Increasing the accuracy with techniques of interpolation

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I am currently pursuing my Masters in electrical engineering and the class I am taking is Image Processing. Last night my professor introduced an interesting idea of ways of "guessing" the color of pixels in a blown up image. In other words, he talked about ways to fill in pixels that were missing once you blow up an image.

He talked about three different types of interpolation (which I've never heard of because I was an electrical engineering undergraduate as well):

  • Natural neighboring (which he described as "the fastest way of getting the wrong answer")
  • Bilinear Interpolation
  • Bicubic Interpolation

All of these have one thing in common: "Guessing" a value based on its surroundings.

How can we make an assumption that by mathematically "guessing", we can get a desirable answer? If it's not the correct answer, how would we know? My guess is that if we want to be more accurate, we would have to increase the area that we're sampling to see more behavior around our targeted area.