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36 2 Basic Relations: Image Sequences – “the World”
nents, the unknowns of which are written below them. The total HTM including
perspective mapping, designated by T t is written
T tot = P R șc R ȥc T Rc T Ro R ȥo . (2.13)
The partial derivative of T t with respect to any component of the state vector x S
(Equation 2.12) is characterized by the fact that each component enters just one of
the HTMs and not the other ones. Applying the chain rule for derivatives of prod-
ucts such as T t/ (x) yields zeros for all the other matrices, so that the overall deriva-
tive matrix, for example, with respect to y gc (abbreviated here for simplicity by just
x) entering the central HTM T Rc is given by
w T t /( ) T ' = R șc R ȥc w T Rc /( ) T Ro R ȥo
w
w
x
x
tx
(2.14)
T
R R ' T R .
șc ȥc Rcx Ro ȥo
Two cases have to be distinguished for the general partial derivatives of the
HTMs containing the variables to be iterated: Translations and rotations.
Translations: These components are represented by the first three values in col-
umn 4 of HTMs. In addition to the nominal value for the transformation matrix T N
as given in Equation 2.2, also the partial derivative matrices for the unknown vari-
ables are computed in parallel. The full set of these matrices is given by
§ 10 0 ǻr · x § 00 0 1· (2.15)
¨ 01 0 ǻr ¸ ¨ 0 000 ¸
r ¨ y ¸ r T r ; w T Rc ¨ ¸ = ' ,
T
1 0 rN 0 rx
¨ 00 1 ǻr ¸ z w (ǻ ) ¨ 0 000¸
r
x
¨ ¸ ¨ ¸ (a)
© 00 0 1 ¹ © 00 0 0 ¹
§ 0000· § 0000·
¨ ¸ ¨ ¸ (2.15)
w T Rc ¨ 000 1 ¸ w T Rc ¨ 0000 ¸
T
ry
rz
w (ǻ ) ¨ 0000¸ = ' ; w (ǻ ) ¨ 000 1¸ T ' . (b)
r
r
y
z
¨ ¸ ¨ ¸
© 0000 ¹ © 0000 ¹
Rotations: According to Equation 2.3 the nominal HTMs for rotation (around the
x-, y- and z- axis respectively) are repeated below: s stands for the sine and c for
the cosine of the corresponding angle of rotation, say D.
1 § 0 0 0· c § 0 s 0· § c s 0 0·
¨ 0 c s 0 ¸ ¨ 0 1 0 0 ¸ ¨ s c 0 0 ¸
R x = ¨ ¸ ; R y = ¨ ¸ ; R z = ¨ ¸ . (2.3)
0 ¨ sc 0¸ s ¨ 0 c 0¸ ¨ 0 0 1 0¸
¨ ¸ ¨ ¸ ¨ ¸
© 0 0 0 1 ¹ © 0 0 0 1 ¹ © 0 0 0 1 ¹
The partial derivatives of the transformation matrices R/D, may be obtained from
d(sin )/ d D D cos D ; (cos )/d D D sin D s . (2.16)
c
d
This leads to the derivative matrices for rotation
§ 0 0 0 0· § s 0 c 0· § s c 0 0·
¨ 0 s c 0 ¸ ¨ 0 0 0 0 ¸ ¨ c s 0 0 ¸
R xx ' = ¨ ¨ 0 c s 0¸ ¸ ; R yy ' = ¨ ¨ c 0 s 0¸ ¸ ; R zz ' = ¨ ¨ 0 0 0 0 ¸ ¸ . (2.17)
¨ ¸ ¨ ¸ ¨ ¸
© 0 0 0 0 ¹ © 0 0 0 0 ¹ © 0 0 0 0 ¹
It is seen that exactly the same entries s and c as in the nominal case are re-
quired, but at different locations and some with different signs. The constant values
1 have disappeared, of course.
