Implementation

One of the critical challenges in CosmoSim is to represent the lens potential and to compute the distortion in both the raytrace model and the roulette model.

• Implementation of Raytrace Support

• Implementation of Roulette Amplitudes

Notation in the Implementation¶

While the implementation aims to stay true to the mathematical models, the notation sometimes differs. Some discrepancies are part of the technical debt, where the code has not been updated to match evolving notation. Other discrepancies are due to ambiguities, reused symbols, and subscripts which are hard to represent in ASCII. We will try to define the mapping between the two systems of notation.

• alpha and beta refer to the roulette amplitudes.

• The actual source position $\beta$ is denoted eta in the C++ code.

  • the value is retrieved as SimulatorModel::getEta().

• einsteinration refers to the Einstein radius $\theta_E$.

  • theta refers to the rotation of an elliptic source.

• The actual position $\theta$ is denoted nu

  • using angular co-ordinates, we identify $\nu$ and $\xi$ with $\theta$, and variables and functions are not consistently named.

  • nu is computed by Lens::getXi() (which may be overridden) as a function of eta ($\beta$).

• referenceXi is the point in which the roulette amplitudes are computed.

  • In the standard mode of operation, this is equal to nu ($\theta$), but it is possible to set it arbitrarily.

• xi is often used as a local variable, referring to an observed point where the light intensity is being computed. It is not fixed point in the model.

Coordinate system¶

Applying machine learning, it is important to view the coordinate system as arbitrary. If the lens is fixed, say, at origin, information is leaked. For this reason, the images are translated to have the origin at the centre of visible light.

As a consequence, CosmoSim has to consider at least two co-ordinate systems: one for calculation and one for presentation.

The critical points in the simulation are

• Lens position $O_{\textrm{lens}}$

• Centre of light $O_{\textrm{light}}$

• Reference point $\xi_{\textrm{ref}}$ for the roulette expansion

• Apparent source position $\nu$

The output CSV file reports the following points:

• (lensX,lensY) is $O_{\textrm{lens}}$

• (reletaX,reletaY) is $\eta$

• (xiX,xiY) is $\nu$ These are reported relative to the centre of the image as simulated. Hence, they depend on the simulator.centred setting. If the image has not been centred, (lensX,lensY) is zero.

In the default mode (when referencexi is true), we have $\xi_{\textrm{ref}}=\nu$.

The (centreX,centreY) columns are deprecated. They represent the centre of light in the original coordinate system, i.e. $-O_{\textrm{lens}}$ in the new system.

Custom reference point¶

The simulator can be set to make the roulettes expansion in an arbitrary point. The output CSV file does not give the actual $\xi_{\mathrm{ref}}$. Instead it gives the source offset (offsetX, offsetY) which suffices to do resimulation in the roulette formalism.

The source offset is defined as $\xi_{\mathrm{ref}} - \nabla\psi(\xi_{\mathrm{ref}}) - \eta$. In other words, it is the difference between the actual source position $\eta$ and the source position deflected at the reference point $\xi_{\mathrm{ref}}$ (relativeXi). This is used in the distort() function in the simulator to get light from the right pixel in the source image.

Here $\xi'$ is the origin of the new co-ordinate system, so that xi? and releta? are $\nu$ and $\eta$ in the the new frame.

The (offsetX,offsetY) pair places referenceXi.

Image coordinates¶

It is important to note that images are indexed as matrices, and not with standard Cartesian co-ordinates. The distortion models are implemented in Cartesian co-ordinates, following the mathematical model, and needs to be converted to image co-ordinates whenever images are references.

This looks like this:

xi = pointCoordinate( cv::Point2d( row, col ), dst ) ;
ij = imageCoordinate( xi, src ) ;

Here xi is the geometric point corresponding to index (row,col) in the image array, and ij is the index (row,col).
Both image and Cartesian co-ordinates are represented as cv::Point2d objects.

The second arguments src and dst are the images to which the co-ordinates refer. In reality, only the size of these images is relevant, but it is important because the origin of the Cartesian system is the centre of the image and not the corner.