Flattening

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Flattening
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An approach to terrain mapping would be to select
a land juncture that represents the center of a
flat map, but to apply a transformation such that
departure of earth-surface junctures from that
point carry with them departure from the simple
mapping rule of proportionality.
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Such a map would look like a picture taken of a
globe with the selected point closest to the
camera.
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It would be an orthographic projection of the
globe onto a flat surface.
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Land forms as you approach the outer rim of such
a map would be deformed or distorted, and so the
map may be of limited use to a human reader.
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But a computer, with the mapping formula stored
in memory, would have no problem transforming
locations of two points on such a map to their
overland separation across the face of the globe.
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A similar accommodation to the camera mapping
from physical space into camera space is what we
here call flattening.
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The kind of distortion that a viewer of an image
treated by flattening would see amounts to
elimination of the perspective effect where
objects in the foreground appear larger compared
to their actual size than do objects in the
background.
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If the original image were a perfect perspective
projection into a plane, and if flattening
eliminated all perspective effects, then the
model would, with the right elements of C,
perfectly express the mapping of the camera.
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Note that whereas terrain mapping is 2D to 2D,
this flattening affects the way that we model and
store the camera/lens mapping of
three-dimensional, physical space into
two-dimensional camera space. It is 3D to 2D.
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Consider the origin o relative to which the 3D
robot-kinematics model specify the forward
kinematics of the robot.
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According to the simple perspective or pinhole
camera model the coordinates xc, yc of point o
will depend upon its location relative to the XYZ
reference frame shown the frame aligned with
the camera according to
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Camera-space coordinates of the ith sample, xci
, yci, are said to be flattened about point o
with application of the following conversion
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Note that the flattened samples represent an
orthographic mapping of physical space since xci
Xi x Constant, yci Yi x Constant, where
Constant f/Zo.
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Note further that the mapping of point o itself
would not change in other words xco xco.
Hence the phrase flattening about point o.
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In practice Zo would be only roughly known.
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Let DXi DYi DZi be the coordinates of point P
relative to point o referred to the cameras XYZ
coordinate frame
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After convergence the fit is much better.
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C1 -0.0000 0.0000 13.0656 -5.4120
-0.0004 0.0001T
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A second camera is added in order to solve for
the joint rotations needed to collocate point P
with point o in each of two camera spaces and
hence also in physical space.
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C2 3.8268 9.2388 9.2388 3.8268
-0.0000 0.0000T Again, flattened about o.
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(x1ct y1ct)(0.0 0.0) (x2ct y2ct)(0.0 0.0)
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Because we have flattened about the target point
o, it is permissible to use for the camera-space
kinematics relationships the Cs flattened about
o, given above.
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J(q1, q2 , q3) xc1t f1x(q1, q2 , q3 )2
yc1t - f1y(q1, q2 , q3 )2 xc2t f2x(q1, q2
, q3 )2 yc2t f2y(q1, q2 , q3 )2
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J(q1, q2 , q3) xc1t f1x(q1, q2 , q3 )2
yc1t - f1y(q1, q2 , q3 )2 xc2t f2x(q1, q2
, q3 )2 yc2t f2y(q1, q2 , q3 )2
q1p/2 q2p/2 q3p/2
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q1p/2 q2p/2 q3p/2 q1-p/2 q20 q3p/2
q1-p/2 q2p q3p/2 q1p/2 q2-p/2 q3-p/2
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The once-iterated-upon C1 and C2, however,
represent an improvement over the strict
orthographic assumption. It is therefore
preferable to use these to reestablish the target
point prior to the next upgrade of C1 and C2
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