Page 558 - Sensors and Control Systems in Manufacturing

SpectRx NIR Technology
A radiometric calibration is a two-step process. First, the two 511
unknowns per detector radiometric gain K(σ) and radiometric offset
M Stray (σ), are determined using an experimental step called a charac-
terisation. Second, the calibration is applied to an un-calibrated meas-
urement to produce a calibrated spectrum.
The gain and offset characterisation requires two measurements,
one hot blackbody measurement and one cold blackbody measure-
ment, ideally both uniformly filling the field of view. This is referred
to as a two-point calibration, as shown in Fig. 10.13. The temperatures
of the calibration blackbody measurements are judiciously set in
order to minimize calibration error.
Applying Eq. (10.17) to the measurements of the hot and cold
blackbodies, we obtain the following equations:
σ
σ
σ
σ
L () = S Measured ()/ K()− M Stray () (10.37)
H
H
σ
σ
σ
σ
L () = S Measured ()/ K()− M Stray () (10.38)
C C
where L (σ) and L (σ) are the theoretically calculated spectral radi-
H C
ances using the emissivity of the blackbody and the Planck function
at the temperature of the blackbody:
C σ 3
σ
σ =
σ
σ =
L () ε () P () ε () 1 (10.39)
x x T x x ⎛ C σ ⎞
exp⎜ 2 ⎟ − 1
⎝ T x ⎠
where ε (σ) = emissivity of the x blackbody
x
x = C for the cold blackbody and H for the hot blackbody
2
–12
C = 1.191 × 10 W cm and is the first radiation constant
1
C = 1.439 K cm and is the second radiation constant
2
–1
σ = wave-number (cm )
T = temperature of the x blackbody (K)
x
The solution to Eqs. (10.18) and (10.19) yields Eqs. (10.21) and
(10.22). It is then simple to solve for the two unknowns K(σ) and
M stray (σ).
σ
σ
σ
H
K() = S Measured ()− S C Measured () (10.40)
L ()− L ()
σ
σ
H C
L ()⋅ S Measured ()− L ()⋅ S Measured (σ)
σ
σ
σ
σ
σ
M Stray () = H C C H (10.41)
−
S Measured σ () S Measured σ ()
H C
The calibrated spectrum is given by:
σ
σ
σ
σ
S Calibrated () = S Measured () K ()− M Stray () (10.42)
−1