How does upsampling work here? In other words, what is the algorithm used to transform $f_2$ to $l_1$?
motoole2
The short answer is that we perform some type of interpolation, i.e., we insert new columns and rows with values determined by its neighbors. But this does need to be done somewhat carefully.
For example, suppose that your image consisted of discrete points representing a continuous sinusoid. If you want to upsample that image, one idea might be to (i) fit a continuous sinusoidal signal to the discrete points, and (ii) use a higher number of discrete samples to represent that same sinusoid. This makes a critical assumption that the signal was not aliased (frequency of the sinusoid is no larger than half of the number of samples used to represent said signal).
When it comes to more general signals, the same idea applies. Provided that the frequency content of the original image is not too high, we can reconstruct that signal exactly by fitting linear combinations of sinusoids to the data.
Note that there are other ways to upsample the data, e.g., through linear interpolation. This, however, is not necessarily going to provide a perfect inverse to the downsampling operation.
How does upsampling work here? In other words, what is the algorithm used to transform $f_2$ to $l_1$?
The short answer is that we perform some type of interpolation, i.e., we insert new columns and rows with values determined by its neighbors. But this does need to be done somewhat carefully.
For example, suppose that your image consisted of discrete points representing a continuous sinusoid. If you want to upsample that image, one idea might be to (i) fit a continuous sinusoidal signal to the discrete points, and (ii) use a higher number of discrete samples to represent that same sinusoid. This makes a critical assumption that the signal was not aliased (frequency of the sinusoid is no larger than half of the number of samples used to represent said signal).
When it comes to more general signals, the same idea applies. Provided that the frequency content of the original image is not too high, we can reconstruct that signal exactly by fitting linear combinations of sinusoids to the data.
Note that there are other ways to upsample the data, e.g., through linear interpolation. This, however, is not necessarily going to provide a perfect inverse to the downsampling operation.