[PD] tabread4~ "broken" interpolation algorithm - was Re: Max Smoother Audio than Pd?

[Roman Haefeli]

Hi Matt

Thanks for the detailed explanation.  I still have troubles getting the
idea of the Lagrange interpolator in the context of [tabread4~]. You
say, that it finds the cubic polynomial which hits all four points. But
what is the advantage of that? As I understand [tabread4~], if the index
is between 5 and 6, it will use the cubic hitting the points at indizes
4, 5, 6 and 7. If the index is between 6 and 7, it will use the cubic
going through the points at 5, 6, 7 and 8. So for the former the fact,
that the curve hits also the points 4 and 7 seems irrelevant and so does
it for the latter for the points 5 and 8, since always only the segment
between floor(n) and ceiling(n) appears in the result. Or is it my
misunderstanding and this is completely wrong?
It seems logical to me, that discontinuities in the first derivative are
avoided in order not to add any partials to the signal. What I don't get
is why it is good to hit all four points, if the segments "outside" the
middle segment are ignored/not used for the result.

I haven't studied those things in school, so please forgive, if I am
asking things with completely wrong assumptions. I am just trying to
understand why [tabread4~] is good for what it is.

Roman
 


On Mon, 2010-03-29 at 16:33 -0400, Matt Barber wrote:

> Miller's is a true implementation of the former -- his is a Lagrange
> interpolator which goes through all points -- it's algebraically
> identical to the cubic interpolator in csound, and so it should have a
> similar "sound" as any of the table-reading opcodes in csound that
> also employ cubic interpolation.
> 
> The latter is an Hermite interpolator which uses the outside points to
> approximate the first derivative -- the resulting curve only passes
> through the middle two points, but doesn't go through the outside two;
> this ensures that as it's pieced together the first derivative will be
> continuous at the junctions.  It's algebraically identical to the
> cubic interpolator in supercollider.
> 
> They're two different approaches -- each has its own frequency
> response, but both are true cubics.  If you want to match all four
> points AND the first derivatives, you have to use a 5th-order
> polynomial.  The formulas are easily derivable using the Gaussian
> method, and it's easy to implement all these as a library of functions
> that can be accessed by the relevant objects, where the user can
> choose which type of interpolation he/she wants to use.
> 
> Matt