Linear solution of overdeterminated seven-parameter transformation

The current applications in Photogrammetry and mobile mapping systems (MMSs) use normally the centric perspective projection to construct the mathematic relationship of images and their corresponding ground points. The relationship is based on a three dimensional seven-parameter similarity transformation which is generally nonlinear. Traditionally, to solve these kinds of transformation problems, we have to first use the linearization method and give the initial values of unknowns together with iterative processing to solve the transformation parameters. Linearized solution procedures need relative good initial values of orientation elements and approximated ground coordinates of object points which may lead to some instable problems and computation-effectiveness. In this paper, we transform the nonlinear seven-parameter similarity transformation to a linear one by a special transformation, namely the Cayley transformation. In this model, we do not need the initial values and linearization. The solution procedures of the linear model here will be suggested by three step stages. The algebra formulation will be given. In this paper, a set of three coordinates is simulated with random errors for analysis. The results of the linear method including Helmert method and Molodensky method will be compared with those of the iteratively linearized method. Finally, we will discuss the application possibility of the linear model for Photogrammetry and MMSs.