The least squares fit of a hyperplane to uncertain data

The authors became interested in this problem because they wanted to calibrate their laser range cameras. Laser range cameras are powerful tools for defining the Cartesian coordinates of surfaces of objects in an environment. The camera scans a laser beam over a scene and determines the distance ({tau}) by the time required for the beam to reach an object and return. The beam is directed back and forth ({phi}) and up and down {theta} by two rotating mirrors. Thus, the data measured by the camera has the form ({tau}, {phi}, {theta}). The geometry of the camera can be used to map the measured data to Cartesian coordinates (x, y, z). Since the geometry of the camera may not be known precisely, the cameras can be calibrated by comparing the calculated surface shapes to the known surface shapes. The most simple surface is a plane and many physical objects have planar surfaces. Thus, an important problem in the calibration of range cameras is to find the best (least squares) fit of a plane to a set of 3D points. For many least squares problems, the uncertainty is in one of the variables [for example, y = f(x) or z = f(xy)]. However, for some problems, the uncertainty is in the geometric transformation from measured data to Cartesian coordinates and all of the calculated variables are uncertain. The authors have formulated a constrained optimization problem to determine the least squares fit of a hyperplane to uncertain data. The first order necessary conditions require the solution of an eigenvalue problem. They have shown that the solution satisfies the second order conditions (the Hessian matrix is positive definite). Thus, the solution satisfies the sufficient conditions for a local minimum. They have performed numerical experiments that demonstrate that the solution is superior to alternative methods