REGISTRATION FOR DIGITAL CAMERA IMAGE OF PAPER MAPDISTORTED BY TRANSPARENT FILM
Takashi YAMAMOTO and Masataka TAKAGIMaster course student, Department of Infrastructure Systems Engineering
Kochi University of Technology
185 Miyanokuti, Tosayamada-Town, Kami-City Kochi 782-8502, Japan
Tel:(81)-887-53-1040 Fax:(81)-887-57-2420
E-mail: 105522k@gs.kochi-tech.ac.jp

KEY WORDS: Map Digitizing, Registration, Refractive index, Collinearity Equation,
Snell’s Law
ABSTRACT: In the past, a large accurate image scanner was used for map digitizing. Theimage scanner is so expensive that it is diﬃcult to use in general. Nowadays, a digital camera
with 12 mega pixels can be used in many ﬁelds. In the future, such digital camera will beable to replace the image scanner. The digital camera will become suitable equipments forcollecting many GIS data by map digitizing. But registration accuracy of digital map by thedigital camera will be still lower than by the image scanner. For example, a paper map isneeded to press by a transparent ﬁlm for correction of wrinkle. Then the digital camera
image distorted by transparent ﬁlm should transformed for registration. In this study,
geometric transformation was established by extended collinearity equation with snell’s law
for correction of refractive index. This geometric transformation was evaluated by using
simulated data. This method can be applied in the case of distortion by accurate opticalglass.

1

BackgroundIn the past, a large accurate image scanner was used for map digitizing. The image scanner
is so expensive that it is diﬃcult to use in general. Nowadays, a digital camera with 12
mega pixels can be used in many ﬁelds. In the future, such digital camera will be able toreplace the image scanner. The digital camera will become suitable equipments for collecting
many GIS data by map digitizing. But registration accuracy of digital map by the digital
camera will be still lower than by the image scanner. For example, a paper map is needed topress by a transparent ﬁlm for correction of wrinkle. Then the digital camera image of thepaper map which covered by transparent ﬁlm should transformed for registration. By using
two-dimensional perspective projection, the registration accuracy will become lower due torefractive index of transparent ﬁlm. When error due to refractive index can be eliminated,registration accuracy will be improved.

2Objectives

111098765432100.5
Distortion unit:mm )Distortion㩷㩷unit:mm :Objective of this study is development of registration method for digital camera image of paper
map distorted by transparent ﬁlm. Geometric transformation was established by extended
collinearity equation with snell’s law for correction of refractive index. Figure1 shows relation
between incidence angle and distorted value using snell’s law when refractive index is 2 andthickness of transparent ﬁlm is 1mm. Figure 2 shows relation between pixel number of digital
camera image and distorted value in same condition with previous ﬁgure. The pixel number iscorresponded with incidence angle. In this case, angle of one pixel in calibrated digital camera
image is 0.00026551 deg. Therefore, when accurate registration such as less than 0.1mm is0.40.30.20.1001020304050607080900200Incidence angel unit:deg

400

600

800

1000

1200

1400

Pixel number unit:pixel

Figure 1: Relation between incidence angle and dis- Figure 2: Relation between pixel number and distorted value
torted value

required, distortion of refractive index must be corrected.

3

Geometric Transformation

A conceptual procedure for registration of digital camera image is illustrated in Figure 3 andFigure 4. In this study, the paper map is assumed XY plane of object space. First, distortion
value of control points by refractive index is calculated by using Snell’s law with equation (1).
{∆Xp = ∆Dp cos κp(1)∆Yp = ∆Dp sin κp∆Dp means distorted value. ∆Xp and ∆Yp are distortion component of control points in map
coordinates. κp is rotation angle of control points alound Z axis of object space. κp can becalculated following equation.
Yi − Y0κp = tan−1
Xi − X0
Xi , Yi and Zi are coordinates of imaging control point in object space. X0 , Y0 and Z0 are
coordinates of projection center in object space. Distortion value can be calculated by followingequation using snell’s law.
∆Dp = thickness(tan θi − tan θr )zo(Xo, Yo, Zo)o(0, 0, 0)
ȁ,Ǿ,ǲ
i(Xi, Yi, Zi)
i(xi, yi, zi)ǰiǰrthicknessǍ Dpp(XP, YP, ZP)
p’(XP+ǍXp, YP+ǍYp, ZP)

ǲyǾ
(Xo, Yo, Zo) O(0, 0, 0)ȁxǰi(Xi, Yi, Zi)(xi, yi, zi)ip’(xp’, yp’, zp’)np
X, Y, Zx, y, z: Coordinates of projection center in object space: Camera coordinates of projection center: Attitude of camera: Coordinates of imaging control point in object space: Camera coordinates of imaging control point: Incident angle: Refractive angle: Thickness of transparent film
: Distorted value: Coordinates of control point in object space: Coordinates including distorted valueof control point in obuect space: Camera coordinates including distorted value of control point: Camera nadir point: Coordinate axis in object space: Camera coordinate axis

ǰiZSnell’s lawǰrYǍ Dpthickness

npp(XP, YP, ZP)

p’(XP+ǍXp, YP+ǍYp, ZP)(xp’, yp’, zp’)X

Figure 3: Principle of registration for digital camera image with transparent film: Control point in object space: Control point including distorted value
in obuect spacenp: Camera nadir point
Ǎ Dp : Distorted valueǍXp : Distorted Value of X-axis direction
ǍYp : Distorted Value of Y-axis direction
ǲp: Rotation angle of control point
alound Z-axis
X, Y : Coordinate axis in object spacepp’

ǍDp
pp’
ǍYp
ǍXpY

ǲp
np

X

Figure 4: Projected Figure in X-Y plane

Z i − Z0csin θiθr = sin−1nir

θi = cos−1thickness is thickness of transparent ﬁlm. θi is incident angle. θr is refractive angle. nir isrefractive index of transparent ﬁlm.
nir =sin θisin θr(Snell’s law)

Xi − X0 , Yi − Y0 and Zi − Z0 can be calculated by following equation. xi , yi and c are camera
coordinates of control points. ω, ϕ and κ are the attitude of the digital camera.
  
Xi − X0
a11 a21 a31
xi
 Yi − Y0  = a12 a22 a32   yi Zi − Z0a13 a23 a33−c 
a11 a12 a13
100cos ϕ 0 sin ϕcos κ − sin κ 0
a21 a22 a23  = 0 cos ω − sin ω   010   sin κ cos κ 0
a31 a32 a330 sin ω cos ω− sin ϕ 0 cos ϕ001
Usually, collinearity equation can be derived from this equation. In this study, this equation
was extended to solve distortion of refractive index as follows.
  
xp′
Xp + ∆Xp − X0
a11 a12 a13
 yp′  = a21 a22 a23   Yp + ∆Yp − Y0 Zp − Z 0zp′
a31 a32 a33
xp′
xi = −czp′
(2)
yp′

yi = −czp′
where xi and yi are camera coordinates of control points; c is focus length; Xp , Yp and Zp arecoordinates of control points in object space. In this extended collinearity equation, unknown
parameters are X0 , Y0 , Z0 , ω, ϕ, κ and nir . To derive these unknown parameters, control
point data should be used. At least 4 control point data are needed. When control point data
(Xp , Yp , Zp ) (xi , yi ) input to equation(2), least square method is applied to derive unknownparameters. This extended collinearity equation is also non linear function(Equation(3)), thatmust be expanded in Taylor series to make linear function.
xp′

− xi
F (X0 , Y0 , Z0 , ω, ϕ, κ, nir ) = −czp ′(3)yp′− yiG(X0 , Y0 , Z0 , ω, ϕ, κ, nir ) = −czp′
Partial diﬀerential coeﬃcients in the Taylor series shows following equations.
ca11 + xi a31∂F=∂X0zp′∂F
ca12 + xi a32=∂Y0zp′ca13 + xi a33∂F=∂Z0zp′

∂Fxi y i=∂ωc∂Fx2i= − cos ω − yi sin ω − c · cos ω∂ϕc∂F∂F∂F=(Yp + ∆Yp − Y0 ) −(Xp + ∆Xp − X0 )
∂κ
∂X0
∂Y0
∂∆Xp∂∆Yp(−ca11 − xi a31 )+ (−ca12 − xi a32 )
∂F
∂nir
∂nir=
∂nirzp′∂G
ca21 + yi a31=
∂X0zp′∂G
ca22 + yi a32=
∂Y0zp′∂G
ca23 + yi a33=
∂Z0zp′∂Gy2=c+ i∂ωc∂Gxi y i= xi sin ω −cos ω∂ϕc∂G∂G∂G=(Yp + ∆Yp − Y0 ) −(Xp + ∆Xp − X0 )∂κ
∂X0∂Y0∂∆Xp∂∆Yp(−ca21 − yi a31 )+ (−ca22 − yi a32 )
∂G
∂nir
∂nir
=∂nirzp′∂∆Xp∂ tan θr= −(cos κp )thickness
∂nir
∂nir
∂∆Yp
∂ tan θr= −(sin κp )thickness
∂nir
∂nir
∂ tan θr
− sin θi=()√()2∂nir
sin θisin θi2−1n2ircos sin1−nirnirWhen initial condition of unknown parameters must be decided, correction value for initialcondition can be calculated by least square method with expanded equation(3). By using
corrected parameters, correction value can be calculated again. This calculation should be
iterated to converge enough accuracy. The program in C language was coded to calculate
unknown parameters based on previous procedure.

4

Evaluation by a simulation

Simulated data those were (Xp , Yp , Zp ) and (xi , yi , zi ) were generated for evaluation. The
simulated data can be used as control point. When parameters those were X0 , Y0 , Z0 , ω, ϕ, κand nir were decided, (Xp , Yp , Zp ) and (xi , yi , zi ) can be calculated by equation(1) and (2).
In this evaluation, 9 control points were generated. Table 1 shows results of iterative solution
using 9 control points of simulated data. Simulated data were (Xp , Yp , Zp ) and (xi , yi , zi ). In
simulated data, ω, ϕ and κ were 0˚, X0 and Y0 were 0mm, Z0 was 700mm and refractive index
was 2. The result shows that unknown parameter of refractive index was solved by around 10times iterative solution when initial condition of refractive index was input over 1.2.
Table 1: Results of iteration solution

Initial condition Iterationof refractive index number
1.0
1.2
1.4
1.6
1.8

513121111

Calculated
results
divergence1.999976
1.999976
1.999976
1.999976

Conclusion

Registration method for digital camera image of paper map distorted by transparent ﬁlm
was successfully developed. This method can be applied in the case of the distortion by
accurate optical glass. When a soft plastic ﬁlm is covered on the paper map, the distortion
will not be eliminated because of undulation on the surface. This registration method must
be demonstrated in real paper map covered by optical glass and plastic ﬁlm in near future.

References[1] Japan Society of Photogrammetry and Remote Sensing, 1997. Analysis of Photogrammetry,pp.46-56 (In Japanese)[2] Hans-Gerd Maas, 2001. New developments in Multimedia Photogrammetry,
URL:http://www.tu-dresden.de/fghgipf/forschung/material/publ maas/O3D95 multimed.pdf[3] Francois Blais, J., 1997. Calibration of an Anamorphic Laser Based 3-D Range Sensor,
URL:http://iit-iti.nrc-cnrc.gc.ca/iit-publications-iti/docs/NRC-40216.pdf