Hi, am I right to say that the higher the order of a B-spline, the more continuous derivatives it has? And that sinc interpolation has infinite continuous derivatives? What about windowed sinc? Is the Lanczos a good choice for an interpolator whose goal is to have as many continuos and smooth derivatives as possible, and to pass through the points? Thanks, Mike
A question about continuous derivatives
Started by ●October 21, 2005
Reply by ●October 21, 20052005-10-21
in article 4358f7d8$0$24646$4fafbaef@reader3.news.tin.it, mike@unknown.no.spam.com at mike@unknown.no.spam.com wrote on 10/21/2005 10:14:> am I right to say that the higher the order of a B-spline, the more > continuous derivatives it has?yes.> And that sinc interpolation has infinite continuous derivatives?yes.> What about windowed sinc?all derivatives are continuous until you get to the place where the window is spliced to silence. then at least *some* derivative at some order has to have a jump discontinuity.> Is the Lanczos a good choice for an interpolator whose goal is to > have as many continuos and smooth derivatives as possible, and to > pass through the points?maybe consider Hermite polynomials or Osculating polynomials. maybe check out http://www.biochem.oulu.fi/~oniemita/dsp/deip.pdf -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
Reply by ●October 22, 20052005-10-22
robert bristow-johnson wrote:> in article 4358f7d8$0$24646$4fafbaef@reader3.news.tin.it, > mike@unknown.no.spam.com at mike@unknown.no.spam.com wrote on 10/21/2005 > 10:14: > > > am I right to say that the higher the order of a B-spline, the more > > continuous derivatives it has? > > yes. > > > And that sinc interpolation has infinite continuous derivatives? > > yes. > > > What about windowed sinc? > > all derivatives are continuous until you get to the place where the window > is spliced to silence. then at least *some* derivative at some order has to > have a jump discontinuity.This is not necessarily the case. Some windows are infinitely often differentiable.
Reply by ●October 22, 20052005-10-22
abariska@student.ethz.ch wrote:> robert bristow-johnson wrote: >> > What about windowed sinc? >> >> all derivatives are continuous until you get to the place where >> the window is spliced to silence. then at least *some* >> derivative at some order has to have a jump discontinuity. > > This is not necessarily the case. Some windows are infinitely > often differentiable.But we would moreover require that the value and all derivatives be zero in the splice location, only true of the all-zero "window". Martin -- Quidquid latine scriptum sit, altum viditur.
Reply by ●October 22, 20052005-10-22
abariska@student.ethz.ch wrote: Are you in China now? Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●October 22, 20052005-10-22
Martin Eisenberg wrote:> abariska@student.ethz.ch wrote: > >>robert bristow-johnson wrote: >> >>>>What about windowed sinc? >>> >>>all derivatives are continuous until you get to the place where >>>the window is spliced to silence. then at least *some* >>>derivative at some order has to have a jump discontinuity. >> >>This is not necessarily the case. Some windows are infinitely >>often differentiable. > > > But we would moreover require that the value and all derivatives be > zero in the splice location, only true of the all-zero "window". > > > Martin >Consider the function f(x) = exp(-1/x^2) for x other than 0 and f(0) = 0. (This choice of value at zero removes the discontinuity in f that would otherwise exist there.) Now f and all its derivatives are easily seen to have value zero at x=0, despite the fact that f is not identically zero. So we can form a splice g like so: let g(x) = 0 if x <= 0 and g(x) = exp(-1/x^2) if x > 0, and g will also be infinitely often (continuously) differentiable. It isn't much more work to concoct a window function which is smooth (i.e. has infinitely many continuous derivatives everywhere), takes on only values between 0 and 1 inclusive, is 0 except on ]-1,1[, is 0 nowhere in ]-1,1[, and is 1 only at 0. Windowing sincs with such functions allows one to interpolate regularly-spaced data points with a smooth interpolant that exactly passes through each data point. But in practice something like a spline will be cheaper to compute, and one seldom needs the interpolant to have infinitely many continuous derivatives. Robert E. Beaudoin
Reply by ●October 22, 20052005-10-22
CH is (still) Switzerland. I'm recuperating from a two-week stroke of pneumonia. Every now and again, I manage to turn on the home laptop.
Reply by ●October 22, 20052005-10-22
in article 1129968325.560607.57340@z14g2000cwz.googlegroups.com, abariska@student.ethz.ch at abariska@student.ethz.ch wrote on 10/22/2005 04:05:> robert bristow-johnson wrote: >> in article 4358f7d8$0$24646$4fafbaef@reader3.news.tin.it, >> mike@unknown.no.spam.com at mike@unknown.no.spam.com wrote on 10/21/2005 >> 10:14: >> >>> am I right to say that the higher the order of a B-spline, the more >>> continuous derivatives it has? >> >> yes. >> >>> And that sinc interpolation has infinite continuous derivatives? >> >> yes. >> >>> What about windowed sinc? >> >> all derivatives are continuous until you get to the place where the window >> is spliced to silence. then at least *some* derivative at some order has to >> have a jump discontinuity. > > This is not necessarily the case.it *is* necessarily the case.> Some windows are infinitely often differentiable.not at where they get spliced to silence. conceptually, you can have windows like the Gaussian that goes on forever and never gets spliced to anything, but practically it is kinda difficult to store and operate on an infinite buffer of data. in practice, *all* windows are eventually spliced to silence. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
Reply by ●October 22, 20052005-10-22
Robert E. Beaudoin wrote:> Consider the function f(x) = exp(-1/x^2) for x other than 0 and > f(0) = 0.Thanks for straightening me out. Martin -- Seek simplicity and mistrust it. --Alfred Whitehead
Reply by ●October 22, 20052005-10-22
