Blobel's Regularized Unfolding: Eigenmode Decomposition and Automatic Smoothing for Inverse Problems in Particle Physics

TL;DR

This paper details Blobel's Regularised Unfolding (BRU), a spline-based method for inverse problems in particle physics that uses eigenmode filtering and automatic regularization to improve distribution estimation.

Contribution

It introduces a modern implementation of Blobel's unfolding method using cubic B-splines and eigenmode filtering, with automatic regularization parameter determination.

Findings

The method employs spline parametrization for smooth distribution modeling.

Eigenmode filtering enhances regularization and stability.

Automatic regularization strength determination adapts to data quality.

Abstract

This document presents a self-contained treatment of regularized unfolding based on cubic B-spline representations and eigenmode filtering, following the original formulation by Blobel and direct translation of the original implementation in Fortran into a modern format. The method, which has been called by several names under its various historical representations, is named here as Blobel's Regularised Unfolding (BRU). This method differs from conventional histogram-based unfolding approaches in that the true distribution is represented as a smooth function parametrised by spline coefficients, and the regularization operates through an eigenmode decomposition of the curvature penalty relative to the statistical precision. This document describes the mathematical structure of the method, the mechanism by which the regularisation strength is determined automatically from the data, and provides a detailed comparison with standard methods including Tikhonov regularisation based methods, Richardson-Lucy iteration, and naive matrix inversion.