Shapelets at work

What you can see in this series of images is the original image data (left), which we will call I(x), and its approximation by a shapelet model (center),

The reconstuction depends on a set of parameters, the scale size β, the maximum order of the shapelet basis functions n_max and the position of the centroid x_c.
The residuals of this model (right),
  
are compatible with the surrounding noise. This can be achieved by defining the goodness-of-fit measure

and optimizing the parameters until the condition χ² < 1 is met.

Because of the noise our knowledge about the true brightness distribution of the galaxy is limited. Therefore, a shapelet model with χ² = 1 contains all information on the morphology of a galaxy we can get out of the noisy image. For achieving this, we do not need many coefficients: In this case the model was made up of 45 coefficients, whereas the image size was 66x66 pixels. Thus, any follow-up analysis of the galaxy properties can be done in the much smaller shapelet space, thereby saving memory and computation time. Furthermore, statistical approaches on galaxy measures (like the concentration, intrinsic ellipticity etc.) become straightforward by investigating trends in the rather compact shapelet space of a galaxy sample.