One method to estimate image values in the presence of noise and discontinuities is to try all possible configurations of discontinuities in a neighborhood, and then select the one that yields the best estimate according to some criterion. We call these methods multiple oriented filters since each configuration is usually a rotated and shifted version of another. Multiple oriented filters work well, but are computationally intensive for the large neighborhood sizes required by high-noise problems. This paper describes a method that optimizes the implementation of multiple oriented filters, reduces the time complexity, and makes the algorithm practical, even when noise levels are high. We do this by exploiting the fact that each neighborhood configuration is nearly the same as several others. We apply the optimization to smoothing estimators with discontinuities and binary restoration of thin sheets, and speculate that the optimization may also apply well to estimation of optical flow with discontinuities.