Edward M. Mikhail
2.2 Trilinearity Model
The geometric relationship between three perspective views is established through the trifocal tensor, which consists of
27 dependent intrinsic elements of a vector T. Previous research has shown that the trifocal tensor provides greater
immunity to degenerate cases that plague the F matrix techniques for dealing with triplets. Four homogeneous
equations that are linear with respect to the elements of T can be written as a function of the image coordinates on three
overlapping photographs and the 27 elements of T. Thus the trifocal tensor can be computed linearly using a set of at
least seven points whose image coordinates are measured on all three photographs. Lack of robustness using this linear
technique for three-photo image transfer and object reconstruction has prompted us to derive constraints among the 27
dependent elements of T to provide stability to the solution, which becomes non-linear due to the constraint equations.
An additional constrained solution is provided, which uses only three of the four dependent equations.
2.2.1 Linear Formulation. This subsection provides a summary of the derivation of the four trilinearity equations.
The trilinearity equations are derived from the projective relationship that exists between an object point in 3D model
space and its associated image coordinates in the 2D image planes of each of the three photographs. Consider the
perspective projection for a single photograph. Using projective geometry, a 3 by 4 camera transformation matrix, P,
linearly relates 3D object coordinates to 2D image coordinates as follows:
B »"ulsPx y z 1f (5)
where x,y are image point coordinates,
X,Y,Z are object point coordinates, and
the symbol " =" implies equality up to a scale factor.
Since we are establishing a relationship between image coordinates, we are not concerned with absolute ground
positions of points for now. Therefore, we can assume camera 1 is fixed, and consider the relative positions and
orientations of cameras 2 and 3 with respect to the fixed camera 1. Then the relative P's (P,,P.,,P.,) of the three
cameras can be expressed as follows:
P,= LI : 0 | P. fa, } P. b, (6)
3x4 x 3x4
3x1 3x4
These are the matrices that relate the homogeneous model coordinates (X;, X», X5, /) and the three pairs of image
coordinates, (x, y), (x^ y, (x", y"), on the three images, respectively. The reference [Hartley, 1996] explains why P;
may be assumed, as shown in equation (6), without loss of generality. Note that there are 24 total unknown elements in
the three P,’s. Because there is more than one possible value for t, the 24 elements in P,;, P,., are replaced by a set of 27
dependent T coefficients, leading to the following four trilinearity equations (see [Shashua, 1997]):
x'(xT, - yT, € T,) — xx" (GT, + yT; £T.) t xQT, t yf, T,) -(xTi + yT,, +T,) =0
Y Gt. T - YOO, YT. X1.) + X(xT,, t yT,, t T, )- GT, t yT, t T. )=0 (7)
x'(xT,, * yT4 * T, )- x y (XT, * YT, + T+ YT, € yT, t T,) - QT, * yT * TL) =0
Y (Ty * yT4 * T4)- Y GT, yT, c1.) y (T t yTu t Ts )- GT4 yT * T„) =0
To establish the relationship between image coordinates on a triplet of photographs, we write four equations (7) per
point as a function of its six image coordinate observations (x,y,x"y"x",y") and 27 trilinearity coefficients (T's), and
solve the linear system of homogeneous equations for the 27 parameters. Note that the minimum number of points
would be seven, since we would write 28 equations to solve for 27 unknowns.
This linear solution has two problems: 1) although the four equations (7) are linearly independent, only three of the
four equations are algebraicly independent [Vieville, et. al., 1993]; and 2) the 27 parameters (T's) are not independent.
Therefore, the solution to these linear equations is used to obtain initial estimates for the parameters to be used as input
586 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.
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