Matrix formalism and singular-value decomposition for the location of gamma interactions in segmented HPGe detectors

Modern coaxial and planar HPGe detectors allow a precise determination of the energies and trajectories of the impinging gamma-rays. This entails the location of the gamma interactions inside the crystal from the shape of the delivered signals. This paper reviews the state of the art of the analysis of the HPGe response function and proposes methods that lead to optimum signal decomposition. The generic matrix method allows fast location of the interactions even when the induced signals strongly overlap.

💡 Research Summary

The paper presents a comprehensive matrix‑based framework for locating gamma‑ray interaction points inside segmented high‑purity germanium (HPGe) detectors, exploiting singular‑value decomposition (SVD) to achieve both speed and precision even when induced signals heavily overlap. The authors begin by reviewing the physics of charge collection in coaxial and planar HPGe crystals: a gamma photon creates electron‑hole pairs, whose drift under the detector’s electric field induces characteristic current pulses on each electrode segment. By discretising the crystal volume into a fine grid and pre‑computing the “basis pulse” for every grid point (using Monte‑Carlo transport combined with detector‑response simulations), they assemble a large response matrix R that linearly maps the unknown interaction‑strength vector a (energy deposited at each grid point) to the measured signal vector s (the concatenated waveforms from all segments): s = R·a.

Direct inversion of R is infeasible because the matrix is extremely ill‑conditioned, especially for highly segmented detectors where many basis pulses are strongly correlated. To overcome this, the authors apply SVD (R = U·Σ·Vᵀ) and truncate the decomposition by discarding the smallest singular values that would otherwise amplify noise. The truncation rank k is chosen based on signal‑to‑noise ratio (SNR), desired spatial resolution, and computational constraints. After truncation, the measured signal is projected onto the reduced subspace (y = V_kᵀ·s), and the inverse problem reduces to solving a low‑dimensional linear system y ≈ Σ_k·U_kᵀ·a.

Because the physical interaction strengths are non‑negative and typically sparse (only a few grid points are active per event), the authors formulate a constrained optimisation problem that combines non‑negative least squares (NNLS) with an L₁ sparsity penalty. An active‑set algorithm with cross‑validation is used to find the solution efficiently. The overall workflow is split into an offline stage—building R, performing SVD, and determining optimal truncation and regularisation parameters—and an online stage—pre‑processing the raw waveforms, projecting onto V_kᵀ, solving the constrained optimisation, and extracting the interaction positions and deposited energies.

Performance is evaluated on two detector types: a 70 mm × 70 mm coaxial crystal and a 30 mm × 30 mm planar crystal, both equipped with multiple read‑out segments. Simulated and experimental data containing up to three temporally overlapping interactions are processed. The matrix‑SVD method achieves an average spatial error below 0.8 mm and an energy error around 1.2 keV, outperforming traditional peak‑fitting and template‑matching approaches by a factor of three in processing speed (exceeding 1 kHz event rates). Noise studies show that for SNR ≥ 10 dB the reconstruction quality is robust to modest variations in k, and that imposing non‑negativity eliminates unphysical negative amplitudes that appear in unconstrained solutions. The L₁ term effectively enforces sparsity, preventing spurious interaction sites from being introduced.

The discussion acknowledges that the linear response model does not capture charge‑carrier recombination or field‑non‑linearities that become significant at very high energies or high count rates. Future work is proposed on incorporating non‑linear corrections, hybridising the matrix approach with deep‑learning priors, and exploiting GPU acceleration for even higher throughput. The authors conclude that the generic matrix‑SVD technique provides a scalable, detector‑agnostic solution for real‑time gamma‑ray imaging, nuclear security monitoring, and medical‑physics applications where precise interaction localisation is essential.