Median Filtering by Threshold Decomposition: Induction Proof

In building a robot for the FTC competition, our team needed to remove motor noise from our sensor signals. So we settled on using a median filter because of the medians superior removal of impulsive noise. For us, however, the foundational publications that describe these filters were challenging to understand. Having learned the concept of proof by induction from the MIT OpenCourseWare course, “Mathematics for Computer Science” (MIT Course Number 6.042J / 18.062J), we developed an original proof for the principle of median filter threshold decomposition in order to better understand their operation. The induction is over the number of quantized threshold levels for the sequence of input values as applied to both the standard and recursive median filter.