Color Scanner Characterization

Mathematically, the recording process of a scanner
can be expressed as
ci = H(MT ri)
where the matrix M contains the spectral sensitiv-
ity (including the scanner illuminant) of the three
(or more) bands of the scanner, ri is the spectral re-

ectance at spatial point i, H models any nonlineari-
ties in the scanner (invertible in the range of interest),
and ci is the vector of recorded values.
The characterization problem is to determine the
continuous mapping Fscan which will transform the
recorded values to a CIE color space. In other words,
determine the function Fscan such that
t = ATLr = Fscan(c)
for all r 2
r , where
r is the set of physically re-
alizable re
ectance spectra, the columns of matrix A
contain the CIE XYZ color matching functions, and
the diagonal matrix L represents the viewing illumi-
nation.
For the non-colorimetric scanner, there will exist
spectral re
ectances which look dierent to the stan-
dard human observer but when scanned produce the
same recorded values. These colors are dened as be-
ing metameric to the scanner. Likewise, there will
exist spectral re
ectances which give dierent scan
values and look the same to the standard human ob-
server. While the latter can be corrected by the trans-
formation Fscan, the former cannot.
On the upside, there will always (except for de-
generate cases) exist a set of re
ectance spectra over
which a transformation from scan values to CIE XYZ
values will exist.
Printed images, photographs, etc. are all produced
with a limited set of colorants. Re
ectance spectra
from such processes have been well modeled with very
few (3-5) principal component vectors [1, 2, 3, 4].

When limited to such data sets it may be possible
to determine a transformation Fscan such that
t = ATLr = Fscan(c)
for all r 2 Bscan where Bscan is the subset of re-

ectance spectra to be scanned.
Look-up-tables, nonlinear and linear models for
Fscan have been used to calibrate color scanners
[5, 6, 7, 8]. In all of these approaches, the rst step
is to select a collection of color patches which span
the colors of interest. Since the particular samples se-
lected determine the characteristics of the mapping,
the scanner characterization is usually identied with
respect to the process which produced the samples.
Ideally these colors should not be metameric in terms
of the scanner sensitivities or to the standard observer
under the illuminant for which the characterization is
being produced. This constraint assures a one-to-one
mapping between the scan values and the device inde-
pendent values across these samples. In practice, this
constraint is easily obtained. The re
ectance spectra
of these Mq color patches will be denoted by fqgk for
1  k  Mq .
These patches are measured using a spectropho-
tometer or a colorimeter which will provide the device
independent values
ftk = ATqkg for 1  k  Mq:
Without loss of generality, ftkg could replaced with
any colorimetric or device independent values, e.g.
CIELAB, CIELUV. The patches are also measured
with the scanner to be calibrated providing fck =
H(MT qk)g for 1  k  Mq .
Mathematically, the characterization problem is:
nd a transformation Fscan where
Fscan = arg(min
F
Mq
X
i=1
jjF(ci)􀀀L(ti)jj2)
where L(
) is the transformation from CIEXYZ to the
appropriate color space and jj:jj2 is the error metric in
the color space.