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

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.

Figures (7)

• Figure 1: Illustration of the inverse problem. Left: The true distribution $f(x)$, a double-peaked spectrum and the measured distribution $g(y)$ demonstrating the effect of the detector smearing. Right: The unfolded result from bru (black points with error bars) compared with the truth (blue histogram).
• Figure 2: Left: Eigenvalue spectrum of the curvature matrix in the Fisher-normalised basis. The dashed line indicates $1/\hat{\tau}$, where $\hat{\tau}$ is the regularization parameter selected automatically by the criterion of Eq. (12); for this example $1/\hat{\tau} = 12$. Right: Filter factors $h_k = 1/(1+\hat{\tau}\, d_k)$. Modes with $h_k \approx 1$ are retained; modes with $h_k \approx 0$ are suppressed.
• Figure 3: Amplitude spectrum of the unregularized solution. The horizontal dashed line marks the noise level. Modes to the right of the vertical dotted line are suppressed by the regularization.
• Figure 4: The first six eigenvectors of the regularization, evaluated in the B-spline basis. Low-index modes describe smooth variations; high-index modes capture progressively finer structure. The eigenvalue $d_k$ is indicated in each panel title.
• Figure 5: Single pseudo-experiment for the double-peaked distribution. The solid histogram shows the truth; points with error bars show the unfolded results from each method.