.. File Position.rst Position and shape parameters derived from the isophotal profile ================================================================ The following parameters are derived from the spatial distribution :math:`\cal S` of pixels detected above the extraction threshold. *The pixel values* :math:`I_i` *are taken from the (filtered) detection image*. **Note that, unless otherwise noted, all parameter names given below are only prefixes. They must be followed by “_IMAGE” if the results shall be expressed in pixel units (see §..), or “_WORLD” for World Coordinate System (WCS) units (see §[astrom])**. For example: THETA :math:`\rightarrow` THETA\_IMAGE. In all cases, parameters are first computed in the image coordinate system, and then converted to WCS if requested. Limits: XMIN, YMIN, XMAX, YMAX ------------------------------ These coordinates define two corners of a rectangle which encloses the detected object: .. math:: \begin{aligned} {\tt XMIN} & = & \min_{i \in {\cal S}} x_i,\\ {\tt YMIN} & = & \min_{i \in {\cal S}} y_i,\\ {\tt XMAX} & = & \max_{i \in {\cal S}} x_i,\\ {\tt YMAX} & = & \max_{i \in {\cal S}} y_i, \end{aligned} where :math:`x_i` and :math:`y_i` are respectively the x-coordinate and y-coordinate of pixel :math:`i`. Barycenter: X, Y ---------------- Barycenter coordinates generally define the position of the “center” of a source, although this definition can be inadequate or inaccurate if its spatial profile shows a strong skewness or very large wings. X and Y are simply computed as the first order moments of the profile: .. math:: \begin{aligned} {\tt X} & = & \overline{x} = \frac{\displaystyle \sum_{i \in {\cal S}} I_i x_i}{\displaystyle \sum_{i \in {\cal S}} I_i},\\ {\tt Y} & = & \overline{y} = \frac{\displaystyle \sum_{i \in {\cal S}} I_i y_i}{\displaystyle \sum_{i \in {\cal S}} I_i}. \end{aligned} In practice, :math:`x_i` and :math:`y_i` are summed relative to XMIN and YMIN in order to reduce roundoff errors in the summing. Position of the peak: XPEAK, YPEAK ---------------------------------- It is sometimes useful to have the position XPEAK,YPEAK of the pixel with maximum intensity in a detected object, for instance when working with likelihood maps, or when searching for artifacts. For better robustness, PEAK coordinates are computed on *filtered* profiles if available. On symmetrical profiles, PEAK positions and barycenters coincide within a fraction of pixel (XPEAK and YPEAK coordinates are quantized by steps of 1 pixel, thus XPEAK_IMAGE and YPEAK_IMAGE are integers). This is no longer true for skewed profiles, therefore a simple comparison between PEAK and barycenter coordinates can be used to identify asymmetrical objects on well-sampled images. 2nd order moments: X2, Y2, XY ----------------------------- (Centered) second-order moments are convenient for measuring the spatial spread of a source profile. In |SExtractor| they are computed with: .. math:: \begin{aligned} {\tt X2} & = \overline{x^2} = & \frac{\displaystyle \sum_{i \in {\cal S}} I_i x_i^2}{\displaystyle \sum_{i \in {\cal S}} I_i} - \overline{x}^2,\\ {\tt Y2} & = \overline{y^2} = & \frac{\displaystyle \sum_{i \in {\cal S}} I_i y_i^2}{\displaystyle \sum_{i \in {\cal S}} I_i} - \overline{y}^2,\\ {\tt XY} & = \overline{xy} = & \frac{\displaystyle \sum_{i \in {\cal S}} I_i x_i y_i}{\displaystyle \sum_{i \in {\cal S}} I_i} - \overline{x}\,\overline{y}, \end{aligned} These expressions are more subject to roundoff errors than if the 1st-order moments were subtracted before summing, but allow both 1st and 2nd order moments to be computed in one pass. Roundoff errors are however kept to a negligible value by measuring all positions relative here again to XMIN and YMIN. Basic shape parameters: A, B, THETA ----------------------------------- These parameters are intended to describe the detected object as an elliptical shape. A and B are its semi-major and semi-minor axis lengths, respectively. More precisely, they represent the maximum and minimum spatial dispersion of the object profile along any direction. THETA is the position-angle of the A axis relative to the first image axis. It is counted positive in the direction of the second axis. Here is how they are computed: 2nd-order moments can easily be expressed in a referential rotated from the :math:`x,y` image coordinate system by an angle +\ :math:`\theta`: .. math:: \label{eq:varproj} \begin{array}{lcrrr} \overline{x_{\theta}^2} & = & \cos^2\theta\:\overline{x^2} & +\,\sin^2\theta\:\overline{y^2} & -\,2 \cos\theta \sin\theta\:\overline{xy},\\ \overline{y_{\theta}^2} & = & \sin^2\theta\:\overline{x^2} & +\,\cos^2\theta\:\overline{y^2} & +\,2 \cos\theta \sin\theta\:\overline{xy},\\ \overline{xy_{\theta}} & = & \cos\theta \sin\theta\:\overline{x^2} & -\,\cos\theta \sin\theta\:\overline{y^2} & +\,(\cos^2\theta - \sin^2\theta)\:\overline{xy}. \end{array} One can find interesting angles :math:`\theta_0` for which the variance is minimized (or maximized) along :math:`x_{\theta}`: .. math:: {\left.\frac{\partial \overline{x_{\theta}^2}}{\partial \theta} \right|}_{\theta_0} = 0, which leads to .. math:: 2 \cos\theta \sin\theta_0\:(\overline{y^2} - \overline{x^2}) + 2 (\cos^2\theta_0 - \sin^2\theta_0)\:\overline{xy} = 0. If :math:`\overline{y^2} \neq \overline{x^2}`, this implies: .. math:: \label{eq:theta0} \tan 2\theta_0 = 2 \frac{\overline{xy}}{\overline{x^2} - \overline{y^2}}, a result which can also be obtained by requiring the covariance :math:`\overline{xy_{\theta_0}}` to be null. Over the domain :math:`[-\pi/2, +\pi/2[`, two different angles — with opposite signs — satisfy ([eq:theta0]). By definition, THETA is the position angle for which :math:`\overline{x_{\theta}^2}` is *max*\ imized. THETA is therefore the solution to ([eq:theta0]) that has the same sign as the covariance :math:`\overline{xy}`. A and B can now simply be expressed as: .. math:: \begin{aligned} {\tt A}^2 & = & \overline{x^2}_{\tt THETA},\ \ \ {\rm and}\\ {\tt B}^2 & = & \overline{y^2}_{\tt THETA}.\end{aligned} A and B can be computed directly from the 2nd-order moments, using the following equations derived from ([eq:varproj]) after some algebra: .. math:: \begin{aligned} \label{eq:aimage} {\tt A}^2 & = & \frac{\overline{x^2}+\overline{y^2}}{2} + \sqrt{\left(\frac{\overline{x^2}-\overline{y^2}}{2}\right)^2 + \overline{xy}^2},\\ {\tt B}^2 & = & \frac{\overline{x^2}+\overline{y^2}}{2} - \sqrt{\left(\frac{\overline{x^2}-\overline{y^2}}{2}\right)^2 + \overline{xy}^2}.\end{aligned} Note that A and B are exactly halves the :math:`a` and :math:`b` parameters computed by the COSMOS image analyser :cite:`1980SPIE..264..208S`. Actually, :math:`a` and :math:`b` are defined in :cite:`1980SPIE..264..208S` as the semi-major and semi-minor axes of an elliptical shape with constant surface brightness, which would have the same 2nd-order moments as the analyzed object. Ellipse parameters: CXX, CYY, CXY --------------------------------- A, B and THETA are not very convenient to use when, for instance, one wants to know if a particular |SExtractor| detection extends over some position. For this kind of application, three other ellipse parameters are provided; CXX, CYY and CXY. They do nothing more than describing the same ellipse, but in a different way: the elliptical shape associated to a detection is now parameterized as .. math:: {\tt CXX} (x-\overline{x})^2 + {\tt CYY} (y-\overline{y})^2 + {\tt CXY} (x-\overline{x})(y-\overline{y}) = R^2, where :math:`R` is a parameter which scales the ellipse, in units of A (or B). Generally, the isophotal limit of a detected object is well represented by :math:`R\approx 3` (:numref:`fig_ellipse`). Ellipse parameters can be derived from the 2nd order moments: .. math:: \begin{aligned} {\tt CXX} & = & \frac{\cos^2 {\tt THETA}}{{\tt A}^2} + \frac{\sin^2 {\tt THETA}}{{\tt B}^2} = \frac{\overline{y^2}}{\overline{x^2} \overline{y^2} - \overline{xy}^2}\\ {\tt CYY} & = & \frac{\sin^2 {\tt THETA}}{{\tt A}^2} + \frac{\cos^2 {\tt THETA}}{{\tt B}^2} = \frac{\overline{x^2}}{\overline{x^2} \overline{y^2} - \overline{xy}^2}\\ {\tt CXY} & = & 2 \,\cos {\tt THETA}\,\sin {\tt THETA} \left( \frac{1}{{\tt A}^2} - \frac{1}{{\tt B}^2}\right) = -2\, \frac{\overline{xy}}{\overline{x^2} \overline{y^2} - \overline{xy}^2}\end{aligned} .. _fig_ellipse: .. figure:: figures/ellipse.* :figwidth: 100% :align: center Meaning of shape parameters. By-products of shape parameters: ELONGATION and ELLIPTICITY [1]_ ---------------------------------------------------------------- These parameters are directly derived from A and B: .. math:: \begin{aligned} {\tt ELONGATION} & = & \frac{\tt A}{\tt B}\ \ \ \ \ \mbox{and}\\ {\tt ELLIPTICITY} & = & 1 - \frac{\tt B}{\tt A}.\end{aligned} Position uncertainties: ERRX2, ERRY2, ERRXY, ERRA, ERRB, ERRTHETA, ERRCXX, ERRCYY, ERRCXY ----------------------------------------------------------------------------------------- Uncertainties on the position of the barycenter can be estimated using photon statistics. In practice, such estimates are a lower-value of the full uncertainties because they do not include, for instance, the contribution of detection biases or contamination by neighbors. Furthermore, |SExtractor| does not currently take into account possible correlations of the noise between adjacent pixels. Hence variances simply write: .. math:: \begin{aligned} {\tt ERRX2} & = {\rm var}(\overline{x}) = & \frac{\displaystyle \sum_{i \in {\cal S}} \sigma^2_i (x_i-\overline{x})^2} {\displaystyle \left(\sum_{i \in {\cal S}} I_i\right)^2},\\ {\tt ERRY2} & = {\rm var}(\overline{y}) = & \frac{\displaystyle \sum_{i \in {\cal S}} \sigma^2_i (y_i-\overline{y})^2} {\displaystyle \left(\sum_{i \in {\cal S}} I_i\right)^2},\\ {\tt ERRXY} & = {\rm cov}(\overline{x},\overline{y}) = & \frac{\displaystyle \sum_{i \in {\cal S}} \sigma^2_i (x_i-\overline{x})(y_i-\overline{y})} {\displaystyle \left(\sum_{i \in {\cal S}} I_i\right)^2}.\end{aligned} :math:`\sigma_i` is the flux uncertainty estimated for pixel :math:`i`: .. math:: \sigma^2_i = {\sigma_B}^2_i + \frac{I_i}{g_i}, where :math:`{\sigma_B}_i` is the local background noise and :math:`g_i` the local gain — conversion factor — for pixel :math:`i` (see §[chap:weight] for more details). Semi-major axis ERRA, semi-minor axis ERRB, and position angle ERRTHETA of the :math:`1\sigma` position error ellipse are computed from the covariance matrix exactly like in [chap:abtheta] for shape parameters: .. math:: \begin{aligned} \label{eq:erra} {\tt ERRA}^2 & = & \frac{{\rm var}(\overline{x})+{\rm var}(\overline{y})}{2} + \sqrt{\left(\frac{{\rm var}(\overline{x})-{\rm var}(\overline{y})}{2}\right)^2 + {\rm cov}^2(\overline{x},\overline{y})},\\ \label{eq:errb} {\tt ERRB}^2 & = & \frac{{\rm var}(\overline{x})+{\rm var}(\overline{y})}{2} - \sqrt{\left(\frac{{\rm var}(\overline{x})-{\rm var}(\overline{y})}{2}\right)^2 + {\rm cov}^2(\overline{x},\overline{y})},\\ \label{eq:errtheta} \tan (2{\tt ERRTHETA}) & = & 2 \,\frac{{\rm cov}(\overline{x},\overline{y})} {{\rm var}(\overline{x}) - {\rm var}(\overline{y})}.\end{aligned} And the ellipse parameters are: .. math:: \begin{aligned} \label{eq:errcxx} {\tt ERRCXX} & = & \frac{\cos^2 {\tt ERRTHETA}}{{\tt ERRA}^2} + \frac{\sin^2 {\tt ERRTHETA}}{{\tt ERRB}^2} = \frac{{\rm var}(\overline{y})}{{\rm var}(\overline{x}) {\rm var}(\overline{y}) - {\rm cov}^2(\overline{x},\overline{y})},\\ \label{eq:errcyy} {\tt ERRCYY} & = & \frac{\sin^2 {\tt ERRTHETA}}{{\tt ERRA}^2} + \frac{\cos^2 {\tt ERRTHETA}}{{\tt ERRB}^2} = \frac{{\rm var}(\overline{x})}{{\rm var}(\overline{x}) {\rm var}(\overline{y}) - {\rm cov}^2(\overline{x},\overline{y})},\\ \label{eq:errcxy} {\tt ERRCXY} & = & 2 \cos {\tt ERRTHETA}\sin {\tt ERRTHETA} \left( \frac{1}{{\tt ERRA}^2} - \frac{1}{{\tt ERRB}^2}\right)\\ & = & -2 \frac{{\rm cov}(\overline{x},\overline{y})}{{\rm var}(\overline{x}) {\rm var}(\overline{y}) - {\rm cov}^2(\overline{x},\overline{y})}.\end{aligned} Handling of “infinitely thin” detections ---------------------------------------- Apart from the mathematical singularities that can be found in some of the above equations describing shape parameters (and which |SExtractor| handles, of course), some detections with very specific shapes may yield quite unphysical parameters, namely null values for B, ERRB, or even A and ERRA. Such detections include single-pixel objects and horizontal, vertical or diagonal lines which are 1-pixel wide. They will generally originate from glitches; but very undersampled and/or low S/N genuine sources may also produce such shapes. For basic shape parameters, the following convention was adopted: if the light distribution of the object falls on one single pixel, or lies on a sufficiently thin line of pixels, which we translate mathematically by .. math:: \label{eq:singutest} \overline{x^2}\,\overline{y^2} - \overline{xy}^2 < \rho^2, then :math:`\overline{x^2}` and :math:`\overline{y^2}` are incremented by :math:`\rho`. |SExtractor| sets :math:`\rho=1/12`, which is the variance of a 1-dimensional top-hat distribution with unit width. Therefore :math:`1/\sqrt{12}` represents the typical minor-axis values assigned (in pixels units) to undersampled sources in |SExtractor|. Positional errors are more difficult to handle, as objects with very high signal-to-noise can yield extremely small position uncertainties, just like singular profiles do. Therefore |SExtractor| first checks that ([eq:singutest]) is true. If this is the case, a new test is conducted: .. math:: \label{eq:singutest2} {\rm var}(\overline{x})\,{\rm var}(\overline{y}) - {\rm covar}^2(\overline{x}, \overline{y}) < \rho^2_e, where :math:`\rho_e` is arbitrarily set to :math:`\left( \sum_{i \in {\cal S}} \sigma^2_i \right) / \left(\sum_{i \in {\cal S}} I_i\right)^2`. If ([eq:singutest2]) is true, then :math:`\overline{x^2}` and :math:`\overline{y^2}` are incremented by :math:`\rho_e`. .. [1] Such parameters are dimensionless and therefore do not accept any _IMAGE or _WORLD suffix .. include:: keys.rst