Specifications
Submitted to Studies in Conservation, March 2006
6
based on the Wyszecki hypothesis that a stimulus can be decomposed into a fundamental
stimulus and a metameric black [13]. Our approach was similar to the Fairman technique of
transforming a parameric pair into a metameric pair [14]. Conceptually, the fundamental
stimulus corresponds to the spectral information that our visual system processes. The metameric
black corresponds to the spectral information that is not processed; hence it has no color and it is
black. The metameric black defines the spectral characteristics that depend on the specific
colorants used to provide selective absorption. Accurate estimation of a fundamental stimulus
results in high colorimetric accuracy. Accurate estimation of a metameric black results in high
spectral accuracy.
Previously [12], the colorimetric transformation consisted of a nonlinear stage that
accounted for stray light, differences in geometry between the reference spectrophotometer and
camera system, and any non-linearities in the camera signal processing. Experimentally, the
transfer functions were nearly linear with a small offset. This allowed the transformation to be
simplified as shown in Eqs. (1) – (5):
M
pinv
= R
Reference
pinv D
Reference
( )
(1)
A =
100
!
S y
diag S
( )
xyz
(2)
T
!E
00
= f
NonLinOpt
(R
Reference
,A, D
Reference
), where T
!E
00
minimizes !E
00
(T
!E
00
D,
"
A R
Reference
)
#
$
%
&
(3)
M
!E
00
= A
"
A A
( )
#1
T
!E
00
+ I # A
"
A A
( )
#1
"
A
( )
M
pinv
(4)
ˆ
R
!E
00
= M
!E
00
D
(5)
where n is the number of wavelengths, i is the number of camera channels, and j is the number of
reference color patches. Matrix M
pinv
is a [n × (i+1)] transformation matrix from digital counts to
spectral reflectance factor computed from R
Reference
, a [n × j] matrix containing the calibration
target reference spectrophotometric measurements ranging from zero to unity and D
Reference
is an
[(i+1) × j] camera digital count matrix with the last row set to unity (homogenous coordinates).
The operation pinv represents the Moore-Penrose singular-value decomposition-based
pseudoinverse function in Matlab [6]. Matrix A is a [n × 3] matrix of tristimulus weights