# De ne bi as 1 a1 i bi 0 i 0 a1 di let 01 1 b2 and

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Unformatted text preview: ne Bi as ,1 A,1 i Bi = 0 i 0 ,A1 di   Let 01 = 1 B2 and 02 = 2 B1 . Let xi ; yi T be the third column of 0i . In 15 it is shown that the epipolar y lines in image Ii make an angle of i = arctan x  with the horizontal axis. To make the epipolar lines horizontal, each image is rotated by an angle of , i , using the matrix: 5.3 Recti ed View Interpolation i Although image interpolation is not always physically valid, interpolation of recti ed monotonic images always produces valid in-between views of a scene. In light of Eq. 3, it su ces to show that recti ed image ^ interpolation preserves monotonicity. Let Vi be an in^1 and V2 . Denote the ^ terpolation of recti ed views V ^^ ^ rst row of 1 , 2 , and i by 1 , 2 , and i respectively. Because epipolar lines are horizontal, i.e., ^ ^ parallel to the image x-axis, X1 and X2 are in the same epipolar plane. Therefore, the position of a scene point P along its epipolar line in I^j is given by j P. Let P and Q be two scene points in the same epipolar plane of V1 and V2 . Suppose that monotonicity is satis ed ^ ^ for I1 and I2 so that 1 P 1 Q and 2 P 2 Q. Then i P , Q = 2 , i1 + i , 12 P , Q = 2 , i1 P , Q + i , 12 P , Q 6 Since both terms on the right of Eq. 6 are strictly positive for 1  i  2, it follows that i P i Q and hence monotonicity is preserved. This result indicates that linear interpolation of corresponding points in two recti ed basis images always i  cos i R, = ,sin i i sin i cos i  To ensure that corresponding epipolar lines share the same numbered scanline, we must vertically scale I2 with respect to I1 . Accordingly, let  R, 0  B = R, 210211 where R, 2 021 is the rst row of R, 2 02 . If B is not invertible, either the two views have parallel optical axes or the reference features are co-planar. In either case, choose instead  01  B = R, 01 1 . It follows that 0 R, 2 02B,1 =  001 0s0  If s 0 it means that the epipolar lines in I2 are horizontal but reversed with respect to I1 . In this case, I2 should be rotated 180 degrees. To understand why 5 ^ Therefore Ii is an orthographic view of the scene ^ ^ with aspect ratio kXi k : 1. In general, 0 kXi k  1, ^ i k decreasing monotonically as j j increases. with kX ^ ^ In particular, if V1 and V2 are within 45 degrees of one ^ another then kXi k is strictly greater than 0:92. In this case, the greatest possible distortion, an 8 horizontal contraction, occurs when i = 1:5, corresponding to a ^ ^ view halfway between I1 and I2 . If is known, this distortion can be avoided alto^ ^ gether by scaling the rows of Ii by 1=kXi k. Although cannot be determined from the two basis images alone, any third view of the scene is su cient to uniquely determine 16 . produces a valid view, assuming that monotonicity holds for the basis images. The signi cance of this result is that 1. Valid views can be produced by simple 2D image operations, without knowledge of either scene structure or camera geometry, and 2. Morphing techniques based on geometric image interpolation will produce physically-valid intermediate images if the basis...
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## This note was uploaded on 06/13/2011 for the course CAP 6938 taught by Professor Staff during the Spring '08 term at University of Central Florida.

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