Comparison of Hybrid and Standard Reconstruction Methods on IXPE Astrophysical Data

Introduction

The Imaging X-ray Polarimetry Explorer (IXPE; [28, 24]), launched in December 2021, has opened a new observational window on the polarized X-ray sky, allowing spatially resolved spectro-polarimetric measurements in the 2-8 keV band for the first time. The instrument consists of three identical detector units, each comprising a mirror module and a Gas Pixel Detector (GPD; [10, 3, 1]), in which the direction of linear polarization of incoming X-rays is encoded in the azimuthal distribution of the photoelectron emission angle $\phi$. The standard event reconstruction method, known as moment analysis (Bellazzini et al. [4]), estimates both the photoelectron emission angle and the photon impact point (IP) from the morphological properties of the ionization track left by the photoelectron in the detector gas.

Cibrario et al. [8] proposed a hybrid analytic-machine learning (ML) approach in which a convolutional neural network (CNN) based on the DenseNet-121 architecture (Huang et al. [18]), incorporating hexagonal convolutional layers (Steppa & Holch [25]), is trained to predict the photon IP from the track image. By substituting this predicted IP from CNN into the standard moment analysis, the hybrid method improves the modulation factor by $\sim1\%$ at 3 keV and $\sim6\%$ at 6 keV on Monte Carlo simulations. A further improvement was achieved by introducing an artificial image-sharpening step prior to CNN inference. Cibrario et al. [7] subsequently validated the algorithm on laboratory calibration data from the IXPE pre-launch campaign ([21, 12]), confirming the expected improvement in modulation factor and demonstrating a significant mitigation of the polarization leakage effect ([6]) for extended sources such as the Pulsar Wind Nebula G21.5-0.9 ([11]).

A critical step in characterizing any new reconstruction method for IXPE is its application to astrophysical sources with well-known polarimetric properties. Laboratory validation of Cibrario et al. [7] used unpolarized and polarized monochromatic spatially uniform beams under controlled conditions.

However, astrophysical observations involve broad-band spectra, source-dependent spatial distributions, and variable background levels that may affect the relative performance of the two methods in ways not captured by laboratory tests. In particular, no systematic comparison with bright point sources (for which the polarimetric sensitivity improvement is most directly measurable) has been done to date.

In this work, we present the first comparison of hybrid and standard reconstruction methods on IXPE astrophysical data, using two point sources that probe complementary polarimetric regimes:

• Her X-1 is a bright and highly polarized accreting X-ray pulsar with a well-characterized polarization degree of $\sim8$-10% across the IXPE band ([15, 16, 17, 29]). Its high count rate, relatively hard spectrum, and large polarization signal make it an ideal quantitative benchmark for comparing the sensitivity of the two reconstruction methods as a function of energy.

• XTE J1701-461 is a transient neutron star low-mass X-ray binary. Its soft state is bright and its 2–8 keV polarimetric signal could not be constrained, with a 99% confidence level upper limit of 1.5% (OBSID 01250701; Cocchi et al. [9]), providing a measurement of residual systematic effects under conditions where the true polarimetric signal is (close to) zero. This is analogous to the analysis of unpolarized beam measurements performed in Cibrario et al. [7] and allows us to characterize the hexagonal-pixel systematic modes ($m = 6$, $m = 12$) that appear in the photoelectron angle distribution of the hybrid method at low energies. We stress that the classification of a source as "unpolarized" is necessarily relative to the statistical precision of the measurement, not an absolute property: as the statistical uncertainty on PD is reduced (e.g. with longer exposures, brighter sources, or more sensitive reconstruction methods), an observationally unpolarized source can in principle reveal either a residual instrumental systematic or a true, low-level polarization.

The paper is organized as follows. § 2 describes the observations, data reduction, and comparison methods. § 3 presents the polarimetric results and the Fourier analysis of the angle distributions. We discuss the implications in § 4.

Observations and methods

Observations and data reduction

Both sources were observed by IXPE with all three detector units (DUs). Table 1 summarizes the observations used in this work.

Source	OBSID	Exposure (ks)	Date
Her X-1	02004001	244.5	2023-02-03
XTE J1701-461	01250701	46.4	2022-10-08

The required software ecosystem as well as the processing pipeline that we used is implemented in a publicly available dedicated container (Silvestri et al. [23])² that can run the full analysis from level 1 to level 2 and also includes an ixpeobssim version with the hybrid response functions. To ensure a consistent comparison between the standard and hybrid reconstructions, we did not use the level 2 files distributed by HEASARC. Instead, we reprocessed the observations starting from the original level 1 event files, since the standard IXPE level 1 to level 2 processing pipeline has undergone updates throughout the mission lifetime. We obtained the standard files by processing the level 1 files distributed by HEASARC with the publicly available tool ixpepipe³. For hybrid reconstruction, we first generated modified level 1 files by replacing the IP determined by the standard moment analysis with the IP predicted by the convolutional neural network. The photoelectron emission angle was then recomputed by running the moment analysis employing the new CNN-predicted IP. The resulting hybrid level 1 files were subsequently processed with ixpepipe following the same procedure adopted for the standard reconstruction. The hybrid files use dedicated spurious modulation correction maps and modulation factor curves generated by applying the hybrid reconstruction pipeline to the IXPE pre-launch laboratory calibration data ([21, 12]), so that they account for the different systematic response of the CNN-predicted impact point relative to the standard moment-analysis estimate. The modulation factor curves are additionally corrected for the GPD gas pressure at the epoch of each observation, following the same procedure employed for the standard analysis. We perform polarimetric analysis of the level 2 event files with the ixpeobssim package (v32.0.1; Baldini et al. [2]) using the PCUBE algorithm. We used for both observations the instrument response functions ixpe:obssim20230101:v01.3, in the case of the hybrid files, we used the improved modulation factor from the hybrid method.

We extract source events from a circular region centered on each source with radius 2.5′. Due to the brightness of the sources considered, no background subtraction or rejection is needed, as prescribed by Di Marco et al. [13]. We perform the analysis in four logarithmically spaced energy bins covering the 2.00-8.00 keV range. We describe the analysis of Level 1 event files, used for the Fourier study of the angle distributions, in § 2.3.

Significance ratio and its uncertainty

The significance of measuring a certain polarization degree PD is $\sigma= \mathrm{PD}/\mathrm{PD}_{\mathrm{err}}$, where PD and its uncertainty $\mathrm{PD}_{\mathrm{err}}$ are derived from the normalized Stokes parameters following Kislat et al. [19]. To quantify the improvement due to the hybrid method, we define the significance ratio $R$:

where $\sigma_{\mathrm{hyb}}$ and $\sigma_{\mathrm{std}}$ are the significance of the hybrid and standard measurement, respectively. A value $R > 1$ indicates that the hybrid method achieves a higher significance for detection in that energy bin. The factor $R_{\mathrm{err}}$ quantifies the improvement in the statistical precision, whereas $R_{\mathrm{PD}}$ depends on the particular realization of the measured polarization degree. Therefore, $R$ can be affected by statistical fluctuations in the central PD values even when the two reconstruction methods have identical expectation values.

The uncertainty in $R$ requires careful treatment. Since $\sigma_{\mathrm{hyb}}$ and $\sigma_{\mathrm{std}}$ are derived from the same photons processed through two different pipelines, they are not independent. The propagation of standard error for a ratio of independent quantities would substantially overestimate $\sigma(R)$. The correct formula, accounting for the covariance between the two estimators, is then:

With $\operatorname{Cov}(\sigma_{\mathrm{hyb}},\sigma_{\mathrm{std}}) = \rho\sigma_{\sigma_{\mathrm{hyb}}}\sigma_{\sigma_{\mathrm{std}}}$ and using the result that in the high-statistic limit (number of effective photons $N_{\mathrm{eff}} \gg\sigma^{2}/2\mu^{2}$, verified for all our energy bins ⁴), the uncertainty on the significance

satisfies $\sigma_{\mathrm{std,err}} \sim\sigma_{\mathrm{hyb,err}} \approx1$ [26], and Eq. (2) simplifies to:

The correlation coefficient $\rho$ is estimated directly from the normalized Stokes parameters $q$ and $u$:

For both sources, we find $\rho\approx0.999$ in all energy bins, as expected since the two pipelines differ only in the IP prediction and share the same underlying photon events. Neglecting this correlation would overestimate $R_{\mathrm{err}}$ by a factor of $\sim1/\sqrt{1-\rho^{2}} \approx20$.

The significance ratio $R$ also allows us to express the improvement in terms of an equivalent increase in the mirror effective area or detector efficiency. Since $\sigma\propto\sqrt{A_{\mathrm{eff}}}$, under the hypothesis that the PD is the same under the two methods, the fractional increase in effective area that would be needed for the standard method to match the hybrid significance is:

with uncertainty:

This quantity provides us with a measure of the gain delivered by the hybrid method that is directly comparable across different energy bands and sources. Because $R$ contains both the change in statistical precision and the realization-dependent change in the measured PD, this quantity should be interpreted as

an observed effective-area-equivalent gain rather than as a direct measurement of an instrumental effective-area increase. A complementary figure of merit that depends only on the measurement precision is the minimum detectable polarizations (MDP, Weisskopf et al. 2010):

where $\mu$ is the modulation factor and $C_{\mathrm{s}}$ the source counts. This quantity, valid for the current case of bright sources, is independent of the particular PD realization and captures both the reduction in statistical uncertainty and the improvement in the modulation factor. We can therefore explicitly define an MDP ratio,

We report both quantities in Table 3 for Her X-1 and Table 5 for XTE J1701-461. We also note that $\sigma= \mathrm{PD}/\mathrm{PD}_{\mathrm{err}}$ has a simple Gaussian interpretation only when $\sigma> 1$. For low-significances it should be understood as a detection-test statistic and all PD values for that source are formally upper limits (see for example § 3.2).

$E$ (keV)	MDP$_{\mathrm{std}}$ (%)	MDP$_{\mathrm{hyb}}$ (%)	$R_{\mathrm{MDP}}$	$\sigma_{\mathrm{std}}$	$\sigma_{\mathrm{hyb}}$	$R$	$(R-1)/R_{\mathrm{err}}$	$\Delta A_{\mathrm{eff}}/A_{\mathrm{eff}}$ (%)
2.00–2.83	2.96	2.93	1.010	9.26	9.27	$1.002\pm0.002$	$1.1\sigma$	$0.4\pm0.4$
2.83–4.00	2.28	2.24	1.018	9.29	10.18	$1.096\pm0.010$	$9.3\sigma$	$20.0\pm2.3$
4.00–5.66	2.73	2.50	1.092	9.73	10.89	$1.119\pm0.015$	$7.7\sigma$	$25.2\pm3.4$
5.66–8.00	3.79	3.55	1.068	6.19	7.30	$1.180\pm0.029$	$6.2\sigma$	$39.3\pm6.9$

Table 5: Detection significance comparison for XTE J1701-461. Energy bins are logarithmically spaced in the 2-8 keV range. MDP$_{\mathrm{std}}$ and MDP$_{\mathrm{hyb}}$ are the minimum detectable polarizations and RMDP their ratio; $R = \sigma_{\mathrm{hyb}}/\sigma_{\mathrm{std}}$; $R_{\mathrm{err}}$ is computed from Eq. (4); $(R - 1)/R_{\mathrm{err}}$ is the deviation from parity; $\Delta A_{\mathrm{eff}}/A_{\mathrm{eff}}$ is the equivalent fractional increase in effective area (Eq. (6)). All significance values are below $3\sigma$; neither method achieves a significant detection.

$E$ (keV)	MDP$_{\mathrm{std}}$ (%)	MDP$_{\mathrm{hyb}}$ (%)	RMDP	$\sigma_{\mathrm{std}}$	$\sigma_{\mathrm{hyb}}$	$R$	$(R-1)/R_{\mathrm{err}}$	$\Delta A_{\mathrm{eff}}/A_{\mathrm{eff}}$ (%)
2.00–2.83	1.62	1.61	1.01	1.41	1.49	1.06 ± 0.14	0.4σ	12.1 ± 29.6
2.83–4.00	1.21	1.18	1.03	1.50	1.49	0.99 ± 0.07	−0.1σ	−1.7 ± 13.2
4.00–5.66	1.66	1.53	1.09	1.83	2.37	1.30 ± 0.17	1.7σ	68.4 ± 44.2
5.66–8.00	3.67	3.41	1.08	2.42	2.74	1.13 ± 0.17	0.8σ	27.8 ± 38.3

Fourier analysis of the photoelectron angle distribution

To characterize systematic effects in the photoelectron angle distribution at the track level, we directly analyze the IXPE level 1 event files. For the unpolarized source XTE J1701-461, in each logarithmically binned energy band, we extract the reconstructed emission angle $\phi$ from the DETPHI2 column, applying an energy selection via event time-matching with the corresponding level 2 files. We combine the three detector units after applying nominal rotations of $0^\circ$, $120^\circ$, and $240^\circ$ to account for the clocking of each DU in the focal plane. We compute the power spectrum of the angle distribution $N(\phi)$ via a discrete Fourier transform after subtracting the mean count rate (i.e. the $m = 0$ component), so that the power spectrum reflects only the periodic modulations of the distribution. The power at mode $m$ corresponds to a periodicity of $360^\circ/m$ in the angle distribution. Mode $m = 2$ carries the polarization signal, while modes $m = 6$ and $m = 12$ are associated with the hexagonal symmetry of the GPD pixel grid [7 Appendix B].

Results

Polarimetric results: Her X-1

We detected polarization from Her X-1 with high significance in all four energy bins with both reconstruction methods. Figure 1 (left column) shows the normalized Stokes parameters $q$ and $u$, PD, and PA as a function of energy for both methods. Both methods yield consistent polarimetric observables values within their respective uncertainties at all energies, with no statistically significant differential systematic offset between the two pipelines. The hybrid method produces noticeably smaller error bars above $\sim3$ keV, reflecting the improved modulation factor. We report the normalized Stokes parameters $q$ and $u$, PD, and PA in Table 2. We report the significance of detection and its comparison between the two methods in Table 3. Figure 2 shows the significance ratio $R = \sigma_{\mathrm{hyb}}/\sigma_{\mathrm{std}}$ as a function of energy.

$E$ (keV)	$q$ (%) Std	$q$ (%) Hyb	$u$ (%) Std	$u$ (%) Hyb	PD (%) Std	PD (%) Hyb	PA (°) Std	PA (°) Hyb
2.00–2.83	$-1.87\pm0.98$	$-1.99\pm0.97$	$8.86\pm0.98$	$8.77\pm0.97$	$9.05\pm0.98$	$8.99\pm0.97$	$51.0\pm3.1$	$51.4\pm3.1$
2.83–4.00	$0.09\pm0.77$	$0.07\pm0.75$	$7.13\pm0.77$	$7.66\pm0.75$	$7.13\pm0.77$	$7.66\pm0.75$	$44.6\pm3.1$	$44.7\pm2.8$
4.00–5.66	$0.24\pm0.93$	$-0.55\pm0.85$	$9.06\pm0.93$	$9.27\pm0.85$	$9.07\pm0.93$	$9.29\pm0.85$	$44.2\pm2.9$	$46.7\pm2.6$
5.66–8.00	$-1.12\pm1.53$	$-1.43\pm1.43$	$9.40\pm1.53$	$10.37\pm1.43$	$9.47\pm1.53$	$10.46\pm1.43$	$48.4\pm4.6$	$48.9\pm3.9$

The hybrid MDP is systematically lower than the standard one at all energies because of the higher modulation factor, yielding $R_{\mathrm{MDP}} > 1$ and $R > 1$ at all energies. At 2.00-2.83 keV the improvement is consistent with zero at $1.1\sigma$, in agreement with the marginal enhancement predicted at low energies Cibrario et al. [7]. Above 2.83 keV the improvement is highly significant: the significance ratio deviates from unity by

9.3, 7.7, and 6.2$\sigma$ in the three higher-energy bins, with $R$ reaching 1.18 ± 0.03 at 5.66–8.00 keV. In terms of equivalent effective area (Eq. (6)), the hybrid method provides gains of 20%±2%, 25%±3%, and 39%±7% in the three bins above 2.83 keV, with a negligible improvement of 0.4%±0.4% at 2.00–2.83 keV. The average improvement in the full 2–8 keV band is of 9.9%±1.4%.

Polarimetric results: XTE J1701-461

XTE J1701-461 in its soft state is consistent with zero polarization [9]. Figure 1 (right column) shows for this source the normalized Stokes parameters, PD, and PA as a function of energy for both methods. All values of $q$ and $u$ are consistent with zero within 1–2$\sigma$ in all energy bins. The formally computed PD values are all sub-percent and statistically consistent with zero: they are not significant detections and should formally be reported as upper limits, and PA is unconstrained. We nonetheless display the computed PD and PA values for both methods to demonstrate their mutual consistency: the two pipelines yield results that agree within uncertainties at all energies. In particular, we find no statistically significant differential systematic bias in the polarimetric observables.

We report the polarimetric parameters in Table 4 and the significance comparison in Table 5. Figure 3 shows the significance ratio $R=\sigma_{\mathrm{hyb}}/\sigma_{\mathrm{std}}$ as a function of energy. The hybrid method achieves a smaller MDP in all the four energy bins for this source, but provides no significant improvement over the standard one in terms of significance or relative effective area.

Table 4: Polarimetric parameters ($q$, $u$, PD, and PA) for XTE J1701-461 as a function of energy, derived with the standard moment analysis (Std) and the hybrid method (Hyb). All quantities are derived with PCUBE. None of the PD measurements are statistically significant: PD values are effectively upper limits and PA values are unconstrained (see text).

$E$ (keV)	$q$ (%) Std	$q$ (%) Hyb	$u$ (%) Std	$u$ (%) Hyb	PD (%) Std	PD (%) Hyb	PA (°) Std	PA (°) Hyb
2.00–2.83	−0.48±0.54	−0.60±0.53	−0.58±0.54	−0.51±0.53	0.75±0.54	0.79±0.53	−64.6±20.4	−69.8±19.2
2.83–4.00	0.55±0.40	0.50±0.39	−0.25±0.40	−0.29±0.39	0.60±0.40	0.58±0.39	−12.2±19.1	−15.1±19.2
4.00–5.66	−0.38±0.55	−0.36±0.50	−0.93±0.55	−1.14±0.50	1.00±0.55	1.19±0.50	−56.0±15.7	−53.7±12.1
5.66–8.00	−2.43±1.21	−1.77±1.12	−1.64±1.21	−2.52±1.12	2.93±1.21	3.08±1.12	−73.0±11.8	−62.5±10.5

None of the individual energy bins resolves $R \ne1$ at more than $1.7\sigma$. The hybrid method cannot be claimed to provide a significant improvement per-bin for this fainter, lower-significance source ($\sigma_{\mathrm{std}} \sim1$-2, compared to $\sigma_{\mathrm{std}} \gtrsim9$ for Her X-1). This is however to be expected for an unpolarized source, so that this result is the astrophysical counterpart of the unpolarized beam analysis of Cibrario et al. [7] and demonstrates that the hybrid pipeline does not introduce spurious polarization signals for unpolarized sources. For a source with no significant detection, the most informative figure of merit is the MDP ratio $R_{\mathrm{MDP}} = \mathrm{MDP}_{\mathrm{std}}/\mathrm{MDP}_{\mathrm{hyb}}$ (reported in Table 5), which quantifies the sensitivity improvement independently of the realized PD. We report $R$ for completeness and to verify that the hybrid pipeline does not artificially inflate the significance for unpolarized sources. Nonetheless, the central values follow the same trend seen for Her X-1: $R > 1$ in three of the four bins, and the 2-8 keV significance increase ($12.0\% \pm13.7\%$) is statistically consistent with the value measured for Her X-1 ($9.9\% \pm1.4\%$). This indicates that the larger relative uncertainty for XTE J1701-461 reflects the much lower per-bin significance available in this source ($N_{\mathrm{eff}}$ and PD both far smaller than for Her X-1), rather than evidence that the hybrid method provides no benefit for fainter or weakly polarized sources; resolving this trend at the bin level for sources in this flux regime would require either longer exposures or a sample of multiple comparably faint sources.

Modulation curves and systematics

In Figure 4 we show the photoelectron angle distributions and in Figure 5 their Fourier power spectra in the 2-5 keV and 5-8 keV bands for XTE J1701-461.

In the low energy 2-5 keV band we observe excess power at modes at $m = 3$, $m = 6$, $m = 12$, and $m = 15$ in the hybrid angle distributions with respect to the expected polarization mode at $m = 2$. To quantify the significance of these excesses, we devise a simple statistical test: for a set of $N$ angles drawn from a flat distribution, the Fourier coefficient at each mode $m > 0$ is the sum of $N$ unit vectors with random phase, so that its squared modulus (the power) follows an exponential distribution, with mean equal to the underlying noise level of the power spectrum. We estimate this noise level, separately for each method and energy band, from the sample mean of the power in modes with no expected signal ($m \in\{4, 5, 7, 8, 9, 10, 11, 13, 14, 16\}$), and use it to assign a $p$-value to the power observed at $m = 2, 3, 6, 12,$ and $15$, applying a conservative Bonferroni correction for the 15 modes tested (i.e. multiplying the raw $p$-value by 15, Bonferroni 1936). Table 6 reports the resulting significance for both methods and both energy bands. As expected, the significance of the mode $m = 2$, which carries the polarization/spurious-modulation signal, is higher for the hybrid method than for the standard one in both energy bands: this is the same effect, measured here at the level of the individual Fourier mode, that produces the significance ratio $R > 1$ for the full PD measurement discussed in Sect. 2.2 and Sect. 3.1. In the 2-5 keV band, the power at $m = 3$, $6$, $12$, and $15$ in the hybrid method exceeds the noise level at significance $> 7\sigma$ (Bonferroni-corrected) for all four modes, while the same modes in the standard method are either fully consistent with noise (all $< 1\sigma$) or only marginally significant. The ratio of hybrid to standard power in these modes ranges between $\sim40$ and $\sim1300$, confirming that these excesses are a genuine feature of hybrid reconstruction. However, despite their high statistical significance, the normalized amplitudes $|A_m|/N = \sqrt{P_m/N}$ of all four modes in the hybrid method remain well below 1% (Table 6), indicating that the modulation of the angle distribution at these periodicities is quantitatively small. These excesses lead to a markedly poorer fit with the $\cos^2$ modulation model in the hybrid method, as seen from Figure 4.

$m$	\multicolumn{4}{c}{2–5 keV}	\multicolumn{3}{c}{5–8 keV}
	$\sigma_{\mathrm{hyb}}$	$\sigma_{\mathrm{std}}$	$P_{\mathrm{hyb}}/P_{\mathrm{std}}$	$\lvert A_m\rvert/N$ (hyb, %)	$\sigma_{\mathrm{hyb}}$	$\sigma_{\mathrm{std}}$	$P_{\mathrm{hyb}}/P_{\mathrm{std}}$
2	5.5	4.3	1.0	0.18	3.4	2.7	1.7
3	7.9	1.7	43.2	0.74	1.7	−	3.7
6	7.9	−	162.2	0.54	−	1.2	0.3
12	7.9	−	1339.8	0.65	−	−	0.8
15	7.9	−	222.0	0.48	−	−	0.1

On the other hand, in the 5-8 keV band none of $m = 3$, $6$, $12$, or $15$ reaches Bonferroni-corrected significance in either method (Table 6), and the ratio of hybrid to standard power in these modes is of order unity or below. This indicates that the systematic excess observed is confined to the energy range where photoelectron tracks are short and only weakly elongated.

As discussed in Cibrario et al. [7], the modes $m = 6$ and $m = 12$ are naturally associated with the hexagonal symmetry of the GPD pixel grid, via the alignment between the barycenter of the horseshoe region selected by the hybrid method and the track barycenter: they are expected to be amplified by the CNN at low energies, where the network tends to predict the IP closer to the track barycenter. This low-energy regime is the same in which we find the modulation factor improvement of the hybrid method to be negligible (§ 3.1, $R = 1.002 \pm0.002$ at 2.00-2.83 keV): since the second moment of the charge distribution about the track barycenter is intrinsically less sensitive to the true emission direction, an IP estimate that remains close to the barycenter carries comparatively little additional polarimetric information relative to the standard method, consistent with both the modulation factor improvement and the $m = 6$, $12$ excess being most prominent in this same regime. The excess is reduced in the 5-8 keV energy band, where tracks are longer and the CNN-predicted IP is generally far from the barycenter of the track.

The $m = 3$ and $m = 15$ modes (corresponding respectively to periodicity of $120^\circ$ and $24^\circ$, the latter being a harmonic of the former) are not discussed in Cibrario et al. [7]. Both modes are absent in the standard analysis and disappear at 5-8 keV in the hybrid one. To assess whether these modes could arise from the same horseshoe-barycenter alignment that produces $m = 6$ and $m = 12$, we repeated the moment analysis substituting the CNN-predicted IP with the track barycenter. This test reproduces significant $m = 6$ and $m = 12$ excesses but yields no excess at $m = 3$ or $m = 15$, ruling out the horseshoe-barycenter alignment as their origin and pointing to a mechanism associated with the CNN inference itself. A plausible explanation is that they reflect the triple-rotation inference procedure used internally by the hybrid CNN: as described in Cibrario et al. [8], for each track the network predicts the IP for three copies of the image rotated by $0^\circ$, $120^\circ$, and $240^\circ$, and averages the three back-rotated predictions. Small residual asymmetries between these three predictions, more likely for the short and nearly circular tracks found at low energy, could imprint a $120^\circ$ periodicity on the angle distribution.

This procedure has $C_3$ symmetry by construction, and could imprint a 120° periodicity on the reconstructed angle distribution when the three predictions differ non-negligibly (as expected for the short, nearly circular tracks found at low energies). Consistently, we verified that $m = 3$ and $m = 15$ excesses are also present in re-analysis of IXPE ground calibration data with the hybrid method, where they had previously gone undetected likely due to the lower signal-to-noise ratio of those measurements relative to the residual spurious modulation and intrinsic beam polarization. This removes the apparent discrepancy between calibration and astrophysical data, and suggests that the $m = 3$ and $m = 15$ modes are a general feature of the hybrid pipeline rather than a property specific to astrophysical observations.

Crucially, none of these modes have any effect on the polarimetric results. The normalized Stokes parameters are defined as $q = (2/N) \sum_i \cos(2\phi_i)$ and $u = (2/N) \sum_i \sin(2\phi_i)$ ([19]): these sums are sensitive only to the $m = 2$ Fourier component of the angle distribution, and all higher-order modes cancel exactly. This is demonstrated by the consistency between the two methods in the polarimetric parameters of § 3.1 and § 3.2, which show no evidence of any bias attributable to the modes discussed above.

Discussion and conclusions

We presented the first systematic comparison of the hybrid ML-analytic reconstruction method of Cibrario et al. [8, 7] and the standard moment analysis on IXPE data of astrophysical point sources.

For the highly polarized source Her X-1, the hybrid method yields a consistently higher detection significance at all energies above $\sim2.83$ keV, with improvements that are statistically significant at the level of $5.8$–$9.3\sigma$ in three of four energy bins. The improvement is negligible at $2.00$-$2.83$ keV ($1.1\sigma$), consistent with the predictions of Cibrario et al. [7]: at low energies the photoelectron tracks are nearly circular, providing little morphological information to improve the IP reconstruction. Above $\sim3$ keV, where the tracks become elongated and asymmetric, the IP predicted by the CNN is significantly more accurate, leading to improved polarimetric sensitivity.

The magnitude of the improvement is up to $\sim18\%$ in $\sigma_{\mathrm{hyb}}/\sigma_{\mathrm{std}}$ at $5.66$-$8.00$ keV. For bright sources near the minimum detectable polarization limit, an improvement of this magnitude can make the difference between a detection and an upper limit. In terms of the equivalent fractional increase in the effective area $\Delta A_{\mathrm{eff}}/A_{\mathrm{eff}} = R^2 - 1$ (Eq. (6)), it ranges from $20\%$ at $2.83$-$4.00$ keV to $39\%$ at $5.66$-$8.00$ keV at no cost of additional observation time.

For the observationally unpolarized source XTE J1701-461, the per-bin significance ratios are all consistent with unity within their uncertainties, as expected given the much lower per-bin significance available for this source ($\sigma_{\mathrm{std}} \sim1$–$2$) compared to Her X-1 ($\sigma_{\mathrm{std}} \gtrsim9$). The source-averaged central value ($12.0\% \pm13.7\%$) is nonetheless consistent with the trend established on Her X-1 ($9.9\% \pm1.4\%$), and the hybrid method achieves a smaller PD uncertainty than the standard one in three of the four energy bins (Table 4). This indicates that the present non-detection of a per-bin improvement for XTE J1701-461 reflects the limited statistics of this fainter source rather than an intrinsic loss of benefit from the hybrid method at low source flux or polarization. We stress that this conclusion is established here for bright point sources only, and tests similar to the ones performed in this work should be performed for any applications of this NN to other data sets.

Both sources analyzed in this work have per-bin significance $\sigma_{\mathrm{std}} \gtrsim1$, and the benefit of the hybrid method is robustly detected only for the brighter of the two; for the unpolarized XTE J1701-461, the same central improvement is present, but not individually resolved in the available statistics (Sect. 3.2).

The central PD values show a mild tendency to be larger for the hybrid reconstruction: across the eight energy bins of the two sources, $\mathrm{PD}_{\mathrm{hyb}} > \mathrm{PD}_{\mathrm{std}}$ in six cases. However, this does not constitute statistically significant evidence for a systematic positive offset. A binomial sign test gives a one-sided probability of $14.5\%$ for obtaining six or more positive differences under the null hypothesis of no preferred direction. Excluding the lowest-energy bin of each source leaves five positive differences out of six, with a corresponding one-sided probability of $10.9\%$. We therefore interpret this pattern as a possible realization-dependent fluctuation rather than evidence for a differential systematic bias. The observed significance ratio contains this effect explicitly. For example, using the values in the $2.83$-$4.00$ keV bin of Her X-1 from Table 2,

so that the observed value $R = 1.096$ contains contributions from both the reduced PD uncertainty and the higher central PD measured with the hybrid reconstruction. If the two methods had returned the same PD, the significance ratio would instead have been $R = 1.027$. The MDP ratio $R_{\mathrm{MDP}}$ provides a complementary measure that is independent of this particular PD realization. For Her X-1, $R_{\mathrm{MDP}}$ is $1.010$, $1.018$, $1.092$, and $1.068$ in the four energy bins, confirming a sensitivity improvement consistent with that expected from the modulation-factor enhancement of the hybrid reconstruction.

Preliminary comparisons on faint extended sources, such as supernova remnants, show no measurable improvement with the hybrid method, with results from the two pipelines being statistically consistent but offering no significant gain in sensitivity. This suggests the existence of a flux (or, more precisely, a per-bin significance) threshold below which the benefit of the improved IP reconstruction is not recoverable, plausibly because the gain in $N_{\mathrm{eff}}$ and $\mu$ documented here for Her X-1 becomes comparable to or smaller than the photon-counting noise once the number of source counts is sufficiently low. A quantitative determination of this threshold, and an explanation of why the method does not yet show a benefit for fainter sources, requires a dedicated analysis of a larger source sample spanning a range of fluxes and is left to future work. All in all, the comparison of the analysis of both sources shows no statistically significant evidence for a differential systematic bias between the two reconstruction methods at the present statistical precision. We nevertheless note a mild tendency for the hybrid reconstruction to yield larger central PD values in the present sample. Given the limited number of independent measurements, this tendency cannot be distinguished from a statistical fluctuation, although a small residual method-dependent contribution cannot be excluded. The Fourier analysis of the level 1 angle distributions reveals the power excess $m = 6$, $m = 12$, $m = 3$ and $m = 15$ in the hybrid method at $2$–$5$ keV, significant at $7.9\sigma$ for all four modes, compared to a standard method that is either consistent with noise or only marginally significant for the same modes (Table 6); at $5$–$8$ keV none of these modes reach significance in either method. This confirms that the excess is a genuine, energy-dependent systematic of the hybrid reconstruction, and one that is fully characterized by the comparative test presented here, regardless of its ultimate physical origin. A barycenter-as-IP test confirms that $m = 6$ and $m = 12$ originate from the horseshoe-barycenter alignment described in Cibrario et al. [7], while $m = 3$ and $m = 15$ require a distinct mechanism, plausibly the $C_3$-symmetric triple-rotation inference procedure of the CNN. We further verified that $m = 3$ and $m = 15$ are present in re-analysis of IXPE calibration data with the hybrid method, confirming that they are a general feature of the hybrid pipeline. We confirm that none of these modes propagates to the Stokes parameters, which is consistent with the analysis of Cibrario et al. [7]. We note that a similar $120^\circ$ periodic artifact has been independently reported for a different CNN-based reconstruction method applied to IXPE ground calibration data: Di Marco et al. [14] find that the deep-ensemble approach of Peirson & Romani [22] introduces $m = 6$ and $m = 12$ harmonics in the modulation curves of polarized calibration data, which they attribute to the network’s processing of the hexagonal track image, itself based on three $120^\circ$ separated rotations to map the hexagonal grid onto a square one (Peirson & Romani 2021, Sect. 2.2). This suggests that $120^\circ$ periodicity artifacts may be a generic consequence of CNN-based approaches that explicitly exploit the threefold rotational structure of the GPD hexagonal grid, rather than a peculiarity of the specific architecture used by Cibrario et al. [8]. Recently, a method using a Graph Neural Network (GNN) trained on null polarization signals successfully resolved the harmonics implied by the hexagonal pattern of the IXPE detector, but without a significant improvement over the standard method in terms of polarimetric performance ([20]).

In summary, the hybrid reconstruction method is ready for a broader application to IXPE astrophysical data of bright point sources. Its adoption as the standard analysis pipeline in the $\gtrsim3$ keV band would provide a sensitivity improvement equivalent to an average increase in the effective area of $\sim20\%$ in the entire $2-8$ keV IXPE energy band, without additional observational cost.

Acknowledgements

Acknowledgements. The Imaging X-ray Polarimetry Explorer (IXPE) is a joint US and Italian mission. The US contribution is supported by the National Aeronautics and Space Administration (NASA) and led and managed by its Marshall Space Flight Center (MSFC), with industry partner Ball Aerospace (contract NNM15AA18C). The Italian contribution is supported by the Italian Space Agency (Agenzia Spaziale Italiana, ASI) through contract ASI-OHBI-2022-13-I.0, agreements ASI-INAF-2022-19-HH.0 and ASI-INFN-2017.13-H0, and its Space Science Data Center (SSDC) with agreements ASI-INAF-2022-14-HH.0 and ASI-INFN 2021-43-HH.0, and by the Istituto Nazionale di Astrofisica (INAF) and the Istituto Nazionale di Fisica Nucleare (INFN) in Italy. This research used data products provided by the IXPE Team (MSFC, SSDC, INAF, and INFN) and distributed with additional software tools by the High-Energy Astrophysics Science Archive Research Center (HEASARC), at NASA Goddard Space Flight Center (GSFC). NC and MN wish to thank Nikita Moriakov for his contribution to the original implementation of hybrid method, first presented in Cibrario et al. [8].