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5. 3D POLYNOMIAL TRANSFORMATION
x x > : o gf 3 z
General form of a polynomial of r^ order with three variables
can be given as follows.
n n n
2 , ba =k
en S au yir
j=1 jalk=l (1)
i+j+ksn
where,
az: Coefficients x,y,z : Coordinates
n : Order of polynomial
A 3D polynomial transformation for scanner calibration can be
defined using three polynomials as follows.
R “AR GR)
G - f (Rs. Gs. Bs) (2)
B = £,(Rs. Gs, Bs)
where,
R.G.B :calibrated values (device independent color space)
Rs Gs Bs : scanned values (device dependent color space)
R,G,B values can be calculated from the CIE XYZ color space
values of the calibration card. Rs, Gs Bs values are obtained by
scanning the calibration card. In case of Q-60 card there are 240
colors. This makes possible to determine the coefficients of
polynomials in (2) by regression. If we select polynomials of
Ist order in (2), the coefficients can be determined as follows.
The polynomials:
R= f, [Gn ay +a Rs + a,Gg + ay Bg
G = f (Rs. Gs, Bs )= ba + bRs +5,Gs +b:B; 3)
B=f (ROS Iis) cy + Rs 6565 t 6B,
If there are m known colors in device color space (scanned RGB
values) and device independent color space (e.g. RGB values
calculated from CIE XYZ values), we have 3*m equations
(3*240=720 equations).
Because the number of known color values are more than
required, the coefficients can be determined with least square
adjustment. After adjustment the RMS (root mean square error)
is also obtained.
R, EUR, GQ A00 909. 009 0 9779
a, 0.500 1 Gu 0 059 094 dv
B, OWL o9 s dol UO S Ph
L=| : |4= 8 À Pg
R, Bap Gi Baia sce 0 8 a0 8 asin Jr fa
6, 9.25507 1871 ouioGuodls iod 9 19 &%
B 00 0 OA OO RO 200]
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004
E (4)
The relationship between the number of coefficients (g) and the
order of polynomial (1) can be expressed as follows.
HET) 5)
k=1
Table 1 shows the number of coefficients according to order of
polynomial.
Order of 3D polynomial
Number of ! 2 3 4 3 n
coefficients
12 30 60 105 | 168 q
Table 1. The number of coefficients
In order to find the most suitable polynomial for a certain
device, starting from the Ist order several polynomials have to
be determined with least square adjustment. The polynomial
with the smallest RMS can be considered to be the most
suitable one.
6. APPLICATION
The aim of the application is to find answers to following
questions. Are 3D polynomials applicable for color calibration?
If yes, what kind of polynomial is the most suitable for this
purpose? Polynomial transformations are applied between
device dependent color space (scanned color values) and device
independent color space (original color values of the test card).
In the application Kodak's IT8 7.2-1993 2001:02 calibration
card is used.
The original color space for the test card is the CIE XYZ color
space. Prior to transformation mentioned above a device
independent RGB color space is required. It is obtained from
CIE XYZ color space by using D65 standard illuminant and 2°
standard observer (ITU-R BT.709-2).
The device color space is constructed by scanning the
calibration card. 228 color parts of the card (A1-L19) are used.
For each part 100 pixels are selected, and the average of them is
calculated. With these average color values 228 colors are
determined. If we consider these colors like points in
geometrical sense, they are control points for 3D
transformation. 228 control points make easily possible to
realize polynomial regression until 4% order.
The scanners selected for the application were flat bed desktop
publishing scanners. Their models and specifications are given
in Table 2. Color calibration card was scanned at the optical
resolution of each scanner. No improvements of the scanned
image were undertaken. The tests were made at the same
conditions (illumination, room temperature, etc.).
Inte
The
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Poly
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R =
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