In the above visualization the colors of the 3D points have a specific meaning, which is explained in the following table:
| Color | Description | Example |
|---|---|---|
| Red | Measured data point | Measured brightness of pixel (x,y) in a digital image |
| Green | Interpolated data point along one axis | Estimated brightness between pixel (x,y) and (x+1,y) Note the change only along the x dimension! |
| Blue | Interpolated data point along two axes | Estimated brightness between pixel (x,y) and (x+1,y+1) Note the change both along the x and the y dimension! |
The math behind the visualiuation is explained in the Wikipedia articles Cubic interpolation and Bicubic interpolation.
The concept of bicubic interpolation hereby refers to problems with two input variables, such as the x and y dimensions in the above example.
In these cases the basic cubic interpolation method is applied two times in a row, first along the x dimension using measured samples, then along the y dimension using the previously obtained estimates.
The details of this procedure a well explained in the corresponding Wikipedia entries.
What you should probably take from this article are two things:
(1) a visual understanding of cubic interpolation and (2) a feeling for when this method is applicable.
Also, if questions are open please do not hesitate to ask, and I will try to make my best to eliminate all certainties ;).