Image Processing Reference
In-Depth Information
planes could be rotated and imaged from a different view than the camera views. On
the right we show the used set of points in the right camera view.
13.5 Searching for Corresponding Points in Stereo
Finding correspondence points in stereo images automatically is a difficult problem.
In this section we discuss how the search for a point can at least be constrained even
if the problem remains a difficult one. Simplified, the underlying logics consists in
translating the knowledge on the stereo system to limit the freedom of a correspond-
ing point, as to where it can be. Without stereo assumption, the corresponding point
can be anywhere in the other camera view, i.e., it has a degree of freedom of 2,
whereas we expect to reduce this freedom to 1 by use of knowledge on the stereo
set-up.
We will utilize an axiom of the 3D world, namely that three points define a plane.
We observe that the normal of a plane is orthogonal to all vectors lying in that plane.
The points O L ,P,O R in a stereo system then define a plane, with its normal vector
given by the cross-product −−−−→
−−−→
O L P , which is in turn orthogonal to −−−→
O R O L
O R P :
×
( −−−→
O R P ) T ( −−−−→
× −−−→
O R O L
O L P )=0
(13.93)
All vectors must be represented in the same frame for this equation to hold. We keep
the same notation as in previous sections to characterize the stereo system, meaning
that
the relative displacement vector t =( −−−−→
O R O L ) R is in the right camera frame,
whereas
the relative rotation matrix R transfers the coordinates of a vector represented in
the left camera to the right camera.
Accordingly, in Eq. (13.93) there is one vector, −−−→
O L P , that can be expressed most nat-
urally in the left (reference) camera frame, whereas the remaining two are normally
expressed in the right camera coordinates. We assume then that the coordinates of all
vectors are w.r.t. the right camera basis
( −−−→
O R P ) R (( −−−−→
( −−−→
O R O L ) R ×
O L P ) R )= 0
(13.94)
so that by the replacement ( −−−→
O L P ) R = R ( −−−→
O L P ) L , we obtain:
( −−−→
O R P ) T ( −−−−→
× ( R ( −−−→
O R O L
O L P ) L )) = 0
(13.95)
where, and below, we have dropped marking the right camera frame because all vec-
tors are in the right camera coordinates unless otherwise mentioned. Note however,
that this does not mean a change of the world frame. The left camera still plays
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