Tutorial 04: Implementing an analytical module for an X-ray component to allow gradient descent

What You Will Need

• Parameters: \(a, b, c, \dots\) to be estimated or optimized.

• An analtyical model: \(S(E; a, b, c, \dots)\) that defines the spectrum or energy response.

What You Will Expect

• How to build an analtyical model?

• Implementing the model using PyTorch for differentiability.

• A step-by-step guide to setting up and testing the interpolation module.

A. Analytical Model of Filter

A1. Background

In X-ray systems, filters are always used to protect the detector and enhance image quality by selectively absorbing low-energy X-rays that contribute to image noise without improving image contrast. According to Beer’s law, the response of a single filter is

\[f\left(E; m, \theta\right) = \mathrm{e}^{-\mu(E, m) \theta}\]

where:

• \(m\) denotes the filter material, which is a discrete parameter with only a limited set of choices.

• \(\mu(E, m)\) is the Linear Attenuation Coefficient (LAC) of material \(m\) at energy \(E\).

• \(\theta\) denotes filter thickness, which is a continuous parameter within a continuous range.

A2. Step-by-Step Implementation

To build an analytical model that supports gradient descent, we need two key functions:

1. ``__init__`` (Initialize the Model)

   • Defines materials and thickness as model parameters.

   • Assigns separate memory for continuous parameters corresponding to each discrete material selection.

   • Enables search over all material combinations, allowing the model to explore different discrete parameter configurations.

2. ``forward`` (Compute Filter Response)

   • Retrieves the current material and thickness for the filter.

   • Calls gen_fltr_res() to compute the X-ray attenuation response using Beer’s Law.

   • Ensures the response is computed for a given set of X-ray energies.

This setup enables efficient spectral modeling and optimization using PyTorch.

The Material class stores the chemical formula and density of a material, ensuring valid input types and allowing it to be used in X-ray modeling and optimization.

The get_lin_att_c_vs_E function calculates the linear attenuation coefficient (LAC) value with density, thickness, and energy vector.

The get_params() function, defined in Base_Spec_Model, retrieves the estimated parameters as a dictionary, applying denormalization and clamping to ensure they remain within valid bounds while maintaining gradient flow for optimization.

import numpy as np
import torch
from xcal.models import Base_Spec_Model
from xcal.defs import Material
from xcal.chem_consts._consts_from_table import get_lin_att_c_vs_E

# Implement the analytical model for filter.
def _obtain_attenuation(energies, formula, density, thickness, torch_mode=False):
    # thickness is mm
    mu = get_lin_att_c_vs_E(density, formula, energies)
    if torch_mode:
            mu = torch.tensor(mu)
            att = torch.exp(-mu * thickness)
    else:
            att = np.exp(-mu * thickness)
    return att

def gen_fltr_res(energies, fltr_mat:Material, fltr_th:float, torch_mode=True):

    return _obtain_attenuation(energies, fltr_mat.formula, fltr_mat.density, fltr_th, torch_mode)

# Gradient descent module.
class Filter(Base_Spec_Model):
    def __init__(self, materials, thickness):
        """
        A template filter model based on Beer's Law and NIST mass attenuation coefficients, including all necessary methods.

        Args:
            materials (list): A list of possible materials for the filter,
                where each material should be an instance containing formula and density.
            thickness (tuple or list): If a tuple, it should be (initial value, lower bound, upper bound) for the filter thickness.
                If a list, it should have the same length as the materials list, specifying thickness for each material.
                These values cannot be all None. It will not be optimized when lower == upper.
        """
        if isinstance(thickness, tuple):
            if all(t is None for t in thickness):
                raise ValueError("Thickness tuple cannot have all None values.")
            params_list = [{'material': mat, 'thickness': thickness} for mat in materials]
        elif isinstance(thickness, list):
            if len(thickness) != len(materials):
                raise ValueError("Length of thickness list must match length of materials list.")
            params_list = [{'material': mat, 'thickness': th} for mat, th in zip(materials, thickness)]
        else:
            raise TypeError("Thickness must be either a tuple or a list.")

        super().__init__(params_list)

    def forward(self, energies):
        """
        Takes X-ray energies and returns the filter response.

        Args:
            energies (torch.Tensor): A tensor containing the X-ray energies of a poly-energetic source in units of keV.

        Returns:
            torch.Tensor: The filter response as a function of input energies, selected material, and its thickness.
        """
            # Retrieves
        mat = self.get_params()[f"{self.prefix}_material"]
        th = self.get_params()[f"{self.prefix}_thickness"]
        energies = torch.tensor(energies, dtype=torch.float32) if not isinstance(energies, torch.Tensor) else energies
        return gen_fltr_res(energies, mat, th)

import matplotlib.pyplot as plt

gt_fltr_th = 2.5 # target thickness in um

psb_fltr_mat = [Material(formula='Al', density=2.702), Material(formula='Cu', density=8.92)]
filter_1 = Filter(psb_fltr_mat, thickness=(gt_fltr_th, 0, 10))

ee = np.linspace(1,150,150)
ff = filter_1(ee)
est_param = filter_1.get_params()
print(f'{est_param}')

plt.plot(ee, ff.data)

{'Filter_1_material': Material(formula='Al', density=2.702), 'Filter_1_thickness': tensor(2.5000, grad_fn=<ClampFunctionBackward>)}

[<matplotlib.lines.Line2D at 0x15ee0fb50>]

B. An Example of Estimating Filter Thickness

max_simkV = 50 # keV
takeoff_angle = 13 # degree
mas_list = 0.01 # Milliampere-seconds

fltr_mat = 'Al' # filter material
fltr_th = 3 # filter thickness in mm

det_mat = 'CsI' # scintillator material
det_density = 4.51 # scintillator density g/cm^3
det_th = 0.33 # scintillator thickness in mm

sample_mats = ['V', 'Al', 'Ti', 'Mg']

ct_info = {
    "SDD": 15,
    "psize": [0.01, 0.01],  # Width and height in mm
}

from xcal import get_filter_response,get_scintillator_response
from xcal.chem_consts._periodictabledata import density
import spekpy as sp

simkV = 50
gt_takeoff_angle = 13
s = sp.Spek(kvp=simkV, th=gt_takeoff_angle, dk=1, z=ct_info['SDD']/10, mas=mas_list, char=True)
k, phi_k = s.get_spectrum(edges=False)
phi_k = phi_k * ((ct_info['psize'][0] / 10) * (ct_info['psize'][1] / 10))  #
src_spec = np.zeros(max_simkV-1)
src_spec[:simkV-1] = phi_k  # Assign spectrum values starting from 1.5 keV

ee = k
gt_src = src_spec
gt_fltr = get_filter_response(ee, fltr_mat, density[fltr_mat], fltr_th)
gt_det = get_scintillator_response(ee, det_mat, det_density, det_th)

import matplotlib.pyplot as plt

# Plot setup
fig, axes = plt.subplots(1, 3, figsize=(15, 4))

# Subplot 1: Source spectra for different voltages
axes[0].plot(ee, src_spec, label=f'{simkV} kVp')
axes[0].set_title('Transmission Source Spectra')
axes[0].set_xlabel('Energy (keV)')
axes[0].set_ylabel('Intensity')
axes[0].legend()
axes[0].grid(True)

# Subplot 2: Filter response
axes[1].plot(ee, gt_fltr, label=f'{fltr_mat} {fltr_th} mm')
axes[1].set_title('Filter Response')
axes[1].set_xlabel('Energy (keV)')
axes[1].set_ylabel('Transmission')
axes[1].legend()
axes[1].grid(True)

# Subplot 3: Scintillator response
axes[2].plot(ee, gt_det, label=f'{det_mat} {det_th} mm')
axes[2].set_title('Scintillator Response')
axes[2].set_xlabel('Energy (keV)')
axes[2].set_ylabel('Absorption Efficiency')
axes[2].legend()
axes[2].grid(True)

# Layout adjustment
plt.tight_layout()
plt.show()

import matplotlib.pyplot as plt
import numpy as np

# Assuming ee, gt_srcs, gt_fltr, gt_det, and vol_list are already defined

# Create a new figure with a specified size
plt.figure(figsize=(10, 6))
gt_spec_list = []
# Plot the efficient spectra for each voltage

# Calculate the efficient spectrum by multiplying with filter and detector responses.
# Then do normalization.
efficient_spectrum = gt_src * gt_fltr * gt_det
gt_spec_list.append(efficient_spectrum)
# efficient_spectrum/=np.trapezoid(efficient_spectrum)
# Plot the efficient spectrum
plt.plot(ee, efficient_spectrum/np.trapezoid(efficient_spectrum), label=f'{simkV} kVp')

# Add labels and title
plt.xlabel('Energy (keV)')
plt.ylabel('Relative Intensity')
plt.title('Efficient Spectra (Source × Filter × Detector)')
plt.legend()
plt.grid(True)
plt.tight_layout()
plt.show()

from xcal.chem_consts import get_lin_att_c_vs_E
sample_mats = ['V', 'Al', 'Ti', 'Mg']
mat_density = [density[formula] for formula in sample_mats]
lac_vs_E_list = [get_lin_att_c_vs_E(den, formula, ee) for den, formula in zip(mat_density, sample_mats)]
sample_thicknesses = np.linspace(0, 1, 100) # mm

# Attenuation Matrix, multiply effective spectrum can obtain tranmission measurement.
# (4*100)*145: 4 materials, 100 different thicknesses, at total 400 measurements and 145 energy bins.
spec_F = np.concatenate([np.exp(-sample_thicknesses[:,np.newaxis]@lac_vs_E[np.newaxis]) for lac_vs_E in lac_vs_E_list],axis=0)

import matplotlib.pyplot as plt

# Create a new figure with a specified size
plt.figure(figsize=(10, 6))

# Plot the data for each sample material
for smid, sample_mat in enumerate(sample_mats):
    plt.plot(sample_thicknesses, spec_F[100*smid:100*(smid+1), 30], label=sample_mat)

# Set the y-axis to a logarithmic scale
plt.yscale('log')

# Add labels and title
plt.xlabel('Sample Thickness (mm)')
plt.ylabel('Intensity at 30 keV')
plt.title('Attenuation vs. Sample Thickness at 30 keV')
plt.legend()
plt.grid(True, which='both', linestyle='--', linewidth=0.5)
plt.tight_layout()
plt.show()

trans_list = []

# Obtain the converted energy, which is proportional to the detected visible light photons by the camera.
# gt_spec is the converted energy without an object.
# Notice that, trapezoid does the energy integration.
trans = np.trapezoid(spec_F * efficient_spectrum, ee, axis=-1) # Object scan
trans_0 = np.trapezoid(efficient_spectrum, ee, axis=-1) # Air scan value
# Add poisson noise.
# The noise level can be adjusted by changing the mas, the current-time product in the beginning of this tutorial.
trans_noise = np.random.poisson(trans).astype(np.float32)
trans_noise /= trans_0

# Store noisy transmission data.
trans_list.append(trans_noise)

import matplotlib.pyplot as plt
import numpy as np

# Determine the number of cases
num_cases = len(gt_spec_list)

# Ensure axes is iterable
if num_cases == 1:
    axes = [axes]

# Plot data for each case

for smid, sample_mat in enumerate(sample_mats):
    # Extract the transmission data for the current sample and case
    transmission_data = trans_list[0][100 * smid : 100 * (smid + 1)]
    plt.plot(sample_thicknesses, transmission_data, label=sample_mat)


# Set the x-axis label for the bottom subplot
plt.xlabel('Sample Thickness (mm)')

# Adjust layout to prevent overlap
plt.tight_layout()
plt.show()

from xcal.models import Reflection_Source, Scintillator
from xcal.defs import Material
from xcal.estimate import Estimate
from T04 import Filter

learning_rate = 0.001 # 0.01 for NNAT_LBFGS and 0.001 for Adam
max_iterations = 5000 # 5000 ~ 10000 would be enough
stop_threshold = 1e-6
optimizer_type = 'Adam' # Can also use Adam.

# Use Spekpy to generate a source spectra dictionary.
takeoff_angles = np.linspace(5,45,11)
src_spec_list = []

for ta in takeoff_angles:
    # Generate the X-ray spectrum model with Spekpy for each voltage.
    s = sp.Spek(kvp=simkV, th=ta, dk=1, z=ct_info['SDD'], mas=mas_list, char=True)
    k, phi_k = s.get_spectrum(edges=False)  # Retrieve energy bins and fluence spectrum [Photons cm^-2 keV^-1]

    # Adjust the fluence for the detector pixel area.
    phi_k = phi_k * ((ct_info['psize'][0] / 10) * (ct_info['psize'][1] / 10))  # Convert pixel size from mm² to cm²

    # Initialize a zero-filled spectrum array with length max_simkV.
    src_spec = np.zeros(max_simkV-1)
    src_spec[:simkV-1] = phi_k  # Assign spectrum values starting from 1.5 keV

    # Add the processed spectrum for this voltage to the list.
    src_spec_list.append(src_spec)

src_spec_list = np.array(src_spec_list)
src_spec_list = src_spec_list.reshape((1,len(takeoff_angles),-1))
source = Reflection_Source(voltage=(50, None, None), takeoff_angle=(gt_takeoff_angle, None, None), single_takeoff_angle=True)
source.set_src_spec_list(ee, src_spec_list, [50], takeoff_angles)


# Not estimate filter and scintillator
psb_fltr_mat = [Material(formula='Al', density=2.702),
                Material(formula='Cu', density=8.92)]
filter_1 = Filter(psb_fltr_mat, thickness=(5, 0, 10))
psb_scint_mat = [Material(formula='CsI', density=4.51)]
scintillator_1 = Scintillator(materials=psb_scint_mat, thickness=(0.33, None, None))

spec_models = [[source, filter_1, scintillator_1]]

Estimator = Estimate(ee)
# For each scan, add data and calculated forward matrix to Estimator.
Estimator.add_data(trans_list[0], spec_F, spec_models[0], weight=None)

# Fit data
Estimator.fit(learning_rate=learning_rate,
              max_iterations=max_iterations,
              stop_threshold=stop_threshold,
              optimizer_type=optimizer_type,
              loss_type='transmission',
              logpath=None,
              num_processes=1) # Parallel computing for multiple cpus.

Number of cases for different discrete parameters: 2

2025-05-25 12:51:53,737  - Start Estimation.
2025-05-25 12:51:53,759  - Initial cost: 2.388144e-03
2025-05-25 12:51:53,842  - Iteration: 50
2025-05-25 12:51:53,844  - Cost: 0.0015051235677674413
2025-05-25 12:51:53,844  - Filter_1_material: Material(formula='Al', density=2.702)
2025-05-25 12:51:53,844  - Filter_1_thickness: 4.509207248687744
2025-05-25 12:51:53,845  - Reflection_Source_1_voltage: 50.0
2025-05-25 12:51:53,845  - Reflection_Source_takeoff_angle: 13.0
2025-05-25 12:51:53,845  - Scintillator_2_material: Material(formula='CsI', density=4.51)
2025-05-25 12:51:53,845  - Scintillator_2_thickness: 0.33000001311302185
2025-05-25 12:51:53,845  -
2025-05-25 12:51:53,936  - Iteration: 100
2025-05-25 12:51:53,938  - Cost: 0.0008214872796088457
2025-05-25 12:51:53,938  - Filter_1_material: Material(formula='Al', density=2.702)
2025-05-25 12:51:53,938  - Filter_1_thickness: 4.061138153076172
2025-05-25 12:51:53,938  - Reflection_Source_1_voltage: 50.0
2025-05-25 12:51:53,938  - Reflection_Source_takeoff_angle: 13.0
2025-05-25 12:51:53,938  - Scintillator_2_material: Material(formula='CsI', density=4.51)
2025-05-25 12:51:53,938  - Scintillator_2_thickness: 0.33000001311302185
2025-05-25 12:51:53,938  -
2025-05-25 12:51:54,028  - Iteration: 150
2025-05-25 12:51:54,029  - Cost: 0.00037792042712680995
2025-05-25 12:51:54,030  - Filter_1_material: Material(formula='Al', density=2.702)
2025-05-25 12:51:54,030  - Filter_1_thickness: 3.6804146766662598
2025-05-25 12:51:54,030  - Reflection_Source_1_voltage: 50.0
2025-05-25 12:51:54,030  - Reflection_Source_takeoff_angle: 13.0
2025-05-25 12:51:54,030  - Scintillator_2_material: Material(formula='CsI', density=4.51)
2025-05-25 12:51:54,030  - Scintillator_2_thickness: 0.33000001311302185
2025-05-25 12:51:54,030  -
2025-05-25 12:51:54,109  - Iteration: 200
2025-05-25 12:51:54,111  - Cost: 0.00014460438978858292
2025-05-25 12:51:54,111  - Filter_1_material: Material(formula='Al', density=2.702)
2025-05-25 12:51:54,111  - Filter_1_thickness: 3.387305974960327
2025-05-25 12:51:54,111  - Reflection_Source_1_voltage: 50.0
2025-05-25 12:51:54,111  - Reflection_Source_takeoff_angle: 13.0
2025-05-25 12:51:54,111  - Scintillator_2_material: Material(formula='CsI', density=4.51)
2025-05-25 12:51:54,111  - Scintillator_2_thickness: 0.33000001311302185
2025-05-25 12:51:54,111  -
2025-05-25 12:51:54,212  - Iteration: 250
2025-05-25 12:51:54,214  - Cost: 5.164837057236582e-05
2025-05-25 12:51:54,214  - Filter_1_material: Material(formula='Al', density=2.702)
2025-05-25 12:51:54,214  - Filter_1_thickness: 3.190600633621216
2025-05-25 12:51:54,214  - Reflection_Source_1_voltage: 50.0
2025-05-25 12:51:54,214  - Reflection_Source_takeoff_angle: 13.0
2025-05-25 12:51:54,214  - Scintillator_2_material: Material(formula='CsI', density=4.51)
2025-05-25 12:51:54,214  - Scintillator_2_thickness: 0.33000001311302185
2025-05-25 12:51:54,214  -
2025-05-25 12:51:54,315  - Iteration: 300
2025-05-25 12:51:54,316  - Cost: 2.4953735191957094e-05
2025-05-25 12:51:54,316  - Filter_1_material: Material(formula='Al', density=2.702)
2025-05-25 12:51:54,316  - Filter_1_thickness: 3.0789196491241455
2025-05-25 12:51:54,316  - Reflection_Source_1_voltage: 50.0
2025-05-25 12:51:54,316  - Reflection_Source_takeoff_angle: 13.0
2025-05-25 12:51:54,316  - Scintillator_2_material: Material(formula='CsI', density=4.51)
2025-05-25 12:51:54,316  - Scintillator_2_thickness: 0.33000001311302185
2025-05-25 12:51:54,316  -
2025-05-25 12:51:54,397  - Iteration: 350
2025-05-25 12:51:54,400  - Cost: 1.9495602828101255e-05
2025-05-25 12:51:54,400  - Filter_1_material: Material(formula='Al', density=2.702)
2025-05-25 12:51:54,400  - Filter_1_thickness: 3.025747776031494
2025-05-25 12:51:54,400  - Reflection_Source_1_voltage: 50.0
2025-05-25 12:51:54,400  - Reflection_Source_takeoff_angle: 13.0
2025-05-25 12:51:54,400  - Scintillator_2_material: Material(formula='CsI', density=4.51)
2025-05-25 12:51:54,400  - Scintillator_2_thickness: 0.33000001311302185
2025-05-25 12:51:54,400  -
2025-05-25 12:51:54,470  - Iteration: 400
2025-05-25 12:51:54,472  - Cost: 1.868593790277373e-05
2025-05-25 12:51:54,472  - Filter_1_material: Material(formula='Al', density=2.702)
2025-05-25 12:51:54,472  - Filter_1_thickness: 3.004319429397583
2025-05-25 12:51:54,472  - Reflection_Source_1_voltage: 50.0
2025-05-25 12:51:54,472  - Reflection_Source_takeoff_angle: 13.0
2025-05-25 12:51:54,472  - Scintillator_2_material: Material(formula='CsI', density=4.51)
2025-05-25 12:51:54,472  - Scintillator_2_thickness: 0.33000001311302185
2025-05-25 12:51:54,472  -
2025-05-25 12:51:54,545  - Iteration: 450
2025-05-25 12:51:54,547  - Cost: 1.8596598238218576e-05
2025-05-25 12:51:54,547  - Filter_1_material: Material(formula='Al', density=2.702)
2025-05-25 12:51:54,547  - Filter_1_thickness: 2.9969093799591064
2025-05-25 12:51:54,547  - Reflection_Source_1_voltage: 50.0
2025-05-25 12:51:54,547  - Reflection_Source_takeoff_angle: 13.0
2025-05-25 12:51:54,547  - Scintillator_2_material: Material(formula='CsI', density=4.51)
2025-05-25 12:51:54,547  - Scintillator_2_thickness: 0.33000001311302185
2025-05-25 12:51:54,547  -
2025-05-25 12:51:54,614  - Iteration: 500
2025-05-25 12:51:54,616  - Cost: 1.8589174942462705e-05
2025-05-25 12:51:54,616  - Filter_1_material: Material(formula='Al', density=2.702)
2025-05-25 12:51:54,616  - Filter_1_thickness: 2.9946932792663574
2025-05-25 12:51:54,616  - Reflection_Source_1_voltage: 50.0
2025-05-25 12:51:54,616  - Reflection_Source_takeoff_angle: 13.0
2025-05-25 12:51:54,616  - Scintillator_2_material: Material(formula='CsI', density=4.51)
2025-05-25 12:51:54,616  - Scintillator_2_thickness: 0.33000001311302185
2025-05-25 12:51:54,616  -
2025-05-25 12:51:54,657  - Stopping at epoch 528 because updates are too small.
2025-05-25 12:51:54,657  - Cost: 1.858877294580452e-05
2025-05-25 12:51:54,657  - Filter_1_material: Material(formula='Al', density=2.702)
2025-05-25 12:51:54,657  - Filter_1_thickness: 2.9942777156829834
2025-05-25 12:51:54,657  - Reflection_Source_1_voltage: 50.0
2025-05-25 12:51:54,657  - Reflection_Source_takeoff_angle: 13.0
2025-05-25 12:51:54,657  - Scintillator_2_material: Material(formula='CsI', density=4.51)
2025-05-25 12:51:54,657  - Scintillator_2_thickness: 0.33000001311302185
2025-05-25 12:51:54,657  -
2025-05-25 12:51:54,659  - Start Estimation.
2025-05-25 12:51:54,662  - Initial cost: 9.644821e-02
2025-05-25 12:51:54,731  - Iteration: 50
2025-05-25 12:51:54,733  - Cost: 0.09464140981435776
2025-05-25 12:51:54,733  - Filter_1_material: Material(formula='Cu', density=8.92)
2025-05-25 12:51:54,733  - Filter_1_thickness: 4.4902663230896
2025-05-25 12:51:54,733  - Reflection_Source_1_voltage: 50.0
2025-05-25 12:51:54,733  - Reflection_Source_takeoff_angle: 13.0
2025-05-25 12:51:54,733  - Scintillator_2_material: Material(formula='CsI', density=4.51)
2025-05-25 12:51:54,733  - Scintillator_2_thickness: 0.33000001311302185
2025-05-25 12:51:54,733  -
2025-05-25 12:51:54,802  - Iteration: 100
2025-05-25 12:51:54,804  - Cost: 0.09225913137197495
2025-05-25 12:51:54,804  - Filter_1_material: Material(formula='Cu', density=8.92)
2025-05-25 12:51:54,804  - Filter_1_thickness: 3.9403140544891357
2025-05-25 12:51:54,804  - Reflection_Source_1_voltage: 50.0
2025-05-25 12:51:54,804  - Reflection_Source_takeoff_angle: 13.0
2025-05-25 12:51:54,804  - Scintillator_2_material: Material(formula='CsI', density=4.51)
2025-05-25 12:51:54,804  - Scintillator_2_thickness: 0.33000001311302185
2025-05-25 12:51:54,804  -
2025-05-25 12:51:54,877  - Iteration: 150
2025-05-25 12:51:54,879  - Cost: 0.08896404504776001
2025-05-25 12:51:54,879  - Filter_1_material: Material(formula='Cu', density=8.92)
2025-05-25 12:51:54,879  - Filter_1_thickness: 3.3324739933013916
2025-05-25 12:51:54,879  - Reflection_Source_1_voltage: 50.0
2025-05-25 12:51:54,879  - Reflection_Source_takeoff_angle: 13.0
2025-05-25 12:51:54,879  - Scintillator_2_material: Material(formula='CsI', density=4.51)
2025-05-25 12:51:54,879  - Scintillator_2_thickness: 0.33000001311302185
2025-05-25 12:51:54,879  -
2025-05-25 12:51:54,954  - Iteration: 200
2025-05-25 12:51:54,956  - Cost: 0.08397167176008224
2025-05-25 12:51:54,956  - Filter_1_material: Material(formula='Cu', density=8.92)
2025-05-25 12:51:54,956  - Filter_1_thickness: 2.643720865249634
2025-05-25 12:51:54,956  - Reflection_Source_1_voltage: 50.0
2025-05-25 12:51:54,956  - Reflection_Source_takeoff_angle: 13.0
2025-05-25 12:51:54,956  - Scintillator_2_material: Material(formula='CsI', density=4.51)
2025-05-25 12:51:54,956  - Scintillator_2_thickness: 0.33000001311302185
2025-05-25 12:51:54,956  -
2025-05-25 12:51:55,028  - Iteration: 250
2025-05-25 12:51:55,030  - Cost: 0.07504528015851974
2025-05-25 12:51:55,030  - Filter_1_material: Material(formula='Cu', density=8.92)
2025-05-25 12:51:55,030  - Filter_1_thickness: 1.82919442653656
2025-05-25 12:51:55,030  - Reflection_Source_1_voltage: 50.0
2025-05-25 12:51:55,030  - Reflection_Source_takeoff_angle: 13.0
2025-05-25 12:51:55,030  - Scintillator_2_material: Material(formula='CsI', density=4.51)
2025-05-25 12:51:55,030  - Scintillator_2_thickness: 0.33000001311302185
2025-05-25 12:51:55,030  -
2025-05-25 12:51:55,098  - Iteration: 300
2025-05-25 12:51:55,100  - Cost: 0.04900381714105606
2025-05-25 12:51:55,100  - Filter_1_material: Material(formula='Cu', density=8.92)
2025-05-25 12:51:55,100  - Filter_1_thickness: 0.7538713216781616
2025-05-25 12:51:55,100  - Reflection_Source_1_voltage: 50.0
2025-05-25 12:51:55,100  - Reflection_Source_takeoff_angle: 13.0
2025-05-25 12:51:55,100  - Scintillator_2_material: Material(formula='CsI', density=4.51)
2025-05-25 12:51:55,100  - Scintillator_2_thickness: 0.33000001311302185
2025-05-25 12:51:55,100  -
2025-05-25 12:51:55,171  - Iteration: 350
2025-05-25 12:51:55,173  - Cost: 0.0006778572569601238
2025-05-25 12:51:55,173  - Filter_1_material: Material(formula='Cu', density=8.92)
2025-05-25 12:51:55,173  - Filter_1_thickness: 0.08765144646167755
2025-05-25 12:51:55,173  - Reflection_Source_1_voltage: 50.0
2025-05-25 12:51:55,173  - Reflection_Source_takeoff_angle: 13.0
2025-05-25 12:51:55,173  - Scintillator_2_material: Material(formula='CsI', density=4.51)
2025-05-25 12:51:55,173  - Scintillator_2_thickness: 0.33000001311302185
2025-05-25 12:51:55,173  -
2025-05-25 12:51:55,260  - Iteration: 400
2025-05-25 12:51:55,262  - Cost: 0.00014357846521306783
2025-05-25 12:51:55,262  - Filter_1_material: Material(formula='Cu', density=8.92)
2025-05-25 12:51:55,262  - Filter_1_thickness: 0.10207980126142502
2025-05-25 12:51:55,262  - Reflection_Source_1_voltage: 50.0
2025-05-25 12:51:55,262  - Reflection_Source_takeoff_angle: 13.0
2025-05-25 12:51:55,262  - Scintillator_2_material: Material(formula='CsI', density=4.51)
2025-05-25 12:51:55,262  - Scintillator_2_thickness: 0.33000001311302185
2025-05-25 12:51:55,262  -
2025-05-25 12:51:55,287  - Stopping at epoch 417 because updates are too small.
2025-05-25 12:51:55,287  - Cost: 0.00014372407167684287
2025-05-25 12:51:55,287  - Filter_1_material: Material(formula='Cu', density=8.92)
2025-05-25 12:51:55,287  - Filter_1_thickness: 0.10162963718175888
2025-05-25 12:51:55,287  - Reflection_Source_1_voltage: 50.0
2025-05-25 12:51:55,287  - Reflection_Source_takeoff_angle: 13.0
2025-05-25 12:51:55,287  - Scintillator_2_material: Material(formula='CsI', density=4.51)
2025-05-25 12:51:55,287  - Scintillator_2_thickness: 0.33000001311302185
2025-05-25 12:51:55,287  -

Estimator.get_params()

{'Reflection_Source_1_voltage': tensor(50.),
 'Reflection_Source_takeoff_angle': tensor(13.),
 'Filter_1_material': Material(formula='Al', density=2.702),
 'Filter_1_thickness': tensor(2.9943, grad_fn=<ClampFunctionBackward>),
 'Scintillator_2_material': Material(formula='CsI', density=4.51),
 'Scintillator_2_thickness': tensor(0.3300)}

import torch
# Get the estimated effective response for each source voltage.
# Make sure to convert to numpy array from tensor before plotting.
est_sp = Estimator.get_spectra()

with torch.no_grad():
    plt.plot(ee, (gt_spec_list[0]/np.trapezoid(gt_spec_list[0],ee)),
            label='Ground Truth')
    es = est_sp[0].numpy()
    es /= np.trapezoid(es,ee)
    plt.plot(ee, es, '--', label='Estimate')

plt.legend()
plt.title(f'{simkV} kV Spectral Response')

fig.suptitle('Comparison of Ground Truth and Estimate')
plt.show()
print(f"Ground Truth: {gt_fltr_th:.2f}, Estimated: {Estimator.get_params()['Filter_1_thickness'].item():.2f}")

Ground Truth: 2.50, Estimated: 2.99