Coincidence-Based Response Matrix (CBRM)

• CBRM is a calibration-derived matrix that corrects charge-sharing distortions in energy-resolved photon-counting detectors using coincidence measurements.
• It employs a calibration-first correction workflow with 3x3 pixel blocks and iterative inversion to accurately recover true spectral counts.
• The framework extends to forward-folding in cross-section measurements and tensor constructions in TIC-based analyses, illustrating its broad applicability.

Searching arXiv for the specified papers to ground the article and verify citation metadata. Coincidence-Based Response Matrix (CBRM) denotes, in its explicit usage, a calibration-derived detector response matrix obtained from coincidence measurements and used to correct charge-sharing spectral distortions in energy-resolved photon-counting detectors (Monaco et al., 7 Aug 2025). A plausible broader interpretation is that CBRM also names a family of coincidence-response constructions in which a matrix, tensor, or operator encodes how micro-level coincidences map into measured or reconstructed quantities. Under that broader reading, closely related formulations appear in response-matrix-centred forward-folding for cross-section measurements (Koch, 2019), in a tensorized coincidence-response framework built on Transcendental Information Cascades (TICs) (Luczak-Roesch, 2019), and in coincidence imaging through first-order field correlation for micro-vibration reconstruction (Liu et al., 2022). The common structure is that coincidence information is not treated as an incidental by-product, but as the central object from which correction, inference, or reconstruction proceeds.

1. Terminological scope and conceptual definition

In the detector-spectroscopy literature, CBRM is explicitly defined as a “model-independent calibration matrix” used to “correct charge-sharing spectral distortions” in pixellated photon-counting detectors. The calibration is based on “the collection of the number of coincidences between a reference pixel and its neighbours for different combinations of energy bins,” from which one computes “a set of charge sharing probabilities which are independent of the input spectrum” and then determines “a detector response matrix” applicable to later acquisitions with the same detector (Monaco et al., 7 Aug 2025).

A broader comparative reading of adjacent literature suggests that the phrase can also be used, as an editor’s organizing label, for response objects built from coincidence structure even when the original papers do not use that exact name. In high-energy physics, the relevant object is a detector response matrix that maps truth-bin expectations into reconstructed-bin yields through forward folding rather than unfolding (Koch, 2019). In the TIC-based emergence framework, the corresponding object is not a single matrix but a system of tensor representations whose decomposed matrices serve as the macroscopic response objects whose eigenvalue statistics determine whether rare coincidences are meaningful (Luczak-Roesch, 2019). In coincidence imaging for micro-vibration reconstruction, the analogous object is the first-order field-correlation operator, whose discretized form would be a coincidence-based response matrix even though the paper keeps it in analytical operator form (Liu et al., 2022).

Domain	Core response object	Function
Photon-counting detectors	Coincidence-based response matrix	Charge-sharing correction
Cross-section measurements	Detector response matrix	Forward folding to reconstructed space
TIC-based emergence analysis	Tensor decompositions and their matrices	Spectral test of meaningful coincidences
Micro-Doppler coincidence imaging	First-order field-correlation operator	Reconstruction of vibration-mode images

This diversity is important because CBRM is not a universally standardized object across fields. The explicit detector formulation is concrete and operational; the other cases are closely related coincidence-response formalisms. A common misconception is therefore to assume that all uses share the same algebraic form. The literature instead supports a family resemblance: coincidence structure is encoded in a response object, and that object is used to infer, correct, or test higher-level behavior.

2. Explicit CBRM in energy-resolved photon-counting detectors

The most direct and named formulation appears in the work on charge sharing in energy-resolved photon-counting detectors. Charge sharing causes “a single photon” to be counted as “multiple lower-energy hits,” moves counts “from the correct energy bin to lower bins,” and distorts the measured spectrum (Monaco et al., 7 Aug 2025). CBRM addresses this by learning detector-specific charge-sharing behavior from coincidence measurements rather than from a fitted transport model.

The calibration uses a $3\times 3$ pixel block consisting of a center reference pixel $C$ and eight surrounding pixels $T_l$, with the neighbors summed into $T_{\text{sum}}$. For threshold pair $(i,j)$, the acquisition stores the counts above threshold $i$ in the reference pixel, $N_i'$, and the coincidence counts $N^{AND}_{i,j}$. Energy-bin counts are then recovered by threshold differencing:

$n_i' = N_i' - N_{i+1}'.$

The coincidence-bin counts are recovered through the two-dimensional analog

$$n^{AND}_{i,j} = N^{AND}_{i,j} - N^{AND}_{i+1,j} - N^{AND}_{i,j+1} + N^{AND}_{i+1,j+1}.$$

The paper distinguishes two transition classes. CS-OUT denotes events originating in the reference pixel and sharing charge to neighbors; CS-IN denotes events originating in one of the neighbors and depositing some charge in the reference pixel. The corresponding probabilities are $C$0 and $C$1, with the paper assuming

$C$2

The measured counts in reference-pixel bin $C$3 are then written as

$C$4

Under uniform irradiation, the flat-field condition is

$C$5

and the coincidence counts simplify to

$C$6

so that the charge-sharing probability can be estimated as

$C$7

This is the central “spectrum-independent” step: a matrix learned from one arbitrary polychromatic flat-field calibration is subsequently reused to correct other spectra acquired with the same detector.

The matrix form is obtained by writing

$C$8

The paper identifies $C$9 as the elements of the CBRM, states that the matrix is triangular, and notes that it can be inverted to recover the true spectrum (Monaco et al., 7 Aug 2025).

3. Calibration, inversion, and reduced-matrix construction

The explicit CBRM workflow is calibration-first and correction-afterward. During calibration, the paper derives the input counts from measured counts and coincidences as

$T_l$0

Once the probabilities $T_l$1 have been estimated, inversion is performed iteratively from the highest-energy bin downward:

$T_l$2

The triangular structure is what makes this practical.

A further practical issue is that calibration may use finer energy binning than routine acquisition. To address this, the paper introduces a reduced response matrix. If the fine calibration uses $T_l$3 bins and the standard acquisition uses $T_l$4 bins with $T_l$5, the fine-bin probabilities are aggregated into coarse-bin probabilities:

$T_l$6

with

$T_l$7

and

$T_l$8

The resulting reduced response equation is

$T_l$9

For uniform illumination this becomes

$T_{\text{sum}}$0

The significance of this reduced form is operational rather than conceptual. It allows one to calibrate at a fine granularity and deploy the correction on ordinary multi-threshold acquisitions without redesigning the electronics. The paper emphasizes that coincidence logic is required during calibration, not during routine imaging, so the method “can afterwards be applied to correct other spectra acquired with the same detector and a conventional multi-comparator electronics, without introducing penalties in terms of processing time” (Monaco et al., 7 Aug 2025).

4. Response-matrix-centred measurement and forward folding

A second, closely related use of response matrices arises in the presentation of differential cross-section measurements. Here the central object is not a coincidence counter between neighboring detector elements, but a detector response matrix that encodes efficiency and smearing from truth bins to reconstructed bins. The paper argues that instead of unfolding reconstructed data back to truth space, one should publish the response matrix and perform forward-folded comparison of theory predictions in reconstructed space (Koch, 2019).

The matrix element $T_{\text{sum}}$1 expresses the contribution from truth bin $T_{\text{sum}}$2 to reconstructed bin $T_{\text{sum}}$3. The expected reconstructed signal is

$T_{\text{sum}}$4

or, in vector notation,

$T_{\text{sum}}$5

In this framework, the measured data remain in reconstructed space, and the model prediction is mapped through the response matrix. The paper explicitly frames this as avoiding the ill-posed inverse problem associated with unfolding and enabling direct comparison to data.

The likelihood is formulated in reconstructed space:

$T_{\text{sum}}$6

with

$T_{\text{sum}}$7

Backgrounds can be included through additive reconstructed-bin terms or through a special truth-space bin for background or out-of-fiducial contributions. Systematic uncertainties are handled by matrix variations and nuisance parameters, with profiling or Bayesian marginalisation.

The paper’s software implementation is ReMU, a Python package that “offers all methods needed to build response matrices from Monte Carlo data sets, use the response matrix to forward-fold truth-level model predictions, and compare the predictions to real data using Bayesian or frequentist statistical inference” (Koch, 2019). A plausible implication is that this response-matrix-centred philosophy belongs to the same broader CBRM family insofar as the measurable object is the response itself and inference proceeds in the forward direction. However, the paper does not use CBRM as its formal term; its own terminology is “response-matrix-centred” and “forward-folding.”

5. Coincidence response beyond matrices: tensors and correlation operators

In the emergence literature, the relevant coincidence-response construction is developed through TICs rather than through a detector matrix. TICs “transform any kind of sequential data into a directed temporal network of recurring low-level tokens of information,” with formal representation

$T_{\text{sum}}$8

where

$T_{\text{sum}}$9

$(i,j)$0

$(i,j)$1

and

$(i,j)$2

The token set for a sequence element is

$(i,j)$3

The paper extends this by constructing multiple TICs from the same source sequence using different extraction methods and then integrating them through three classes of 3-tensors: a tensor combining feature vector spaces, a tensor $(i,j)$4 combining adjacency matrices, and a set of tensors $(i,j)$5 derived from edge-weighted TIC variants. All share the time dimension $(i,j)$6. Coincidence is then defined as

“sets of co-occurring information tokens from any $(i,j)$7 that are situated in the tail of the respective recurrence and co-occurrence rank-size distributions ... and that are not part of a causal macroscopic pattern in $(i,j)$8 or $(i,j)$9.”

The macroscopic response objects are obtained by applying canonical polyadic decomposition and Tucker decomposition to the tensors. These decompositions yield matrices, and in the Tucker case also a core tensor, that are taken as low-dimensional representations of the macroscopic system state. The proposed invariant is “the statistical properties of the eigenvalues of the matrices of the tensor decompositions,” and the central hypothesis is that a rare event that does not change those invariant properties can be considered random, whereas a rare event that alters them is meaningful (Luczak-Roesch, 2019).

The paper therefore provides a coincidence-response model in which the “response matrix” is implicit in the decomposition outputs rather than specified as a single calibrated matrix. It is also explicit that the work is preliminary and conceptual: it “does not yet provide a completed formal theorem, a fully specified CBRM object, or empirical validation.” Meaningful coincidence is operationally defined as an “observable and non-random” impact on the macroscopic state, and universality remains a future claim rather than a demonstrated theorem (Luczak-Roesch, 2019).

A further operator-level analogue appears in micro-Doppler coincidence imaging. There, the key object is the first-order field correlation

$i$0

which acts as the forward operator from vibration-modulated target reflectivity to reconstructed image (Liu et al., 2022). After compensation, the static-target result is

$i$1

For discrete targets, the reconstruction can isolate the spatial distribution associated with a selected vibration mode:

$i$2

while for continuous targets the method yields either a full modal superposition or a selected principal mode under two interval-sampling strategies. The paper does not write an explicit matrix $i$3, but it states a functionally equivalent operator relation of the form $i$4 as an interpretive mapping. This suggests that a discretized version of the field-correlation operator is naturally read as a CBRM-like object (Liu et al., 2022).

6. Validation, performance, applications, and limitations

The explicit CBRM detector method was validated in both simulation and experiment. The simulation used a Geant4 model of a “1 mm thick CdTe detector,” with “9×9 pixels,” pixel sizes from “100 to 300 µm,” spectra generated using “Spektr,” $i$5 events per condition, and “10 independent noise realizations.” The experiment used a “300 µm silicon sensor,” “55 µm pitch,” bonded to a “Timepix4 ASIC,” with monochromatic beams from “8.5 to 40 keV” at the “SYRMEP beamline” and polychromatic measurements from a “50 kVp W-anode X-ray tube” in the “PEPI laboratory” (Monaco et al., 7 Aug 2025).

Performance was assessed with mean absolute percentage error,

$i$6

excluding bins with counts below $i$7 of the average bin count. For a simulated 120 kVp W-tube spectrum, the reported MAPE values were “ACS: 8.0%” and “CBRM: 12.0%”; for an attenuated 120 kVp spectrum they were “ACS: 18.8%” and “CBRM: 20.9%.” If the CBRM was calibrated using the same attenuated data rather than the unattenuated calibration, “MAPE worsened to 28.9%,” which the paper uses to support the claim that calibration should be performed with good low-energy coverage (Monaco et al., 7 Aug 2025). In Timepix4 experiments, CBRM was compared with “3×3 offline clustering.” For the 50 kVp spectrum, MAPE versus clustering was “3.6% without absorber” and “6.2% with absorber,” and the “Ag K-edge dip at 25.5 keV” remained visible in both reconstructed spectra (Monaco et al., 7 Aug 2025).

The same paper is careful about limitations. It notes sensitivity to low-energy noise thresholds, reduced accuracy for very small pixels where charge sharing extends beyond the assumed $i$8 neighborhood, dependence on the small-bin-width assumption, and the fact that validation was carried out under flat-field illumination with pileup not included. These caveats matter because the claimed spectrum-independence pertains to detector response parameters under the stated calibration conditions, not to arbitrary departures from the acquisition model (Monaco et al., 7 Aug 2025).

The other CBRM-like literatures carry their own limitations. The response-matrix-centred cross-section strategy depends on binning choice and on the fidelity with which a binned matrix captures detector effects; it is presented as an alternative or complement to unfolding, not as an elimination of detector simulation or measurement expertise (Koch, 2019). The TIC-based emergence program depends on tokenization and extraction methods and remains a proposed research program rather than an empirically completed theory (Luczak-Roesch, 2019). The micro-vibration coincidence-imaging method assumes ideal coherence, known reference fields, spatially incoherent uniform illumination, ideal-resolution approximations, phase-only micro-vibration encoding, and sufficient ensemble averaging; its response object is therefore an idealized analytical forward operator rather than a general calibrated measurement matrix (Liu et al., 2022).

Taken together, these works show that CBRM has both a narrow and a broad meaning in contemporary research. Narrowly, it is an invertible, calibration-derived response matrix built from pixel-neighbor coincidences for correcting charge-sharing distortions in photon-counting spectra (Monaco et al., 7 Aug 2025). More broadly, it denotes a methodological stance in which coincidence structure is encoded in a response object—matrix, tensor, or operator—and used for forward folding, correction, or reconstruction rather than discarded through aggregation or treated only as noise.