[Numpy-discussion] Solving a NLLSQ Problem by Pieces?

Wayne Watson sierra_mtnview@sbcglobal....
Sat Jun 26 12:12:25 CDT 2010

```The context here is astronomy and optics. The real point is in the last
paragraph.

I'm looking at a paper that deals with 5 NL (nonlinear) equations and 8
unknown parameters.
A. a=a0+arctan((y-y0)/(x-x0)
B. z=V*r+S*e**(D*r)
r=sqrt((x-x0)**2+(y-y0)**2)
and
C.  cos(z)=cos(u)*cos(z)-sin(u)*sin(ep)*cos(b)
sin(a-E) = sin(b)*sin(u)/sin(z)

He's trying to estimate parameters of a fisheye lens which has taken
star images on a photo plate. For example, x0,y0 is the offset of the
center of projection from the zenith (camera not pointing straight up in
the sky.) Eq. 2 expresses some nonlinearity in the lens.

a0, xo, y0, V, S, D, ep, and E are the parameters. It looks like he uses
gradient descent (NLLSQ is nonlinear least squares in Subject.), and
takes each equation in turn using the parameter values from the
preceding one in the next, B. He provides reasonable initial estimates.

A final step uses all eight parameters. He re-examines ep and E, and
assigns new estimates. For all (star positions) on the photo plate, he
minimizes SUM (Fi**2*Gi) using values from the step for A and B, except
for x0,y0. He then does some more dithering, which I'll skip.

What I've presented is probably a bit difficult to understand without a
good optics understanding, but my question is something like is this
commonly done to solve a system of NLLSQ? It looks a bit wild. I guess
if one knows his subject well, then bringing some "extra" knowledge to
the process helps. As I understand it, he solves parameters in A, then
uses them in B, and so on. I guess that's a reasonable way to do it.

--

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Web Page:<www.speckledwithstars.net/>

```