146 
| 
© 
{ 2'"'(a1x + a9Y + as) + x"'x'(a4x + œ5y + œ6) + &' (are + agy + ag) + aor + a11y + Q12 (1) 
V'(B1x + Boy + 03) + y x'(Bax + Bsy + Be) + x'(Brx + Bay + Bo) + Brox + Bi1Y + Pi2 = 0 
where a; z f, 7 |...6. 
This relation is demonstrated in [Sha 94] using projective invariants. Another kind of demonstration can be 
found in [Har 94], who uses en explicit projective reconstruction : given the matrices of projection Mi, M» and 
M3 in the 3 images, and the matched points p and p', projections of a physical point P in the two first images 
(reference views), we can algebraically compute the coordinates of P in the projective space. We then reproject 
P onto the third image, via M3, and we obtain the coordinates (z", y") of P as a function of : the coordinates 
(z, y) and (z', y') of p and p’, and the coefficients of the matrices Mi, M»; and Ma. If we develop the equations, 
the 17 parameters of Shashua's relation appear as combinations of the coefficients of the three M;. 
Another way to recover the trilinearity constraints has been recently proposed by [Lon 95]. If P 2 (X, Y, Z, Tyr 
projects in the three images via the three projection matrices Mi — (/3«3 0), Mo — Asx4 and Ma — Bax4, we 
can write : 
f-—z 
y—z 
wat — ra! 
vai - Pr =CP =0 (2) 
g'bT — bT 
y bT ve zz" 
The rank of C cannot be more than 3. As a consequence, all 4 x 4 determinants are vanishing. Developping 
the equations leads to the bilinear and trilinear constraints. 
3.2 Results 
In our tests, we were given the third image (to recover). We tracked points between the 3 images to compute 
the 17 parameters of the trilinear relation (for the house, we used 48 matched points, and a least squares 
minimization). 
These parameters define the position of the virtual camera. Not every configuration of the parameters 
describe a physical situation ; they only define a perspective projection of the projective model of the scene onto 
the plane of the virtual camera. 
The only case where all forms of the trilinear relations are degenerated is for points P lying on the line 
joining the optical centers. For these points, it is impossible to recover depth anyway. 
See figure 3 the reprojected house we obtained. The “holes” in the synthesized image arise from missing 
points in the matching process, and from a stretching phenomenon : two adjacent pixels in the reference views 
can be transfered at quite different (non-adjacent) places in the third view. There are numerous ways to avoid 
the holes ; we can also simply filter the synthesized image (figure 4). 
4 Discussion 
We presented an algorithm to, given two 2D views of the same scene, build any other 2D view, while avoiding 
the burden of a 3D reconstruction. This work has strong connections with [Lav 94], and also with [Wer 94]. 
The process works on real images. The house pictures were taken from a distance of about 1.1 m with a 
standard video camera (Pulnix TM-6EX with a Schneider-Kreunach Xenoplan 17 mm lens ; pixels are about 
10 um wide) and digitized with a FG150 frame-grabber from Imaging Technologies. 
We also successfully experimented on outdoor scenes, taken with a home video camera (interlaced video). 
Figure 5 and 6 shows the result obtained on such a scene. Difficulties arise from the matching process, especially 
on the leaves of the trees in the foreground, or on the waves of the river. 
IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop “From Pixels to Sequences”, Zurich, March 22-24 1995