The `frames` pipeline also provides several characterizations of the shape and
morphology of an object.

`frames`'s ``Type'' Determination

The `frames` pipeline provides a simple star/galaxy separator in its `type`
parameters (provided separately for each band) and its `
objc_type` parameters (one value per object); these are set to 3
(galaxy) or 6 (star).
In particular, lupton01a show that the following simple cut
works at the 95% confidence level for our data to and even
somewhat fainter:

If satisfied, `type` is set to `GALAXY` for that band;
otherwise, `type` is set to `STAR`. The global type `
objc_type` is set according to the same criterion, applied to the
summed fluxes from all bands in which the object is detected.

Experimentation has showed that simple variants on this scheme, such
as defining galaxies as those objects classified as such in any two of
the three high signal-to-noise ratio bands (namely, *g*, *r*, and
*i*), work better in some
circumstances. This scheme occasionally fails to distinguish pairs of
stars with separation small enough (<2'') that the deblender does
not split them; it also occasionally classifies Seyfert galaxies with
particularly bright nuclei as stars.

Further information to refine the star-galaxy separation further may be used, depending on scientific application. For example, scranton01 advocate applying a Bayesian prior to the above difference between the PSF and exponential magnitudes, depending on seeing and using prior knowledge about the counts of galaxies and stars with magnitude.

The `frames` pipeline extracts an azimuthally-averaged radial
surface brightness profile. In the catalogs, it is given as the
average surface brightness in a series of annuli (whose outer radii
are listed in Table 7). This quantity is in units of
``maggies'' per square arcsec, where a *maggie* is a linear
measure of flux; one maggie has an AB magnitude of 0 (thus a surface
brightness of 20 mag/square arcsec corresponds to maggies
per square arcsec). In the class `Profile`
(Table 6), the number of annuli for which there is a
measurable signal is listed as `nprof`, the mean surface
brightness is listed as `profMean`, and the error is listed as
`profErr`. This error includes both photon noise, and the
small-scale ``bumpiness'' in the counts as a function of azimuthal
angle.

When converting the `profMean` values to a local surface
brightness, it is *not* the best approach to assign the mean
surface brightness to some radius within the annulus and then linearly
interpolate between radial bins. Do not use smoothing
splines, as they will not go through the points in the cumulative
profile and thus (obviously) will not conserve flux. What `frames`
does, e.g., in determining the Petrosian ratio, is to fit a taut spline to the
*cumulative* profile and then differentiate that spline fit,
after transforming both the radii and cumulative profiles with asinh
functions. We recommend doing the same here.

Surface Brightness and Inverse Concentration Index

The `frames` pipeline also reports the radii containing 50% and 90% of the Petrosian
flux for each band, `petroR50` and `petroR90` respectively.
The usual characterization of surface-brightness in the target
selection pipeline of the SDSS is the mean surface brightness within `
petroR50`.

It turns out that the ratio of `petroR50` to `petroR90`, the
so-called ``inverse concentration index'', is correlated with
morphology shimasaku01,strateva01. Galaxies with a de
Vaucouleurs profile have an inverse concentration index of around 0.3;
exponential galaxies have an inverse concentration index of around
0.43. Thus, this parameter can be used as a simple morphological
classifier.

An important caveat when using these quantities is that they
are *not* corrected for seeing. This causes the surface
brightness to be underestimated, and the inverse concentration index
to be overestimated, for objects of size comparable to the PSF. The
amplitudes of these effects, however, are not yet well characterized.

Model Fit Likelihoods and Parameters

In addition to the model and PSF magnitudes described above,
the likelihoods `deV_L`, `exp_L`, and `star_L` are also
calculated by `frames`. These are the probabilities of achieving the
measured for the de Vaucouleurs, exponential, and PSF fits,
respectively. If one wishes to make use of this trinary scheme to
classify objects, calculation of the fractional likelihoods is recommended:

and similarly for and .
A fractional likelihood greater than 0.5 for any of these three profiles
is generally a good threshold for object classification. This works
well in the range ; at the bright end, the
likelihoods have a tendency to underflow to zero, which makes them less
useful. In particular, `star_L` is often zero for bright stars.
For future data releases we
will incorporate improvements to the model fits to give more
meaningful results at the bright end.

The model fits yield an estimate of the axis ratio and position angle
of each object, but it is useful to have model-independent measures of
ellipticity. In the data released here, `frames` provides two further
measures of ellipticity, one based on second moments, the other based
on the ellipticity of a particular isophote. The model fits do
correctly account for the effect of the seeing, while the methods
presented here do not.

The first method measures flux-weighted second moments,
defined as:

In the case that the object's isophotes are self-similar ellipses, one
can show:

where *a* and *b* are the semi-major and semi-minor axes, and
is the position angle. *Q* and *U* are `Q` and `U` in the
class `PhotoObj`
and are referred to as ``Stokes parameters.'' They can be used to
reconstruct the axis ratio and position angle, measured relative to row
and column of the CCDs. This is equivalent to the normal definition of
position angle (East of North), for the scans on the Equator.
The performance of the Stokes parameters are not ideal at
low S/N.
For future data releases, `frames` will also output variants
of the adaptive shape measures used in the weak lensing analysis of
fischer00, which are closer to optimal measures of shape for
small objects.

A second measure of ellipticity is given by measuring the ellipticity
of the 25 magnitudes per square arcsecond isophote (in all bands). In detail, `frames`
measures the radius of a particular isophote as a function of angle
and Fourier expands this function. It then extracts from the
coefficients the centroid, major and minor axis, position angle, and average radius of
the isophote in question. It also reports the derivative of each of these
quantities with respect to isophote level, necessary to recompute
these quantities if the photometric calibration changes.

Thu Jan 30 11:15:34 EST 2003