[Paper Review] Perspectives on the Calibration of CNN Energy Reconstruction in Highly Granular Calorimeters

N. Akchurin, C. Cowden+3 more|arXiv (Cornell University)|Aug 24, 2021

Particle Detector Development and Performance2 references7 citations

TL;DR

This paper investigates the calibration of convolutional neural network (CNN)-based energy reconstruction in highly granular calorimeters under miscalibration, bias, and noise. It demonstrates that CNN responses remain linear across 500 GeV despite input perturbations, enabling reliable post-model calibration via multiplicative and additive corrections, with resolution behavior transitioning from traditional to noise-dominated when noise and bias exceed thresholds.

ABSTRACT

We present a study which shows encouraging stability of the response linearity for a simulated high granularity calorimeter module reconstructed by a CNN model to miscalibration, bias, and noise effects. Our results also show an intuitive, quantifiable relationship between these factors and the calibration parameters. We trained a CNN model to reconstruct energy in the calorimeter module using simulated single-pion events; we then observed the response of the model under various miscalibration, bias, and noise conditions that affected the model input. From these data, we estimated linear response models to calibrate the CNN. We also quantified the relationship between these factors and the calibration parameters by regression analysis.

Motivation & Objective

• To assess the feasibility of calibrating CNN-based energy reconstruction in highly granular calorimeters under realistic detector non-idealities.
• To determine whether CNN responses remain linear under miscalibration, bias, and noise, enabling practical calibration.
• To quantify the relationship between input perturbations (miscalibration, noise, bias) and resolution parameters (constant, stochastic, noise).
• To identify critical thresholds where resolution behavior transitions from traditional to noise-dominated regimes.
• To provide a framework for post-model calibration using linear response models derived from perturbed input data.

Proposed method

• Trained a CNN on simulated single-pion events in a Cu/Si calorimeter module with 2×2×4 cm³ segmentation.
• Applied controlled miscalibration, bias, and noise to input cell energies using Equation 1.1: $\tilde{E_{ij}} = C_i E_{ij} + b + \epsilon_{ij}$.
• Used beta distributions ($B_{2,2}$) transformed via Equations 1.2 and 1.3 to generate miscalibration ($C$) and noise ($\epsilon$) with tunable scale factors $s_c$, $s_\epsilon$, and $s_b$.
• Applied zero suppression at 0.6 MeV to simulate realistic thresholds and preserve signal integrity.
• Fitted linear models to CNN outputs to estimate calibration parameters and used regression to relate perturbations to resolution parameters.
• Defined an indicator $I_{ND}$ (Equation 2.3) to identify high noise and bias regimes where resolution behavior changes.

Experimental results

Research questions

• RQ1 Does the CNN energy reconstruction response remain linear under miscalibration, bias, and noise perturbations?
• RQ2 Can the CNN’s output be reliably calibrated using post-model corrections?
• RQ3 How do miscalibration, noise, and bias scale factors affect the resolution parameters (constant, stochastic, noise)?
• RQ4 At what threshold do resolution behavior transition from traditional to noise-dominated regimes?
• RQ5 Is there a quantifiable, transferable relationship between input perturbations and calibration parameters?

Key findings

• The CNN response remains linear up to 500 GeV, even though training data only extended to 125 GeV, indicating strong extrapolation capability.
• The response linearity supports post-model calibration via simple multiplicative and additive corrections, enabling uniform energy reconstruction.
• When noise and bias exceed thresholds ($s_\epsilon > 0.5$ or $s_b \geq 0.15$ with $s_\epsilon \geq 0.5$), the resolution is dominated by the noise term $\mathcal{N}$, indicating a shift from Poisson-like to constant-variance behavior.
• In high noise-bias regimes, the constant ($\mathcal{C}$) and stochastic ($\mathcal{S}$) terms are suppressed, while $\mathcal{N}$ becomes the primary resolution contributor.
• The resolution parameters follow predictable trends: $\log{\mathcal{C}}$, $\log{\mathcal{S}}$, and $\log{\mathcal{N}}$ are linearly dependent on $s_c$, $s_\epsilon$, and $s_b$, with significant suppression in the $I_{ND}=1$ region.
• The study identifies a transition region where non-signal effects (noise and bias) significantly degrade resolution, especially when combined with zero suppression.