Inversion module ‘general’

This lens inversion module can handle a variety of input data: image data, null space grids, time delay info, etc. Based on the precise input that was provided, one or more fitness measures will be used and the genetic algorithm will try to optimize these. The currently implemented fitness measures are:

  • extendedimageoverlap: for multiply imaged extended sources, this fitness measure calculates the fractional overlap of back-projected images as described in Liesenborgs et al (2006) [6]. Point sources can be included as well, in that case the average size of the extended images is used as the distance scale for determining how well the back-projected points overlap.

  • pointimageoverlap: for strongly lensed point sources, this is a measure for how well the back-projected point images overlap in the source plane, as described in Zitrin et al (2010) [7]. By default the size of the area encompassing all backprojected points is used as a distance scale in determining the overlap, but the median of absolute deviations MAD can be used as well (see the section about ‘module parameters’).

  • pointgroupoverlap: an experimental fitness measure, which backprojects specific points and estimates the RMS in the image plane. Either the average source position is used, or each backprojected image is used separately as a source position estimate (see the section about ‘module parameters’). The magnification matrix is used to map differences in the source plane to differences in the image plane.

  • extendedimagenull: this fitness measure takes the null space into account, the region where no images should be found, as described in Liesenborgs et al (2007) [8]. An extension was made so allow it to add a penalty when multiple images are generated when only a single image should be allowed.

  • pointimagenull: similar to the null space fitness measure for extended images but now for point images, also described in Zitrin et al (2010) [7]. Here too the extension was made to allow singly imaged sources to be used as a contraint.

  • timedelay: when time delay information is available for one or more multiply-imaged sources, a fitness measure will be calculated as described in Liesenborgs et al (2009) [9] or as in Liesenborgs et al (2020) [10]. You can select which type to use using the fitness_timedelay_type parameters (see the section about ‘module parameters’).

  • causticpenalty: also described in Liesenborgs et al (2009) [9], it is also possible to add a fitness measure to try to prevent a caustic from intersecting a reconstructed source.

  • weaklensing: if shear information is provided, a \(\chi^2\)-like fitness measure will be provided for this, as described in Liesenborgs et al (2020) [10]. By default, the shear data will be interpreted as averaged ellipticities, but if desired (for testing purposes for example), real shear, or actual reduced shear can be specified as well (see the section about ‘module parameters’). For each data point, a weight can be specified.

  • kappathreshold: if it’s certain that the convergence (\(\kappa\)) values at certain locations should not exceed a certain threshold, this fitness measure is added.

  • bayesweaklensing: TODO, bayesian, per galaxy

In general, when more than one fitness measure is used, at the end of the algorithm a set of mass maps is found in which no single one will be better than another with respect to all fitness measures. While you can specify that all of these solutions need to be retrieved, usually you’ll just want to keep a single solution. To choose this solution, the order of importance of the fitness measures can be left to the default or you can specify it yourself. See the section about ‘module parameters’ for more information.

Input images data list

In InversionWorkSpace.addImageDataToList, you specify the image type, as well as (optionally) extra parameters. The images data set type name describes what kind of data is contained in this entry, and can be one of the following:

  • pointimages: the images data set describes point images that originate from a single source (also a point of course). Alternatively, extended images may be provided as well, but in this case point groups specifying which points are really images of the same source plane point must be present. In general in a strong lensing scenario there will be more than one image, but single images are allowed as well. If this is the case, the image can’t be used directly because it originates from a single point source by definition, but it can still be useful for a null space constraint to inform the algorithm that this source should not lie in a region that produces multiple images.

    For the overlap fitness, two options are possible: if the same groupname is used for more multiple point images, the overlap of the points will be measured relative to the scale of only these back-projected points. By default, the scale of all backprojected points is used. If usescalefrom is present, then for those points the scale set by another set of points, identified by a specific groupname.

    In case time delay information was added to the images data set, by default a time delay fitness measure will be calculated as well. This can still be disabled by specifying a boolean extra parameter timedelay that is set to False.

    The fitness measures to which this type of data is relevant, are pointimageoverlap, pointimagenull and timedelay.

  • pointgroupimages: an image data set with type is used in the pointgroupoverlap fitness measure, which calculates the RMS in the image plane. The input should either be point image data, or extended images in which point groups are used to indicate which points belong together.

    In case time delay information was added to the images data set, by default a time delay fitness measure will be calculated as well. This can still be disabled by specifying a boolean extra parameter timedelay that is set to False.

    The fitness measures to which this type of data is relevant, are pointgroupoverlap, pointimagenull and timedelay.

  • pointnullgrid: the images data set should contain only one ‘image’, a triangulated grid, and is interpreted as a null space grid. It refers to the previous images data set of pointimages type and the algorithm will use this grid to estimate the number of extra images that are generated by a trial solution.

    One extra parameter can be specified: weight, which should be a real and positive value. The estimated amount of images is multiplied by this number, and can be used to adjust the relative importance of some null spaces. This can be useful to provide extra weight to singly imaged sources because they may otherwise provide only a small penalty.

    The only fitness measure to which this type of data is relevant, is pointimagenull.

  • extendedimages: in this case the images data set describes the extended images originating from a single source. Therefore, in principle each image should consist of at least three points. A set of point images may be provided as well, but since each separate point image does not provide a distance scale to measure overlap with, the average size of the other backprojected (extended images) is used as the distance scale instead. A single image containing exactly one point is allowed as well. In this case it will not be used as a constraint regarding the overlap of back-projected images in the source plane, but it can be used as a constraint in the null space, indicating that the corresponding source plane position should produce only one image.

    For extended images, the extra parameters userectangles and usepointgroups (both taking a boolean value) can be useful. By default, the overlap of surrounding back-projected rectangles is always used, but can be disabled with the first option. If point groups (points in different images that correspond to each other) are available in the images data set, they are used by default as well. The second option can disable their use.

    In case time delay information was added to the images data set, by default a time delay fitness measure will be calculated as well. This can still be disabled by specifying a boolean extra parameter timedelay that is set to no.

    The fitness measures to which this type of data is relevant, are extendedimageoverlap, extendedimagenull``, pointimagenull and timedelay.

  • extendednullgrid: similar to pointnullgrid, the images data set should contain only one ‘image’, a triangulated grid, and is interpreted as a null space grid. Here, typically the regions containing observed images or regions that may harbor an as yet undetected image are removed from the grid. The images data set refers to the previous images data set of extendedimages type and the algorithm will use this grid to calculate the sizes of the regions in the image plane that contain additional images.

    One extra parameter can be specified: weight, which should be a real and positive value. The total size of the additional images is multiplied by this number, and can be used to adjust the relative importance of some null spaces. This can be useful to provide extra weight to singly imaged sources because they may otherwise provide only a small penalty.

    The only fitness measure to which this type of data is relevant, is extendedimagenull.

  • sheardata: use this type to provide shear measurements to the algorithm. The images data set should contain only one ‘image’, a set of points in the image plane for which shear components have been specified. One extra parameter (a real number) called threshold must be provided: this contains a threshold value for \(|1-\kappa|\). Only when at a certain point the value for \(|1-\kappa|\) exceeds the specified threshold, will it be included in the \(\chi^2\)-like calculation.

    The only fitness measure to which this type of data is relevant, is weaklensing.

  • bayesellipticities: TODO, galaxy ellipticities.

    The only fitness measure to which this type of data is relevant, is bayesweaklensing.

  • kappathresholdpoints: this data set should only contain a single ‘image’, a set of points at which the convergence \(\kappa\) is calculated. The mandatory extra (real valued) parameter threshold specifies if a penalty is needed for a certain point: if its convergence is below the threshold, no penalty is added to the fitness measure, otherwise the amount by which the threshold is exceeded is added.

    The only fitness measure to which this type of data is relevant, is kappathreshold.

  • causticgrid: when an images data set of this type is specified, it should contain a triangulated grid, which will be used to estimate the caustics. The algorithm will calculate the length of the caustics that intersect the estimated source, and this source estimate is based on the previously encountered images data set marked as extendedimages.

    The only fitness measure to which this type of data is relevant, is causticpenalty.

  • singlyimagedpoints: the images data set of this type should contain only one ‘image’ entry, which may consist of several points. Each point is assumed to have only a single image, i.e. it should not lie in a multiply imaged region. These data are therefore intended to be used together with a null space grid of type pointnullgrid. The points could also be specified separately as pointimages, but this way is more convenient if several points are needed for which the same null space grid can be used.

    The only fitness measure to which this type of data is relevant, is pointimagenull.

Module parameters

Extra parameters for this module can be set using the fitnessObjectParameters argument in e.g. inversion.invert. The defaults can be obtained using the command inversion.getDefaultModuleParameters, and are listed in the following table:

Parameter name

Value

priority_causticpenalty

100

priority_pointimagenull

200

priority_extendedimagenull

200

priority_pointgroupoverlap

250

priority_pointimageoverlap

300

priority_extendedimageoverlap

300

priority_timedelay

400

priority_weaklensing

500

priority_bayesweaklensing

500

priority_kappathreshold

600

scalepriority_pointimageoverlap

100

scalepriority_extendedimageoverlap

100

scalepriority_pointgroupoverlap

200

scalepriority_pointimagenull

-1

scalepriority_extendedimagenull

-1

scalepriority_weaklensing

300

scalepriority_timedelay

-1

scalepriority_kappathreshold

-1

scalepriority_causticpenalty

-1

scalepriority_kappagradient

-1

scalepriority_bayesweaklensing

300

fitness_pointgroupoverlap_rmstype

‘AllBetas’

fitness_pointimageoverlap_scaletype

‘MinMax’

fitness_timedelay_type

‘NoSrc’

fitness_weaklensing_type

‘AveragedEllipticities’

fitness_bayesweaklensing_zdist_values

None

fitness_bayesweaklensing_zdist_range

None

fitness_bayesweaklensing_zdist_numsamples

16

fitness_bayesweaklensing_b_over_a_distribution

None

fitness_bayesweaklensing_sigmafactor

3.0

fitness_bayesweaklensing_sigmasteps

7

In case input is provided with pointgroupimages type, the pointgroupoverlap fitness calculation is used, which estimates the RMS in the image plane. It does this by projecting the image points onto the source plane, determining a source position based on these points, and using the magnification matrix to convert differences in the source plane to difference in the image plane. By default, each backprojected image point is used as a possible source position, corresponding to the value AllBetas of fitness_pointgroupoverlap_rmstype. To use the averaged source position instead, you can set it to AverageBeta.

If input is provided of pointimages type, the pointimageoverlap fitness calculation will be activated. It projects the image points onto the source plane, and uses the differences between the backprojected points to base the fitness measure on. The distance scale with which these differences are measured depends on all backprojected points and by default the size of the entire area is used. This corresponds to the setting fitness_pointimageoverlap_scaletype to MinMax. In case the median of absolute deviations should be used instead, this option can be set to MAD.

The fitness_timedelay_type can also be Paper2009, to use the older one from the 2009 article [9].

For the weak lensing fitness, the data is by default interpreted as (averaged) ellipticity measurements. This corresponds the default value of AveragedEllipticities for fitness_weaklensing_type. In case true shear is supplied, or the actual reduced shear, the value can also be RealShear or RealReducedShear respectively. This is mainly meant for testing purposes.

As the names suggest, the options that start with priority_ describe priorities for fitness measures. These values do not have any effect on the way the genetic algorithm operates, only on the final solution that is chosen from the set of ‘best’ solutions.

Here, the lower the priority value, the more important it is considered to be, so if for example both the extendedimageoverlap and extendedimagenull fitness measures are used based on the provided images data sets, there may be more than one ‘best’ solution found by the genetic algorithm. You may decide to save all these solutions by setting returnNds to True in the invert function, but typically you’ll want to go on using a single solution.

It is based on the priorities that were specified, that one solution will be chosen. Using the default settings in our example, first the solution(s) will be chosen with the lowest value for the extendedimagenull fitness value, and then for extendedimageoverlap. You can force this order to be turned around by overriding some of these values in the fitnessObjectParameters argument.

As you can see, some priorities have the same value. In case two fitness measures with the same priority are used, the number of images is typically used as a tie breaker. For example, if you’ve specified both point images and extended images as input, the fitness measures pointimageoverlap and extendedimageoverlap will be used. When choosing a single final solution, the fitness value that corresponds to most images will have the best priority for the default settings. So if there are more point images than extended images, the point image criterion will be considered first.

References