The detection of a source and the measurement of its position in an image begins with convolving the image with its PSF (as determined from a set of isolated, well exposed stars in the image). The peaks above a defined threshold in the convolved image give the approximate locations of sources; subsequent fitting with models of the source brightness distributions, in which the peak positions of the model sources are allowed to vary, yield the source positions to sub-pixel accuracy. Two models are used for point-sources: a 2-D Gaussian for the GaussianCentroid, and a symmetrized version of the PSF model for the SdssCentroid. Which centroid is best depends on the type of source.
Measurement of the most accurate source fluxes in an image consists of fitting models to the brightness distribution; the resulting flux is the integral of that model, exclusive of flagged pixels. The most appropriate model (or composite model) depends upon the true brightness distribution of the source, which is different for sources that are unresolved (stars) and those that are resolved (galaxies). Some sources are too complex (e.g., nebulae) to model in a production system. Others, while complicated because they consist of a superposition of sources of different types (e.g., a supernova within a galaxy), are nonetheless tractable (see Multi-Fit Algorithm below) in production.
While the pipeline makes a rough characterization of the nature of each source (location, shape, extendedness, etc.), generally all implemented measurements of flux are performed on each source so that the most appropriate one can be selected by downstream analysis software.
Uncertainties in the flux measurements are derived by propagating the uncertainty captured in the image's variance plane into the measured fluxes, assuming Gaussian errors. Systematic errors (from e.g. PSF modeling or background subtraction) are not included, but may be in the future.
Some photometry applications estimate the local background near a source when measuring a flux. The approach for LSST is distinctly different, and more practical for large sky surveys. The background is characterized with a function (the parameters for which are preserved), and subtracted from the image. The specific steps are:
- Compute a statistic (clipped-mean) to estimate background in image sub-sections
- currently, 256 or 512 pixels on a side
- ignore masked pixels and footprints of detected sources
- Construct a spatial fit (currently, Akima spline) to the background values across the image
- Evaluate and subtract the background from all pixels in the image
- Detect and measure sources, de-blending as necessary
- Iterate steps 1-4.
The resulting 2-D characterization usually characterizes the non-astrophysical background well. This approach will likely be improved (methods are TBD) for measurements on images with complex astrophysical backgrounds (e.g., stars embedded in nebulosity), and for images containing sources with apparent diameters that are larger than a single sensor.
The following discussion of photometric measurements assumes that the background has already been subtracted correctly.
The optimal measure of total flux from isolated stars consists of fitting a model of the point-spread function (PSF), and applying that model at the location of the source to fit the intensity profile. LSST pipeline processing of an image consists in part of characterizing the PSF in two dimensions. PSF modelling works well when bright point sources in the image are sufficiently numerous and well distributed to support a robust fit to the model at different positions in the image. These stars will also be used to compute aperture corrections (i.e., a correction to the flux as measured through a fixed aperture to derive the total source flux) for all positions in the image.
Currently the PSF models are based upon eigen-images (which is similar to the SDSS algorithm), but this basis is likely to be replaced in the future.
PSF photometry works very well for stars or barely resolved objects. It produces a higher signal-to-noise measurement at the faint limit, since the PSF is typically very well characterized in the core using bright stars; the number of degrees of freedom in the PSF fit is comparable to an aperture flux. It is also well suited to new, moving, or variable point sources that are identified in the Image Difference pipeline. However, PSF photometry may be inferior to large aperture photometry for very bright, isolated sources if systematic errors in the PSF model (e.g., in the extended wings) are significant. A good aperture correction (once implemented) can correct PSF fluxes so that they match a large aperture photometry for bright stars. This will allow PSF fluxes to be used for both faint and bright point-sources.
PSF photometry will also be applied to sources identified in the Image Difference pipeline, which will most commonly be solar-system bodies, variable stars, and transients. Solar system bodies will also be fit with a trailed source model, which is a 2-D model PSF that is trailed in one direction, to measure accurate fluxes of sources with high angular velocity.
For bright, relatively isolated stars the classic aperture photometry can yield the best result. With this technique, fluxes are measured within a circularly elliptically symmetric aperture about the centroid of a source. The accuracy of this technique is highest with large apertures (ensuring all the light is collected), yet the signal-to-noise is highest (and contamination from field sources is lowest) with small apertures. The standard refinement of this technique is to derive a (field dependent) correction to account for light not captured in the small aperture. This correction is derived for each image using a range of apertures to characterize the brightness profile of very well exposed, isolated stars. This correction is applied to source fluxes measured with small apertures.
The aperture correction implementation in the Stack is currently broken, which will compromise the quality of both the aperture and PSF photometry.
The LSST Camera will sample the expected PSF well, but many cameras in use today are closer to critical sampling. Bickerton & Lupton (2013) have developed a technique for precise aperture photometry that overcomes small-aperture limitations arising from point-sampling. This method, which is implemented in the LSST Stack, numerically integrates over the aperture, using sinc interpolation to reconstruct values between pixel centers.
Measuring fluxes for galaxies is more difficult than for stars because galaxies do not all have the same radial surface brightness profile, they have no sharp edges, and the profiles may even be asymmetric. Galaxies brightness profiles may consist of multiple components, and some include superimposed stars or companion galaxies.There are a variety of approaches to galaxy photometry, some of which will be featured in the LSST Stack, but for the moment the implementation is somewhat rudamentary and is likely to change.
Galaxy photometry, like stellar photometry, consists of fitting models to source intensity distributions. Our current plan is to fit a linear combination of two elliptical Sersic models, corresponding to a bulge (de Vaucouleur profile, Sersic n=4) and disk (exponential profile, Sersic n=1). These are approximated as sums of Gaussians to enable fast convolution with the PSF, which is similarly approximated using multiple elliptical shapelet expansions (see Hogg & Lang 2013; Bosch 2010). While we are thus using shapelets in the implementation, this approach is much more similar to standard Sersic fitting approaches than to other shaplet-based image analysis.
At present, this multiShapelet code can be invoked for model-based fluxes when running the Stack, but has known problems when measuring faint galaxies and should be considered a proof-of-concept only.
Adaptive moments are the second moments of the source intensity distribution, measured using a particular scheme designed to have near-optimal signal-to-noise ratio. Moments are measured using a radial weight function iteratively adapted to the shape (ellipticity) and size of the object. This elliptical weight function has a signal-to-noise advantage over axially symmetric weight functions. In principle there is an optimal (in terms of signal-to-noise) radial shape for the weight function, which is related to the light profile of the object itself. In practice an elliptical Gaussian with size matched to that of the object is used, and is nearly optimal. Details can be found in Bernstein & Jarvis (2002).
The LSST Stack features a prototype of a sophisticated technique for model fitting, Multi-Fit, which attempts a simultaneous fit of multi-component source models to multiple exposures of the same region, taken at different times. There is considerable flexibility in configuring the algorithm, which enables customizations for measuring time variability in sources, such as variable stars, moving objects, and optical transients. Such an approach, when successful, could provide a more robust characterization of object properties while avoiding problems sometimes encountered with stacked images (e.g., images obtained in very different seeing conditions).
Multi-Fit is currently in an experimental stage of development, and at present only fits simpler models to one or more overlapping sources in a single field.
At mid- to high-Galactic latitude and magnitudes fainter than mV=23, the spatial density of galaxies begins to exceed that of stars. Such galaxies also tend to be angularly small, making them difficult to distinguish from stars. The LSST Stack attempts to distinguish them using an extendedness measure, which is a simple cut on the ratio of the galaxy model flux to the PSF flux. Its usefulness is thus limited by the current poor quality of the galaxy model fluxes.
Photometry of sources that include flagged pixels (i.e., pixels affected by some sort of pathology identified in the image or during the photometric measurement) will be flagged in the output catalogs. The conditions that are flagged, though their encodings are for the time being subject to change. (See
PixelFlags.h in the meas_algorithms package.) Presently, the conditions flagged during photometric measurements are:
|Cosmic ray detected in the source footprint|
|Cosmic ray detected in the source center|
|Could not use the full model in the fit because of proximity to the exposure border|
|Interpolated pixel (e.g., for bad columns) in the source footprint|
|Interpolated pixel in the source center|
|Saturated pixel in the source footprint|
|Saturated pixel in the source center|
|Bad pixel in the source footprint|
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