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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
central perspective
projection
-
‘
Central Perspective Model Orthogonal Projection Model
Figure 1: Conceptual diagram of orthogonal projection
model
as:
q 111 0129 4433 X — Xo
y =A ann am 43 Yo (1)
eC Q31 032 33 7-4.
where À is a scale factor, c is a principal distance, a;; is
components of rotation matrix and ( X o. Yo. Zo) is perspec-
tive center.
The value of A changes in proportion to distance to object
points. By substituting A by constant scale parameter m,
Equation (1) is described as :
Ta zT
Ya = AT (2)
—m/Ac —C
Gyr G12 013 X X
= mM | Gs 429 0023 Y-Y,
431 Q32 433 7 Zo
By transposing (X0. Yo, Zo) to left site, equation (2) is de-
scribed as following.
Samo ay; 012 | 013 X.
Va 7 Yo —-mií| 4931 022 23 y
—m/ Ac — zo 31 432 Q33 Z
(3)
where
Yo tir {21013 Xo
yo. ]z-ml.an. am ax Yo
Zo (0314 032 — (133 Zo
The orthogonal projection with contraction is expressed by
the first and second equations of (3).
Tyg = mia; X + a12Y Eis d13Z] + Io (4)
Yo míagiX - a23Y +aZ}+ Yo
The number of independent parameters is six. They consist
0f z,, y,, m and three rotation angles.
1011
Mathematically m is an arbitrary constant, which is in-
volved with image coordinates r,,y,. From a practical
standpoint, 1n is adjusted so as to scale down the average
photographing distance to be same length as principal dis-
tance c (Figure 2).
contraction
m
Figure 2: Constant scale parameter m
Let H be the average photographing distance to Z direction
(H = Z — Ze), M is described as following.
G33C G33C
m= — = (5)
Z — Lo H
Equation (1) is reversely transformed as following.
X — Xo 411 021 3| T
É 1
y a = |. Gisa 6» 0 y (6)
Ze 0434, d23 4033 =e
Taking notice of the third equation of (6), A is expressed
by:
0133: 7231] — 033€
À = 033471 d33 = Case (7)
Z — Zs
By substituting (5) and (7) into (2), the equations of trans-
formation from central perspective image coordinates to
orthogonal projection ones are derived.
DZ — Lo A33C
Has = d
: IT (33€ — 0132. — 02313 (8)
0 Q33C y T
Ya ‘
te HU "age aT ay
Both of equations (4) and (8) are derived by the central
perspective model without approximation. In this sense,
the model consisting of (4) and (8) is as rigorous as the
central perspective model.
2.3 Generalization of Orthogonal Projection Model
By simply generalizing equations (4), collinearity equa-
tions of affine projection model are derived.
dao = Ar X + AY =k AaZ + A4 (9)
Ya 7 As X + AgY == Ar Z e As
Addition of constraints for orthogonal projection to (9) leads
to the generalized orthogonal projection model. Because
