Let p and q be two scene points on the same epipolar

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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 in-between V1 and V2 . We say that a third view V3 is in-between 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 cross-sections 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 in-between views, is quite powerful and is su cient to completely predict the appearance of the visible scene from all viewpoints in-between 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 in-between view V3 w...
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