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Photometric stereo and depth-map estimation provide a way to construct a depth map from images of an object under one viewpoint but with varying illumination directions. While estimating surface normals using the Lambertian model of reflectance is well established, depth-map estimation is an ongoing field of research and dealing with image noise is an active topic. Using the zero-mean Gaussian model of image noise, this paper introduces a method for maximum likelihood depth-map estimation that accounts for the propagation of noise through all steps of the estimation process. Solving for maximum likelihood depth-map estimates involves an independent sequence of nonlinear regression estimates, one for each pixel, followed by a single large and sparse linear regression estimate. The linear system employs anisotropic weights, which arise naturally and differ in value to related work. The new depth-map estimation method remains efficient and fast, making it practical for realistic image sizes. Experiments using synthetic images demonstrate the method's ability to robustly estimate depth maps under the noise model. Practical benefits of the method on challenging imaging scenarios are illustrated by experiments using the Extended Yale Face Database B and an extensive data set of 500 reflected light microscopy image sequences.