# Aperture photometry¶

Besides isophotal, PSF and model-fitting flux estimates, SExtractor can currently perform two types of flux measurements: fixed-aperture and adaptive-aperture. For every FLUX_ measurement, an error estimate FLUXERR_, a magnitude MAG_ and a magnitude error estimate MAGERR_ are also available: see Fluxes and magnitudes.

An estimate of the error is available for each type of flux. For aperture fluxes, the flux uncertainty is computed using

(1)${\tt FLUXERR} = \sqrt{\sum_{i\in{\cal A}}\, (\sigma_i^2 + \frac{p_i}{g_i})}$

where $${\cal A}$$ is the set of pixels defining the photometric aperture, and $$\sigma_i$$, $$p_i$$, $$g_i$$ respectively the standard deviation of noise (in ADU) estimated from the local background, $$p_i$$ the measurement image pixel value subtracted from the background, and $$g_i$$ the effective detector gain in $$e^- / \mbox{ADU}$$ at pixel $$i$$. Note that this error estimate provides a lower limit of the true uncertainty, as it only takes into account photon and detector noise.

## Fixed-aperture flux: FLUX_APER¶

FLUX_APER estimates the flux from the measurement image above the background inside a circular aperture. The diameter of the aperture in pixels is defined by the PHOTOM_APERTURES configuration parameter. It does not have to be an integer: each “regular” pixel is subdivided in $$5\times 5$$ sub-pixels before measuring the flux within the aperture. If FLUX_APER is provided as a vector FLUX_APER[n], at least $$n$$ apertures must be specified with the PHOTOM_APERTURES configuration parameter.

## Automatic aperture flux: FLUX_AUTO¶

FLUX_AUTO provides an estimate of the “total flux” by integrating pixel values within an adaptively scaled aperture. SExtractor’s automatic aperture photometry routine derives from Kron’s “first moment” algorithm [31]:

1. An elliptical aperture is defined by the second order moments of the object’s light distribution, with semi-major axis $$a={\tt A\_IMAGE}$$, semi-minor axis $$b={\tt B\_IMAGE}$$, and position angle THETA_IMAGE.
2. The ellipse’s major and minor axes are multiplied by 6 (which corresponds roughly to twice the size of the isophotal footprint on each axis).
3. Inside this elliptical aperture $${\cal E}$$, an analog of Kron’s “first moment” is computed:
(2)$r_{\rm Kron} = \frac{\sum_{i\in\cal E} r_i\,p^{(d)}_i}{\sum_{i\in\cal E} p^{(d)}_i},$

where $$p^{(d)}_i$$ is the pixel value in the detection image. $$r_i$$ is what we shall call the “reduced pseudo-radius” at pixel $$i$$

(3)$r_i \equiv \sqrt{{\tt CXX\_IMAGE} \times \Delta x_i^2 + {\tt CYY\_IMAGE} \times \Delta y_i^2 + {\tt CXY\_IMAGE} \times \Delta x_i \Delta y_i},$

where $$\Delta x_i$$ and $$\Delta y_i$$ are the pixel coordinates relative to the detection centroid:

\begin{split}\begin{aligned} \Delta x_i & = x_i - {\tt X\_IMAGE}\\ \Delta y_i & = y_i - {\tt Y\_IMAGE}. \end{aligned}\end{split}

[31] and [5] have shown that for stars and galaxy profiles convolved with Gaussian seeing, $$\ge 90\%$$ of the flux is expected to lie inside a circular aperture of radius $$k r_{\rm Kron}$$ with $$k = 2$$, almost independently of the magnitude. Experiments have shown [8] that this conclusion remains unchanged if one replaces the circular aperture with the “Kron elliptical aperture” $${\cal K}$$ with reduced pseudo-radius $$k r_{\rm Kron}$$.

FLUX_AUTO is the sum of pixel values from the measurement image, subtracted from the local background, inside the Kron ellipse:

(4)${\tt FLUX\_AUTO} = \sum_{i\in\cal K} p_i.$

The quantity $$k r_{\rm Kron}$$, known as the Kron radius (which in SExtractor is actually a “reduced pseudo-radius”) is provided by the KRON_RADIUS. $$k = 2$$ defines a sort of balance between systematic and random errors. By choosing a larger $$k = 2.5$$, the mean fraction of flux lost drops from about 10% to 6%, at the expense of SNR in the measurement. Very noisy objects may sometimes end up with a Kron ellipse being too small, even smaller that the isophotal footprint of the object itself. For this reason, SExtractor imposes a minimum size for the Kron radius, which cannot be less than $$r_{\rm Kron,min}$$. The user has full control over the parameters $$k$$ and $$r_{\rm Kron,min}$$ through the PHOT_AUTOPARAMS configuration parameters. PHOT_AUTOPARAMS is set by default to 2.5,3.5.

Hint

Aperture magnitudes are sensitive to crowding. In SExtractor v1, MAG_AUTO measurements were not very robust in that respect. It was therefore suggested to replace the aperture magnitude by the corrected-isophotal one when an object is too close to its neighbors (2 isophotal radii for instance). This was done automatically when using the MAG_BEST magnitude: MAG_BEST=MAG_AUTO when it is sure that no neighbor can bias MAG_AUTO by more than 10%, and $${\tt MAG\_BEST} = {\tt MAG\_ISOCOR}$$ otherwise. Experience showed that the MAG_ISOCOR and MAG_AUTO magnitude would loose about the same fraction of flux on stars or compact galaxy profiles: around 0.06 % for default extraction parameters. The use of MAG_BEST is now deprecated as MAG_AUTO measurements are much more robust in versions 2.x of SExtractor. The first improvement is a crude subtraction of all the neighbors that have been detected around the measured source (MASK_TYPE BLANK option). The second improvement is an automatic correction of parts of the aperture that are suspected to be contaminated by a neighbor. This is done by mirroring the opposite, cleaner side of the measurement ellipse if available (MASK_TYPE CORRECT option, which is also the default).

## Petrosian aperture flux: FLUX_PETRO¶

Similar to FLUX_AUTO, FLUX_PETRO provides an estimate of the “total flux” by integrating pixel values within an adaptively scaled elliptical aperture. FLUX_PETRO’s algorithm derives from Petrosian’s photometric estimator [32][33][34]:

1. An elliptical aperture is defined by the second order moments of the object’s light distribution, with semi-major axis $$a={\tt A\_IMAGE}$$, semi-minor axis $$b={\tt B\_IMAGE}$$, and position angle THETA_IMAGE.
2. The ellipse’s major and minor axes are multiplied by 6 (which corresponds roughly to twice the size of the isophotal footprint on each axis).
3. Within this elliptical aperture $${\cal E}$$, the Petrosian ratio $$R_{\rm P}(r)$$ is computed:
(5)$R_{\rm P}(r) = \frac{\sum_{0.9r < r_i < 1.1r} p^{(d)}_i}{\sum_{r_i < r} p^{(d)}_i} \frac{N_{r_i < r}}{N_{0.9r < r_i < 1.1r}},$

where $$p^{(d)}_i$$ is the pixel value in the detection image. $$r_i$$ is the “reduced pseudo-radius” at pixel $$i$$ as defined in (3). The Petrosian ellipse $${\cal P}$$ is the ellipse with reduced pseudo-radius $$N_{\rm P}r_{\rm P}$$, where $$r_{\rm P}$$ is defined by

(6)$R_{\rm P}(r_{\rm p}) \equiv 0.2$

The quantity $$N_{\rm P}r_{\rm P}$$ is called Petrosian radius in SExtractor[1] and is provided by the PETRO_RADIUS catalog parameter. The Petrosian factor $$N_{\rm P}$$ is set to 2.0 by default. Very noisy objects may sometimes end up with a Petrosian ellipse being too small. For this reason, SExtractor imposes a minimum size for the Petrosian radius, which cannot be less than $$r_{\rm P,min}$$. The user has full control over the parameters $$N_{\rm P}$$ and $$r_{\rm P,min}$$ through the PHOT_PETROPARAMS configuration parameters. PHOT_PETROPARAMS is set by default to 2.0,3.5.

The Petrosian flux is the sum of pixel values from the measurement image, subtracted from the local background, inside the Petrosian ellipse:

(7)${\tt FLUX\_PETRO} = \sum_{i\in\cal P} p_i.$
 [1] Some authors prefer to define the Petrosian radius as $$r_{\rm P}$$ instead of $$N_{\rm P}r_{\rm P}$$.

## Photographic photometry¶

In DETECT_TYPE PHOTO mode, SExtractor assumes that the response of the detector, over the dynamic range of the image, is logarithmic. This is generally a good approximation for photographic density on deep exposures. Photometric procedures described above remain unchanged, except that for each pixel we apply first the transformation

(8)$I = I_0\,10^{D/\gamma},$

where $$\gamma$$ (MAG_GAMMA) is the contrast index of the emulsion, $$D$$ the original pixel value from the background-subtracted image, and $$I_0$$ is computed from the magnitude zero-point $$m_0$$:

(9)$I_0 = \frac{\gamma}{\ln 10} \,10^{-0.4\, m_0}.$

One advantage of using a density-to-intensity transformation relative to the local sky background is that it corrects (to some extent) large-scale inhomogeneities in sensitivity (see [35] for details).

## Local background¶

Almost all SExtractor measurements are done using background-subtracted pixel values. In crowded fields, or in images where the background is irregular, the background model may be significantly inaccurate, locally creating biases in photometric estimates.

The user has the possibility to force SExtractor to correct, for every detection, the background used to compute fluxes by setting the BACKPHOTO_TYPE configuration parameter to LOCAL (GLOBAL is the default). In LOCAL mode, a mean background residual level is estimated from background-subtracted pixel values within a “rectangular annulus” around the isophotal limits of the object. The user can specify the thickness of the annulus, in pixels, with the BACKPHOTO_SIZE configuration parameter. The default thickness is 24 pixels. The residual background level computed in LOCAL mode is used by SExtractor to correct all aperture photometry measurements, as well as basic surface brightness estimations such as FLUX_MAX. However in practice the BACKPHOTO_TYPE LOCAL option has not proven as being really beneficial to photometric accuracy, and it is generally advised to leave BACKPHOTO_TYPE set to GLOBAL.

In both LOCAL and GLOBAL modes, the BACKGROUND catalog parameter gives the value of the background estimated at the centroid of the object.