What are reasons people might want to choose one of these methods vs another? Is one considered generally best? I know that, to me, the bi-linear one looks better.
@silentQ Something I noticed with the piecewise constant is that it approximates nearby pixels but becomes lost. Think of an application that needs to "de-noise" a picture to read text, like a license plate reader. To read a license from far away, the piecewise constant function can obtain a clearer reading of the image by tracing the representation of the numbers and letters from far away. A Piecewise bi-linear on the other hand can be used to blur objects in images or even create a gradient look (i.e. object shadows, sunsets).
@dvanmali what do you mean by "de-noise" a picture? Why is there less noise in the third image than in the second one?
@siminl Sorry if I was unclear, by "de-noise" I meant the removal of blurry edges, or a better word would be to "sharpen" an image. Here is a paper to describe this more: https://link.springer.com/content/pdf/10.1155/S1110865704404041.pdf. Figure 6 shows what applying this to an image is like.
Also for clarity, there is not really a sense of more-or-less noise between the second or third images. I wanted to point out the usage of the of the Piecewise constant function vs. the Piecewise bi-linear function. Here is a paper between Piecewise Constant and Piecewise bi-linear: https://www.ipol.im/pub/art/2011/g_lmii/article.pdf. Figure 24 shows what happens to an image with increasing bi-linearity.