Threshold dynamics is well known as a popular algorithm for simulating the motion of interfaces; nowadays, it's broadly used in image segmentation and solid dewetting. The original version of the algorithm, which is only first-order accurate in time in the two-phase setting, was proposed by Merriman, Bence, and Osher in 1992. Since then, many extensions of the algorithm have been given, for instance, to multiphase mean curvature motion, where it has proven particularly useful and flexible. There have also been high-order accurate versions of the algorithm proposed in several previous studies. Our goal is to take a step toward providing more accurate versions of threshold dynamics with nice properties, for example, monotonicity, which respects the comparison principle of the exact evolution. We'll also introduce our recent finding: the connection between threshold dynamics and the median filter, which is an elegant, monotone discretization of the level set formulation of motion by mean curvature. This results in a new level set method for multiphase mean curvature motion that allows locating the interface via interpolation and enforces the correct junction condition at the free boundaries, at the generality demanded by applications.