Geometric reconstruction problems in computer vision can be solved by minimizing the maximum of reprojection errors, i.e., the L\infty-norm. Unlike L2-norm (sum of squared reprojection errors), the global minimum of L\infty-norm can be efficiently achieved by quasiconvex optimization. However, the maximum of reprojection errors is the meaningful measure to minimize only when the measurement noises are independent and identically distributed at every 2D feature point and in both directions in the image. This is rarely the case in real data, where the positional noise not only varies at different features, but also has strong directionality. In this paper, we incorporate the directional uncertainty model into a quasiconvex optimization framework, in which global minimum of meaningful errors can be efficiently achieved, and accurate geometric reconstructions can be obtained from feature points that contain high directional uncertainty.