Accounting for PSF Effects in 2D Image Fitting
Sherpa Threads (CIAO 4.15 Sherpa)
In this thread, we fit 2-D image data using a Point-Spread-Function (PSF) image as the convolution kernel.
Last Update: 9 Dec 2022 - Updated for CIAO 4.15, typos fixed.
- Getting Started
- Reading & Filtering Image Data
- Defining the Instrument Model with a PSF Image
- Defining a Multi-Component Source Model Expression
- Modifying Statistic Setting
- Examining Fit Results
- Scripting It
Please follow the "Obtaining data used in Sherpa threads" thread.
Reading & Filtering Image Data
In this thread, we fit 2-D image data from the FITS datafile center_box_0.25pix.fits. This is the image created from an event file by binning over a region to 0.25 ACIS pixel size with dmcopy; for example:
unix% dmcopy "event.fits[sky=region(center_box.reg)][bin x=::0.25,y=::0.25]" \ center_box_0.25pix.fits
This dataset is input into Sherpa with the load_data command:
Now the dataset may be displayed:
The input data image is shown in Figure 1.
Figure 1: Input data image
Image data may be filtered in Sherpa interactively with DS9. After the data have been displayed, go to the Region menu in DS9 and choose a designated region shape. In this example, we use the box shape. When the box is displayed, the size of the box can be changed with the cursor or within the information window which is opened via the Region→Get Information menu item, as shown in Figure 2.
Figure 2: Selecting box region in DS9
After the desired region size is set, the region can be used to filter the data as follows:
sherpa> image_getregion() 'box(88.16875,75.8625,70.416667,68.508334,0);' sherpa> notice2d("rotbox(88.16875,75.8625,70.416667,68.508334,0)") dataset 1: Field() -> Box(88.1688,75.8625,70.4167,68.5083)
In this case, the dataset is using image coordinates, so we have to make sure that DS9 is set to interact with Sherpa in physical coordinates by setting Analysis→CIAO→Coords→Image.
sherpa> show_all() Data Set: 1 Filter: Box(88.1688,75.8625,70.4167,68.5083) name = center_box_0.25pix.fits x0 = Float64 x1 = Float64 y = Float64 shape = (147, 169) staterror = None syserror = None sky = physical crval = [4075. ,4039.25] crpix = [0.5,0.5] cdelt = [0.25,0.25] eqpos = world crval = [248.6211, 70.531 ] crpix = [4096.5,4096.5] cdelt = [-0.0001, 0.0001] crota = 0 epoch = 2000 equinox = 2000 coord = logical
The same filter can be set on the command line with the region definition if it is already known. Note that Sherpa requires the Image coordinates (and not Physical coordinates) to be used in region definitions:
Defining the Instrument Model with a PSF Image
An instrument model, named psf0, may be established using an image of a PSF. The PSF image is loaded with the load_psf command, then added to the list of models in the session as a PSF instrument model with set_psf.
sherpa> load_psf("psf0", "psf_f1_norm_0.25pix.fits") sherpa> set_psf("psf0") sherpa> show_psf() PSF Model: 1 name = psf_f1_norm_0.25pix.fits x0 = Float64 x1 = Float64 y = Float64 shape = (256, 256) staterror = None syserror = None sky = physical crval = [4064.5,4026.5] crpix = [0.5,0.5] cdelt = [0.25,0.25] eqpos = world crval = [248.6211, 70.531 ] crpix = [4096.5,4096.5] cdelt = [-0.0001, 0.0001] crota = 0 epoch = 2000 equinox = 2000 coord = logical sherpa> print(psf0) psfmodel.psf0 Param Type Value Min Max Units ----- ---- ----- --- --- ----- psf0.kernel frozen psf_f1_norm_0.25pix.fits psf0.size frozen (256, 256) (256, 256) (256, 256) psf0.center frozen (128, 128) (128, 128) (128, 128) psf0.radial frozen 0 0 1 psf0.norm frozen 1 0 1
Note that the PSF is automatically renormalized to 1. Renormalization is done by summing over all image pixels, regardless of the setting of xsize and ysize. To learn more about 2-D PSF instrument models, please read the ahelp file by typing:
sherpa> ahelp "set_psf"
The PSF image file for this example was created using the CIAO tool mkpsf, which is no longer available as of the CIAO 4.5 release in Decemeber 2012. The best PSF image can be created with The Chandra Ray Tracer (ChaRT). The file center_box_0.25pix.fits was used as input to mkpsf in order to match the binning of the resulting PSF image file (psf_f1_norm_0.25pix.fits). Sherpa requires that the binning of the PSF image file match the binning of the input data image.
To view the PSF image, load it into DS9 with the Sherpa image_psf command.
The PSF image is shown in Figure 3.
Figure 3: PSF image
2-D PSF instrument model terms and parameters
PSF: The point spread function (PSF) is defined by the full (unfiltered) PSF image loaded into Sherpa (or the PSF model expression evaluated over the full range of the dataset; both types of PSFs are established with the load_psf command). The PSF may be displayed with the show_psf command, and visualized with image_psf, plot_psf, and contour_psf.
kernel: The kernel is the sub-section of the PSF image (or model) which is used to convolve the data. This sub-section is created from the PSF when the size and center of the kernel are defined by set_psf. The PSF kernel information may be displayed with the show_kernel command, and visualized with image_kernel, plot_kernel, and contour_kernel.
The PSF image provided may be much larger than the instrument PSF size (in number of pixels), and therefore larger than needed. In order to speed the convolution process—by restricting the number of points within the PSF that Sherpa must evaluate—a square, sub-image kernel is specified. In this example, the input PSF image file has 256x256 pixels, but the portion that will be used for the convolution is only the center 32x32 pixels sub-image (note that the x and y size parameters need not be equal; rectangular PSFs are supported).
sherpa> psf0.size = [32,32] sherpa> psf0.center = [128,129] sherpa> show_kernel() psfmodel.psf0 Param Type Value Min Max Units ----- ---- ----- --- --- ----- psf0.kernel frozen psf_f1_norm_0.25pix.fits psf0.size frozen (32, 32) (32, 32) (32, 32) psf0.center frozen (128, 129) (128, 129) (128, 129) psf0.radial frozen 0 0 1 psf0.norm frozen 1 0 1
Note that the sub-image will be empty if the PSF centroid is located outside the sub-image.
In some cases, a larger sub-image will be needed, such as when the source is located off-axis, or when including the PSF wings is important for the analysis. We may update the parameters to increase the PSF fraction included in the sub-image:
sherpa> psf0.size = [72, 72] sherpa> print(get_psf()) psfmodel.psf0 Param Type Value Min Max Units ----- ---- ----- --- --- ----- psf0.kernel frozen psf_f1_norm_0.25pix.fits psf0.size frozen (72, 72) (72, 72) (72, 72) psf0.center frozen (128, 129) (128, 129) (128, 129) psf0.radial frozen 0 0 1 psf0.norm frozen 1 0 1 sherpa> image_psf()
The instrument PSF sub-images may be displayed with the image_psf command, as shown in Figure 4.
Figure 4: PSF sub-image for convolution
The left panel shows the smaller sub-image of 32x32 pixels, and the right panel shows an expanded to 72x72 pixels sub-image. Notice that the larger sub-image contains a significant amount of the PSF wings structure.
The kernel centroid must always be at the center of the extracted sub-image. Otherwise, systematic shifts will occur in best-fit positions of point sources, etc. The parameter center=[center0,center1] determines the location of the original PSF image on which the sub-image is centered.
The PSF image model array is renormalized to 1 by default, unless the parameter 'norm' is 0; norm=0 produces the functionality of a 2-D kernel model. Renormalization is done by summing over all image pixels, regardless of the x and y size values.
A Sherpa PSF model includes an optional, hidden 'origin' attribute which allows the user to choose which pixel in the PSF kernel to mark as the origin; Sherpa sets the origin to the brightest pixel in the kernel by default. The psf.origin parameter of a PSF model is set using the (x,y) coordinates of a chosen pixel in the PSF kernel, but decremented by one in both x and y directions. For example, if DS9 was used to find that the brightest pixel is at (122, 131), psf.origin should be set equal to '(121, 130)'. This is necessary because Sherpa internally stores data, PSF, and kernel information as NumPy arrays—which start counting from element 0—whereas DS9 uses pixel coordinates, with the first pixel having a value of (1, 1).Caution
In order to appropriately use the psf.origin attribute, the axes of the PSF image must both be even numbers. That is, if the axes of the PSF image are [244, 262] pixels, then there would be no problems setting the psf.origin; but for axes lengths of [243, 262], [244, 261], or [243, 261] pixels, the set psf.origin value gets shifted by one-pixel from the intended origin of the kernel.
When using the Sherpa 2-D PSF model, the user should ensure that the PSF file contains an image where the length of each axis is an even number.
If psf.origin is not set by the user—leaving Sherpa to automatically locate the position of the brightest pixel in the kernel and set this as the origin—Sherpa will find the correct location, provided that a) the brightest pixel should coincide with the origin in the analysis, and b) that there is only one pixel that has that brightest value.
The radial parameter of the PSF model is not applicable to 2-D PSFs; to learn about its use in 1-D PSF models, refer to the set_psf ahelp file.
Defining a Multi-Component Source Model Expression
Do not fit 2-D models with world coordinates in Sherpa. 2-D fitting should only be done in image or physical coordinates. After a fit is completed, then the user may convert the fitted position to RA and Dec, and convert the other parameters (e.g. FWHM) to the desired units (e.g. arcsec).
Now we will define a source model expression using a two-dimensional Gaussian function (gauss2d) and a constant (const2d), which will be convolved by the PSF model using Fast Fourier Transform (FFT). We will then define reasonable initial parameter values to begin the fitting (note that the Sherpa guess command may also be used).
sherpa> set_source(const2d.c1 + gauss2d.g1) sherpa> c1.c0 = 1.0 sherpa> g1.fwhm = 6 sherpa> g1.xpos = 90 sherpa> g1.ypos = 80 sherpa> g1.ampl = 100 sherpa> show_source() Model: 1 (const2d.c1 + gauss2d.g1) Param Type Value Min Max Units ----- ---- ----- --- --- ----- c1.c0 thawed 1 0 3.40282e+38 g1.fwhm thawed 6 1.17549e-38 3.40282e+38 g1.xpos thawed 90 -3.40282e+38 3.40282e+38 g1.ypos thawed 80 -3.40282e+38 3.40282e+38 g1.ellip frozen 0 0 0.999 g1.theta frozen 0 -6.28319 6.28319 radians g1.ampl thawed 100 -3.40282e+38 3.40282e+38
The model component c1 is interpreted as the constant background contribution to the data. We can also see how the source model is convolved with the PSF by using the show_model command:
sherpa> show_model() PSF Model: 1 name = psf_f1_norm_0.25pix.fits x0 = Float64 x1 = Float64 y = Float64 shape = (256, 256) staterror = None syserror = None sky = physical crval = [4064.5,4026.5] crpix = [0.5,0.5] cdelt = [0.25,0.25] eqpos = world crval = [248.6211, 70.531 ] crpix = [4096.5,4096.5] cdelt = [-0.0001, 0.0001] crota = 0 epoch = 2000 equinox = 2000 coord = logical Model: 1 psfmodel.psf0((const2d.c1 + gauss2d.g1)) Param Type Value Min Max Units ----- ---- ----- --- --- ----- c1.c0 thawed 1 -3.40282e+38 3.40282e+38 g1.fwhm thawed 6 1.17549e-38 3.40282e+38 g1.xpos thawed 90 -3.40282e+38 3.40282e+38 g1.ypos thawed 80 -3.40282e+38 3.40282e+38 g1.ellip frozen 0 0 0.999 g1.theta frozen 0 -6.28319 6.28319 radians g1.ampl thawed 100 -3.40282e+38 3.40282e+38
Modifying Statistic Setting
Since the data being fit has low counts, we will change the fit statistic to cash:
Note that truncation is turned on when the Cash statistic is used; this setting prevents negative model-predicted data values from affecting the convergence process.
Further details about the Cash statistic method are available by typing:
sherpa> ahelp "cash"
First, the data is fit assuming a constant, frozen background component. We also want to change the fit optimization method from the default method of Levenberg-Marquardt (levmar) to Nelder-Mead Simplex, since the former is not robust with our choice of statistics.
sherpa> freeze(c1.c0) sherpa> set_method("neldermead") sherpa> set_method_opt("iquad", 0) sherpa> set_method_opt("finalsimplex", 0) sherpa> fit() Dataset = 1 Method = neldermead Statistic = cash Initial fit statistic = 7648.86 Final fit statistic = 4066.78 at function evaluation 296 Data points = 4899 Degrees of freedom = 4895 Change in statistic = 3582.08 g1.fwhm 2.80105 g1.xpos 88.6612 g1.ypos 77.227 g1.ampl 166.653
The fit is run again with the background component thawed:
sherpa> thaw(c1) sherpa> fit() Dataset = 1 Method = neldermead Statistic = cash Initial fit statistic = 4066.78 Final fit statistic = -4457.71 at function evaluation 551 Data points = 4899 Degrees of freedom = 4894 Change in statistic = 8524.49 c1.c0 0.0027774 g1.fwhm 3.73819 g1.xpos 88.5798 g1.ypos 77.2406 g1.ampl 113.113
Now, we will get the confidence level of the fit, and in doing so, recalculate the best-fit parameters using the confidence method.
sherpa> conf() g1.ypos lower bound: -0.0679715 g1.xpos lower bound: -0.068848 g1.ypos upper bound: 0.0679715 g1.xpos upper bound: 0.068848 g1.ampl lower bound: -8.9519 g1.ampl upper bound: 10.2147 c1.c0 lower bound: -0.00125664 c1.c0 upper bound: 0.00146963 g1.fwhm lower bound: -0.151732 g1.fwhm upper bound: 0.151732 Dataset = 1 Confidence Method = confidence Iterative Fit Method = None Fitting Method = neldermead Statistic = cash confidence 1-sigma (68.2689%) bounds: Param Best-Fit Lower Bound Upper Bound ----- -------- ----------- ----------- c1.c0 0.0027774 -0.00125664 0.00146963 g1.fwhm 3.73819 -0.151732 0.151732 g1.xpos 88.5798 -0.068848 0.068848 g1.ypos 77.2406 -0.0679715 0.0679715 g1.ampl 113.113 -8.9519 10.2147
Note that it is possible to simultaneously fit multiple data sets where each is fit using a different PSF model. In order to do this, the load_psf command would be called for each separate PSF model required for fitting, and then set_psf would be used to assign the appropriate PSF model to each data set.
sherpa> load_psf("psf1", "gauss_psf.fits") sherpa> set_psf(psf1) sherpa> set_source(const2d.c1 + gauss2d.g1) sherpa> load_psf("psf2", beta2d) sherpa> set_psf(2, psf2) sherpa> set_source(2, const2d.c2 + gauss2d.g2 + beta2d.b2) sherpa> fit()
In the example above, PSF model 'psf1' is convolved with the source model for data set 1, and PSF model 'psf2' is convolved with the source model for data set 2, when 'fit()' is called to simultaneously fit data sets 1 and 2.
Examining Fit Results
The fit results may be examined with the image_fit command:
This command displays the data (upper left), best fit model (upper right), and residuals (lower left) in three DS9 frames, as shown in Figure 5.
Figure 5: Data, best-fit model, and residuals
This montage of images can be used to see how well the model describes the data.
The residuals may also be examined using a contour plot:
sherpa> contour_resid() sherpa> plt.gca().set_aspect('equal') sherpa> plt.xlim(55, 120) sherpa> plt.ylim(45, 110)
The resulting contour plot is shown in Figure 6.
Figure 6: Contour plot of the residuals
The image model and residual data (in counts) may be written to external FITS image files with the following commands:
sherpa> save_model("2dpsf_model_cnt.fits") sherpa> save_resid("2dpsf_resid_cnt.fits")
The residuals file "2dpsf_resid_cnts.fits" may be analyzed for any significant difference between the model and the data. It can also be smoothed with aconvolve for better visual inspection.
Note that the effects of 2-D blurring in a 2-D image cannot be reproduced by convolving the radial profile of the PSF with a profile of the model.
The file fit.py is a Python script which performs the primary commands used above; it can be executed by typing %run -i fit.py on the Sherpa command line.
The Sherpa script command may be used to save everything typed on the command line in a Sherpa session:
sherpa> script(filename="sherpa.log", clobber=False)
(Note that restoring a Sherpa session from such a file could be problematic since it may include syntax errors, unwanted fitting trials, et cetera.)
|14 Dec 2004||reviewed for CIAO 3.2: no changes|
|21 Dec 2005||reviewed for CIAO 3.3: minor changes to fit results|
|29 Jun 2006||added () = evalfile("sherpa_plotfns.sl"); command in Examining Fit Results section|
|01 Dec 2006||reviewed for CIAO 3.4: no changes|
|03 Dec 2008||reviewed for CIAO 4.1: updated for Sherpa 4.1|
|03 Apr 2009||added note on 2D blurring|
|24 Apr 2009||information on PSF models is in the "set_psf" ahelp file (previously was "ahelp psf2d")|
|29 Apr 2009||new script command is available with CIAO 4.1.2|
|07 Jan 2010||updated for CIAO 4.2|
|21 Mar 2010||added a note about the CIAO 4.2 bug associated with rectangular PSF models|
|08 Jun 2010||updated to include save_model and save_resid functions for saving 2-D fit results|
|29 Jun 2010||updated for CIAO 4.2 Sherpa v2: rectangular 2-D PSF models are now supported; it is possible to simultaneously fit multiple data sets where each independent source model is convolved by a different PSF model; show_psf() suppresses header information for PSFs loaded from file; removal of S-Lang version of thread.|
|03 Sep 2010||figures moved inline with text|
|06 Jan 2012||reviewed for CIAO 4.4: 2-D PSF kernels can now be larger than the source data (previously, the kernel was restricted to be equal to or smaller than the source).|
|08 Feb 2012||added information on the optional, hidden 'origin' attribute of a Sherpa PSF model|
|07 Apr 2014||added warnings about PSF image sizes to be used with psf.origin parameter and WCS should not be used for 2-D fitting.|
|24 Mar 2015||updated for CIAO 4.7, no content change.|
|10 Dec 2015||reviewed for CIAO 4.8, no content change.|
|01 Nov 2016||reviewed for CIAO 4.9, added admonition block about Python 3 and bytes-strings|
|01 Jun 2018||reviewed for CIAO 4.10, no content change.|
|13 Dec 2019||Updated for CIAO 4.12: switched from ChIPS to Matplotlib.|
|29 Mar 2022||reviewed for CIAO 4.14, no content change.|
|09 Dec 2022||Updated for CIAO 4.15, typos fixed.|