My understanding is that the reason we specify the input values of both P and its derivative is because we want to create a curve that is a smooth interpolation of discrete samples (e.g. pixels at 0 and 1). Intuitively, one reason a cubic function is needed here (in addition to the 4 degrees of freedom mentioned in the next slide) is that a lesser degree polynomial would have a derivative that is either linear or just a constant (for quadratic or linear polynomial respectively), which wouldn't give us the complexity required for smooth interpolation.
mishywangiepie
If we are using the middle point only to determine derivative value by a simple slope calculation, does this imply that cubic hermite interpolation only fits every other point since the middle point is not guaranteed to fit within the curve?
Pinbat
It's not every other point, but there is interpolation error and it can be minimized: https://en.wikipedia.org/wiki/Polynomial_interpolation#Interpolation_error
qqqube
I find it interesting that we only consider four input constraints; it turns out that a higher degree polynomial interpolation does not necessarily always produce more accurate results, known as Runge's phenomenon. This would exemplify overfitting; as the degree of the polynomial increases the curves become unnecessarily oscillatory.
kavimehta
What would be the trade offs to sampling at the endpoints at sampling at other locations, maybe the middle?
Michael-hsiu
I think sampling at the endpoints prevents the resulting interpolated polynomial from going to infinity or negative infinity at the endpoints. I guess when we generalize to multiple adjacent pairs of points, we need to force the endpoints of adjacent polynomials to have the same value, which may be hard to represent mathematically without the endpoints.
My understanding is that the reason we specify the input values of both P and its derivative is because we want to create a curve that is a smooth interpolation of discrete samples (e.g. pixels at 0 and 1). Intuitively, one reason a cubic function is needed here (in addition to the 4 degrees of freedom mentioned in the next slide) is that a lesser degree polynomial would have a derivative that is either linear or just a constant (for quadratic or linear polynomial respectively), which wouldn't give us the complexity required for smooth interpolation.
If we are using the middle point only to determine derivative value by a simple slope calculation, does this imply that cubic hermite interpolation only fits every other point since the middle point is not guaranteed to fit within the curve?
It's not every other point, but there is interpolation error and it can be minimized: https://en.wikipedia.org/wiki/Polynomial_interpolation#Interpolation_error
I find it interesting that we only consider four input constraints; it turns out that a higher degree polynomial interpolation does not necessarily always produce more accurate results, known as Runge's phenomenon. This would exemplify overfitting; as the degree of the polynomial increases the curves become unnecessarily oscillatory.
What would be the trade offs to sampling at the endpoints at sampling at other locations, maybe the middle?
I think sampling at the endpoints prevents the resulting interpolated polynomial from going to infinity or negative infinity at the endpoints. I guess when we generalize to multiple adjacent pairs of points, we need to force the endpoints of adjacent polynomials to have the same value, which may be hard to represent mathematically without the endpoints.