Very interesting to see that these are independent of a,b,c,d
moridin22
They actually aren't really independent of a,b,c, and d, since those four variables also depend on the inverse matrix in the middle; instead, the basis functions offer an alternative viewpoint of interpolation so that instead of having the arbitrary coefficients a,b,c, and d, the coefficients become the two points and their derivatives.
kevinliu64
It's interesting to note that the lines seem to be mirrors around the halfway mark (except for h2 and h3) which makes sense as we are trying to smooth out the curve using the given values at the endpoints. In addition, this reminds me of bilinear interpolation and barycentric coordinates because we're trying to get a value in the middle with only certain points. Very cool!
SeungjinYang
The functions make some intuitive sense. H0 raises the y value of start point by 1, H1 does it for endpoint, H3 raises the derivative of the startpoint by 1 (without doing anything for endpoint), and H4 does it for the endpoint.
sandykzhang
If we have prior knowledge about how the derivatives should be distributed (i.e. we don't want the derivatives to vary smoothly), how can we incorporate this knowledge into our solution?
killawhale2
@sandykzhang I guess you would change the inputs and add some bias towards whatever prior distribution of the derivates.
GitMerlin
@sandykzhang In my opinion, we can achieve this by, when we chain cubic Hermite functions together (in a later slide), giving two different derivatives for the place where we don't want smooth derivatives. One derivative is used when we pair this value with the value & derivative to its left, and another with that to its right. In that way, we can create a sharp derivative change.
Very interesting to see that these are independent of a,b,c,d
They actually aren't really independent of a,b,c, and d, since those four variables also depend on the inverse matrix in the middle; instead, the basis functions offer an alternative viewpoint of interpolation so that instead of having the arbitrary coefficients a,b,c, and d, the coefficients become the two points and their derivatives.
It's interesting to note that the lines seem to be mirrors around the halfway mark (except for h2 and h3) which makes sense as we are trying to smooth out the curve using the given values at the endpoints. In addition, this reminds me of bilinear interpolation and barycentric coordinates because we're trying to get a value in the middle with only certain points. Very cool!
The functions make some intuitive sense. H0 raises the y value of start point by 1, H1 does it for endpoint, H3 raises the derivative of the startpoint by 1 (without doing anything for endpoint), and H4 does it for the endpoint.
If we have prior knowledge about how the derivatives should be distributed (i.e. we don't want the derivatives to vary smoothly), how can we incorporate this knowledge into our solution?
@sandykzhang I guess you would change the inputs and add some bias towards whatever prior distribution of the derivates.
@sandykzhang In my opinion, we can achieve this by, when we chain cubic Hermite functions together (in a later slide), giving two different derivatives for the place where we don't want smooth derivatives. One derivative is used when we pair this value with the value & derivative to its left, and another with that to its right. In that way, we can create a sharp derivative change.