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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004 
  
  
  
  
  
  
  
  
  
  
  
  
  
. Scanning | Color | Gray Optical as 
Scanner ; resolution | Size 
method depth | depth ; 
(dpi) 
um single 
ScanJet | 2 24 bit | 8bit 300 A4 
sn pass 
Mustek single ; . 
ce 2 9 ^ 
A3 Sp pass 36 bit } 12 bit 300 A3 
Acer single 
Prisa Si 30 bit | 8 bit 300 A4 
310p pass 
hp single 
ScanJet = 48 bit | 16 bit 2400 A4 
5470c pars 
Mussk ] sse | bitlos bi 600 A4 
Sp Plus pass 3 
  
  
Table 2. Scanner models and specifications 
The criterion for determining the best fitting polynomial for 
each scanner is the RMS error obtained after least square 
adjustment. The polynomial transformation with smallest RMS 
error can be considered as the best fitting function that maps the 
device dependent color space to device independent color 
space. For each scanner in Table 2, bilinear, 2™, 3™ and 4" 
order polynomials are determined by least square adjustment. 
General equations of these polynomials are as follows. 
Bilinear polynomial with 21 coefficients: 
R=a,+a,Rs +a;Gg +a, Bg +asRgGg +agRg Bs +a,G Bs 
G=b +b,Rs +b,Gs +b, Bs +b; RG +b Rg Bg +b,G Bs (6) 
BzctcoR.tOGg FC Bg FK C,R,GS Feo Rs Bs t C; GS B, 
Polynomial of 2™ order with 30 coefficients: 
R=a,+a,Rs +a,Gs +a,Bg +asR; tag Rs Gs 
*ta,Gita,R;Bg + a9G5Bs + a, B2 
G=b +b,Rs +b,Gg +b, Bg +bR3 +b RG mn 
+5,G5 +b,R5 B5 + b,Gs B5 + 5,0 82 
B-c *cRs 6G, tc Bg Fe, Ri 4 e, R.G, 
+2 ^ 2 
+C,G5 *Cg RS Ds. 4 C4 GS B, + co Bs 
Polynomial of 3™ order with 60 coefficients: 
R = a, + a,R, + a,Gs + a,Bs + asR$ da ROG. 
* 4G] + a, Ro By + 0, By +, BE HRS 
Pa RE + 0 RG + AGs Fo RBs 
tas RGB, + GR, + ay Ra: 
+000, 87 + eB 
G =b + byRs + b,Gy + b,Bg + bsRg + byRsG + b,G2 +... 
(8) 
> 2 ~ 2 
B=c +c,R, *OUs te Bs Fes Rs reU RSS tr eG 
Polynomial of 4" order with 105 coefficients: 
Rz-za ta,R.taGg t a,B. *as R2 tag RS Gg + a, 
tGygRSBS Fa GS Bg au Bi ra Ria R2G, 
tag RSG cau Gi cas RÍB; ra RS0,B, 4a, 62B, 
+a Rs BS tay Gs B; +ay By +a RS +a, RIG 
VG ras RAO asi Va RB. ra REG BR: (9) 
tag RSGLBs cag Gi Bg ta R2B; + a5 RG BE 
+ GEBE tay Rs B: tau Gs B; tas B; 
G=b+b,Rs +b,G4 +b,Bs +bsRI +b, RG +... 
2 
B=c +C3Rs +¢3Gs +04 Bg +05 RS +c RG +... 
The RMS errors after least square adjustment are shown in 
Table 3. It is evident from Table 3 that the RMS errors of 4" 
order polynomials are the smallest ones for each scanner. 
Higher order polynomials cannot easily be determined because 
the coefficients matrix A is getting ill conditioned (see Eq.4). 
  
  
  
  
  
  
  
  
  
  
  
Scanner 2 order | 3 order 4" order bilinear 
HP pw 3.813920 | 3.385246 | 2.621954 | 4.326551 
Mer ^3 | 5278431 | 3.774867 | 337716 | 10.313035 
A TUN 4.342248 | 3.296468 | 2.970317 | 9.013698 
HE Scanlet 3.988234 | 3.646822 | 2.812947 | 4.858807 
5470c 
Meer SP | 5.899126 | 5.939940 | 4.674509 | 9.852291 
  
Table 3. RMS errors after adjustment 
7, CONCLUSION 
In digital color imaging it should be known how aécurate the 
scanners are. It considers the geometric accuracy and color 
accuracy. Employing mathematical methods can reduce the 
color distortions. In this study 3D polynomial transformations 
are applied for this problem. The tests show evident 
applicability of this transformation method for reducing color 
distortions of desktop publishing scanners. The transformations 
make possible to map device dependent color space to device 
independent color space. The results for a certain scanner model 
can be generalized, and a device profile for a color managing 
system can be defined. In the application the 4" order 
polynomial transformation is found to be most suitable for each 
scanner. In the appendix, original colors, scanned color and 
transformed colors (by 4” order polynomial) are shown. Based 
on the RMS errors in Table 3, the tested scanners can be 
ordered according to color accuracy; HP ScanJet 5p, Mustek A3 
Sp, Acer Prisa 310p, HP ScanJet 5470c, Mustek Sp Plus. The 
colors in figures in the appendix confirm these ordering.