Unformatted text preview: ation procedure, linear interpolation of corresponding points in images produces new inbetween
views of the scene. unit length. Clearly, V1:5 does not correspond to an
inbetween view, as de ned in Section 4.1, so monotonicity may not be preserved and correctness of the
interpolated image cannot be ensured. Furthermore,
there are cases where interpolation degenerates, such
as when I2 is a 180 degree rotation of I1 . In this case,
the morph collapses to a point, with all points mapping
to the origin in I1:5 . In short, image interpolation will
generally not produce valid views. Fortunately, however, these problems can be corrected by appropriately
aligning the two images before performing the interpolation see Fig. 3b, as demonstrated in the remainder
of this section. 5.1 Image Interpolation Morphing techniques combine a geometric warp
with a cross dissolve to interpolate two images. A set
of corresponding userspeci ed control points is provided in each image to guide the interpolation. As
these points are typically sparse, the correspondence
must be extended so that every pixel has a wellde ned
path. For analytical purposes, we assume for the moment that a correct and complete correspondence is
provided between pixels of the two images. This constraint is relaxed in the next section to require correspondences between but not within uniform regions of
the two images. We consider the common case where
linear interpolation of corresponding point positions is
used to create intermediate images. In other words, if
p1 and p2 are corresponding points in images I1 and
I2 respectively, the corresponding point in image Ii ,
1 i 2 is
pi = 2 , ip1 + i , 1p2
If images I1 and I2 are represented by arrays of corresponding points, then image interpolation is expressed
by the following equation:
Ii = 2 , iI1 + i , 1I2
2
Image interpolation has a direct physical interpretation in terms of views, a connection that was recognized by Ullman and Basri 5 in the general context of
linear combinations of views. Here we present a simple geometric interpretation that makes the underlying
principles more explicit.
Consider two views V1 and V2 of a scene S. By
Eqs. 1 and 2,
Ii = 2 , i1 + i , 12 S
= i S
3
where i = 2 , i1 + i , 12 . Ii represents what
the scene would look like from a new viewpoint if every
feature visible in I1 and I2 were also visible in Ii . The
axes and o set of the new viewpoint are interpolations
of the corresponding vectors of V1 and V2 .
Eq. 3 provides a simple link between interpolation
of images in 2D and of views in 3D. In spite of this result, image interpolations do not account for changes in
visibility and often correspond to very unintuitive view
interpolations. Fig. 3a graphically depicts the interpolation of views V1 and V2 . Although both V1 and V2
are normal to the epipolar plane E12 , the interpolated
view V1:5 is tilted by 45 degrees with respect to E12 .
In addition, the axes of V1 and V2 are orthonormal,
whereas the axes of V1:5 are neither orthogonal nor of 5.2 Image...
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 Spring '08
 Staff
 Computer Graphics, Orthographic Projection, Monotonicity, view interpolation

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