We now describe the minimization process, making explicit use of the same notation adopted in FSLY99.
Given the model profiles (the selected model profiles if the threshold tf has been used) of each object Pi(m,n) the minimization procedure is necessary to obtain the scaling factors Fi of each object that best reproduce the measure image I. The best fit solution will be found by minimizing the χ2, that reads:
is the sum of all profiles, Bi is the background (in the measure image) of each objects, σ(m,n) is the r.m.s. of the measure image and m and n run over the pixels.
The best-fit solution is found by solving the linear system:
whose Hessian matrix is
and right-hand term is:
Since the model profiles are non-zero only within the area provided by the (dilated) segmentation map, most of the terms are Aij actually null (non-null terms will correspond to overlapping objects), such that the matrix A is very sparse. The solution of sparse linear systems can be very efficiently performed with the linear biconjugate gradient method (Press et al. 1992), such that the solution can be obtained even for a large number of free parameters (that is, of detected objects) that typically occurr in deep HST exposures.
Finally, the statistical uncertainties on the fitted parameters Fi are the diagonal terms of the inverse of the Hessian matrix Aij.