Cylindrical Geometry Correction for Transmission Data

Important

Status: design document, not yet implemented. This page derives the Beer-Lambert (exponential) correction with per-TOF μ estimation. The correction currently implemented in neutron_geomcorr is a linear chord-length division (see Introduction), which is valid only for inputs proportional to path length (attenuation, −ln(T), or thickness maps) — not for raw transmission data. Reconciling the two — implementing this derivation or re-scoping the package documentation — is tracked in issue #57. The algorithm below is prototyped in notebooks/00_development.ipynb.

This document presents the methodology for applying cylindrical geometry correction to transmission data, specifically in the context of neutron or X-ray imaging where Beer-Lambert law governs the attenuation of intensity through a material.

Background: Beer-Lambert Law for Transmission

The transmitted intensity \( I(x, y) \) at a position \((x, y)\) through a material is described by the Beer-Lambert law:

\[ I(x, y) = I_0 \cdot e^{-\mu \cdot L(x, y)} \]

where:

\( I(x, y) \) is the transmitted intensity at position \((x, y)\),
\( I_0 \) is the incident (background) intensity,
\( \mu \) is the linear attenuation coefficient (accounting for both absorption and scattering),
\( L(x, y) \) is the chord length through the material at position \((x, y)\).

For a vertical cylindrical sample, the system can be simplified by considering only the horizontal coordinate \( x \), assuming uniformity along \( y \):

\[ I(x) = I_0 \cdot e^{-\mu \cdot L(x)} \]

where \( L(x) \) is the chord length through the cylinder at position \( x \).

Transmission and Attenuation Coefficient

The normalized transmission \( T(x) \), also known as the normalized radiograph, is defined as the ratio of transmitted intensity to incident intensity:

\[ T(x) = \frac{I(x)}{I_0} = e^{-\mu \cdot L(x)} \]

From this relation, the attenuation coefficient \( \mu \) can be calculated as:

\[ \mu = - \frac{\ln(T(x))}{L(x)} \]

Practical Considerations

In practice, ideal background measurements \( I_0 \) may not be available or may be imperfect, resulting in transmission values \( T(x) > 1 \), which leads to negative or non-physical values for \( \mu \). To address this, two robust methods for estimating \( \mu \) are presented below.

Method 1: Discrete Method (Two-Column Approach)

This method leverages transmission values at two distinct positions \( x_1 \) and \( x_2 \) on the cylinder to estimate a physically meaningful attenuation coefficient.

Given:

\[ T(x_1) = e^{-\mu \cdot L(x_1)} \quad \text{and} \quad T(x_2) = e^{-\mu \cdot L(x_2)} \]

Taking the ratio:

\[ \frac{T(x_1)}{T(x_2)} = \frac{e^{-\mu \cdot L(x_1)}}{e^{-\mu \cdot L(x_2)}} = e^{-\mu \cdot (L(x_1) - L(x_2))} \]

Solving for \( \mu \):

\[ \mu = -\frac{1}{L(x_2) - L(x_1)} \cdot \ln\left(\frac{T(x_1)}{T(x_2)}\right) \]

Interpretation: This approach ensures that the calculated attenuation coefficient \( \mu \) is always positive and physically meaningful, as it relies on relative transmission differences rather than absolute values. The choice of \( x_1 \) and \( x_2 \) should correspond to positions with reliable transmission data.

Method 2: Iterative Method (Global Regression with Nuisance Term)

An alternative approach involves performing a global regression across all spatial positions \( x \) to estimate \( \mu \) while accounting for nuisance parameters (such as background fluctuations or instrumental effects). This iterative method fits the model:

\[ T(x) \approx e^{-\mu \cdot L(x)} \cdot N(x) \]

where \( N(x) \) represents a nuisance term that captures deviations from ideal background normalization.

The regression optimizes the parameters \( \mu \) and \( N(x) \) to minimize the discrepancy between measured and modeled transmissions across the entire dataset. This method is particularly useful when background measurements are noisy or non-uniform.

Correction Application per Wavelength (TOF Bin)

It is important to note that the attenuation coefficient \( \mu \) and the corresponding correction factor are wavelength-dependent (i.e., vary with Time-of-Flight (TOF) bins). Therefore, the correction must be applied separately for each wavelength bin to accurately account for energy-dependent attenuation.

Cylindrical Sample Thickness Correction

Once \( \mu \) is determined, the goal is to transform the measured transmission \( T(x) \) corresponding to the chord length \( L(x) \) at position \( x \) to a uniform thickness corresponding to the cylinder diameter \( D \).

Given:

\[ T(x) = e^{-\mu \cdot L(x)} \]

We define the corrected transmission for uniform thickness \( D \) as:

\[ T^D(x) = e^{-\mu \cdot D} \]

The correction factor \( C(x) \) to be applied to the measured transmission is:

\[ C(x) = \frac{T^D(x)}{T(x)} = e^{-\mu \cdot (D - L(x))} \]

This factor adjusts the transmission data as if the sample thickness were uniform and equal to the diameter \( D \).

Note: Since the attenuation coefficient \( \mu \) is derived using differences in chord lengths, the correction factor \( C(x) \) is dimensionless and physically consistent.

This methodology provides a robust framework for correcting cylindrical sample transmission data, enabling accurate attenuation coefficient estimation and uniform thickness correction across all wavelengths.