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Over-determined linear systems with noise in all measurements are common in computer vision, and particularly in motion estimation. Maximum likelihood estimators have been proposed to solve such problems, but except for simple cases, the corresponding likelihood functions are extremely complex, and accurate confidence measures do not exist. This paper derives the form of simple likelihood functions for such linear systems in the general case of heteroscedastic noise. We also derive a new algorithm for computing maximum likelihood solutions based on a modified Newton method. The new algorithm is more accurate, and exhibits more reliable convergence behavior than existing methods. We present an application to affine motion estimation, a simple heteroscedastic estimation problem.