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a plane parallel 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part BS. Istanbul 2004 
  
to the image plane at a distance c from it, is also parallel to the 
image plane; when projected orthogonally onto it, the circle de- 
fines the principal point locus ('ppl' in Fig. 2). 
On the other hand, it is worth mentioning that the circle formed 
as section of the sphere and the image plane itself (also seen in 
Fig. 2) is in fact the locus of the *isocentre'. This image point — 
known in photogrammetric literature, thanks to its properties as 
regards measurement of angles on flat terrain (Church & Quinn, 
1948) — lies on the bisector of the angle formed by the image 
rays to the principal point and to the third vanishing point (ie. 
the vertical direction in the case of an oblique aerial image). Re- 
cently, Hartley & Silpa-Anan (2001) have used this point, as the 
‘conformal point’, for measuring angles on the projected plane. 
A further interesting geometrical aspect is that Eq. (1), after the 
substitutions, takes the following form: 
(%1—Xo0)X2 +(¥1 7 Yo)Y2 = XoX1 = Yo¥1 + Xo? + Yo? +¢2=0 (3) 
This equation represents the line (vanishing line), which passes 
through the second vanishing point V» (x»,y;), assuming that va- 
nishing point V, and interior orientation parameters are known. 
Eq. (3) is the main equation of the calibration process to follow. 
2.2 Consideration of the aspect ratio 
All preceding equations and geometric interpretations hold for 
the case of a ‘normal camera’, whose only parameters are the ca- 
mera constant and the principal point, under the assumption of 
square image pixels. Yet, there is the possibility of a non-square 
pixel, especially in CCD cameras. In these cases, introduction of 
an additional parameter is needed, namely of the camera aspect 
ratio a, which grasps the relative scaling of the two camera axes 
(for a normal camera a = 1). However, addition of this parame- 
ter does not change the basic geometric interpretation. Let a « 1 
be the scale factor for the vertical (y) coordinates of the image 
points which, thus scaled, are the orthogonal projections of the 
initial ‘normal’ points onto an image plane, tilted by the angle p 
(cos = a) about the x-axis. The calibration sphere still holds for 
the normal plane but the projection of the ‘isocentre circle’ is an 
ellipse on the transformed (affine) image plane (see Fig. 4). The 
same is also true for the locus of the principal point (which, of 
course, coincides with the isocentre locus for ¢ = 0). 
  
  
  
  
  
Figure 4. The isocentre ellipse (ice) is the orthogonal projection 
of the isocentre circle (icc) from the normal image plane (nip) 
onto the affine image plane (aip).The angle p between these two 
planes is related to the aspect ratio (a) as B = cos la. 
Calibration equations Eq. (1) and Eq. (3) become, respectively: 
2 7f ; 2 E 
(xq — Xm y Hz) zi Yu). +c? = R? (4) 
A 
  
: 1 36 1 12 
with R= J(xi-x2) sq (yı-yz) 
2 a*} : 
and 
(yı-yo)y2 YoY Vo? 
(xj xoxo don ) -XoX - 22 E *xo4£9-4c2-0 (5) 
a+ a? a? 
2.3 Calibration sphere and image of the absolute conic 
As mentioned above, recovery of interior orientation is possible 
through an estimation of the image œ of the absolute conic from 
three orthogonal vanishing points (Liebowitz et al., 1999). The 
conic wis defined as: 
w=K-TK-! (6) 
where K is the calibration matrix (Hartley & Zisserman, 2000). 
Each pair of orthogonal vanishing points V,, V; supplies a linear 
constraint on the entities of c of the form 
VitoV, =0 (7) 
whereby V,, V, are in homogeneous representation. The two va- 
nishing points are said to be conjugate with respect to w. Three 
such pairs suffice for the estimation of the calibration matrix of 
a normal camera. Yet w is an imaginary conic, and the geome- 
tric relation of the vanishing points with the camera internal pa- 
rameters is not obvious. Ignoring aspect ratio (and skewness), ® 
may be written as: 
1 0 —Xo 
2 D n 
a Vo* ber 
Introducing Eq. (8) into (7), with V, 7 (xi, yi, 131, Vs exo, y» " 
in inhomogeneous representation, yields Eq. (1), the equation of 
the “calibration sphere’. This means that constraint (7) and the 
sphere equation (1) are equivalent for inhomogeneous notation 
of vanishing points. 
Next it will be seen that the calibration sphere is also relevant in 
the case of plane-based calibration, a process which estimates 
interior orientation using homographies H between a plane with 
known Euclidean structure and its images (Zhang, 2000). The 
two basic equations of plane-based calibration are: 
hj! oh» -0 (9) 
hi! oh; = ha" wh> (10) 
where h; = [H,,, Ha, Hal]. h> = [H;2, Hz, Hs] are the first two 
columns of the H matrix. Each homography H provides two such 
constraints (9), (10) on the elements of c Gurdjos et al. (2002) 
put the problem of plane-based calibration into a more intuitive 
geometric framework, by proving that the solution is equivalent 
to intersecting circles (‘centre circles"). The centre circle is the 
locus of the projection centre when space-to-image homography 
is known and can be obtained as intersection of a sphere (centre 
sphere) and a plane (centre plane). In photogrammetric termino- 
logy, this plane — typically defined in aerial images as the verti- 
cal plane containing the camera axis — is the ‘principal plane’. 
In fact, Eq. (9) is equivalent to the calibration sphere presented 
here. Using Eqs. (9) and (1), it is set: