It is shown that after a simple recti cation

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Unformatted text preview: ation procedure, linear interpolation of corresponding points in images produces new in-between views of the scene. unit length. Clearly, V1:5 does not correspond to an in-between 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 user-speci 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 well-de 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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