From this perspective you can look at it as comparable to the dot product concept of 'how much' of this thing is in that thing. You can also think of it as a 'r

From this perspective you can look at it as comparable to the dot product concept of 'how much' of this thing is in that thing. You can also think of it as a 'rotation' operation - a change of basis into the frequency domain. That insight I garnered from graphics pioneer James Blinn. A related funky thing is that if you choose the right functions you can engineer the analysis to require only shifts and adds which was a boon for making bespoke compression schemes on limited hardware. An example is the Slant transform used in an early Intel video codec. IIRC the Hadamard transform was also similarly useful where the coefficients were all 1s and -1s. I also seem to recall that some modern codecs have modes that use transforms that avoid the traditional DCT. One last thing, I might have dreamt this but I recall reading a NASA tech report from the 70s or 80s outlining a video compression scheme that used a non DCT approach that could be calculated with simpler hardware without multiplication. I remember thinking, wow they were out front but never commercialized that.