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From davis Tue May 18 15:09:59 1993
Received: by tucana.tuc.noao.edu (4.1/SAG.tucana.12)
id AA26431; Tue, 18 May 93 15:09:56 MST; for sites
Date: Tue, 18 May 93 15:09:56 MST
From: davis (Lindsey Davis)
Message-Id: <9305182209.AA26431@tucana.tuc.noao.edu>
To: belkine@mesiob.obspm.circe.fr
Subject: RE: geomap
Cc: sites
Igor,
The following is a copy of a mail message I sent to another user who made
the same request regarding geomap. I hope this is of use to you.
Lindsey Davis
###############################################################################
Jeannette forwarded your request for a detailed description of the
geomap output format to me. This format was originally intended to be
for the internal use of geomap only, but the following should help you
decode it.
1. For simple linear geometric transformations you will see the
following two entries in the fit record. Surface1 describes the linear
portion of the fit; surface2 describes the residual distortion map
which is always 0 for linear fits.
surface1 11
surface(xfit) surface(yfit) (surface type 1=cheb, 2=leg, 3=poly)
xxorder(xfit) yxorder(yfit) (always 2)
xyorder(xfit) yyorder(yfit) (always 2)
xxterms(xfit) yxterms(yfit) (always 0)
xmin(xfit) xmin(yfit) (geomap input or data)
xmax(xfit) xmax(yfit) (geomap input or data)
ymin(xfit) ymin(yfit) (geomap input or data)
ymax(xfit) ymax(yfit) (geomap input or data)
a d
b e
c f
surface2 0
This above describes the following linear surfaces.
xfit = a + b * x + c * y (polynomial)
yfit = d + e * x + f * y
xfit = a + b * xnorm + c * ynorm (chebyshev)
yfit = d + e * xnorm + f * ynorm
xfit = a + b * xnorm + c * ynorm (legendre)
yfit = d + e * xnorm + f * ynorm
xnorm = (2 * x - (xmax + xmin)) / (xmax - xmin)
ynorm = (2 * y - (ymax + ymin)) / (ymax - ymin)
Xnorm and ynorm are the input x and y values normalized between -1.0
and 1.0.
2. For a higher order fit, say xorder=4 yorder=4 and xterms=yes,
the format is more complicated. The second surface is computed by fitting
the higher order surface to the residuals of the first fit. The geomap
output will look something like the following.
surface1 11
surface(xfit) surface(yfit) (surface type 1=cheb, 2=leg, 3=poly)
xxorder(xfit) yxorder(yfit) (always 2)
xyorder(xfit) yyorder(yfit) (always 2)
xxterms(xfit) yxterms(yfit) (always 0)
xmin(xfit) xmin(yfit) (geomap input or data)
xmax(xfit) xmax(yfit) (geomap input or data)
ymin(xfit) ymin(yfit) (geomap input or data)
ymax(xfit) ymax(yfit) (geomap input or data)
a d
b e
c f
surface2 24
surface(xfit) surface(yfit) (surface type 1=cheb, 2=leg, 3=poly)
xxorder(xfit) yxorder(yfit) (4)
xyorder(xfit) yyorder(yfit) (4)
xxterms(xfit) yxterms(yfit) (1 in this case)
xmin(xfit) xmin(yfit) (geomap input or data)
xmax(xfit) xmax(yfit) (geomap input or data)
ymin(xfit) ymin(yfit) (geomap input or data)
ymax(xfit) ymax(yfit) (geomap input or data)
C00(xfit) C00(yfit)
C10(xfit) C10(yfit)
C20(xfit) C20(yfit)
C30(xfit) C30(yfit)
C01(xfit) C01(yfit)
C11(xfit) C11(yfit)
C21(xfit) C21(yfit)
C31(xfit) C31(yfit)
C02(xfit) C02(yfit)
C12(xfit) C12(yfit)
C22(xfit) C22(yfit)
C32(xfit) C32(yfit)
C03(xfit) C03(yfit)
C13(xfit) C13(yfit)
C23(xfit) C23(yfit)
C33(xfit) C33(yfit)
where the Cmn are the coefficients of the polynomials Pmn, and the Pmn
are defined as follows
Pmn = x ** m * y ** n (polynomial)
Pmn = Pm(xnorm) * Pn(ynorm) (chebyshev)
P0(xnorm) = 1.0
P1(xnorm) = xnorm
Pm+1(xnorm) = 2.0 * xnorm * Pm(xnorm) - Pm-1(xnorm)
xnorm = (2 * x - (xmax + xmin)) / (xmax - xmin)
P0(ynorm) = 1.0
P1(ynorm) = ynorm
Pn+1(ynorm) = 2.0 * ynorm * Pn(ynorm) - Pn-1(ynorm)
ynorm = (2 * y - (ymax + ymin)) / (ymax - ymin)
Pmn = Pm(xnorm) * Pn(ynorm) (legendgre)
P0(xnorm) = 1.0
P1(xnorm) = xnorm
Pm+1(xnorm) = ((2m + 1) * xnorm * Pm(xnorm) - m * Pm-1(xnorm))/
(m + 1)
xnorm = (2 * x - (xmax + xmin)) / (xmax - xmin)
P0(ynorm) = 1.0
P1(ynorm) = ynorm
Pn+1(ynorm) = ((2n + 1) * ynorm * Pn(ynorm) - n * Pn-1(ynorm))/
(n + 1)
ynorm = (2 * y - (ymax + ymin)) / (ymax - ymin)
Hopefully I have copied this all down correctly. The main points to remember
is that the mangitudes of the coefficients reflect both the function type
(polynomial, chebyshev, or legendre) and the normalization (xmin, xmax,
ymin, ymax).
Hope this helps you out and write back if you have more questions.
Lindsey Davis
=======================================
# <Date>
begin <name>
task fitcoords
axis 1 # Axis of fitted value
surface 24 # The number of following parameters/coefficients
surface # surface type 1=chebyshev, 2=legendre
xorder # X order
yorder # Y order
xterms # Cross terms? 0=no, 1=yes (always 1 for fitcoords)
xmin # Minimum x value in fit - usually 1
xmax # Maximum x value in fit - usually image dimension
ymin # Minimum y value in fit - usually 1
ymax # Maximum y value in fit - usually image dimension
C00 # Coefficients (shown for xorder=4 and yorder=4)
C10
C20
C30
C01
C11
C21
C31
C02
C12
C22
C32
C03
C13
C23
C33
The fit is a sum of the form:
fit = sum(m=0 to xorder-1) sum(n=0 to yorder-1) {Cmn*Pm(x')*Pn(y')}
where the cross-terms may or may not be included depending on the xterms
parameter. Cross-terms are always used in FITCOORDS.
The coefficients are defined in terms of normalized independent variables
in the range -1 to 1. If x and y are actual values then the normalized
variables, x' and y', are defined using the data range parameters as:
x' = (2 * x - (xmax + xmin)) / (xmax - xmin)
y' = (2 * y - (ymax + ymin)) / (ymax - ymin)
The Pi(z), where z is either x' or y', are defined iteratively as follows:
# Chebyshev
P0(z) = 1.0
P1(z) = z
Pi+1(z) = 2.0 * z * Pi(z) - Pi-1(z)
# Legendre
P0(z) = 1.0
P1(z) = z
Pi+1(z) = ((2i + 1) * z * Pi(z) - i * Pi-1(z)) / (i + 1)
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