Optionally, the software produces fits with the same x and y scales for both coordinate systems and/or fits with no rotation between the two coordinate systems. It has also been recently upgraded to perform optional outlier rejection.
The software is an extension of source code developed by Howard McCallon for the 2MASS program.
x'=[(1+a1)x+a2][cos(t0)+a0sin(t0)]+[(1+a3)y+a4][sin(t0)-a0cos(t0)]+x0 y'=-[(1+a1)x+a2][sin(t0)-a0cos(t0)]+[(1+a3)y+a4][cos(t0)+a0sin(t0)]+y0by adjusting the parameters {a0, a1, a2, a3, a4} using the method of least squares (LSQ). Thus the model allows for small additional differences in rotation angle, and x and y offsets, as well as small changes in x and y scale between the two coordinate systems.
Because, as an approximation, the data model is linear in its fit parameters, a good fit will not be obtained unless the correction rotation angle (a0), x and y offsets (a2, a4), and x and y scale changes (a1, a3) are sufficiently small. To partly make up for the deficiencies of the linear approximation, the LSQ routine is iterated to improve the fit.
The image registration data {t0, x0, y0} are zero for the present application; estimates of these quantities for the general case may be obtained using rotlsqdrv2.f, where t0=theta, x0=delta", and y0=gamma".
Only rotations about (x=0, y=0) are currently handled by lsqtrsdrv.f.
At least three sources (three sets of {x, y, x', y'} pixel coordinates) are required for the data fitting of all five parameters (translation, rotation, and plate scales). A minimum of two sources are required when data fitting with either no rotation or no plate-scale changes. A minimum of one source is requred when data fitting with both no rotation and no plate-scale changes.
The optional outlier-rejection capability of the software is enabled by specifying a greater-than-zero minimum tolerable reduction in reduced chi2 of the fit to the data with outlier candidates excluded, relative to the case where no sources are rejected. Outlier candidates are selected on the basis of whether their radial position deviations from the fit are greater than a specified tolerance, or whether their x or y position deviations from the fit are greater than a specified non-integral multiple of the fit standard deviation with the source in question excluded from the estimate of the fit standard deviation.
The software is now under both CVS version control (/proj/wire/cvsroot/da/russ/acs), and make control (makefile.f). The executables have been delivered to /proj/wire/tst/bin. The source codes are located in /proj/wire/russ/acs.
The source positions and weights are read from standard input, one source per line, where each line includes {x, y; x', y'; w} in that order.
The weighted average standard deviation of the data is only used to normalize the goodness of fit measure, chi2, which is included with the outputs. Its value affects how outlier rejection is done (but it does not affect the convergence of the iterated linear least-squares fitting).
The table below describes the available command-line inputs.
| Command-line input | Definition |
|---|---|
| -x0 value | Estimate of x offset (pixels). Default is zero. |
| -y0 value | Estimate of y offset (pixels). Default is zero. |
| -t0 value | Estimate of rotation angle (degrees). Default is zero. |
| -siginp or -sig value | Weighted average standard deviation (pixels). Default is 0.1 pixels. |
| -chi2tol or value | Minimum tolerable reduction in reduced chi2 for acceptable fit with outlier candidates excluded versus no sources rejected. Must be greater than zero to enable outlier-rejection processing. Default is 0.0. |
| -npostol or value | Number of x or y fit standard deviations that are used as maximum tolerable x or y deviations between data and fit in the outlier-rejection processing. Default is 5.0. |
| -radtol or value | Maximum tolerable radial position deviations between data and fit, used as a basis for outlier rejection. Default is 0.25 pixels. |
| -noscale or -nos | Switch that causes the data fitting to be done with no scale change between the two coordinate systems. |
| -norotation or -nor | Switch that causes the data fitting to be done with no rotation between the two coordinate systems. |
| -verbose or -v | Verbose mode. Default is off. |
| -test | Execute in test mode? Default is no test mode. Five test source positions and weights, which are embedded within the software, are used. |
a2 (pixels)
a4 (pixels)
a1 (dimensionless)
a3 (dimensionless)
a0 (degrees)
Corrected x offset (pixels)
Corrected y offset (pixels)
Corrected rotation angle (degrees)
Number of sources kept for final data fitting
Number of outliers rejected
chi2 (dimensionless)
# of degrees of freedom
x residual (pixels)
x standard deviation (pixels)
y residual (pixels)
y standard deviation (pixels)
Note that the corrected x and y offsets and rotation angle are given in terms of theta, delta", and gamma" as defined for rotlsqdrv2.f.
lsqtrsdrv -test -v
In this case, five parameters are computed. The corresponding output is
$Id: lsqtrsdrv.f,v 1.10 1998/04/06 14:40:15 laher Exp $
lsqtrsdrv.f
Date - Time: 04/06/1998(98096) - 07:46:06 | Cpu: 0.140 0.050u 0.090s
All parameters for command /proj/wire/tst/bin/lsqtrsdrv:
t0=0.;x0=0.;y0=0.;xpivot=0.;ypivot=0.;siginp=0.1;chi2tol=0.0;npostol=5.0;
radtol=0.25;verbose=true;debug=false;noscale=false;test=true;
Solving for 5 parameters: dx, dy, dt, dsx, and dsy
TESTING: Running with 5 test points.
Input for test case:
t0 = 215.00000000000
x0 = 402.00000000000
y0 = 103.000000000000
dx = -6.8793509513911
dy = -11.462308364455
dtheta= -5.2000000000000
dsx = 5.0000000000000D-02
dsy = -8.0000000000000D-02
x y x' y' w
41.18000 46.43000 353.85902 102.58388 2.00000
24.74000 86.17000 343.41214 63.35384 1.50000
69.13000 109.95000 293.48325 76.77087 1.00000
73.46000 74.21000 311.29978 104.93936 0.50000
25.25000 115.94000 325.25586 42.69128 0.25000
Fitting input data...
Output from lsqtrsdrv.f (lsqtrs2):
Iter dxout dyout dtout dsxout dsyout chi2
1 -8.166E+00 -1.110E+01 -5.009E+00 4.883E-02 -8.117E-02 1.282E+03
2 -6.902E+00 -1.157E+01 -5.199E+00 5.024E-02 -7.961E-02 2.789E+00
3 -6.900E+00 -1.156E+01 -5.199E+00 5.024E-02 -7.961E-02 2.467E+00
4 -6.900E+00 -1.156E+01 -5.199E+00 5.024E-02 -7.961E-02 2.467E+00
Iterated linear LSQ routine converged after 4 iterations.
Final outputs:
dxout = -6.9001318568390
dyout = -11.561123163140
dsxout = 5.0239558133034D-02
dsyout = -7.9605503783698D-02
dtout = -5.1986277857829
delx = 414.78361741244
dely = 107.39804871681
theta = 220.19862778578
scalex = 1.0502395581330
scaley = 0.92039449621630
# kept = 5
# rejected = 0
chi2 = 2.4670984176760
# degs frdm = 5
x residual = 5.30773E-02
x std. dev. = 6.88436E-03
y residual = 4.46616E-02
y std. dev. = 7.10467E-03
Example of how to execute lsqtrsdrv.f to solve for only x and y shifts and a rotation angle (keeping the x and y scales fixed):
lsqtrsdrv -test -v -nos -siginp 0.05
For this case, the input sigma is set to the same value used by rotlsqdrv2.f for fitting the same data using a different technique. The corresponding output is
$Id: lsqtrsdrv.f,v 1.10 1998/04/06 14:40:15 laher Exp $
lsqtrsdrv.f
Date - Time: 04/06/1998(98096) - 07:47:38 | Cpu: 0.120 0.050u 0.070s
All parameters for command /proj/wire/tst/bin/lsqtrsdrv:
t0=0.;x0=0.;y0=0.;xpivot=0.;ypivot=0.;siginp=0.05;chi2tol=0.0;npostol=5.0;
radtol=0.25;verbose=true;debug=false;noscale=true;test=true;
Solving for 3 parameters: dx, dy, and dt
TESTING: Running with 5 test points.
Input for test case:
t0 = 69.000000000000
x0 = -10.0000000000000
y0 = 102.000000000000
dx = -2.4129448887140
dy = -0.97655814427490
dtheta= -2.1958000000000
dsx = 0.
dsy = 0.
x y x' y' w
41.18000 46.43000 45.53000 79.94000 2.00000
24.74000 86.17000 77.87000 108.36000 1.50000
69.13000 109.95000 114.63000 73.98000 1.00000
73.46000 74.21000 82.23000 58.30000 0.50000
25.25000 115.94000 106.12000 117.37000 0.25000
Fitting input data...
Output from lsqtrsdrv.f (lsqtrs2):
Iter dxout dyout dtout dsxout dsyout chi2
1 -2.479E+00 -9.387E-01 -2.195E+00 0.000E+00 0.000E+00 2.516E+01
2 -2.445E+00 -1.032E+00 -2.197E+00 0.000E+00 0.000E+00 4.573E+00
3 -2.445E+00 -1.032E+00 -2.197E+00 0.000E+00 0.000E+00 4.573E+00
4 -2.445E+00 -1.032E+00 -2.197E+00 0.000E+00 0.000E+00 4.574E+00
5 -2.445E+00 -1.032E+00 -2.197E+00 0.000E+00 0.000E+00 4.573E+00
6 -2.445E+00 -1.032E+00 -2.197E+00 0.000E+00 0.000E+00 4.573E+00
7 -2.445E+00 -1.032E+00 -2.197E+00 0.000E+00 0.000E+00 4.573E+00
Iterated linear LSQ routine converged after 7 iterations.
Final outputs:
dxout = -2.4452442699382
dyout = -1.0321920116430
dsxout = 0.
dsyout = 0.
dtout = -2.1968080150192
delx = -11.766586855050
dely = 103.983473445258
theta = 71.196808015019
scalex = 1.0000000000000
scaley = 1.0000000000000
# kept = 5
# rejected = 0
chi2 = 4.5735325777658
# degs frdm = 7
x residual = -2.36911E-03
x std. dev. = 3.57262E-02
y residual = 2.86211E-03
y std. dev. = 2.32810E-02
The results from lsqtrsdrv.f and rotlsqdrv2.f for this case agree to within 0.07 pixels (~1 arcsecond for WIRE pixels) for the offsets, and to within 0.001 degrees for the rotation (which is negligible).
Example of how to execute lsqtrsdrv.f on the test case with outlier rejection enabled:
lsqtrsdrv -test -v -chi2tol 0.8
The corresponding output is
$Id: lsqtrsdrv.f,v 1.10 1998/04/06 14:40:15 laher Exp $
lsqtrsdrv.f
Date - Time: 04/06/1998(98096) - 07:42:30 | Cpu: 0.130 0.020u 0.110s
All parameters for command /proj/wire/tst/bin/lsqtrsdrv:
t0=0.;x0=0.;y0=0.;xpivot=0.;ypivot=0.;siginp=0.1;chi2tol=0.8;npostol=5.0;
radtol=0.25;verbose=true;debug=false;noscale=false;test=true;
Solving for 5 parameters: dx, dy, dt, dsx, and dsy
TESTING: Running with 5 test points.
Input for test case:
t0 = 215.00000000000
x0 = 402.00000000000
y0 = 103.000000000000
dx = -6.8793509513911
dy = -11.462308364455
dtheta= -5.2000000000000
dsx = 5.0000000000000D-02
dsy = -8.0000000000000D-02
x y x' y' w
41.18000 46.43000 353.85902 102.58388 2.00000
24.74000 86.17000 343.41214 63.35384 1.50000
69.13000 109.95000 293.48325 76.77087 1.00000
73.46000 74.21000 311.29978 104.93936 0.50000
25.25000 115.94000 325.25586 42.69128 0.25000
Fitting input data...
Output from lsqtrsdrv.f (lsqtrs2):
Iter dxout dyout dtout dsxout dsyout chi2
1 -8.166E+00 -1.110E+01 -5.009E+00 4.883E-02 -8.117E-02 1.282E+03
2 -6.902E+00 -1.157E+01 -5.199E+00 5.024E-02 -7.961E-02 2.789E+00
3 -6.900E+00 -1.156E+01 -5.199E+00 5.024E-02 -7.961E-02 2.467E+00
4 -6.900E+00 -1.156E+01 -5.199E+00 5.024E-02 -7.961E-02 2.467E+00
Iterated linear LSQ routine converged after 4 iterations.
Performing outlier rejection...
Fit measure: chi2,#degfreedom,chi2reduce = 2.4670984176760 5 0.493420
Number of sources considered = 5
Number of sources to keep = 3
MAXPTS,MINPTS,npts= 5000 3 5
verb,e2tol= 1 0.800000
npostol,radtol= 5.00000 0.250000
theta0,delta,gamma= -3.7524578917878 402.00000000000 103.000000000000
---With source # 1 rejected...
poserrrad, radtol = 8.08821E-02 0.250000
poserrx, n*sigx = 6.06275E-02 3.59857E-02
poserry, n*sigy = 5.35371E-02 2.80677E-02
---With source # 4 rejected...
poserrrad, radtol = 6.84432E-02 0.250000
poserrx, n*sigx = 4.76596E-02 4.54653E-02
poserry, n*sigy = 4.91226E-02 4.74108E-02
---With source # 2 rejected...
poserrrad, radtol = 6.68736E-02 0.250000
poserrx, n*sigx = 5.44698E-02 5.22077E-02
poserry, n*sigy = 3.87959E-02 4.79859E-02
---With source # 5 rejected...
poserrrad, radtol = 5.57319E-02 0.250000
poserrx, n*sigx = 4.72983E-02 4.48699E-02
poserry, n*sigy = 2.94774E-02 4.24879E-02
---With source # 3 rejected...
poserrrad, radtol = 5.46820E-02 0.250000
poserrx, n*sigx = 4.00082E-02 2.80001E-02
poserry, n*sigy = 3.72755E-02 4.54961E-02
Source # 1 is a candidate outlier
Source # 4 is a candidate outlier
Found 2 candidate outliers in rejection processing.
Fitting data with candidate outliers excluded...
Output from lsqtrsdrv.f (lsqtrs2):
Iter dxout dyout dtout dsxout dsyout chi2
1 -8.615E+00 -1.058E+01 -5.286E+00 4.612E-02 -8.388E-02 1.018E+03
2 -6.918E+00 -1.158E+01 -5.199E+00 4.972E-02 -7.976E-02 3.451E+00
3 -6.919E+00 -1.158E+01 -5.199E+00 4.972E-02 -7.976E-02 3.639E+00
4 -6.919E+00 -1.158E+01 -5.199E+00 4.972E-02 -7.976E-02 3.639E+00
Iterated linear LSQ routine converged after 4 iterations.
chi2,degfreedom= 3.6386776485356 1.00000
chi2minreduce,chi2reduce= 0.493420 3.6386775970459
chi2(reduced)error,e2(reduced)tol= -6.3744068145752 0.800000
Fit with NO outliers rejected has lowest reduced chi2.
Final outputs:
dxout = -6.9001318568390
dyout = -11.561123163140
dsxout = 5.0239558133034D-02
dsyout = -7.9605503783698D-02
dtout = -5.1986277857829
delx = 414.78361741244
dely = 107.39804871681
theta = 220.19862778578
scalex = 1.0502395581330
scaley = 0.92039449621630
# kept = 5
# rejected = 0
chi2 = 2.4670984176760
# degs frdm = 5
x residual = 5.30773E-02
x std. dev. = 6.88436E-03
y residual = 4.46616E-02
y std. dev. = 7.10467E-03
For this case no sources were rejected. This is because the reduced chi2 of the fit with identified candidate outliers excluded was not lower than the reduced chi2 of the fit with no sources excluded.
Last revised: September 29, 1998
Software developer: Russ Laher (laher@ipac.caltech.edu)
URL: http://spider.ipac.caltech.edu/staff/laher/lsqtrsdrv.html