Three-dimensional position reconstruction of orthogonal-strip planar high-purity germanium detectors using maximum likelihood estimation

Hao Ma; Mingxin Yang; Ming Zeng; Peng Zhang; Qiuli Zhang; Wenhhan Dai; Yang Tian; Zhi Zeng

REVIEW 3 major objections 5 minor 18 references

Three-dimensional position reconstruction of orthogonal-strip planar high-purity germanium detectors using maximum likelihood estimation

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A maximum-likelihood fit over four integral pulse parameters reconstructs 3D interaction positions with near-zero bias and sub-mm resolution in simulated strip HPGe detectors.

desk verdict Sound simulation study of a jointly-reconstructed multi-parameter MLE for HPGe strip detectors, but the headline bias/resolution are in-sample and the method is unproven for off-grid positions. read the letter →

arxiv 2507.18222 v1 pith:6YF2CVDH submitted 2025-07-24 physics.ins-det nucl-ex

Qiuli Zhang , Peng Zhang , Wenhhan Dai , Mingxin Yang , Yang Tian , Ming Zeng , Hao Ma , Zhi Zeng This is my paper · ORCID

classification physics.ins-detnucl-ex PACS 29.40.Wk

keywords HPGedetectorspositionreconstructionpulseshapeanalysismaximumlikelihoodestimationintegral-basedparametersorthogonal-stripgamma-rayimagingComptontelescope

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

An orthogonal-strip planar HPGe detector can localize a photon interaction in three dimensions from the shapes of the charge signals it induces on crossed strip electrodes. The paper claims that the standard approach—reconstructing depth from collection-time difference and lateral position from image-signal amplitudes, each axis independently—carries systematic bias because each parameter also depends on the other two coordinates. It replaces this with a maximum-likelihood estimator that treats four integral-based pulse parameters jointly, modeling their noise-corrupted values as a multivariate normal whose mean and covariance are mapped over the pixel volume. In pulse-shape simulations of 100 keV interactions under 1 keV RMS electronic noise, the claimed result is near-zero bias (Z bias reduced from 0.4 mm to 0.02 mm centrally and from 2 mm to 0.15 mm near electrodes; X/Y bias from 0.4 mm to 0.016 mm) and sub-millimeter resolution (0.07–0.16 mm in Z, 0.07–0.44 mm in X/Y). If these numbers hold on a real detector, the method would give Compton imagers and gamma-ray telescopes precise interaction locations without finer pixel segmentation.

What carries the argument

The load-bearing mechanism is a maximum-likelihood position estimator built from a precomputed parameter-to-position map. For every simulated interaction site, $N$ noisy realizations of the four integral parameters give a sample mean $\hat{\mu}(x,y,z)$ and covariance $\hat{\Sigma}(x,y,z)$; the estimated position is the grid point maximizing the multivariate Gaussian density $P(A \mid \hat{\mu},\hat{\Sigma})$. The parameters themselves matter: replacing collection-time difference by the net areas $S_C$ (collection signals) and $S_I$ (image signals) keeps depth information while averaging out high-frequency noise, and replacing image-signal amplitudes by the normalized absolute-area asymmetries $S_{AX}$, $S_{AY}$ does the same for the lateral coordinates. The joint likelihood is what removes the systematic cross-dimensional bias, because each candidate position is scored against all four parameters simultaneously instead of one axis at a time.

What would settle it

Place a collimated 100 keV source on a real orthogonal-strip planar HPGe detector of the same geometry, acquire pulses with known beam positions, extract the four integral parameters, and run the paper's MLE map; if the measured biases exceed roughly 0.5 mm or the resolutions exceed roughly 0.5 mm at 1 keV equivalent noise—well above the simulated 0.02–0.16 mm—then the simulation-to-detector transfer is the failure point, not the estimator.

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Extended reading notes

Core claim

The central discovery is that cross-dimensional interference—the fact that a depth parameter like collection-time difference also shifts with the lateral position of the interaction—can be absorbed, rather than averaged away, by fitting all coordinates at once. The paper constructs, on a 0.1 mm grid inside a central pixel, a position-dependent multivariate normal model for the parameter set $\{S_C, S_I, S_{AX}, S_{AY}\}$, where $S_C$ and $S_I$ are net-area integrals of collection and image signals and $S_{AX},S_{AY}$ are normalized absolute-area asymmetries of neighboring image signals. A measured parameter vector is assigned the position that maximizes this likelihood. In simulation, this reduces maximum Z bias from 0.4 mm to 0.02 mm in the central region and from 2 mm to 0.15 mm near the electrodes, reduces X/Y maximum bias from 0.4 mm to 0.016 mm, and yields position resolutions of 0.07–0.16 mm (Z) and 0.07–0.44 mm (X/Y) at 1 keV RMS noise. It also shows that the integral-based parameters degrade approximately linearly with increasing noise, whereas the amplitude-based parameters degrade exponentially.

Load-bearing premise

The load-bearing premise is that the simulated pulse shapes—computed from the assumed impurity profile, electric field, charge-carrier drift, and RC response—match the real detector at the sub-0.1 mm level; if they do not, the tiny reported biases are biases of the fit to the simulator rather than of the detector.

Editorial extensions

If this is right

Sub-millimeter 3D interaction positions from a standard orthogonal-strip HPGe detector would sharpen gamma-ray track reconstruction, improving Compton imaging angular resolution without smaller strips.
Because the integral-based parameters keep resolution nearly linear in electronic noise, the method should maintain usable positioning at higher noise levels or lower signal thresholds than amplitude-based methods.
The same joint-likelihood mapping can be rebuilt for other segmented semiconductor detectors such as CdZnTe, extending unbiased 3D positioning beyond germanium.
The near-electrode Z degradation (0.15 mm bias, 0.16 mm resolution) remains the main limitation inside the pixel volume, showing where additional parameters or field shaping would help most.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: the multivariate likelihood's covariance structure could be used to detect simulation mismatch in experiment—if residuals between measured and simulated parameter vectors are correlated across $S_C, S_I, S_{AX}, S_{AY}$, that is a signature that the simulated electric field or drift model is off, not just random noise.
Beyond the paper: a self-calibrating variant is testable—use Compton-scatter kinematics or a collimated beam to label a subset of real events, then fit the parameter-to-position map from data instead of simulation, turning the simulator into a prior rather than the source of truth.
Beyond the paper: the near-cathode X/Y resolution loss (0.44 mm) suggests that adding a second-neighbor or corner image signal to the parameter set would recover lateral information in that zone; the likelihood formalism makes the information gain of such an addition directly measurable.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

I’ll cut to it: the paper tests a good idea — joint MLE over four integral-based pulse-shape parameters for 3D position reconstruction in orthogonal-strip HPGe — but the headline numbers are in-sample, and the practical claim of continuous sub-100-micron reconstruction is not supported by the evidence as presented.

What’s genuinely new: prior work reconstructs X/Y and Z independently from amplitude-based parameters (CTD, A_asy). Here the authors build a single multivariate Gaussian likelihood over S_C, S_I, S_AX, S_AY, learning the parameter-position mapping from simulated pulses. The integral parameters are well-motivated, and the noise-robustness comparison against amplitude-based parameters is useful and physically credible. Figures 12 and 14 show real cross-dimensional interference, so the joint treatment should indeed reduce systematic bias in simulation. The authors are also honest about the simulation-only scope and explicitly flag that the method relies on simulation accuracy.

The soft spot is structural and independent of simulator fidelity. The likelihood is defined on a 0.1 mm grid (Section 2.4), and Section 3 evaluates bias and resolution on that same grid. The arg max in Eq. 2.8 is over the discrete grid, so at a training point the estimate can sit on the true grid point even if the model is completely wrong for any off-grid location. Reported X/Y bias of 0.016 mm and resolution of 0.07 mm are then essentially measures of grid quantization and the frequency of jumps to neighboring points — not evidence of unbiased continuous reconstruction. The paper never describes an interpolation scheme or any off-grid test. This is a load-bearing gap: for a real event, the interaction lands between grid points, and the method as written would quantize to the closest grid point, giving a bias floor of at least half the spacing and a resolution floor that is not captured by the reported numbers.

Missing details (N, M, maximization scheme) and no code/data release make the numbers hard to verify, though these are minor and fixable.

Who benefits: this is for detector simulation groups and HPGe instrument teams, particularly those working on Compton telescopes or small-animal SPECT. It deserves a serious referee — the method is plausible and the paper is clear — but the revision needs to address off-grid evaluation (e.g., leave-one-out on the grid or adding a jittered test set) and describe the arg max procedure. I’d accept to review, not desk-reject.

Referee Report

3 major / 5 minor

Summary. The manuscript proposes a maximum-likelihood-based three-dimensional position reconstruction method for orthogonal-strip planar HPGe detectors. The authors simulate pulse shapes from a central pixel on a 0.1 mm grid using SolidStateDetectors.jl, extract four integral-based parameters (S_C, S_I, S_AX, S_AY), model their per-position multivariate normal distribution from repeated noise realizations, and reconstruct positions by maximizing the likelihood. The paper reports that, under 1 keV RMS electronic noise for 100 keV events, the maximum Z bias drops from 0.4 mm to 0.02 mm centrally and from 2 mm to 0.15 mm near electrodes, maximum X/Y bias drops to 0.016 mm, and position resolution is 0.07-0.16 mm in Z and 0.07-0.44 mm in X/Y, with improved noise robustness relative to amplitude-based parameters. The central methodological claim is that joint modeling of multiple pulse-shape parameters removes the systematic bias caused by cross-dimensional interference.

Significance. If the reported performance holds for continuous, off-grid interaction positions and for a real detector, the method would be a useful advance for HPGe position reconstruction, particularly because it replaces noise-sensitive amplitude parameters with integral-based ones and addresses cross-dimensional interference. The paper benefits from a coherent simulation pipeline that includes realistic physical effects (drift, diffusion, self-repulsion, RC response, Gaussian noise), and from explicit comparisons against a conventional independent reconstruction baseline. However, the current evidence is entirely in-sample: both the likelihood model and the evaluation use the same 0.1 mm simulation grid, so the headline bias and resolution numbers do not yet establish the continuous-reconstruction claim. Experimental validation or at least a perturbed-simulation cross-check would be needed to transfer the method from simulation to practice.

major comments (3)

[Section 2.2, Section 2.4, Section 3.1] The manuscript does not specify how the arg max in Eq. (2.8) is computed: is it restricted to the discrete set of 0.1 mm grid points, or is it optimized over continuous coordinates? If the latter, the likelihood function is only defined at grid points and the interpolation method is missing. This is load-bearing because the central claim is three-dimensional position reconstruction for arbitrary interaction sites, not only for the sampled grid. Please clarify the implementation and, if grid-only, add an off-grid evaluation or an interpolation scheme.
[Section 2.2 and Conclusion] While the paper is presented as a simulation study, the abstract and introduction suggest practical applicability to medical imaging and gamma-ray astronomy. The absence of any experimental data or model-perturbation study is a limitation that should be stated more prominently, and a concrete cross-validation test is necessary to support the extrapolation from simulated to real detector performance.
[Section 2.4, Eq. (2.8), and Section 3] Several implementation parameters are not reported, which prevents reproducibility. The number of noise realizations N used for estimating the mean vector and covariance matrix in Eqs. (2.6)-(2.7) is not stated, nor is the number of repeated reconstructions M in Eq. (2.9). The mismatch of indices in Eq. (2.9) (the summation runs to N while the normalization uses M-1) also makes the exact definition ambiguous. Please specify N and M, correct the indices, and report the statistical uncertainty (e.g., the standard error of the mean bias) so that differences such as 0.02 mm versus 0.03 mm are meaningful.

minor comments (5)

[Eq. (2.9)] Equation (2.9) contains a typo: the sum over j runs from 1 to N while the normalization is 1/(M-1), and the definition of \bar{\hat{x}} also uses N. Both should be indexed by M to match the text describing M repeated reconstructions.
[Figure 10 caption] The caption reads "S_AX (left) as a function of the X postion" and "S_AY (right) as a function of the Y postion"; "postion" should be "position."
[Section 2.2] The manuscript assigns an energy of 100 keV to each simulated interaction event but does not clarify whether these are single-site energy deposits or full photoelectric absorption events. Since HPGe photon interactions at 100 keV typically involve a single photoelectron, this is likely fine, but a sentence defining the simulated event type (point-like single energy deposit) would avoid ambiguity.
[Section 2.2 and Figure 6] The integration window is fixed at 300 ns (900-1200 ns) for all events. The authors state it covers the entire rise edge, but drift times vary with interaction depth and can approach or exceed this window near the cathode under the adopted field. A justification or a position-dependent window would strengthen the parameter definitions.
[Section 2.2] The impurity type is not specified (n-type vs p-type). The sign of the impurity gradient affects the electric field profile and hence the pulse shapes; stating the polarity would improve reproducibility.

Circularity Check

1 steps flagged · score 3.0 of 10

Headline bias/resolution numbers are in-sample: the MLE lookup is built and evaluated on the same 0.1 mm grid, so sub-grid biases reflect the fitted nodes rather than genuine continuous-position reconstruction.

fitted input called prediction [Section 2.2 (sampling grid) and Section 2.4 Eq. (2.8) with Section 3.1 evaluation]
"Within this central pixel, a set of evenly spaced three-dimensional sampling points with 0.1 mm spacing is generated, with each point corresponding to a simulated interaction event."

The likelihood lookup in Eq. (2.8) is built at exactly these 0.1 mm grid points, and the Section 3.1 bias evaluation uses 'all interaction events in the corresponding X-Y plane'—the same simulated grid events. The true position is therefore always one of the discrete argmax candidates. Under a correctly specified likelihood at a grid point, the expected estimate is that grid point, so the reported near-zero biases (0.016 mm X/Y, 0.02 mm central Z) are in-sample properties of the fitted mapping, partly forced by construction. An off-grid event would suffer up to half-grid quantization bias (or require an interpolation/continuous-optimization scheme that is neither described nor validated). Thus the headline numbers do not yet demonstrate unbiased continuous 3D reconstruction.

full rationale

The paper's core algorithmic content—joint MLE with integral-based parameters—is not itself circular: it builds a well-defined statistical lookup from a simulator (SolidStateDetectors.jl, an external package) and compares against a conventional independent-dimension method. The self-citations ([5], [14]) are background references and are not load-bearing. The conclusion explicitly concedes that effectiveness depends on pulse-shape simulation accuracy, which is an external-validity limitation rather than a circular derivation. The main circularity concern is the evaluation protocol: training and test events coincide on the 0.1 mm grid, so the reported bias and resolution figures measure the fitted lookup at its own nodes rather than its generalization to arbitrary continuous positions. This warrants a moderate score; the comparison with the conventional method is still informative within the simulation, but the headline near-zero bias should be read as an in-sample statistical check, not a validated predictive claim.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

Every result in the paper flows from the pulse-shape simulation: the parameters, the MLE mapping, the bias, and the resolution. No new physical entity is introduced. The main inputs are the detector geometry (30 mm diameter, 14 mm thick, 3 mm strip pitch), operating conditions (-1000 V bias), an assumed impurity profile, the RC preamp time constant (14 us), and a Gaussian noise model. These are domain assumptions inherited from the detector and the simulation toolchain, not fitted to achieve the reported numbers. The hand-chosen integration window and the unreported sample sizes N and M are the effective free parameters of the reported performance.

free parameters (3)

Integration window for net/absolute area parameters = 300 ns fixed window from 900 ns to 1200 ns
Hand-chosen in Section 2.3; enters every parameter and therefore every reconstruction. The paper asserts it covers the full rise edge but does not verify this for events at all depths.
Number of noise realizations N (Eqs. 2.6-2.7) = not stated
The per-position Gaussian mean and covariance depend on N; without N the statistical precision of the mapping is unknown.
Number of repeated reconstructions M (Eq. 2.9) = not stated
The resolution sigma is computed over M reconstructions; Eq. 2.9 sums over N while normalizing by M-1, an index inconsistency that leaves the effective sample size ambiguous.

assumptions (5)

standard math Shockley-Ramo theorem correctly predicts induced charge on strip electrodes (Eq. 2.1)
Foundational electrostatics result; cited from [16-18]; not re-derived.
domain assumption SolidStateDetectors.jl faithfully computes electric field, weighting potentials, drift, diffusion, and self-repulsion for this geometry and bias
All waveforms, and therefore all results, are produced by this package. If its physics is inaccurate at the sub-0.1 mm level, the claimed biases are artifacts (Section 2.2; acknowledged in the Conclusion).
domain assumption The impurity concentration 7.0e9 cm^-3 with -2.86% linear gradient matches the real detector
This profile determines the electric field and charge drift; it is taken from detector specification rather than measured in this work.
domain assumption Single-site 100 keV interactions in one central pixel represent the events of interest
Edge pixels, multi-site events, charge sharing, and other energies are excluded (Section 2.2, Figure 3).
domain assumption For each position, the parameter vector A is multivariate Gaussian for arbitrary noise realizations
The MLE derivation (Eq. 2.8) assumes joint normality of S_C, S_I, S_AX, S_AY; non-Gaussian tails would bias the estimator.

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Cite this review

Pith. "Pith review of Three-dimensional position reconstruction of orthogonal-strip planar high-purity germanium detectors using maximum likelihood estimation." pith.science (2026). https://pith.science/paper/6YF2CVDH

@misc{pith2026250718222,
  author       = {Pith},
  title        = {Pith review of: Three-dimensional position reconstruction of orthogonal-strip planar high-purity germanium detectors using maximum likelihood estimation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6YF2CVDH}},
  note         = {Machine review of arXiv:2507.18222}
}

read the original abstract

Orthogonal-strip planar high-purity germanium (HPGe) detectors can reconstruct three-dimensional (3D) positions of photon interactions through analysis of parameters extracted from multiple charge signals. The conventional method independently reconstructs positions in each dimension using amplitude-based parameters, leading to noise sensitivity and systematic biases. In this study, we propose a multi-parameter-joint reconstruction method based on maximum likelihood estimation (MLE) which establishes a mapping between pulse shape parameters and corresponding 3D positions. To mitigate the effects of electronic noise, we employ integral-based parameters. The reconstruction performance was evaluated using pulse shape simulations. For 100 keV photons under 1 keV root-mean-square (RMS) electronic noise, the maximum Z reconstruction bias was reduced from 0.4 mm to 0.02 mm in the central region and from 2 mm to 0.15 mm near the electrodes. The maximum reconstruction bias in the X/Y directions was reduced from 0.4 mm to 0.016 mm. Furthermore, the use of integral-based parameters mitigated the rapid degradation of resolution under high-noise conditions. The achieved position resolution ranged from 0.07 mm to 0.16 mm in the Z directions and from 0.07 mm to 0.44 mm in the X/Y direction. This method offers a promising approach to 3D position reconstruction with HPGe detectors for applications such as medical imaging and gamma-ray astronomy.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Boston, A

H. Boston, A. Boston, R. Cooper, J. Cresswell, A. Grint, A. Mather et al.,Characterisation of the smartpet planar germanium detectors,Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment579(2007) 104

work page 2007

[2] [2]

Johnson, D.L

L.C. Johnson, D.L. Campbell, E.L. Hull and T.E. Peterson,Characterization of a high-purity germanium detector for small-animal spect,Physics in Medicine & Biology56(2011) 5877

work page 2011

[3] [3]

Siegert, S.E

T. Siegert, S.E. Boggs, J.A. Tomsick, A.C. Zoglauer, C.A. Kierans, C.C. Sleator et al.,Imaging the 511 kev positron annihilation sky with cosi,The Astrophysical Journal897(2020) 45

work page 2020

[4] [4]

Tomsick, S.E

J.A. Tomsick, S.E. Boggs, A. Zoglauer, D. Hartmann, M. Ajello, E. Burns et al.,The compton spectrometer and imager, 2023

work page 2023

[5] [5]

J. Yang, Y. Tian, W. Dai, M. Yang, L. Jiang, J. Wen et al.,A feasibility study of multi-electrode high-purity germanium detector for 76ge neutrinoless double beta decay searching,Journal of Instrumentation18(2023) P05025

work page 2023

[6] [6]

Amman, P

M. Amman, P. Luke and S. Boggs,Amorphous-semiconductor-contact germanium-based detectors for gamma-ray imaging and spectroscopy,Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment579(2007) 886

work page 2007

[7] [7]

P. Luke, M. Amman, B. Phlips, W. Johnson and R. Kroeger,Germanium orthogonal strip detectors with amorphous-semiconductor contacts,IEEE Transactions on Nuclear Science47(2000) 1360

work page 2000

[8] [8]

Sharma, R

A. Sharma, R. Palit, T. Habermann, J. Gerl, I. Kojouharov, H. Schaffner et al.,Performance test of a position sensitive planar germanium detector for phase-iii despec experiments,Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment1051(2023) 168233

work page 2023