Non Parametric Models#

Non parametric models use a pixelizations reconstruct a galaxy’s light on a pixel-grid.

Unlike LightProfile’s, they are able to reconstruct the light of non-symmetric and irregular galaxies.

We will demonstrate this using a complex galaxy with multiple star forming clumps:

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Rectangular Example#

To fit this image with an Inversion, we first mask the Imaging object:

mask = al.Mask2D.circular(
   shape_native=dataset.shape_native, pixel_scales=dataset.pixel_scales, radius=3.0
 )

dataset = dataset.apply_mask(mask=mask)

To reconstruct the galaxy on a pixel-grid, called a mesh, we simply pass it the Mesh class we want to reconstruct its light on.

We also pass a Regularization scheme which applies a smoothness prior on the source reconstruction.

Below, we use a Rectangular pixelization with resolution 50 x 50 and a Constant regularization scheme:

pixelization = ag.Pixelization(
    mesh=ag.mesh.Rectangular(shape=(50, 50)),
    regularization=ag.reg.Constant(coefficient=1.0),
)

galaxy = ag.Galaxy(redshift=1.0, pixelization=pixelization)

Now that our galaxy has a Pixelization, we are able to fit the data using it in the same way as before, by simply passing the galaxy to a Plane and using this Plane to create a FitImaging object.

plane = ag.Plane(galaxies=[galaxy])

fit = ag.FitImaging(dataset=dataset, plane=plane)

Here is what our reconstructed galaxy looks like:

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Note how the reconstruction is irregular and has multiple clumps of light, these features would be difficult to represent using analytic light profiles!

Positive Only Solver#

All pixelized source reconstructions use a positive-only solver, meaning that every source-pixel is only allowed to reconstruct positive flux values. This ensures that the source reconstruction is physical and that we don’t reconstruct negative flux values that don’t exist in the real source galaxy (a common systematic solution in lens analysis).

It may be surprising to hear that this is a feature worth pointing out, but it turns out setting up the linear algebra to enforce positive reconstructions is difficult to make efficient. A lot of development time went into making this possible, where a bespoke fast non-negative linear solver was developed to achieve this.

Other methods in the literature often do not use a positive only solver, and therefore suffer from these unphysical solutions, which can degrade the results of lens model in general.

Why Use Pixelizations?#

From the perspective of a scientific analysis, it may be unclear what the benefits of using an inversion to reconstruct a complex galaxy are.

When I fit a galaxy with light profiles, I learn about its brightness (intensity), size (effective_radius), compactness (sersic_index), etc.

What did I learn about the galaxy I reconstructed? Not a lot, perhaps.

Inversions are most useful when combined with light profiles. For the complex galaxy above, we can fit it with light profiles to quantify the properties of its bulge and disk components, whilst simultaneously fitting the clumps with the inversion so as to ensure they do not impact the fit.

The workspace contains examples of how to do this, as well as other uses for pixelizations.

Wrap-Up#

This was a brief overview of Inverion’s.

There is a lot more to using Inverion’s then presented here, which is covered in chapters 4 of the HowToGalaxy, specifically:

  • How the inversion’s reconstruction determines the flux-values of the galaxy it reconstructs.

  • The Bayesian framework employed to choose the appropriate level of Regularization and avoid overfitting noise.

  • Unphysical model solutions that often arise when using an Inversion.