Hexrd: Distortion

[donald-e-boyce]

Distortion in Hexrd

The abstract base class for distortion is DistortionABC, located in the hexrd.distortion package.
A distortion class has apply and apply_inverse methods. The apply method applies the distortion
correction, and apply_inverse applies the distortion to a corrected field.
Each class also has a maptype attribute that gives a name to the particular subclass and is used
in a dictionary to lookup distortion classes by name.

The apply method is used in convert Cartesian coordinate to angles, and apply_inverse is used in convertings
angles back to Cartesian coordinates.
The distortion is used in a few other places as well, such as computing pixel angles and sizes and simulating
a Laue pattern.

GE41RT

The main distortion class available in hexrd is the GE41RT.
This class models the distortion on the GE detector as a barrel distortion.
Barrel distortion is a radial rescaling, but the GE41RT modifies this somewhat.

The model for distortion correction is this.
If $(x, y)$ is an input point and $(\bar{x}, \bar{y})$ are the corrected values, then the rescaling is given by

$$(\bar{x}, \bar{y}) = f(r, \eta) (x, y),$$

where

$$f(r, \eta) = 1 + c_1 \rho^{\alpha_1} \cos(2\eta) + c_2 \rho^{\alpha_2}\cos(4\eta) + c_3\rho^{\alpha_3}$$

In the above equation, $\rho = r/r_{max}$ with $r = \sqrt{x^2 + y^2}$; $r_{max}$ is the maximum value of $r$
for the detector; $\eta$ is the polar angle of $(x, y)$; and $c_1, c_2, c_3, \alpha_1, \alpha_2,$ and $\alpha_3$ are model parameters.

Barrel Distortion

Barrel distortion is a type of distortion that arises when a lens magnifies less as you move radially from the center,
while pincushion distortion is when the lens magnifies more as you move radially from the center.

There are several models for barrel distortion, but a standard one is $(\bar{x}, \bar{y}) = f(r) (x, y)$, where

$$f(r) = 1 + c_1 r^2 + c_2 r^4 + ... c_n r^n$$

Note that this is an even order polynomial in $(x, y)$ since $r^2 = x^2 + y^2$.
Usually a small number of terms are used, maybe up to fourth degree.

Also note that some consider the division model $(\bar{x}, \bar{y}) = (x, y)/f(r)$ to be an improvement.

References

1. wikipedia
2. Brown's distortion model
3. Drap, Lefevere, 2016

Discussion

Now compare the barrel distortion with the GE41RT.

In the GE41RT the powers $\alpha_i$ are input parameters, but are usually taken to be 2.
As Zack noted, this produces a cubic equation for the inverse, which can be
solved much faster than the general case. It seems that allowing $\alpha_i$ to be arbitrary
complicates the code while not improving modeling capability significantly.

The GE also has terms involving the polar angle $\eta$. These allow for the radial scaling to vary with the polar angle. The particular terms of $\cos(2\eta)$ and $\cos(4\eta)$ were chosen to model the variation, but the overall expressions are not very general.

In the end, the physics of the detector distortion is not the same as the barrel distortion due to
variable lens magnification, but it may be reasonable to model the detector distortion as a radial
scaling depending on polar angle.

Suggestion

I suggest the following model as a cleaner version.

For the radial scaling dependency on $r$, use the polynomial model above,
and for the polar angle dependency, use a truncated Fourier series with mean value 1.
The full model would look like:

$$(\bar{x}, \bar{y}) = f(r)g(\eta) (x, y),$$

where

$$f(r) = c_0 + c_1 r^2 + c_2 r^4 + ... c_n r^n$$

and

$$g(\eta) = 1 + d_{11} \cos(\eta) + d_{12}\sin(\eta) + d_{21} \cos(2\eta) + d_{22}\sin(2\eta) + ... + d_{m1}\cos(m\eta) + d_{m2}\sin(m\eta)$$

Some comments.

1. Allowing $c_0 \ne 1$ allows us to assume the center is not necessarily distortion free.
2. If we want to antipodal symmetry in the scaling (same scaling for x and -x) then we
   take $d_{1j} = 0$ and any other odd degree coefficients.
3. The constant term in the $\eta$ series is taken to be 1 because otherwise it would be a redundant term;
   changing it could be compensated by changing all the $c_i$ proportionally.
4. We can increase the approximation levels of both terms independently.
5. A simple model with 4 parameters would be $f(r) = c_0 + c_1 r^2$ and $g(\eta) = 1 + d_{21}\cos(2\eta) + d_{22}\sin(2\eta) $.
6. Another example (7 parameters) is $f(r) = c_0 + c_1 r^2 + c_2 r^4$ and $g(\eta) = 1 + d_11\cos(\eta) + d_12\sin(\eta) + d_{21}\cos(2\eta) + d_{22}\sin(2\eta)$.