2.1.2. Linear transformation matrix
The proposal to replace the CROTAi keywords of Wells et
with a general linear transformation matrix dates
from Hanisch & Wells
although the details of its
implementation have undergone considerable evolution. The main point
of divergence has been whether the matrix should completely replace or
simply augment the CDELTi, but there are also important
differences relating to the default values of the matrix elements.
In defining a nomenclature which augments the CDELTi we
have been guided by the following considerations:
- Where possible, standards should grow by
generalizing existing usage rather than developing a separate
parallel usage. Augmenting the existing CDELTi with a
separate transformation matrix that defaults to unity makes old
headers equivalent to new ones that omit the keywords that define
the transformation matrix. In any case, the "once FITS,
always FITS" rule means that FITS readers must continue to
interpret CDELTi, so it makes sense for
CDELTi to retain its original function.
- The transformation matrix then replaces the
poorly defined CROTAi with a nomenclature that allows
for both skew andfully general rotations. We do not consider this
replacement and the consequent deprecation of the
CROTAi to be
inconsistent with the aim of generalizing existing usage
since, to our knowledge, the CROTAi have had no formal
definition other than the "AIPS convention" (Greisen
Both Wells et al.
and Hanisch et al.
state that "users of this option should
provide extensive explanatory comments." Paper II describes
the translation of the AIPS interpretation of CROTAi to
the new formalism.
- A large fraction of WCS representations, perhaps
the great majority, will not require the general linear
transformation. FITS writers may continue to use
FITS-writing software need not be rewritten to conform to the new
formalism unless it needs the new features.
- The physical units of a general image may differ by
many orders of magnitude, from frequencies of 1010 Hz (or more)
to angles of 10-3 degrees (or less). If the physical units
enter into the linear transformation matrix, then the elements of
that matrix will have very different magnitudes. These issues pose
difficulties both in computing and in understanding, and it may be
simpler to defer application of physical units until the
multiplication by CDELTi.
- These difficulties are compounded when correcting
for the distortions present in real instruments. Paper IV will show
that some instruments require distortion corrections before, and
others after, the linear transformation matrix. Such corrections
may need to be expressed in terms directly related to pixel
coordinates. If the physical units enter into the linear
transformation matrix, then the distortion corrections which come
after the matrix would have to compensate for the physical units
applied by it, effectively undoing and then redoing a multiplication
by CDELTi. Furthermore, commensurability problems may
arise when recording the maximum distortion correction for a WCS
representation that mixes pre-, and post-corrections between axes.
- A widely used formalism that discards CDELTi
was developed by the Space Telescope Science Institute for the Hubble Space
Telescope and was incorporated generally in the IRAF data analysis system.
We therefore support this as an alternative method.
In the PCi_j formalism, the matrix elements
mij are encoded in
header cards, and si as CDELTi. The
i and j
indices are used without leading zeroes, e.g. PC1_1 and
CDELT1. The default values for PCi_j
are 1.0 for i = j and
0.0 otherwise. The PCi_j matrix must not be
singular; it must have an
inverse. Furthermore, all CDELTi must be non-zero. In other
words, invertibility means that transformations which project from an
initial coordinate system of dimensionality WCSAXES to a world
coordinate system of dimensionality less than WCSAXES are
In the CDi_j formalism Eqs. (1) and (2) are
keywords encode the product
simij. The i and j
indices are used without leading zeroes, e.g. CD1_1. The
CD i_j matrix must not be singular; it must have
CDELTi and CROTAi are allowed to
coexist with CDi_j as an
aid to old FITS interpreters, but are to be ignored by new readers.
The default behavior for CDi_j differs from that
for PCi_j; if one
or more CDi_j cards are present then all
unspecified CDi_j default
to zero. If no CDi_j cards are present then the
header is assumed to
be in PCi_j form whether or not any
PCi_j cards are present since
this results in an interpretation of CDELTi consistent
with Wells et al.
We specifically prohibit mixing of the PCi_j and
in any FITS header data unit. With this restriction, translation from
the CDi_j formalism to the
PCi_j formalism is effected simply in the
keyword parsing stage of header interpretation; the
CDi_j should be considered equivalent to the
to the considerations for default values noted above and with
CDELTi set to unity. Similarly,
CDi_j can be calculated from
PCi_j and CDELTi following Eq. 3.