Mean and median filters bad

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OK its a crap Figure. I spent ages getting the trace above each new one to show the moving window used to compute the output of each stage.

Left panel: running mean. Stage (scale) 1 window of 3. Stage 2 window of 5 on the previous stage. And so forth. Each stage simplifies the signal until, at the bottom, it fades away. BUT it does not preserve scale-space and that is bad. Right panel: running median. Median filters had many followers. It was thought that they preserved the edges in a meaningful way. Stage (scale) 1 window of 3. The filter removes extrema of length (scale) 1. Stage 2 window of 5 on the previous stage and this filter removes extrema of scale 2. And so forth. Each stage simplifies the signal until, at the bottom, it fades away. It is tempting to imagine that extrema at each scale are removed in a nice regular way BUT no, it all falls apart at larger scales it does not preserve scale-space and that is bad.