The way to extract structural information of galaxies in 2-D images
traditionally is to fit analytic functions for the surface brightness
profile (e.g. de Vaucouleurs, Sérsic, exponential, Nuker, etc.)
that are ellipsoidal in shape. Ellipses are convenient for this purpose
because by-and-large galaxies do look roughly elliptical in projection,
and because it is easy to understand what parameters like size, axis ratio
and position angle mean for an elliptical function. If the goal is to
quantify basic properties, ellipsoidal shapes often would suffice. On the
other hand, an ellipse does not capture a lot of information present in a
galaxy. Going beyond it might be interesting for some science objectives,
one of which might be to quantify a galaxy substructure, where it is
necessary to match the detailed shape as well as possible so that the
residuals do not affect the extraction. Another area of growing interest
is to quantify the degree of asymmetry or lopsidedness as a way to relate
how ``relaxed'' a galaxy might be, for which ellipsoidal functions are not
up to the task. Even for doing simple photometry, one might question how
meaningful it is to use an ellipsoid model to describe irregular galaxies.

GALFIT 3.0 (Peng et al. 2010, AJ, 139, 2097) tries to address some of issues raised above in new and flexible ways, while retaining the most useful traditional qualitative sense of galaxy fitting. What makes parametric fitting highly useful traditionally are the fact that we can easily intuit what the fits mean when they are described by analytic functions (e.g. Sérsic, Nuker, Gaussian, etc.), and the profile parameters can be easily compared against other galaxies in a statistical fashion. Therefore, a key requirement in devising a more flexible scheme is to retain those key elements even when things deviate greatly from an ellipse, e.g. the meaning of the concentration index should not change when a profile is lopsided. GALFIT 3.0 now allows even more diverse shapes than previously possible in 2-D, and is able to fit a wide range of isophotal shapes, axial twists, and spiral structures of many spiral galaxies. All the while, the radial profile parameters of each component can be thought of in the traditional way, no matter how complex they may get.

How GALFIT accomplishes the intended objectives is to use the concept of
Fourier modes and coordinate rotation functions, which are two different
ideas. When first hearing about Fourier modes, a natural tendency is to
think about ``shapelet'' or ``wavelet'' decomposition. ** However, it is
important to realize that GALFIT does not do shapelet, wavelet, or even
Fourier decomposition on an image in pixel space. ** Delving into this
a little bit, the idea of shapelets/wavelets analysis is very much
analogous to Fourier transforming an image into an infinite series sum of
basis functions. But instead of Sines and Cosines as the bases, they use
more complicated mathematical patterns called shapelets or wavelets (for
the purpose of reducing the number of terms needed to characterize an
object). Because orthogonality is required to make a mathematical
transform unique, those basis functions often do not look like the object
or components that one is modeling. The shapelet/wavelet technique is not
a ``fit'' in the sense of galaxy fitting (although some iteration might be
required to find the center of a galaxy); shapelets/wavelets are a
mathematical inversion of an image just like Fourier transform. In the
limit of infinite shapelets/wavelets, one can model everything in an
image, down to the noise. The basis functions, and their linear
combination, often do not have a straight forward association with
identifiable physical subcomponents. Whether this is a benefit or not
depends on what you want to do (as is true with all techniques). This is
the principle difference between shapelet/wavelet analysis and 2-D
parametric fitting: in 2-D parametric fitting, the GALFIT model components
are themselves * priors * about the presence of bars, disks, bulges,
nuclear star clusters, an AGN, or what not.

In GALFIT3.0 the radial profile parameters still mean exactly the same as
what we used to understand. The only difference is that the azimuthal *
shape * can be more complicated than a simple ellipse. Although more
abstract, this is not that unusual of an idea since the venerable ellipse
used in traditional 2-D fitting (i.e. GALFIT prior to 3.0) is itself
modified from a perfect circle using the axis ratio parameter, q. The
novelty is that instead of a closed analytic form, the * shape * can
be thought of as a series expansion, with the most useful terms being the
lowest order ones. Also, the terms are very meaningful and intuitive once
one gets used to the idea. In fact, the ellipsoid axis ratio term is
really just a special case of Fourier modes (i.e. mostly the second mode).
The motivation for going to higher modes is to model the radial profile and
shape of each component as accurately as possible, as though each *might*
represent some kind of physical reality (e.g. bulge, disk, spiral, AGN,
rings). This makes the components easier to interpret, in contrast to the
shapelet basis functions described above. In a single component fit, the
model takes on a shape that is flux weighted average over all the radii,
minimizing residuals in the least-squares sense.

The other new idea deals with isophotal twists and spiral structures. Most
objects have some sort of isophotal twists, even elliptical galaxies. In
spiral galaxies isophotal twists are mostly attributed to spiral arms.
What most people do in galaxy bulge-to-disk decomposition is to sweep the
large residuals from spiral structures under-the-rug. This is fine in some
cases, and may not be in others. In elliptical galaxies isophotal twists
may not necessary mean there are two distinct physical components. This is
often a problem for really nearby galaxies but not so much at *z*
> 0.3 or so, because isophotal twist happens mostly toward the center
and can't be resolved when galaxies are too far away to be resolved
clearly. To deal with isophotal twists, GALFIT uses a coordinate rotation
technique to rotate the isophotes as a function of radius. For
details, see
here. Below are some examples that show show the idea does seem to
work.

Both the Fourier and coordinate rotation techniques will be described in detail in Peng et al. 2010 (AJ, 139, 2097). Below are some examples of the new techniques applied to real galaxies. All the images were provided to me to analyze by colleagues whose names are listed below.

- NGC 3741 (original data courtesy of Janice Lee) -- a dwarf
galaxy fitted by a SINGLE COMPONENT Sérsic profile. Instead of
the usual ellipsoidal shape, the shape is modified by Fourier modes to
make it lopsided and slightly distorted. One can still think about the
radial profile parameters (i.e. mag, Re, n, PA, etc.) in exactly the
same way as before. And the amplitudes of the radial deviations are
directly related to the asymmetry of the galaxy.
- NGC 4190 (original courtesy of Janice Lee) -- another dwarf
galaxy fitted by TWO COMPONENT Sérsic profiles: the first
component fits the more irregular looking central part of the galaxy,
and a second component is used to fit the smoother looking outer
component.
- VII Zw 403 (original data courtesy of Janice Lee) -- An
irregular dwarf galaxy fitted, again, by a single Sérsic
component, modified by Fourier modes. Notice that because we are
using only a single component to fit the entire galaxy, in a flux
weighted manner, the shape is an *average* shape over the galaxy that
minimizes the residuals, in a way that also seems fairly sensible.
- II Zw 40 (original data courtesy of Janice Lee) -- One can
also add multiple components together like the traditional GALFIT.
And each component can have its own high order Fourier corrections.
This example shows an irregular looking galaxy fitted by a three
Sérsic components: bulge and two linear arms. Each component
has 6-10 Fourier modes.
- A grand design spiral M51 galaxy. Each galaxy is fitted by
3 Sérsic components. For the main galaxy, two spiral arms and
a bulge are fitted, while the upper one is fitted by a spiral arm and
two bulge-like components. Fourier modes are used to modify the
traditional ellipses to better fit some subtle features in the spiral
structures. The total number of free parameters used in the fit is a
record for me at 144 -- the vast majority of which are higher order
Fourier perturbations.
- Another majestic, flocculent spiral, NGC 3184 (original data courtesy
of Janice Lee), fitted by a bulge and 2 spiral arm components. The
bulge naturally converged to a de Vaucouleurs profile without any
assistance. A traditional two component B/D decomposition using two
Sérsic profiles yields large residuals and a bulge that is
unphysical. Constraining the bulge to be n=4, the model is similar to
the spiral arm fit, except 30% fainter and smaller.
- NGC 3275 (original data courtesy of Ho, Barth, & Seigar) --
an SB(r)ab galaxy. This galaxy has 5 components fitted: bulge,
spiral, disk, and 2 nuclear components composed of a nuclear bar and a
nuclear disk.
- NGC 1512 (original data courtesy of Janice Lee) -- another
SB(r)ab. Notice the degree of detail that can be modeled, as well as the
limitations.
- NGC 4050 (original data courtesy of Ho, Seigar, & Barth).
The nucleus is saturated.
- NGC 4462 (original data courtesy of Ho, Seigar, & Barth)
- Another pair of spiral galaxies (original data courtesy of
Hector Hernandez) Take a close look at the spiral structure. The fork
at the end of the top spiral arm is modeled with higher order modes.
- Yet another pair of spiral galaxies (original data courtesy
of Hector Hernandez). Each spiral is fitted by two spiral arms. The
lower left galaxy has a mini-spiral close-in to the bulge. You can
almost see it in the residuals.

Chien Peng ()