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REVIEW 3 major objections 3 minor 16 references

Properties of the Center of Gravity as an Algorithm for Position Measurements

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Centroiding error is one Fourier series—and crosstalk can kill it

desk verdict A correct and useful derivation of the COG discretization error, with one real definitional gap in the 'ideal detector' section that should be fixed before this is relied upon. read the letter →

arxiv 1908.04447 v1 pith:GK2BBGDF submitted 2019-08-13 physics.ins-det

Gregorio Landi This is my paper · ORCID
classification physics.ins-det PACS 07.05.Kf06.30.Bp42.30.Sy
keywords centerofgravitycentroidingpositionmeasurementsdiscretizationerrorcrosstalkidealdetectorFouriertransformsamplingtheorem
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

The center-of-gravity (COG) algorithm estimates a particle's impact point as the signal-weighted average of sensor positions. This paper derives, from first principles, exactly how that estimate deviates from the true position when the signal is collected by an infinite periodic row of integrating detectors. The deviation is a Fourier sine series in the impact position $\varepsilon$, with coefficients given by the Fourier transform $\Phi(2\pi k/\tau)$ of the average signal distribution at the sampling harmonics. That closed form turns the usual empirical centroid correction into a calculable, invertible relation, and it isolates the special cases where the error vanishes: band-limited signals, particular rectangular or triangular signal shapes, and sensors with a triangular crosstalk of range twice the pitch. The same Fourier machinery gives explicit formulas for finite clusters of sensors, crosstalk, noise, and fluctuations, so the math can be used both to correct measurements and to simulate realistic detectors.

What carries the argument

The load-bearing object is the sampled signal train $s_\varepsilon(x)=f(x-\varepsilon)\sum_n\delta(x-n\tau)$, whose Fourier transform is evaluated by the Poisson identity $S_\varepsilon(\omega)=\tau^{-1}\sum_k F_\varepsilon(\omega-2\pi k/\tau)$. Because the integrating detector converts the signal to $f(x-\varepsilon)$ through a convolution with a rectangular window, $F_\varepsilon(\omega)=[2\sin(\omega\tau/2)/\omega]\Phi(\omega)e^{-i\varepsilon\omega}$, and the COG formula $x_g=i S_\varepsilon'(0)/S_\varepsilon(0)$ turns the derivative into the sine series of Eq. (6). The same machinery is reused in two further forms: the dual Shannon/WKS series expresses $\Phi$ through the form factors $\Phi(2\pi n/D)$, and a contour-residue summation converts the resulting double series into the practical expression Eq. (13). For crosstalk, the response function $p$ enters through its Fourier transform $P$ and its derivative, and the condition for an ideal detector is stated directly as a vanishing weighted sum of shifted response copies.

What would settle it

Scan a known, sharp source across one pitch of a strip or pixel detector, measure both the mean signal shape and the centroid $x_g(\varepsilon)$, and compare the residual $x_g-\varepsilon$ with the series in Eq. (6) using the independently measured Fourier transform $\Phi$. Agreement within noise supports the model; systematic position-dependent deviations that do not follow the predicted coefficients would falsify it. A separate direct test: a detector with triangular crosstalk of range $2\tau$ should show $x_g=\varepsilon$ to within noise for every signal distribution; any periodic wiggle in the residual rules out the ideal-detector claim.

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Extended reading notes

Core claim

On the paper's own terms, the central result is Eq. (6): for an infinite periodic array of ideal integrating sensors with pitch $\tau$, the COG estimate is $x_g = \varepsilon + (\tau/\pi)\sum_{k\ge1}(-1)^k k^{-1}\sin(2\pi k\varepsilon/\tau)\Phi(2\pi k/\tau)$, where $\varepsilon$ is the true impact position and $\Phi$ is the Fourier transform of the mean signal distribution. The discretization error is therefore not a statistical nuisance but a deterministic function of the signal spectrum at the sampling frequencies. The paper further proves that the error vanishes when $\Phi(2\pi k/\tau)=0$ for all $k>0$, a condition broader than the Whittaker-Kotelnikov-Shannon band limit, and exhibits finite-support shapes such as rectangular signals with integer-multiple widths and triangular signals with even-multiple widths that satisfy it. It then extends the analysis to crosstalk: if the detector response $p(x,\tau_1)$ satisfies $\sum_n (x-n\tau)p(x-n\tau,\tau_1)=0$, in particular triangular crosstalk of range $2\tau$, the COG equals $\varepsilon$ for any signal distribution. Finite clusters of sensors are treated too, and they produce discontinuities in $x_g(\varepsilon)$ at the points where the set of sensors changes.

Load-bearing premise

The whole Fourier-series correction rests on treating the detector as an infinite periodic row of identical sensors whose collected signal is exactly a shift-invariant convolution of the mean signal with a fixed response function; real detectors with finite size, varying strip widths or gains, calibration errors, or unread gaps only approximate this model.

Editorial extensions

If this is right

  • For any detector that matches the infinite periodic model, the systematic position error can be removed by inverting Eq. (6) or using Eq. (13), without per-detector Monte Carlo tuning.
  • A detector engineered to have triangular crosstalk of range $2\tau$, or any response satisfying $\sum_n (x-n\tau)p(x-n\tau,\tau_1)=0$, would be unbiased for every signal distribution.
  • Cluster-size cuts create discontinuities in $x_g(\varepsilon)$ at sensor boundaries; subtracting a fixed bias mixes incompatible two-sensor and three-sensor algorithms and cannot be fixed by a smooth polynomial correction.
  • Given a large sample of equivalent hits with a uniform distribution of true positions, the histogram of $x_g$ yields $\varepsilon(x_g)$ by Eq. (25), and when the signal support fits within one pitch it also reconstructs the signal shape.
  • The explicit formulas for crosstalk, noise, inter-strip calibration errors, cracks, and off-axis crystals make the same equations a compact simulation tool for real strip and crystal calorimeters.

Reading between the lines

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

  • The ideal-detector condition suggests a concrete design target for silicon strip and pixel detectors: tune inter-strip capacitive coupling toward a triangular profile with range exactly twice the pitch, so centroids need no correction at all; this is an engineering consequence the paper states mathematically but does not develop as a fabrication recipe.
  • Because only the values $\Phi(2\pi k/\tau)$ enter the correction, the scheme should transfer directly to imaging and star-tracker centroiding wherever the point-spread function is known, not just to particle detectors.
  • The predicted infinite peak in the $x_g$ probability density when the signal support $D\le\tau$ is a testable signature: a histogram of centroids from a source narrower than one pixel should show a sharp spike near zero, distinguishing this model from generic smoothing.
  • The boundary-discontinuity analysis implies that any empirical bias curve subtracted from cluster centroids is only valid for a fixed sensor count; switching cluster sizes in the same analysis introduces uncorrectable systematic jumps.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper analyzes the center-of-gravity (COG) algorithm for one-dimensional position measurements in periodic detector arrays. The central result of Section 3.3 is an explicit formula, Eq. (6), for the systematic discretization error of the COG as a Fourier series whose coefficients are values of the Fourier transform of the average signal distribution at multiples of the sampling frequency. The paper then studies conditions under which this error vanishes, derives a resummed form of the series for finite-support signals in Section 5.3, generalizes the analysis to crosstalk and non-perfect spatial integrators in Section 6, treats finite sensor sets, noise, and fluctuations, and concludes with a claimed construction of an 'ideal detector' whose crosstalk eliminates the discretization error for any signal distribution.

Significance. If the results hold, the paper provides a useful analytical framework for correcting COG systematic errors in position-sensitive detectors, with direct applications to calorimeters and silicon strip trackers. The derivation of Eq. (6) is clean and is benchmarked against known special cases, including the Euler sine series in the delta-function limit and the Lau-Pyrlik conjecture. The paper also gives explicit simulation formulas. However, the 'ideal detector' claim, which is one of the advertised headline results, is under-specified as stated: the vanishing-moment condition alone is insufficient, and the paper's own derivation in Section 6.6 relies on an additional uniformity condition that is not included in the abstract or in the standalone definition. This is fixable, but it must be corrected before the central claim is accepted as stated.

major comments (3)
  1. [Section 6.6 and abstract] The definition of an ideal detector as a crosstalk function satisfying sum_n (x - n tau) p(x - n tau, tau_1) = 0 is insufficient for the claimed conclusion x_g(epsilon) = epsilon for any signal distribution. The derivation on the page with Eq. (14) explicitly uses the uniformity condition sum_n p(x - n tau, tau_1) = 1 to cancel the term involving x' after interchanging the sum and integral. Without uniformity, the zero-first-moment condition does not imply x_g = epsilon: for symmetric p supported on [-tau, tau] of the form p(x) = (tau - x) r(x) on [0, tau] with r symmetric about tau/2 but not constant, one has sum_n (x - n tau) p(x - n tau) = 0 identically, but M_0(x) = sum_n p(x - n tau) = tau r(x) is not constant, and the resulting COG is a weighted average of y M_0(y) rather than epsilon. The uniformity condition should be added to the definition of ideal detector and to the abstract.
  2. [Section 5.3, Eq. (13)] The summation leading to Eq. (13) is compressed, and the sawtooth function is defined inconsistently: the text states Theta(xi) = xi - floor(xi + 1/2), while Eq. (13) states Theta(xi) = xi - floor(xi - 1/2). These differ on half-integers and affect numerical implementation. The authors should specify the exact convention used in Eq. (13) and, if both forms are acceptable up to integer shifts, state that explicitly and verify that the residues are unchanged.
  3. [Section 6.6] The reference to 'Eq. (24)' in the ideal-detector derivation is incorrect: Eq. (24) is the probability-density transformation formula, not the expression for x_g(epsilon). The derivation uses the expression for x_g following from Eq. (14), so the equation numbering should be corrected and the derivation should be made self-contained.
minor comments (3)
  1. [Section 5.1] The condition 'phi(x) = 0 for |x| <= D/2' should read 'phi(x) = 0 for |x| > D/2', or equivalently 'phi(x) != 0 for |x| < D/2', to match the intended finite-support statement.
  2. [Introduction] The section numbering in the introductory text does not match the actual structure of the paper; for example, the text refers to 'Section 2' and 'Section 3' in ways that do not correspond to the displayed section headings.
  3. [Section 7.2] The phrase 'the average distribution omega (x)' appears to be a typo for phi(x), and 'Molier radius' should be 'Moliere radius' throughout the figure captions and text.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: Eq. (6) is derived from standard Fourier/Poisson identities and benchmarked against independent results; self-citations are motivational only.

full rationale

Eq. (6) is derived in Section 3.3 by differentiating Sε(ω) in Eq. (3), applying the Poisson identity Eq. (4), and using the explicit Fourier transform Fε(ω) = [2 sin(ωτ/2)/ω] Φ(ω) e^{-iεω}; no parameter is fitted, and the error series is determined by the Fourier transform Φ exactly as claimed. The checks against the Euler sine series and the Lau-Pyrlik conjecture use independent external results, and the η-function discontinuity [16] is a comparison, not an input. The 2019 preface cites several of the author's own later papers (arXiv:1606.03051, arXiv:1808.06708, INSTRUMENTS 2018 2 22, arXiv:1404.1968) but only for motivation, applications, and context; those citations do not supply any assumption used to derive Eq. (6) or the crosstalk theorems. The only notable defect is the 2019-added ideal-detector definition, which states only Σ_n (x−nτ)p(x−nτ,τ1)=0 while the proof in Section 6.6 uses both uniformity (Σ_n p(x−nτ,τ1)=1) and the vanishing first lattice moment; this is an under-specified definition or overclaim, not a circular reduction. Hence no circularity score above the minor self-citation level is warranted.

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

The paper introduces no new physical entity. It relies on standard Fourier theorems and on modeling assumptions about detector geometry, shift invariance, and linear crosstalk. The 'ideal detector' is a mathematical condition on the crosstalk response, not a new particle or force.

assumptions (5)
  • domain assumption The average signal distribution is shift invariant: for impact point epsilon it is exactly phi(x - epsilon), with phi real, symmetric, normalized and having a continuous, derivable Fourier transform.
    Stated as assumptions (a)-(c) in Section 3.1; without translation invariance the Poisson-shift derivation leading to Eq. (6) fails.
  • domain assumption The detector is an infinite regular array of identical ideal integrators of width tau, with no gaps or position-dependent losses.
    Used in Eqs. (1)-(4) to represent sampled signals as f(x - epsilon) sum delta(x - n tau); finite arrays are treated later as a modification.
  • standard math Poisson summation formula and the WKS sampling theorem.
    Invoked in Eqs. (4), (11), and (27); these are standard results, not proved in the paper.
  • domain assumption Crosstalk is a linear, shift-invariant convolution p(x - x', tau1) of finite range.
    Eq. (14) defines the generalized response; the ideal-detector construction depends on this linear response model.
  • domain assumption For the monotonicity proof in Section 6.5, phi and f are positive, continuous, derivable, single-maximum functions with D > tau.
    The paper states this restriction before proving dxg/depsilon >= 0; rectangular and delta-function limits are excluded from that proof.

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Cite this review

Pith. "Pith review of Properties of the Center of Gravity as an Algorithm for Position Measurements." pith.science (2026). https://pith.science/paper/GK2BBGDF

@misc{pith2026190804447,
  author       = {Pith},
  title        = {Pith review of: Properties of the Center of Gravity as an Algorithm for Position Measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GK2BBGDF}},
  note         = {Machine review of arXiv:1908.04447}
}
abstract

The center of gravity $x_{g}= \sum_{i}E_{i} x_{i}/\sum_{i} E_{i}$ as an algorithm for position measurements is carefully analyzed. Many mathematical consequences of discretization are extracted. The origin of the systematic error of the algorithm is shown to be connected to the absence of band limits in the Fourier Transform of the signal distributions, which, owing to the intrinsic properties of the measuring devices, must have a finite supports. However, special signal distributions exist among the finite support functions which are free from the discretized error. In the presence of crosstalk, it is proved that some crosstalk spreads are able to eliminate the discretization error for any shape ({\em ideal detector}). For all other cases, analytical expressions and prescriptions are given to correct the error and to efficiently simulate various experimental situations.

Figures

Figures reproduced from arXiv: 1908.04447 by the authors.

Figure 1
Figure 1. φ(x) is the signal distribution with zero impact point, f(x) is its convolution with an interval function. The dots are the signals collected by the detectors centered at x=-3,-2,-1,0,1,2,3 and are sam￾ples of f(x). φ(x− ε), f(x− ε) and the diamonds are as above for an impact point ε. Dashed lines are the borders of the detectors. and, defining Fε (ω) as the FT of fε (x), the Poisson identity [9] relates Sε (ω) with… view at source ↗
Figure 2
Figure 2. Plots of (∆/τ) √ 12 versus D/τ for the linear combination of two shapes with D’/D=1/20, β=0,2,4,8 and four shapes: solid lines for rectangles, dot-dash lines for cylinders, dashed lines for triangles, and dotted lines for cones. the COG error of this kind of signal distributions, we will simulate them with a linear combination of two identical functions differing in size D and D’: ϕ(x) = [ϕD(x) +βϕD′(x)]/(1+β) (8) T… view at source ↗
Figure 3
Figure 3. xg versus ε for D=1.8, D’/D=1/7, β=3.5, τ=1 and the graphical conventions as in [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: a), Plots of xg versus ε for a linear combination of two triangles with sizes D=4, D’/D=1/3, β = 3, solid line for a two-sensor COG, dot-dash line for a three-sensor COG, and dotted line for xg = ε. The discontinuities at ε = 0 for the two-sensors COG, and at ε = ±0.5τ…
Figure 5
Figure 5. Figure 5: Probability distributions of xg for the two- and three-sensors COG. Solid lines are the analytical calculations. Dot-dash lines join the midsection of a scaled histogram bins generated on a sample of xg-data (bin size xg/100). The differences are concentrated in the cu…
Figure 6
Figure 6. Figure 6: Signal distribution of a set of random segments generated as explained in the text. Dotted line indicates a random sample; solid line is the average of 150 samples. The horizontal scale is in unit of Molier radiuses. Their origins and zenith angles are Gaussian functio…
Figure 7
Figure 7. Figure 7: Efficiency plots in function of ε/τ for three-sensor (solid line) and five-sensor (dashed line) signal collection, with effective loss of 0.002 at the sensor borders. The sensor size (τ) is 1.1 Molier radius. −0.5 0 0.5 −0.5 0 0.5 x g /τ ε /τ (a −0.5 0 0.5 −0.1 −0.05 0…
Figure 8
Figure 8. Figure 8: The same plots as [PITH_FULL_IMAGE:figures/full_fig_p026_8.png]

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