ISPRS Commission III, Vol.34, Part 3A ,Photogrammetric Computer Vision“, Graz, 2002
tance f and two parameters (a 8) modelling affine image
deformations, then the central perspective image point p
- as a homogenous vector p - of an object point P can be
computed using the projection matrix P, (equation (1)).
pc, "RI. [Ens Os] PP. P (1)
l o —X0
C, = 0 B Yo
0 0. —*
Esxs = diag(1, 1; 1)
The three columns of C, together with Ry represent the
affine direction vectors of the so-called principal rays TA.
ry2, Tys (lines through the projection center parallel to
the image’s coordinate axes) as Ry, - Cy. These three lines
span three planes - the so-called principal planes 71, T2,
73 [Papadopoulo, Faugeras 1998].
An image line £ can also be represented by a homogenous
vector £. If the line / contains the point p it holds: £' IP —
0. The line £ defined by two points p and q is given by:
Í ^ [p],; q. With [p], being the so-called axiator:
0 Az —Qy
axb=[a]| b jal. =| —a,;. 0 az (2)
d=4-¢ — dj, = Pet À " ic) A * 3
2.2 A few basics on tensor calculus
A tensor is an indexed system of numbers. There are
two kinds of indices: sub-indices are called co-variant and
super-indices contra-variant. A tensor with contra-variant
valence p and co-variant valence q has n?*! components
with n being the dimension of the underlying vector-space;
i.e. each index runs from 1 to n. Using these indices and
Einstein’s convention of summation, certain mathematical
relations can be expressed in a very efficient way. This
convention says that a sum is made up of all the same
indices appearing as co- and contra-variant. So, for ex-
ample, the scalar product s(x, y) — x! - y of two vectors
X — (z12223)! and y — (y192 ys) | can be written in a
shorter way as: s(x, y) = zi; y'. The product A - B — C of
two matrices A and B can be written as A5 B7 — Cj. The
contra-variant indices relate to the rows and the co-variant
ones to the columns.
2.3 The trifocal tensor
If the orientation of three images 1, 7e, 1/3 is formulated
according to equation (1) (with O4 = 0 and R; — Es.)
then the TFT can be represented in the following way
(equation (4)); c.f. [Ressl 2000].
T?* — (93)? Bf — (931) 4 (4)
using:
ACRI vaux -C; BR: - On . (5)
B-C RC, Va = C5 RO; (6)
Vzy is the epipole of image 1, in image 1, i.e. the image of
O, in image ;. With ^ it is emphasized that this epipole
is represented as a homogenous vector but in a specific
scale (observe the equality-symbol). The other epipoles
are:
Yız=C,.. O2 $13 =Cr'. Os (7)
$9: 2C; RI. Os + V21 (8)
93 2C; Rl. O2 + Ÿ31 (9)
3 THE TENSORIAL SLICES
One can imagine the trifocal tensor T" formed as a 3x3x3
cube of numbers and the cube's edges related to the indices
i, j, k. If we keep one index fixed, we slice a 3 x 3 matrix
out of the tensor. Since we have three indices, we get
three different kinds of matrices - different also in their
geometrical meaning. For didactical reasons we will start
with the j and k index.
3.1 The homographic slices J, and K,
If we keep the j-index in equation (4) fixed as j = x €
{1,2,3}, we get the following matrix J, (e being the z'^
column of E33):
Je=el Vo: B- Vz 0]. A (10)
J, describes a mapping (a collineation) of points p; in im-
age 1/1 to points ps in image v via the principal plane 72;.
In [Shashua, Werman 1995] this mapping is termed homog-
raphy. The homography matrices J, are distinguished by
the properties shown in Table 1.
Analogously, if we keep the k-index fixed, we get a matrix
K,, which describes a homography from image 1; to image
/2 via the principal plane 73;.
K.z 9 el-B-erl.95-4 (11)
3.2 The correlation slices I,
If we keep the i-index fixed, we get analogously a 3x3 ma-
trix I,. For the homographic matrices J, resp. K, their
form resulted directly from the co-variant (i) and contra-
variant (k resp. j ) indices in equation (4). When the
i-index is fixed, only the contra-variant indices (j, k) re-
main, and therefore one of them has to be chosen for the
columns. We choose the j-index.
1, -B 6e 0] «vf er A (12)
I; describes a mapping (a dual correlation) of lines £; in
image v2 to points pa in image 13 via the principal ray
Tic. IT would map the lines @3 in image 13 to points pa
in image v» via the principal ray ri.
Due to [Papadopoulo, Faugeras 1998] the correlation ma-
trices I, are distinguished by the properties shown in Ta-
ble 2. Since the correlation slices are the basic input for
the constraints and parameterization to be presented, they
are investigated in more detail in section 5.1.
Note: The relations between the homographic and corre-
lation slices are the following: The y'^ column of J, resp.
K, is the z'^ column of I, resp. T
4 PREVIOUS CONSTRAINTS AND
MINIMAL PARAMETERIZATIONS
Two sets of constraints and two minimal parameterizations
(i.e. having 18 DOF) were discussed in the literature so far.
[Torr, Zisserman 1997] present a minimal parameteriza-
tion for the TFT. By assigning projective canonical co-
ordinates to the image and the space points, they show,
that it is possible to compute the tensor from six homolo-
gous point triples across three images. Of the 36 observed
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