Incomplete total least squares

Fitting data points with some model function such that the sum of squared orthogonal distances is minimized is well-known as TLS, i.e. as total least squares, see Van Huffel (1997). We consider situations where the model is such that there might be no perpendiculars from certain data points onto the model function and where one has to replace certain orthogonal distances by shortest ones, e.g. to corner or border line points. We introduce this considering the (now incomplete) TLS fit by a finite piece of a straight line. Then we study general model functions with linear parameters and modify a well-known descent algorithm (see Seufer (1996), Seufer/Spath (1997), Spath (1996), Spath (1997a) and Spath (1997b)) for fitting with them. As applications (to be used in computational metrology) we discuss incomplete TLS fitting with a segment of a circle, the area of a circle in space, with a cylinder, and with a rectangle (see also Gander/Hrebicek (1993)). Numerical examples are given for each case. Mathematics Subject Classification (1991): 65D10.