The Roulette Formalism

Computation of the Roulette Amplitudes

The reference points are calculated in the following order

1. $\eta$ is the actual source position as given by the problem.
2. $\xi$ is the apparent position in the lens plane. This is calculated by inverting the ratrace equation.
3. $\nu=\xi/\chi$ is the apparent position in the source plane.
4. $\nu’$ is the centre of light in the distorted image. This is calculated from the output image, and it serves as an objective reference point which can be recalculated from the image regardless of shifts and cropping.
5. $\xi’=\chi\nu$ scales the centre point back to the lens plane.
6. $\eta’$ is the source point corresponding to $\xi’$ according to the raytrace equation. In principle, any simulator model can compute this, but raytrace is most efficient and there is no reason not to use this.

RouletteModel: Forward Simulation in the Roulette Formalism

• updateApparentAbs() calculates inferred parameters when all input parameters have been set.
  • It calls lens-<updatePsi() to make the lens consistent
  • it gets $\xi$ (referenceXi) from lens->getXi($\chi\eta$)
• getXi() is implemented differently in
  • The superclass Lens, using raytrace logic
    • \xi = \chi\eta + \nabla\psi(\chi\eta)
  • SampledLens which uses fix point iteration
  • PointMass, but this needs review (TODO)

RouletteRegenerator: Simulation from Roulette Amplitudes

To make the reconstruction work, we need the following geometrical points.

1. $\xi$ is the apparent position and the reference point for the roulette model in the original formulation. Unfortunately it cannot easily be recovered from a distorted image.
2. $\nu’=\xi’/\chi$ is the centre of luminence in the distorted image, and can be recovered from the image. Thus this will be used as the reference point and the centre of the image in the regeneration. This is stored in the CSV file as (centreX,centreY).
3. $\eta$ and $\eta’$ are the source position corresponding respectively to $\xi$ and $\xi’$.
   • $\eta$ is input to the original simulation, but it cannot be recovered without knowledge of $\xi$
   • $\eta’$ is computed from $\xi’$ using raytracing.
4. $\Delta\eta=\eta’-\eta$ is the difference between the centre of the source ($\eta$) and the reference point $\eta’$ corresponding to the centre of the roulette convergence ring. It is calculated by LensModel::getOffset$(\nu’)$ and stored in the CSV output from datagen.py as (offsetX,offsetY).
5. $\eta’’=\eta-\nu’$ is the source position ($\eta$) in the co-ordinate system centred at $\nu’$. This is calculated by LensModel::getRelativeEta$(\nu’)$, and stored in the CSV output from datagen.py as (reletaX,reletaY).

Concepts in the original co-ordinate system centred at the lens.

Concept (Ray)	Lens Plane	Source Plane	Data file
Apparent source position	$\xi$	$\nu$
Actual source position	($\chi\eta$)	$\eta$
Centre of light	$\xi’$	$\nu’$	(centreX,centreY)
Source of Centre of light	($\chi\eta’$)	$\eta’$
Offset		$\Delta\eta=\eta’-\eta$	(offsetX,offsetY)

Concepts in the image co-ordinate system. We assume here that the image has been centred at $\nu’$. If the image is not centred, then $\nu’=(0,0)$ for the purpose of calculation.

Concept (Ray)	Image Plane	Data file
Apparent source position	$\nu-\nu’$	(xiX,xiY)
Actual source position	$\eta’’=\eta-\nu’$	(reletaX,reletaY)

Note that $\eta$ and $\eta’$ are defined in the source plane and $\xi$ and $\xi’$ in the lens plane. Working in the source image, which is also in scale with the images, we have $\nu=\xi/\chi$ and $\nu’=\xi’/\chi$.

Generate the Roulette Data Set

The datagen.py script generates the data set.

1. Normal image generation
2. Centre the image and record the image centre $\nu’=\xi’/\chi$.
3. Find $\xi’=\chi\nu’$
4. Find the roulette amplitudes in $\xi’$ using getAlpha and getBeta
5. Get $\eta’’$ from getRelativeEta()
6. Get $\Delta\eta$ from getOffset()
7. Write CSV
   • original data
   • $\eta’’$
   • $\Delta\eta$
   • amplitudes

Reconstruction from the Roulette Data Set

Reconstruction is different from the regular roulette simulation. Both $\xi$ and $\eta$ are set explicitly and the lens location is unknown. In theory the lens location could be inferred, but as a free variable it leaves $\xi$ and $\eta$ to be set independently. This is implemented in RouletteRegenerator which does not use a separate lens model.

1. The setCentre() method will
   1. $\nu := 0$ (centre of distorted image); consequently $\xi=0$
   2. Set $\eta := \eta’’$ as read from the CSV file.
   3. set the etaOffset to $\Delta\eta$ as read from the CSV file
2. The roulette amplitudes are set using setAlphaXi and setBetaXi.
3. Now the simulation can be run using standard methods inherited from LensModel and RouletteModel.

Functions

1. Masking methods
2. setLens()
3. getDistortedPos(r,theta)
4. The distort() function is inherited from LensModel.
   • It uses etaOffset + getDistortedPos(r,theta) to find the source pixel
5. updateApparentAbs()
   • uses $\eta$ as stored in the model object itself, and requests a $\xi$ to be calculated by the Lens object.
   • then it calls setNu() which is inherited from LensModel
   • this also sets etaOffset = 0
6. setXi(xi1)
   • set $\xi$ to the given value
   • calculate $\eta’$ corresponding to $\xi$ using raytrace
   • set $\Delta\eta=\eta’-\eta$

Special for resimulation from roulette amplitudes:

• setXiEta() to be implemented

Masking

Masking is important for the Roulette Simulator, because computation is expensive. The current code is wrangled with technical debt, and masking is done twice.

• For Roulette Models, the distort() function only calculates pixels inside the convergence ring, if maskMode is set. This speeds up calculation.
• The maskImage() functions allows masking after calculation. It takes an optional scale parameter, and the radius of the mask is scale times the radius of the convergence ring.
• RouletteRegenerator has a default masking radius of zero, but the radius can be explicitly set with setMaskRadius(), which will give masking as for other Roulette Models.
• Raytrace does not mask during computation, but maskImage() works as it does for Roulette Models. This is done to facilitate comparison between Raytrace and Roulette, and the mask is only meaningful in relation to Roulette.

TODO

• When are the roulette amplitudes calculated?