Lecture 7: Intro to Geometry, Splines, and Bezier Curves (36)
CarlQGan
Does Hermite Basis bring any sort of advantage towards computation, like speed-ups or simplifications?
NKJEW
@CarlQGan I'd assume that operating in the Hermite Basis does result in a computational speedup, since it only requires one matrix multiply to get from the inputs (h0, h1, etc) to the final polynomial, whereas the more "direct" method involves solving a system of linear equations (as shown on slide 35). Mainly, the simplification seems to stem from the fact that these Hermite Basis functions are independent of the inputs, so they can be precomputed for any pair of arbitrary points (or possibly even hardcoded for ease of access). Then when you actually go about doing the math, your input variables can simply be plugged in without any change of basis.
mcallisterdavid
This Hermite Basis reminds me a lot of JPEG compression and Fourier Series. Establishing function bases then performing a linear combination to restore the full function allows for a really tight representation of the signal/function!
nobugnohair
The Hermite Basis functions seem to be able to increase the calculation efficiency in large-scale linear transforms as they greatly reduce the cost of matrix calculation in interpolations.
nobugnohair
The Hermite Basis functions seem to be able to increase the calculation efficiency in large-scale linear transforms as they greatly reduce the cost of matrix calculation in interpolations.
cchendyc
Once we have the Hermite Basis, we can reuse it afterward with other continuous points h4h5h6h7 for a higher level interpolation or simply switch intervals. Using the direct basis will only give info about the degree of the polynomial
Does Hermite Basis bring any sort of advantage towards computation, like speed-ups or simplifications?
@CarlQGan I'd assume that operating in the Hermite Basis does result in a computational speedup, since it only requires one matrix multiply to get from the inputs (h0, h1, etc) to the final polynomial, whereas the more "direct" method involves solving a system of linear equations (as shown on slide 35). Mainly, the simplification seems to stem from the fact that these Hermite Basis functions are independent of the inputs, so they can be precomputed for any pair of arbitrary points (or possibly even hardcoded for ease of access). Then when you actually go about doing the math, your input variables can simply be plugged in without any change of basis.
This Hermite Basis reminds me a lot of JPEG compression and Fourier Series. Establishing function bases then performing a linear combination to restore the full function allows for a really tight representation of the signal/function!
The Hermite Basis functions seem to be able to increase the calculation efficiency in large-scale linear transforms as they greatly reduce the cost of matrix calculation in interpolations.
The Hermite Basis functions seem to be able to increase the calculation efficiency in large-scale linear transforms as they greatly reduce the cost of matrix calculation in interpolations.
Once we have the Hermite Basis, we can reuse it afterward with other continuous points h4h5h6h7 for a higher level interpolation or simply switch intervals. Using the direct basis will only give info about the degree of the polynomial