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Unformatted text preview: Recti cation The odd view trajectories obtained from image interpolations arise because linear interpolation of views
does not amount to linear interpolation of gaze directions. Two views V1 and V2 each de ne a direction
of gaze, Z1 and Z2 . Intuitively, we might expect the
direction of gaze to follow the most direct path between Z1 and Z2 during a smooth transition between
V1 and V2 . However, this is generally not the case
in view interpolation, as Fig. 3a illustrates, due to
the nonlinear relationship between plane and normal
transformations1.
A morph can be made to interpolate gaze direction and to generate valid inbetween views by rst
aligning the coordinate axes of the two views. This
is accomplished by means of a simple image recti cation procedure that aligns epipolar lines in the two
images. The result of recti cation is that corresponding points in the two recti ed images will appear on
the same scanline. In other words, a point x1 ; y in
the rst image will correspond to point x2 ; y in the
second. The technique is a variant of the recti cation
procedure described in 15 .
We assume that a set of at least four reference image
features is provided and that their positions in each
image are known. The centroid of the reference features is chosen to be the origin of each image, i.e., if
r1 ; : : : ; rk are the positions of the reference features in
i
i
Ii then
k
X
j =1 r= 0
0
j
i Let Ti denote the coordinates of the topleft corner of
Ii in this reference frame. Denote the image coordinates of rj as xj ; yij . De ne the measurement matrix
i
i
as
2 M=6
4 x1
1
1
y1
x1
2
1
y2 :::
:::
:::
::: xk
1
k
y1
xk
2
k
y2 3
7
5 Singular value decomposition yields the following factorization:
M = UV
Normals are transformed by the inverse transpose of the
plane coordinate transformation.
1 4 V1 V 1.5 V 1.5
Y Y Y X X Z
X V1 V2 Y X X Y
X Z
Z Z Z Y E12 V2 Z E12 a b Figure 3: Views Generated by Image Interpolation. a Interpolating the X and Y axes of V1 and V2 produces
a view that is skewed and tilted with respect to the epipolar plane E12 . b Recti cation remedies the problem
by aligning the view coordinate systems prior to interpolation.
the last column of R,2 02 B,1 is of this form, note
that the images have been rotated so that epipolar
lines are horizontal. Therefore, the y coordinate of a
point in one image depends only upon the y coordinate
of the corresponding point in the other image. If the
two original images are weak perspective projections,
s is the scaling factor of I2 with respect to I1 . In
particular, if orthographic images are used, s is 1. The
recti cation process is completed by applying a scale
1
0
matrix Hs = 0 1=s . To summarize, two images
I1 and I2 are recti ed by the following sequence of
image transformations:
^
I1 = R,1 I1 + T1
4
^2 = Hs R,2 I2 + T2
I
5 Let U0 be the matrix formed by the rst 3 columns of
U. The nonhomogeneous a ne projection matrices
1 and 2 are the consecutive 2 3 blocks of U0 :
1 = U0
2
The direction of epipolar lines in I1 and I2 can
be determined from 1 and 2 as follows: partition
i = Ai j di where Ai is 2 2 and di is 2 1.
De...
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 Spring '08
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