Direct Linear Transformation
The Direct Linear Transformation (DLT), originally proposed by Abdel-Aziz and Karara (1971), can be very useful to determine approximate values of the camera parameters. Rearranging the terms in the collinearity equations leads to the DLT equations
LjX + L2Y + L3Z + L4 — (x+ dx)(L9X + L10Y + LuZ +1) = 0
L5X + L6Y + L7Z + L8 — (y+ dy)(L9X + L10Y + L11Z +1) = 0 ‘ (5’5)
The DLT parameters L1y ■ ■ • L11 are related to the camera exterior and interior orientation parameters (ю, ф,к, ХC, YC, ZC) and (c, x,y ) (McGlone 1989).
Unlike the standard collinearity equations Eq. (5.1), Eq. (5.5) is linear for the DLT parameters when the lens distortion terms dx and dy are neglected. In fact, the DLT is a linear treatment of what is essentially a non-linear problem at the cost of introducing two additional parameters. The matrix form of the linear DLT equations for M targets is BL = C, where L = (L1y L11 )T,
C = (xlyy1y xM, yM )T, and B is the 2Mxll configuration matrix that can be directly obtained from Eq. (5.5). A least-squares solution for L is formally given by L = (BTB) 1BTC without using an initial guess. The camera parameters can be extracted from the DLT parameters from the following expressions
Because of its simplicity, the DLT is widely used in both non-topographic photogrammetry and computer vision. When dx and dy cannot be ignored, however, iterative solution methods are still needed and the DLT loses its simplicity. In general, the DLT can be used to obtain fairly good values of the exterior orientation parameter and the principal distance, although it gives a poor estimate for the principal-point location (xp, yp) (Cattafesta and Moore 1996).
Therefore, the DLT is valuable since it can provide initial approximations for more accurate methods like the optimization method discussed below for comprehensive camera calibration.