π c 4 - 3 8 K 2 π 3 c 6 - c 2 / 2 π + ( 9 K 2 π 3 ) - 1 0 c ρ [ δ W ( ρ ) ] 2 d ρ 0.
3 8 K 2 π 3 c 6 - 3 4 K π c 4 + c 2 / 2 π > ( 9 K π 3 ) - 1 .
f ( x ) x 2 ( x 4 - ( 2 / π ) x 2 + 4 / 3 π 2 ) ,
( 3 / 8 K ) f ( x ) > ( 9 K π 3 ) - 1 .
W ( ρ ) = 2 K π circ ( ρ c ) , c = 1 π 1 2 K 0.23 K ,             D 0.25 K π 2 .
W ( ρ ) = 4 π K ( 1 - 2 π 1 3 K ρ ) ,             ρ < c = 0 ,             otherwise , c = 1 2 π 3 K 0.28 K ,             D 0.23 K π 2 .
W ( ρ ) = 16 K π π + 2 cos ( 4 π K ( π - 2 ) π + 2 ρ ) ,             ρ c = 0 ,             otherwise , c = 1 8 π + 2 K ( π - 2 ) 0.27 K ,             D 0.22 + K π 2 > D 0 .
W ( ρ ) = 4 π ( π 2 - 8 ) 3 π 2 - 28 K ( 1 - sin { π ( π 2 - 8 ) K 3 π 2 - 28 ρ } ) ,             ρ < c = 0 ,             otherwise , c = 1 2 ( π 2 - 8 ) 3 π 2 - 28 K 0.34 K ,             D 0.24 K π 2 .