Methods, apparatuses, and systems for determining a binary mask pattern from a pixelated mask pattern include: determining, by a processor, based on a fast marching method (FMM), arrival values for pixels of a portion of the pixelated mask pattern; determining the binary mask pattern based on the arrival values; and updating at least one of the arrival values based on a comparison between a design pattern corresponding to the pixelated mask pattern and a substrate pattern simulated based on the binary mask pattern.