Abstract

A least-squares approach to the planimetric analysis of color scenes is furnished. The technique involves parallel processing in that all image points are involved simultaneously at every step of the processing. It is assumed that each object in the color scene has a unique spectral (color) signature. The end result of the processing is a vector whose components are the areas covered by the different objects in the scene. The technique requires only a simple optical system and can be easily automated. Procedures for finding the signatures that minimize the error variance are investigated. The theory is illustrated with a laboratory example.

© 1975 Optical Society of America

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References

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  1. K. S. Fu, D. A. Landgrebe, and T. L. Phillips, IEEE Proc. 57, 639 (1969).
    [CrossRef]
  2. H. Stark, R. C. Barker, and D. Lee, Appl. Opt. 11, 2540 (1972).
    [CrossRef] [PubMed]
  3. M. R. Holter, in Remote Sensing (National Academy of Sciences, Washington, D. C., 1970), Ch. 3.
  4. As one reviewer pointed out, Eq. (1) must be modified in the case of high-altitude imagery such as is obtained with ERTS observation satellites. In that case, the modification takes the formWT(λ)=S(λ)ρ(λ)τ(λ)+WA(λ),where WT(λ) is the total irradiance and WA(λ) is a component of irradiance scattered into the sensor by the atmosphere between the sensor and the subject. Techniques for obtaining WA(λ) are furnished by R. H. Rogers and K. Peacock in Vol. I of Symposium on Significant Results Obtained from ERTS-1, NASA SP-327, March1973, pp. 1115–1122. Provided that WA(λ) is relatively free from statistical fluctuations, we can write W(λ) = WT(λ) − WA(λ) = S(λ) ρ(λ) τ(λ) and apply the theory developed in this paper. If WA(λ) contains strong statistical fluctuations, then the average bias can be subtracted from WT(λ) and the zero-mean fluctuations can be lumped with the other zero-mean fluctuations in W(λ). Processing of ERTS-type data must take into account the seasonal variations in spectral reflectances, and variations in atmospheric and illuminating conditions.
  5. R. Deutsch, Estimation Theory (Prentice–Hall, Englewood Cliffs, N. J., 1965), p. 59.
  6. Reference 5, p. 62.
  7. R. A. Holmes and R. B. McDonald, IEEE Proc. 57, 629 (1969).
    [CrossRef]

1972 (1)

1969 (2)

K. S. Fu, D. A. Landgrebe, and T. L. Phillips, IEEE Proc. 57, 639 (1969).
[CrossRef]

R. A. Holmes and R. B. McDonald, IEEE Proc. 57, 629 (1969).
[CrossRef]

Barker, R. C.

Deutsch, R.

R. Deutsch, Estimation Theory (Prentice–Hall, Englewood Cliffs, N. J., 1965), p. 59.

Fu, K. S.

K. S. Fu, D. A. Landgrebe, and T. L. Phillips, IEEE Proc. 57, 639 (1969).
[CrossRef]

Holmes, R. A.

R. A. Holmes and R. B. McDonald, IEEE Proc. 57, 629 (1969).
[CrossRef]

Holter, M. R.

M. R. Holter, in Remote Sensing (National Academy of Sciences, Washington, D. C., 1970), Ch. 3.

Landgrebe, D. A.

K. S. Fu, D. A. Landgrebe, and T. L. Phillips, IEEE Proc. 57, 639 (1969).
[CrossRef]

Lee, D.

McDonald, R. B.

R. A. Holmes and R. B. McDonald, IEEE Proc. 57, 629 (1969).
[CrossRef]

Peacock, K.

As one reviewer pointed out, Eq. (1) must be modified in the case of high-altitude imagery such as is obtained with ERTS observation satellites. In that case, the modification takes the formWT(λ)=S(λ)ρ(λ)τ(λ)+WA(λ),where WT(λ) is the total irradiance and WA(λ) is a component of irradiance scattered into the sensor by the atmosphere between the sensor and the subject. Techniques for obtaining WA(λ) are furnished by R. H. Rogers and K. Peacock in Vol. I of Symposium on Significant Results Obtained from ERTS-1, NASA SP-327, March1973, pp. 1115–1122. Provided that WA(λ) is relatively free from statistical fluctuations, we can write W(λ) = WT(λ) − WA(λ) = S(λ) ρ(λ) τ(λ) and apply the theory developed in this paper. If WA(λ) contains strong statistical fluctuations, then the average bias can be subtracted from WT(λ) and the zero-mean fluctuations can be lumped with the other zero-mean fluctuations in W(λ). Processing of ERTS-type data must take into account the seasonal variations in spectral reflectances, and variations in atmospheric and illuminating conditions.

Phillips, T. L.

K. S. Fu, D. A. Landgrebe, and T. L. Phillips, IEEE Proc. 57, 639 (1969).
[CrossRef]

Rogers, R. H.

As one reviewer pointed out, Eq. (1) must be modified in the case of high-altitude imagery such as is obtained with ERTS observation satellites. In that case, the modification takes the formWT(λ)=S(λ)ρ(λ)τ(λ)+WA(λ),where WT(λ) is the total irradiance and WA(λ) is a component of irradiance scattered into the sensor by the atmosphere between the sensor and the subject. Techniques for obtaining WA(λ) are furnished by R. H. Rogers and K. Peacock in Vol. I of Symposium on Significant Results Obtained from ERTS-1, NASA SP-327, March1973, pp. 1115–1122. Provided that WA(λ) is relatively free from statistical fluctuations, we can write W(λ) = WT(λ) − WA(λ) = S(λ) ρ(λ) τ(λ) and apply the theory developed in this paper. If WA(λ) contains strong statistical fluctuations, then the average bias can be subtracted from WT(λ) and the zero-mean fluctuations can be lumped with the other zero-mean fluctuations in W(λ). Processing of ERTS-type data must take into account the seasonal variations in spectral reflectances, and variations in atmospheric and illuminating conditions.

Stark, H.

Appl. Opt. (1)

IEEE Proc. (2)

R. A. Holmes and R. B. McDonald, IEEE Proc. 57, 629 (1969).
[CrossRef]

K. S. Fu, D. A. Landgrebe, and T. L. Phillips, IEEE Proc. 57, 639 (1969).
[CrossRef]

Other (4)

M. R. Holter, in Remote Sensing (National Academy of Sciences, Washington, D. C., 1970), Ch. 3.

As one reviewer pointed out, Eq. (1) must be modified in the case of high-altitude imagery such as is obtained with ERTS observation satellites. In that case, the modification takes the formWT(λ)=S(λ)ρ(λ)τ(λ)+WA(λ),where WT(λ) is the total irradiance and WA(λ) is a component of irradiance scattered into the sensor by the atmosphere between the sensor and the subject. Techniques for obtaining WA(λ) are furnished by R. H. Rogers and K. Peacock in Vol. I of Symposium on Significant Results Obtained from ERTS-1, NASA SP-327, March1973, pp. 1115–1122. Provided that WA(λ) is relatively free from statistical fluctuations, we can write W(λ) = WT(λ) − WA(λ) = S(λ) ρ(λ) τ(λ) and apply the theory developed in this paper. If WA(λ) contains strong statistical fluctuations, then the average bias can be subtracted from WT(λ) and the zero-mean fluctuations can be lumped with the other zero-mean fluctuations in W(λ). Processing of ERTS-type data must take into account the seasonal variations in spectral reflectances, and variations in atmospheric and illuminating conditions.

R. Deutsch, Estimation Theory (Prentice–Hall, Englewood Cliffs, N. J., 1965), p. 59.

Reference 5, p. 62.

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Figures (5)

FIG. 1
FIG. 1

Optical-digital configuration for planimetric analysis of color images.

FIG. 2
FIG. 2

(a) Laboratory-prepared color scene shown in black and white. (b) Designation and location of colors in the scene.

FIG. 3
FIG. 3

Spectral signatures of the five objects in the scene in Fig. 2.

FIG. 4
FIG. 4

Sample-averaged squared errors for LS and approximate LMSE estimates. The dashed lines represent the LS estimates and the solid lines the approximate LMSE estimates.

FIG. 5
FIG. 5

Sample-averaged squared error vs ϕi*.

Tables (4)

Tables Icon

TABLE I Identification of numerically designated filters.

Tables Icon

TABLE II Comparison of actual area with least-squares area estimate for the five objects in the scene.

Tables Icon

TABLE III Identification of groups and maximum number of estimates within group over which averaging was done.

Tables Icon

TABLE IV Optimum filter selections according to minimax criterion and their performance as indicated by LS and MV squared errors.

Equations (41)

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W ( λ ) = S ( λ ) ρ ( λ ) τ ( λ ) ,
D j = i = 1 N W i j K i ,
D = W K ,
K = ( W T W ) 1 W T D .
R LS = D W K ˆ 2 ,
K ˆ LS = ( W T W ) 1 W T D ,
D = W K + v .
R = E [ K K ˆ 2 ] ,
K ˆ MV = ( W T Λ 1 W ) 1 W T Λ 1 D ,
W ˜ i = W i + Δ i ,
Δ i = ( δ i 1 , , δ i L ) T
D = W ˜ K = W K + v ,
Λ = E [ v v T ] = E [ Δ K K T Δ T ]
σ 2 = E [ ( K K ˆ ) T ( K K ˆ ) ] .
σ LS , MV 2 = i j x i j E ( υ i υ j ) = E ( v T X v ) ,
X = X LS = W ( ( W T W ) 1 ) T ( W T W ) 1 W T ,
X = X MV = ( Λ 1 ) T W ( ( W T Λ 1 W ) 1 ) T ( W T Λ 1 W ) 1 W T Λ 1 .
σ LS , MV 2 = K T E [ Δ T X Δ ] K ,
{ E [ Δ T X LS Δ ] } i j = k = 1 L l = 1 L x l k δ i L δ j k
{ E [ Δ T X LS Δ ] } i i = k = 1 L l = 1 L x l k δ i l δ i k
σ LS 2 = i = 1 N K i 2 ( k = 1 L l = 1 L x l k δ i l δ i k ) .
σ LS 2 = L = 1 N K i 2 ( k = 1 L l = 1 L x l k c l k ) .
k = 1 L l = 1 L x l k c l k .
σ LS 2 = L = 1 N K i 2 c i ( Tr X LS ) .
σ LS 2 = i = 1 N K i 2 ϕ i ,
ϕ i k = 1 L l = 1 L x l k δ i l δ i k .
( σ LS 2 ) max = ϕ i * ,
ϕ i * = max i ϕ i ,
ϕ MM = min W ϕ i * ,
R = E [ K K ˆ 2 ]
E ( K K ˆ ) = 0 .
K ˆ = M D
E ( K K ˆ ) = ( I M W ) K = 0 ,
R A = E K - K ˆ 2 + λ T ( I M W ) K ,
R M = 0 = 2 M V V T λ K T W T .
K = ( W T Λ 1 W ) 1 W T Λ 1 D ,
i = 1 N K i = 1 , K i 0 , i = 1 , , N .
ϕ i * = max i ϕ i ϕ i , i = 1 , , N .
σ LS 2 = K i 2 ϕ i ϕ i * K i 2 ϕ i * ( K i ) 2 = ϕ i * .
σ LS 2 = i = 1 N K i 2 ϕ i = K i * 2 ϕ i * = ϕ i * = ( σ LS 2 ) max .
WT(λ)=S(λ)ρ(λ)τ(λ)+WA(λ),