PanoTools mailing list archive

Mailinglist:proj-imim
Sender:Richard Moore
Date/Time:2001-Jan-06 04:42:58
Subject:Re: Number of Control Points, was Re:

Thread:


proj-imim: Re: Number of Control Points, was Re: Richard Moore 2001-Jan-06 04:42:58
Helmut Dersch wrote:

> John Blommers wrote:
> >
> >
> > Haw! Your are right. Each control point represents two variables to be
> > optimized.
>
> No. Each control point represents one equation.

It seems to me a single control point could represent two equations.

If the error function to be optimized is the Cartesian distance between the

two mapped locations of the control point, then yes, the cost is
represented
by a single equation.

However, if you are minimizing that distance with respect to two variables
you
get two partial derivatives that we want to find the zeros of. This should
work
if you hold r and v constant and optimize y and p -- the transformations
are linear.

More generally, we would want to optimize r and v as well as y and p. Now
the
equations become non-linear because of the rotation, but if you consider a
small
region the curves should be close enough to linear within that region. So
unless the
initial estimates of the variables are wildly wrong, I think adding a
second control
point is sufficient to give a unique solution.

Certainly if you add the lens correction factors into the mix, the
non-linearity
gets to be more of a factor. If we want to optimize a, b, and c, I suspect
three
points really are needed because one or two points doesn't give a unique
minimum.
That is, if we set b to 0 and optimize a for a single point, then change b
and
again optimize a, we should find a different value of a that gives the same
error.
This would be like looking for the lowest point in a trough that has a flat
bottom.
Add a second point, and we can optimize a and b with c held constant (now
we're
looking for the intersection of two troughs, and hope they're not
parallel). A third
point would allow us to find c.

Incidentally, I use panotools for single- and double-row panos of less than
full
360-degree coverage, with no fisheye lens (yet), so I find a rough fixed
estimate
of the correction factors a, b, and c to be adequate. Therefore, I do not
optimize
these parameters.

 -- Richard

P.S. -- This kind of thinking takes me way back. It's been a long time
since college for me.





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