Geometric Transformation

Story

I want a way to quickly and efficiently go from un-dispersed to dispersed images reference frame. It is therefore an (x,y) ↔ (dx,dy) transformation but wavelength factors in this, as well as where on the detector we are. It is therefore, in a more general way a transformation from (x,y,lambda) ↔ (dx,dy,lambda), where (x,y) is the location on the detector to use to account for any field dependency.
I can thing of several things I would use this for:

• Simulating: In this case I want to be able to compute the offsets (dx,dy) coordinates in the dispersed image where light originating from (x,y) in the undispersed image and at a particular wavelength lambda ends up.
• Extracting: This is a reverse operation. I know the coordinates in a dispersed spectrum (x',y') and therefore (dx,dy) and  I want to know what the wavelength of that light is since I know it originates from the pixel (x,y) in the direct image
• In between cases: Some times, I want to be able to go back and forth from (x,y,lamba) to (dx,dy,lambda) with missing values of x, y or lambda, or dx,dy,lambda so I need functions that give me the flexibility to solve any of the in between problems, equally as fast.

I envision that I will need to compute this several millions on times so I want to keep things as fast as possible. It could be a 2D polynomial solution, which can be inverted either analytically or numerically in the case of higher polynomial orders (as in UVIS).

I want this system to be flexible and extendable to highly distorted spectra so I do not want the calibration to be dependent on the x or y coordinates, or the pathlength along the trace (which is difficult to compute and invert) or lambda directly. The choise of x or y is a bad one for example in cases where the spectra curve dramatically and either of these coordinates become a poor ways to measure where the trace is. Instead I want a system where each of this transformation is given as a function of an intermediate variable t which can then be defined as best fit a particular disperser. Hence the functions will be dx = fx(x,y,t) dy= fy(x,y,t) lambda=fl(x,y,t) and the inverse functions x = fx'(dx,dy,t), y = fy'(dx,dy,t), lambda=fl'(dx,dy,t). The variable t can be defined as t = x in the case of a simple x-direction disperser. It can also be defined as t = lambda, or t = (lambda-l0)/(l1-l0) where l0 an dl1 and the minimum and maximum wavelength in the dispersed spectrum. The latter results in 0<t<1 which could be beneficial to some.

Inputs

• A grism configuration file that described the transformations
• A set of coordinates and wavelenghts, etc...

Outputs

• As set of coordinates and wavelengths, etc...

Computations

• Computations are handles by a small module which simply implements the equation described above, using parameters values determined during the course of the instrument calibration.

Drawbacks

• The use of the variable t provides more flexibility but some bad choices can still be made. How to parametrized t should be defined by balancing adequacy, speed, as well as the ability of the instrument team to calibrate the grism in that manner