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MPFITELLIPSE

MPFITELLIPSE

## Author

Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
craigm@lheamail.gsfc.nasa.gov
UPDATED VERSIONs can be found on my WEB PAGE:
http://cow.physics.wisc.edu/~craigm/idl/idl.html

## Purpose

Approximate fit to points forming an ellipse

## Major Topics

Curve and Surface Fitting

## Calling Sequence

parms = MPFITELLIPSE(X, Y, start_parms, [/TILT, WEIGHTS=wts, ...])

## Description

MPFITELLIPSE fits a closed elliptical or circular curve to a two
dimensional set of data points. The user specifies the X and Y
positions of the points, and an optional set of weights. The
ellipse may also be tilted at an arbitrary angle.
IMPORTANT NOTE: this fitting program performs simple ellipse
fitting. It will not work well for ellipse data with high
eccentricity. More robust answers can usually be obtained with
"orthogonal distance regression." (See FORTRAN package ODRPACK on
The best fitting ellipse parameters are returned from by
MPFITELLIPSE as a vector, whose values are:
P Ellipse semi axis 1
P Ellipse semi axis 2 ( = P if CIRCLE keyword set)
P Ellipse center - x value
P Ellipse center - y value
P Ellipse rotation angle (radians) if TILT keyword set
If the TILT keyword is set, then the P is meant to be the
semi-major axis, and P is the semi-minor axis, and P
represents the tilt of the semi-major axis with respect to the X
axis. If the TILT keyword is not set, the P and P represent
the ellipse semi-axes in the X and Y directions, respectively.
The returned semi-axis lengths should always be positive.
The user may specify an initial set of trial parameters, but by
default MPFITELLIPSE will estimate the parameters automatically.
Users should be aware that in the presence of large amounts of
noise, namely when the measurement error becomes significant
compared to the ellipse axis length, then the estimated parameters
become unreliable. Generally speaking the computed axes will
overestimate the true axes. For example when (SIGMA_R/R) becomes
This unreliability is also pronounced if the ellipse has high
eccentricity, as noted above.
Users can weight their data as they see appropriate. However, the
following prescription for the weighting may serve as a good
starting point, and appeared to produce results comparable to the
typical chi-squared value.
WEIGHTS = 0.75/(SIGMA_X^2 + SIGMA_Y^2)
where SIGMA_X and SIGMA_Y are the measurement error vectors in the
X and Y directions respectively. However, this has not been
robustly tested, and it should be pointed out that this weighting
may only be appropriate for a set of points whose measurement
errors are comparable. If a more robust estimation of the
parameter values is needed, the so-called orthogonal distance
regression package should be used (ODRPACK, available in FORTRAN
at www.netlib.org).

## Inputs

X - measured X positions of the points in the ellipse.
Y - measured Y positions of the points in the ellipse.
START_PARAMS - an array of starting values for the ellipse
parameters, as described above. This parameter is
optional; if not specified by the user, then the
ellipse parameters are estimated automatically from
the properties of the data.

## Returns

Returns the best fitting model ellipse parameters. Returned
values are undefined if STATUS indicates an error condition.

## Keywords

** NOTE ** Additional keywords such as PARINFO, BESTNORM, and
STATUS are accepted by MPFITELLIPSE but not documented
here. Please see the documentation for MPFIT for the
CIRCULAR - if set, then the curve is assumed to be a circle
instead of ellipse. When set, the parameters P and
P will be identical and the TILT keyword will have
no effect.
PERROR - upon return, the 1-sigma uncertainties of the returned
ellipse parameter values. These values are only
meaningful if the WEIGHTS keyword is specified properly.
If the fit is unweighted (i.e. no errors were given, or
the weights were uniformly set to unity), then PERROR
will probably not represent the true parameter
uncertainties.
If STATUS indicates an error condition, then PERROR is
undefined.
QUIET - if set then diagnostic fitting messages are suppressed.
Default: QUIET=1 (i.e., no diagnostics]
STATUS - an integer status code is returned. All values greater
than zero can represent success (however STATUS EQ 5 may
indicate failure to converge). Please see MPFIT for
the definitions of status codes.
TILT - if set, then the major and minor axes of the ellipse
are allowed to rotate with respect to the data axes.
Parameter P will be set to the clockwise rotation angle
of the P axis in radians, as measured from the +X axis.
P should be in the range 0 to !dpi.
WEIGHTS - Array of weights to be used in calculating the
chi-squared value. The chi-squared value is computed
as follows:
CHISQ = TOTAL( (Z-MYFUNCT(X,Y,P))^2 * ABS(WEIGHTS)^2 )
Users may wish to follow the guidelines for WEIGHTS
described above.

## Example

; Construct a set of points on an ellipse, with some noise
ph0 = 2*!pi*randomu(seed,50)
x = 50. + 32.*cos(ph0) + 4.0*randomn(seed, 50)
y = -75. + 65.*sin(ph0) + 0.1*randomn(seed, 50)
; Compute weights function
weights = 0.75/(4.0^2 + 0.1^2)
; Fit ellipse and plot result
p = mpfitellipse(x, y)
phi = dindgen(101)*2D*!dpi/100
plot, x, y, psym=1
oplot, p+p*cos(phi), p+p*sin(phi), color='ff'xl
; Fit ellipse and plot result - WITH TILT
p = mpfitellipse(x, y, /tilt)
phi = dindgen(101)*2D*!dpi/100
; New parameter P gives tilt of ellipse w.r.t. coordinate axes
; We must rotate a standard ellipse to this new orientation
xm = p + p*cos(phi)*cos(p) + p*sin(phi)*sin(p)
ym = p - p*cos(phi)*sin(p) + p*sin(phi)*cos(p)
plot, x, y, psym=1
oplot, xm, ym, color='ff'xl

## References

MINPACK-1, Jorge More', available from netlib (www.netlib.org).
"Optimization Software Guide," Jorge More' and Stephen Wright,
SIAM, *Frontiers in Applied Mathematics*, Number 14.

## Modification History

Ported from MPFIT2DPEAK, 17 Dec 2000, CM
More documentation, 11 Jan 2001, CM
Example corrected, 18 Nov 2001, CM
Change CIRCLE keyword to the correct CIRCULAR keyword, 13 Sep
2002, CM
Add error messages for SYMMETRIC and CIRCLE, 08 Nov 2002, CM
Found small error in computation of _EVAL (when CIRCULAR) was set;
sanity check when CIRCULAR is set, 21 Jan 2003, CM
Convert to IDL 5 array syntax (!), 16 Jul 2006, CM
Move STRICTARR compile option inside each function/procedure, 9
Oct 2006
ellipses, 17 Sep 2007, CM
Clarify documentation of TILT angle; make sure output contains
semi-major axis first, followed by semi-minor; make sure that
semi-axes are always positive (and can handle negative inputs)
17 Sep 2007, CM
Output tilt angle is now in range 0 to !DPI, 20 Sep 2007, CM
Some documentation clarifications, including to remove reference
to the "ERR" keyword, which does not exist, 17 Jan 2008, CM
Swapping of P and P only occurs if /TILT is set, 06 Nov
2009, CM
Document an example of how to plot a tilted ellipse, 09 Nov 2009, CM
Check for MPFIT error conditions and return immediately, 23 Jan 2010, CM

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