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Re: Bivariate Pchip
Posted:
Jan 18, 2013 4:46 PM
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"Bruno Luong" <b.luong@fogale.findmycountry> wrote in message <kdccnj$kcc$1@newscl01ah.mathworks.com>...
> "John D'Errico" <woodchips@rochester.rr.com> wrote in message <kdbk43$afl$1@newscl01ah.mathworks.com>...
>
> >
> > pchip in 2-d as a tensor product form has been shown
> > NOT to be adequate for the general desired behavior.
> > (It sometimes will produce an acceptable result, but in
> > general, it is not adequate.)
> >
>
> John, I must admit that I don't know what is the adequate definition of shape preserving interpolation in 2D.
>
> I just though at least the tensorial product would preserve at least the monotonic in any cut parallel to the two axis. Unless I'm mistaken, this also implies that the interpolation within a patch will necessary be bounded by the four corner data, thus never overshoot.
>
> May be these characteristics are not enough for OP, but it is better than nothing. No?
>
> Bruno
Yes, monotonicity is not a trivial thing to discuss
when you move to more than one independent
dimension.
It has been many years since I looked at it, but I
recall the statement that a simple tensor product
version of pchip is not adequate here. And I don't
recall under which circumstances that interpolant
fails.
The case of a linear tensor product interpolant
(often known as a bilinear interplant, as used by
photoshop) is a good one to study, as that may
offer an idea. Consider the function f(x,y), where
it is defined at the 4 corners of the unit square,
and we will use bilinear interpolation over that
domain.
f((0,0) = 1
f(0,1) = 0
f(1,0) = 0
f(1,1) = 1
Within the unit square, the tensor product linear
interpolant reduces to
f(x,y) = 1 - y - x + 2*x*y
which is clearly not linear. If we hold either x or y
fixed, then of course it is again linear. But if we
interpolate along some other path, perhaps a
diagonal one from corner to corner, then the
interpolant will be a quadratic polynomial with
min or max at the center of the square. This is
the classic problem with a tensor product
interpolant when applied in three dimensions
for color science problems, and it is why that
method is avoided.
What I don't know (without some study) is if the
tensor product pchip can still have non-monotonic
behavior parallel to an axis. (Perhaps tomorrow
if I have some energy, as I think I know how to
show if this can happen.)
John
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