Page 560 - Sensors and Control Systems in Manufacturing

SpectRx NIR Technology
apply Eq. (10.17) to the measurements of multiple blackbodies, we 513
get the following system of equations:
σ
σ
σ
L () = S Measured ()/ K()− M Stray ()
σ
1 1
σ
σ
σ
−
σ
K
L () = S 2 Measured ()/ () M Stray () (10.43)
2
M
σ
−
σ
=
σ
σ
K
L () S Measured ()/ ()− M Stray ()
n n
The preceding equations form a system of linear equations. The
general solution to such a system for n equations is as follows:
⎛
n∑ Measured () − ∑ Measured σ ⎞ 2
σ
2
S
i ⎜ i ⎝ S i () ⎟ ⎠
σ
i
K() = n∑ Measured σ L ()−∑ Measured ()∑ σ (10.44)
σ
σ
S
i () i S i L ()
i
i i i
(
)
n∑ S Measured ()∑ S Measured () L () − ∑ S Measured () ∑ L () ∑ L ()
2
σ
σ
σ
σ
σ
σ
i i i i i i
σ
M Stray () = i i i i − i
(
n ∑ S Measured () − ∑ S Meeasured () ) 2 n
σ
σ
2
2
n
i i
i i
(10.45)
Eqs. (10.25) and (10.26) can be used to solve for systems of equa-
tions in the form of Eq. (10.24). In the case illustrated in Fig. 10.14, n
equals 5 and i runs from 1 to 5. The calibrated spectrum is then given
by Eq. (10.23) as before.
10.9.4 Nonlinear Multiple Point Calibrations
Nonlinear multiple point calibrations are used to correct for nonlinear-
ity in the response function of the FT spectroradiometer, Fig. 10.15.
Scene Spectral Radiance 2 3 4 5 6
1
Spectral Power at Detector
2 point calibration using 1 and 6
Multiple point linear calibration
Multiple point quadratic calibration
FIGURE 10.15 The nonlinear relationship between the object view radiance and the
detector response.