On Thu, Jan 06, 2011 at 04:45:26AM -0800, Bart van Andel wrote:
>
> On Thursday, January 6, 2011 5:09:06 AM UTC+1, Tom Sharpless wrote:
> >
> > > It has been pointed out that the polynomial should only consist of
> > > even numbered powers, both for speed and for mathematical soundness.
> > >
> > I'm not convinced there is anything wrong with odd powers in a radial
> > correction function; since R is always >= 0 the odd symmetry is never
> > manifested.
> >
>
> Isn't there? There is a sharp discontinuity in (the derivative of) any
> odd-powered polynomial, when mirrored around the Y-axis. Since R can only
> have positive values, this discontinuity will happen at the image "center"
> which is used to compute R from. Therefore, it makes sense to avoid odd
> powers.
Ok. So the function y = x has exponent 1, and is not suitable. There
is a sharp corner at x=0, which mathematically manefests itself as a
discontinuity of the derivative:
dy/dx = 1 where x > 0
and = -1 where x < 0 (and undefined for x=0)
Now the derivative of
y = x ^3
is
y = 3 x^2
which after mirroring in x=0 becomes....
dy/dx = 3 x^2 where x > 0
and = 3 (-x)^2 where x < 0
which simplifies to
dy/dx = 3 x^2 for all x.
Even if we're talking about even powers, where the simplification
doesn't work, the value of the derivative is always zero at x=0, so
the derivative won't be discontinuous.
So I understand why the first power is bad, but why the third and
fifth?
(I'm thinking that this is for correcting lens distortions of the form
where pixels belonging at r,theta end up at r',theta on the sensor,
with r' = f (r), and f (x) = x + .... and we're talking about the
.... part Correct?)
Roger.
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