Abstract

A technique is presented for estimating spectral reflectance curves from multispectral image data even if the spectral samples are obtained from channels whose spectral responsivity is not narrowband. It is demonstrated that these reflectance estimates can be written as a linear combination of the spectral samples and that, analogous to Shannon’s sampling theorem, if the spectral reflectance is a natural cubic spline, it can be estimated exactly provided the number of spectral channels is sufficiently large. Simulation results suggest that the accuracy of the spectral reflectance estimates is quite good and very insensitive to the spectral responsivity shapes.

© 1977 Optical Society of America

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References

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  1. F. O. Huck et al., Space Sci. Instrum. 1, 189 (May1975).
  2. F. O. Huck, S. D. Wall, Appl. Opt. 151748 (1976).
    [CrossRef] [PubMed]
  3. W. K. Pratt, C. E. Mancill, Appl. Opt. 15, 73 (1976).
    [CrossRef] [PubMed]
  4. P. M. Prenter, Splines and Variational Methods (Wiley, New York, 1975).
  5. E. D. Nering, Linear Algebra and Matrix Theory (Wiley, New York, 1970).
  6. C. Shannon, Bel J. l Syst. Tech. 27, 379 (1948).
  7. S. K. Park, F. O. Huck, “A Spectral Reflectance Estimation Technique Using Multispectral Data from the Viking Lander Camera,” NASA TN D -8292 (1976).

1976

1975

F. O. Huck et al., Space Sci. Instrum. 1, 189 (May1975).

1948

C. Shannon, Bel J. l Syst. Tech. 27, 379 (1948).

Huck, F. O.

F. O. Huck, S. D. Wall, Appl. Opt. 151748 (1976).
[CrossRef] [PubMed]

F. O. Huck et al., Space Sci. Instrum. 1, 189 (May1975).

S. K. Park, F. O. Huck, “A Spectral Reflectance Estimation Technique Using Multispectral Data from the Viking Lander Camera,” NASA TN D -8292 (1976).

Mancill, C. E.

Nering, E. D.

E. D. Nering, Linear Algebra and Matrix Theory (Wiley, New York, 1970).

Park, S. K.

S. K. Park, F. O. Huck, “A Spectral Reflectance Estimation Technique Using Multispectral Data from the Viking Lander Camera,” NASA TN D -8292 (1976).

Pratt, W. K.

Prenter, P. M.

P. M. Prenter, Splines and Variational Methods (Wiley, New York, 1975).

Shannon, C.

C. Shannon, Bel J. l Syst. Tech. 27, 379 (1948).

Wall, S. D.

Appl. Opt.

Bel J. l Syst. Tech.

C. Shannon, Bel J. l Syst. Tech. 27, 379 (1948).

Space Sci. Instrum.

F. O. Huck et al., Space Sci. Instrum. 1, 189 (May1975).

Other

S. K. Park, F. O. Huck, “A Spectral Reflectance Estimation Technique Using Multispectral Data from the Viking Lander Camera,” NASA TN D -8292 (1976).

P. M. Prenter, Splines and Variational Methods (Wiley, New York, 1975).

E. D. Nering, Linear Algebra and Matrix Theory (Wiley, New York, 1970).

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Figures (8)

Fig. 1
Fig. 1

A system transfer function which is not adequately approximated by an impulse.

Fig. 2
Fig. 2

System transfer functions.

Fig. 3
Fig. 3

System characteristic functions.

Fig. 4
Fig. 4

The relative standard deviation of the reflectance estimate.

Fig. 5
Fig. 5

Actual (○) and estimated (—) reflectances of four geologic materials.

Fig. 6
Fig. 6

System characteristic functions for the idealized system.

Fig. 7
Fig. 7

The relative standard deviation of the reflectance estimate for the idealized system.

Fig. 8
Fig. 8

Actual (○) and estimated (—) reflectances of four geologic materials for the idealized system.

Equations (47)

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N ( λ ) = ( 1 / π ) S ( λ ) τ a ( λ ) ρ ( λ ) ϕ ( ι , , g ) ,
V i = A i B i k i 0 N ( λ ) τ c ( λ ) R i ( λ ) d λ ,             i = 1 , 2 , , m ,
V i = c i 0 T i ( λ ) ρ ( λ ) d λ ,             i = 1 , 2 , , m ,
c i = ( 1 / π ) A i B i k i t i ϕ ( ι , , g )             i = 1 , 2 , , m ,
T i ( λ ) = 1 t i S ( λ ) τ a ( λ ) τ c ( λ ) R i ( λ )             i = 1 , 2 , , m .
0 T i ( λ ) d λ = 1.
b i = 0 T i ( λ ) ρ ( λ ) d λ             i = 1 , 2 , , m .
b i = ( V i ) / ( c i )             i = 1 , 2 , , m .
b i = 0 δ ( λ - λ i ) ρ ( λ ) d λ = ρ ( λ i )             i = 1 , 2 , , m .
b i = ρ ( λ i ) + ρ ( λ i ) Λ i T i ( λ ) ( λ - λ i ) d λ + 1 2 ρ ( λ i ) Λ i T i ( λ ) ( λ - λ i ) 2 d λ ,
λ i = Λ i T i ( λ ) λ d λ .
b i = ρ ( λ i ) + 1 2 ρ ( λ i ) Λ i T i ( λ ) ( λ - λ i ) 2 d λ ,
ρ ( λ ) = j = 1 n x j h j ( λ )
( λ ) = ρ ( λ ) - ρ ( λ ) .
e i = b i - j = 1 n a i j x j ,             i = 1 , 2 , , m ,
a i j = 0 T i ( λ ) h j ( λ ) d λ ,
e i = 0 T i ( λ ) ( λ ) d λ .
1 m i = 1 m e i 2 .
b i = j = 1 m a i j x j             i = 1 , 2 , , m .
λ ¯ j = λ ¯ 1 + ( j - 1 ) Δ             j = 2 , 3 , , m ,
h j ( λ ) = C ( λ - λ ¯ j )             j = 1 , 2 , , m ,
C ( λ ) = { 1 6 Δ 3 [ Δ 3 + 3 Δ 2 ( Δ - λ ) + 3 Δ ( Δ - λ ) 2 - 3 ( Δ - λ ) 3 ]             λ Δ 1 6 Δ 3 ( 2 Δ - λ ) 3             Δ < λ < 2 Δ 0             λ 2 Δ .
ρ ( λ ) = j = 1 m x j C ( λ - λ ¯ j )
a i j = 0 T i ( λ ) C ( λ - λ ¯ j ) d λ ,
ρ ( λ ) = j = 0 m + 1 x j C ( λ - λ ¯ j ) .
b i = j = 0 m + 1 a i j x j             i = 1 , 2 , , m ,
x 0 - 2 x 1 + x 2 = 0
x m - 1 - 2 x m + x m + 1 = 0.
A = [ 1 - 2 1 0 0 0 0 0 a 1 , 0 a 1 , m + 1 A a m , 0 a m , m + 1 0 0 0 0 0 1 - 2 1 ] .
Ax = b ,
ρ ( λ ) = i = 1 m b i f i ( λ ) ,
ρ ( λ ) = x T h ( λ ) ,
x = A - 1 b ,
ρ ( λ ) = b T ( A - 1 ) T h ( λ ) .
f ( λ ) = ( A - 1 ) T h ( λ ) .
ρ ( λ ) = b T f ( λ ) = i = 1 m b i f i ( λ ) ,
1 = i = 1 m f i ( λ )             λ ¯ 1 λ λ ¯ m .
s ( λ ) = j = 0 n + 1 s j C ( λ - λ ¯ j ) = s T h ( λ ) ,
b i = 0 T i ( λ ) s ( λ ) d λ             ( i = 1 , 2 , , m = n ) = j = 0 n + 1 s j [ 0 T i ( λ ) C ( λ - λ ¯ j ) d λ ] = j = 0 n + 1 a i j s j .
s ( λ ) = ( As ) T ( A - 1 ) T h ( λ ) = s T h ( λ ) = s ( λ ) .
ρ ( λ ) ¯ = i = 1 m b ¯ i f i ( λ ) ,
σ ρ 2 ( λ ) = i = 1 m σ i 2 f i 2 ( λ ) .
σ ρ 2 ( λ ) = σ 2 F 2 ( λ ) ,
F 2 ( λ ) = i = 1 m f i 2 ( λ ) .
λ ¯ j = 0.33 + j Δ             j = 0 , 1 , 2 , , 7 ,
b i = ρ ( λ ¯ i )             i = 1 , 2 , , 6
a i j = C ( λ ¯ i - λ ¯ j ) = { j = i j = i - 1 , i + 1 0 otherwise .

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