XMF documentation

Welcome to the XMF documentation!

In this documentation, you will find information about the XMF project for X-ray mirror surface shape fitting, including its modules and functionalities in Python and MATLAB.

Contents:

Framework

XMF is a framework for the X-ray mirror surface shape fitting, particularly in the context of convex and concave shapes.

[mirror surfaces]

Some typical X-ray mirror surfaces

It provides data analysis and visualization tools for fitting measurement data to various geometric shapes, including elliptic cylinders, hyperbolic cylinders, ellipsoids, and hyperboloids.

[framework]

X-ray mirror surface shape fitting framework (XMF)

The framework is composed of four main layers:

  • Layer 1: Standard shape from expressions. It contains the standard mathematical expressions of various off-axis mirror shapes.

  • Layer 2: Surface generation. It generates surface shapes in height or slope, by taking into account the rotation and translation of the shapes.

  • Layer 3: Optimization. It optimizes the user-selected fitting parameters.

  • Layer 4: Function wrapper. It offers convenient functions to fit measurement data for specific shapes.

Example codes

Python example for fitting a concave ellipsoid height map:

import numpy as np
import xmf

# 1. Define lateral coordinates
x_range = 200e-3
y_range = 20e-3
x_num = 201
y_num = 21

x1d = np.linspace(-x_range/2, x_range/2, x_num)
y1d = np.linspace(-y_range/2, y_range/2, y_num)
x2d, y2d = np.meshgrid(x1d, y1d)

# 2. Set mirror parameters
# 2.1. Shape parameters
abs_p = 30
abs_q = 0.3
theta = 30e-3

# 2.2. Pose parameters
x_i = -1e-3
y_i = -2e-4
z_i = 3e-7
alpha = 2e-6
beta = 1e-5
gamma = 0.5e-3

# 2.3. True parameters as dictionary
true_params_dict = {
    'p': abs_p,
    'q': abs_q,
    'theta': theta,
    'x_i': x_i,
    'y_i': y_i,
    'z_i': z_i,
    'alpha': alpha,
    'beta': beta,
    'gamma': gamma
}

# 3. Set measurement noise
height_measurement_noise_std = 0.5e-9
slope_measurement_noise_std = 100e-9

# 4. Demonstarte the fitting
# 4.1. Set input parameters as dictionary
input_params_dict = {
    'p': abs_p,
    'q': abs_q,
    'theta': theta
}

# 4.2. Set the tolerance dictionary
tol_dict = {
    'p': 0,
    'q': 0,
    'theta': 0
}

# 4.3. Generate the surface
z2d = xmf.generate_2d_curved_surface_height(xmf.standard_concave_ellipsoid_height, x2d, y2d, abs_p, abs_q, theta, x_i, y_i, z_i, alpha, beta, gamma)
# 4.4. Adding noise to mimic the measured data
z2d_measured = z2d + np.random.randn(z2d.shape[0], z2d.shape[1])*height_measurement_noise_std
# 4.5. Fit the surface shape
z2d_res, z2d_fit, opt_params_dict, opt_params_ci_dict, _ = xmf.fit_concave_ellipsoid_height(x2d, y2d, z2d_measured, input_params_dict, tol_dict)
# 4.6. Show fitting results
xmf.fig_show_2d_fitting_map(x2d, y2d, z2d_measured, z2d_fit, z2d_res, true_params_dict, opt_params_dict, opt_params_ci_dict,'Concave Ellipsoid')

MATLAB example for fitting a concave elliptic cylinder height map:

% 1. Define lateral coordinates
x_range = 200e-3;
y_range = 20e-3;
x_num = 201;
y_num = 21;
x1d = linspace(-x_range/2, x_range/2, x_num);
y1d = linspace(-y_range/2, y_range/2, y_num);
[x2d, y2d] = meshgrid(x1d, y1d);

% 2. Set mirror parameters
% 2.1. Shape parameters
abs_p = 30;
abs_q = 0.3;
theta = 30e-3;

% 2.2. Pose parameters
x_i = -1e-3;
y_i = -2e-4;
z_i = 3e-7;
alpha = 2e-6;
beta = 1e-5;
gamma = 0.5e-3;

% 2.3. True parameters as structure
true_params_struct.p = abs_p;
true_params_struct.q = abs_q;
true_params_struct.theta = theta;
true_params_struct.x_i = x_i;
true_params_struct.y_i = y_i;
true_params_struct.z_i = z_i;
true_params_struct.alpha = alpha;
true_params_struct.beta = beta;
true_params_struct.gamma = gamma;

% 3. Set measurement noise
height_measurement_noise_std = 0.5e-9;
slope_measurement_noise_std = 100e-9;

% 4. Demonstarte the fitting
% 4.1. Set input parameters as structure
input_params_struct.p = abs_p;
input_params_struct.q = abs_q;
input_params_struct.theta = theta;

% 4.2. Set the optimization flag structure
opt_struct.p = false;
opt_struct.q = false;
opt_struct.theta = false;

% 4.3. Generate the surface
z2d = generate_2d_cylinder_height(@standard_concave_elliptic_cylinder_height, x2d, y2d, abs_p, abs_q, theta, x_i, z_i, alpha, beta, gamma);
% 4.4. Adding noise to mimic the measured data
z2d_measured = z2d + randn(size(z2d))*height_measurement_noise_std;
% 4.5. Fit the surface shape
[z2d_res, z2d_fit, opt_params_struct, opt_params_ci_struct] = fit_concave_elliptic_cylinder_height(x2d, y2d, z2d_measured, input_params_struct, opt_struct);
% 4.6. Show fitting results
fig_show_2d_fitting_map(x1d, y1d, z2d_measured, z2d_fit, z2d_res, true_params_struct, opt_params_struct, opt_params_ci_struct, 'Concave Elliptic Cylinder');

Indices and tables