Unformatted text preview: be two scene points
on the same epipolar plane that are visible in both
images. Geometrically, the constraint dictates that the
line through P , Q may not intersect the line segment
Z1 Z2 joining the tips of the two view normals.
A useful property of monotonicity is that it extends
to cover a range of views inbetween V1 and V2 . We
say that a third view V3 is inbetween V1 and V2 if its
normal Z3 intersects Z1 Z2 . Because the line through
P and Q intersects Z1 Z2 if and only if it intersects 3 Viewing Geometry Under an orthographic projection model e.g., weak
perspective, paraperspective, a ne, a view represents
a plane onto which the scene projects to produce an
image. Therefore, a view V can be speci ed as a tuple
V = hX; Y; oi where the 3D vectors X and Y represent the coordinate axis of the image plane and the 2D
vector o speci es the o set of the image origin from the
projected world origin. The view projection matrix is
denoted = XT
YT o 1 and the projection p = x; y of a homogeneous scene
point P = X; Y; Z; 1 is given by p = P. The image
plane unit normal, also known as the optical axis or
direction of gaze of V is denoted Z. Under strict orthographic projection, X and Y are constrained to be
orthonormal, whereas in a general a ne model 11 X
2 ZZ 12 Z Z Z s1 2 3 1 s2
s3 P−Q Q
P Ι1
Ι2 Ι3 l1
p 1 q 1 l3 p 3 q 3 l2 p 2 q l1 2 E 12 l3 Figure 1: Monotonic Viewing Geometry. If P appears
to the left of Q in images I1 and I2 then it must also
in I3 , providing Z3 intersects Z1 Z2 . Monotonicity requires that line P , Q does not intersect Z1 Z2 . l2 Figure 2: Correspondence Under Monotonicity. Top
view of projection of three surface crosssections into
corresponding epipolar lines of images I1 , I2 , and I3 .
Although the projected intervals in l1 and l2 do not
provide enough information to reconstruct S1 , S2 , and
S3 , they are su cient to predict the appearance of l3 . either Z1 Z3 or Z3 Z2 , monotonicity of I1 and I2 implies monotonicity of I1 and I3 as well as I3 and I2 .
That is, any two points on E12 must appear in the
same order on corresponding epipolar lines of all three
images. This property, that monotonicity applies to
inbetween views, is quite powerful and is su cient to
completely predict the appearance of the visible scene
from all viewpoints inbetween V1 and V2 . Fig. 1 illustrates the impact of the monotonicity constraint on
view synthesis.
The monotonicity condition imposes a strong visibility constraint on the scene. Intuitively, monotonicity of I1 and I2 means that the same scene points are
visible in the range of views between V1 and V2 . Because monotonicity is needed for view interpolation,
this condition limits the set of views that can be interpolated. Nevertheless, monotonicity is satis ed at
least locally for a wide range of interesting scenes. the interval endpoints are determined from this correspondence by triangulation. We will refer to these
scene points as visible endpoints of S1 , S2 , and S3 .
Now consider an inbetween view V3 w...
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 Spring '08
 Staff
 Computer Graphics, Orthographic Projection, Monotonicity, view interpolation

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