XMF

Welcome to the X-ray Mirror surface shape Fitting (XMF)!

In the XMF documentation (https://nsls2omf.github.io/xmf), we introduce the modules and functionalities of XMF in Python and MATLAB.

Framework

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

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

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 gallery

Example gallery shows example Python scripts demonstrating how to use our code.

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');