STEPC: Uniformity Error Calibration in PCCT

• STEPC is a calibration framework that corrects detector nonuniformity in photon-counting CT using a nonlinear multi-energy polynomial model.
• It estimates ideal projections via per-bin 2D polynomial fitting and predicts pixel-wise errors from the full multi-energy signal to suppress ring artifacts.
• Evaluations show that STEPC outperforms traditional single-energy and linear multi-energy methods, especially in challenging multi-material and iodine-enhanced scenarios.

Searching arXiv for the provided paper to ground the article in the cited source. Signal-to-Uniformity Error Polynomial Calibration (STEPC) is a projection-domain, pixel-wise nonuniformity correction framework for photon-counting CT (PCCT) designed for multi-material and contrast-enhanced imaging scenarios, including configurations with only two energy thresholds. It was introduced to address detector-response nonuniformity that appears as view- and pixel-dependent errors in projections and as ring artifacts in reconstructed slices. The framework combines per-bin 2D polynomial estimation of ideal projections with a nonlinear multi-energy polynomial model that predicts pixel-wise nonuniformity error from the full multi-energy signal vector, using calibration data acquired from homogeneous slab phantoms of PMMA, aluminum, and iodinated contrast material (Zhou et al., 20 Jul 2025).

1. Problem setting and theoretical basis

In photon-counting detectors, each pixel has its own energy response function because of variations in semiconductor crystal properties and electronic processing, including charge transport, trapping, threshold dispersion, and pulse pile-up. For the $j$-th detector pixel and energy bin $E_k$, the measured counts are modeled by an integral over the X-ray spectrum, object attenuation, and the pixel-specific detector response function $R_j(E',E)$. Under an ideal pixel-independent response, one obtains corresponding ideal counts $N_{k,j}^{\text{ideal}}$. The pixel-wise nonuniformity error is then defined as

$$N_{k,j}^{\text{error}} = N_{k,j} - N_{k,j}^{\text{ideal}},$$

with the error expressed through the response deviation $\Delta R_j = R_j - R_j^{\text{ideal}}$ (Zhou et al., 20 Jul 2025).

A central point of the formulation is that the nonuniformity error is a nonlinear functional of both the spectral distribution $S(E)e^{-\int_l \mu(E,\vec{x})\,dl}$ and the pixel-wise response deviation $\Delta R_j(E',E)$. This means the error depends on the incident spectrum, and the incident spectrum changes with material composition and thickness. PMMA, aluminum, iodine, and bone therefore generate different error behaviors. An error model calibrated from a single material is, accordingly, generally not valid for other materials or multi-material mixtures (Zhou et al., 20 Jul 2025).

This spectral dependence explains why detector nonuniformity manifests not merely as a fixed gain defect but as object-dependent structure in projection space and as concentric ring artifacts centered on the rotation axis in reconstructed slices. This also explains why the difficulty becomes more severe in low-energy bins and in contrast-enhanced imaging, where beam hardening and K-edge effects are stronger. A plausible implication is that correction methods that neglect spectral state are structurally limited in heterogeneous PCCT settings.

2. Relation to prior correction strategies

STEPC was proposed against a background of correction methods that the paper groups into three broad classes: single-energy per-bin correction, multi-energy material-decomposition methods, and linear multi-energy models. The reported limitation across these classes is insufficient generalizability in complex multi-material scenarios, particularly when iodine is present and only two thresholds are available (Zhou et al., 20 Jul 2025).

Single-energy methods such as flat-field correction (FF) and Signal-to-Equivalent Thickness Calibration (STC) treat each energy bin independently and typically assume a single calibration material, for example PMMA. They therefore cannot correctly model changes in nonuniformity that arise when spectral shape changes because additional materials such as aluminum or iodine are present. Multi-energy methods based on material decomposition, represented here by Polynomial Material Decomposition Calibration (PMDC), perform dual-material decomposition from two energy bins. They work reasonably for two materials, such as PMMA plus aluminum, but cannot correctly handle a third component such as iodine with only two thresholds. They also require accurate thickness measurements of phantoms and introduce additional noise through signal-to-thickness conversion. Linear multi-energy models such as Affine Transformation Calibration (ATC) use all energy bins jointly but assume a linear mapping in count space and therefore cannot capture the nonlinear spectral dependence of the error, especially at low energies (Zhou et al., 20 Jul 2025).

Method	Core model	Limitation reported
FF	Air-based gain correction	Ignores object-dependent spectral hardening
STC	Per-bin exponential thickness model	Single-material calibration
ATC	Linear multi-energy mapping in counts	Cannot capture nonlinear spectral dependence
PMDC	Two-material decomposition and forward synthesis	Limited to two materials with two thresholds
STEPC	Nonlinear multi-energy polynomial error model	Designed for multi-material and iodine cases

The comparison clarifies a common misconception: the practical restriction in dual-threshold PCCT is not simply the number of image bins, but the adequacy of the calibration model relative to the spectral variability encountered during imaging. STEPC does not claim to recover explicit multi-material thicknesses beyond the threshold count; rather, it models the signal-to-error relationship directly.

3. Core framework and mathematical construction

The STEPC pipeline consists of four stages: ideal projection estimation, nonuniformity error computation, empirical multi-energy polynomial calibration, and application to new data (Zhou et al., 20 Jul 2025).

In the first stage, flat-field-corrected multi-energy projections of slab phantoms are converted to log projections,

$P(x,y,E_k) = -\log I_E^{\text{FF}}.$

For each energy bin $E_k$, an ideal reference projection is then estimated by fitting a second-order 2D polynomial surface over detector coordinates $E_k$0,

$E_k$1

Expanded explicitly, the model contains the terms $E_k$2, $E_k$3, $E_k$4, $E_k$5, $E_k$6, and $E_k$7. The coefficients $E_k$8 are estimated by least squares using MATLAB’s regress function. The role of this surface is to produce a smooth ideal projection that captures non-uniform incident flux and geometric effects, including varying path length through the slab due to beam divergence and small misalignments, while excluding pixel-specific detector response deviations (Zhou et al., 20 Jul 2025).

In the second stage, the pixel-wise nonuniformity error in projection space is defined as the residual

$E_k$9

These residuals are the supervision targets for calibration. They encode pixel-specific deviations but remain spectrum dependent, which is why STEPC uses calibration data spanning many material and thickness conditions (Zhou et al., 20 Jul 2025).

In the third stage, STEPC fits, for each energy bin $R_j(E',E)$0, a nonlinear multi-energy polynomial that predicts the residual error from the full multi-energy projection vector

$R_j(E',E)$1

The predicted error is

$R_j(E',E)$2

where $R_j(E',E)$3 is a multi-index of non-negative integers and $R_j(E',E)$4 is the polynomial order. In the main experiments, $R_j(E',E)$5, though the paper reports that second order often suffices. Calibration minimizes the mean squared error between $R_j(E',E)$6 and $R_j(E',E)$7 over all pixels and slab conditions (Zhou et al., 20 Jul 2025).

For the dual-threshold system used in the study, three bins are available—Low, High, and Total—so $R_j(E',E)$8. With polynomial order $R_j(E',E)$9, the number of monomial terms is

$N_{k,j}^{\text{ideal}}$0

Each energy bin therefore has 20 coefficients, for a total of 60 across three bins. The coefficients are global rather than pixel-specific; the pixel-wise behavior emerges from applying the same calibrated polynomial to each pixel’s measured multi-energy signal vector (Zhou et al., 20 Jul 2025).

In the fourth stage, the model is applied to new phantom or mouse data. After flat-field correction and log transformation, the polynomial predicts $N_{k,j}^{\text{ideal}}$1, and corrected projections are obtained as

$N_{k,j}^{\text{ideal}}$2

These corrected projections are then reconstructed.

4. Calibration design, imaging system, and implementation context

The experiments were performed on a custom Micro Photon Counting CT system with translate-rotate architecture, a stationary object, and a rotating gantry. The X-ray source operated at 80 kV and 200 $N_{k,j}^{\text{ideal}}$3A with a 0.5 mm Al filter. The detector was a photon-counting flat panel with effective matrix $N_{k,j}^{\text{ideal}}$4 pixels after cropping invalid pixels and pixel size $N_{k,j}^{\text{ideal}}$5. Two independently adjustable energy thresholds were set to 15 keV and 30 keV. The Total bin was 15–80 keV, the High bin 30–80 keV, and the Low bin 15–30 keV, computed as Total minus High (Zhou et al., 20 Jul 2025).

Calibration used homogeneous slab phantoms of PMMA, aluminum, and iodixanol solution. PMMA slabs had thicknesses 0, 5, 10, 15, 20, 30, and 40 mm. Aluminum slabs had thicknesses 0, 0.5, 1, 1.5, 2, 3, 4, 5, 6, and 8 mm. Iodixanol solution slabs, placed in containers with 2 mm PMMA wall and 6 mm solution core, had concentrations 0, 5, 10, 15, 20, 30, 50, 70, 100, 150, and 250 mg/cm$N_{k,j}^{\text{ideal}}$6. Non-contrast calibration used combinations of PMMA and aluminum. Iodine-enhanced calibration used the same PMMA-plus-aluminum combinations plus iodixanol slabs combined with PMMA/aluminum pairs such as $N_{k,j}^{\text{ideal}}$7, $N_{k,j}^{\text{ideal}}$8, $N_{k,j}^{\text{ideal}}$9, $$N_{k,j}^{\text{error}} = N_{k,j} - N_{k,j}^{\text{ideal}},$$0, $$N_{k,j}^{\text{error}} = N_{k,j} - N_{k,j}^{\text{ideal}},$$1, and $$N_{k,j}^{\text{error}} = N_{k,j} - N_{k,j}^{\text{ideal}},$$2 mm. For each slab combination, 600 projection frames were acquired and averaged (Zhou et al., 20 Jul 2025).

The calibration philosophy is explicit: PMMA approximates soft tissue, aluminum approximates bone, and iodixanol introduces iodine-specific spectral conditions. PMMA-plus-aluminum combinations represent non-contrast biological spectra; addition of iodinated slabs broadens the spectral training set to iodine-rich cases. This multi-material calibration is the basis for the claimed generalization to both non-contrast and contrast-enhanced imaging (Zhou et al., 20 Jul 2025).

Test objects included cylindrical phantoms of outer diameter 30 mm with 8 mm inserts and euthanized mice scanned to remove motion. Phantom compositions included PMMA only, 200 mg/mL CaCl$$N_{k,j}^{\text{error}} = N_{k,j} - N_{k,j}^{\text{ideal}},$$3 in a PMMA cylinder with 2 mm wall, 50 mg/mL iodixanol in a PMMA cylinder with 2 mm wall, CaCl$$N_{k,j}^{\text{error}} = N_{k,j} - N_{k,j}^{\text{ideal}},$$4 inserts at 100, 200, 400, and 600 mg/mL in PMMA, and mixed inserts containing iodixanol at 20, 50, and 100 mg/mL plus CaCl$$N_{k,j}^{\text{error}} = N_{k,j} - N_{k,j}^{\text{ideal}},$$5 at 200 and 400 mg/mL. Mouse experiments included a non-contrast head scan and an iodine-enhanced kidney scan after intravenous injection of 0.3 mL iodixanol at 300 mg/mL (Zhou et al., 20 Jul 2025).

Acquisition for phantoms used continuous mode with 1440 views per rotation, $$N_{k,j}^{\text{error}} = N_{k,j} - N_{k,j}^{\text{ideal}},$$6 per view, field of view 50 mm, source-to-isocenter distance 90 mm, and source-to-detector distance 325 mm. Mouse scans used the same settings except field of view 35 mm and source-to-isocenter distance 74 mm. Reconstruction used the FDK algorithm through the TIGRE toolbox to volumes $$N_{k,j}^{\text{error}} = N_{k,j} - N_{k,j}^{\text{ideal}},$$7, followed by HU calibration with a water phantom (Zhou et al., 20 Jul 2025).

At deployment time, the computation is light: per-pixel evaluation of the polynomial requires approximately $$N_{k,j}^{\text{error}} = N_{k,j} - N_{k,j}^{\text{ideal}},$$8 operations per energy bin, which the paper characterizes as negligible relative to 3D reconstruction. No iterative optimization is required at runtime.

5. Quantitative evaluation and comparative performance

STEPC was evaluated in both projection space and image space. Projection uniformity was measured by mean local standard deviation (MLSD). A sliding window of size $$N_{k,j}^{\text{error}} = N_{k,j} - N_{k,j}^{\text{ideal}},$$9 pixels moves across the projection; the standard deviation is computed in each window and then averaged over all windows. Lower MLSD indicates higher projection uniformity (Zhou et al., 20 Jul 2025).

For PMMA plus aluminum calibration, the reported MLSD values $\Delta R_j = R_j - R_j^{\text{ideal}}$0 were as follows. In the Low bin: FF 5.09, STC 1.58, ATC 1.31, PMDC 0.50, STEPC 0.38. In the High bin: FF 2.28, STC 1.10, ATC 0.31, PMDC 0.19, STEPC 0.18. In the Total bin: FF 1.40, STC 0.41, ATC 0.26, PMDC 0.24, STEPC 0.19. For PMMA plus aluminum plus iodixanol calibration, the Low-bin values were FF 5.31, STC 1.09, ATC 0.90, PMDC 1.81, STEPC 0.56; the High-bin values were FF 1.13, STC 0.76, ATC 0.35, PMDC 0.75, STEPC 0.25; and the Total-bin values were FF 1.34, STC 0.29, ATC 0.24, PMDC 0.82, STEPC 0.21. The reported pattern is that STEPC achieved the lowest or near-lowest MLSD, with a particular advantage in the PMMA-plus-aluminum-plus-iodine case where PMDC degraded strongly (Zhou et al., 20 Jul 2025).

Ring artifact severity in reconstructed images was quantified by ring artifact deviation (RAD), following Rodesch et al. The method converts a reconstructed slice $\Delta R_j = R_j - R_j^{\text{ideal}}$1 into polar coordinates $\Delta R_j = R_j - R_j^{\text{ideal}}$2, computes the angular average $\Delta R_j = R_j - R_j^{\text{ideal}}$3, fits a second-order polynomial in radius to remove cupping artifacts and global trends, and then computes the standard deviation of the residuals over a specified radial region and over slices. Smaller RAD corresponds to less ring artifact (Zhou et al., 20 Jul 2025).

The paper reports Total-bin RAD values in HU for several representative objects. For a PMMA phantom, the values were FF 122.26, STC 17.12, ATC 29.25, PMDC 39.17, and STEPC 16.93. For a 200 mg/mL CaCl$\Delta R_j = R_j - R_j^{\text{ideal}}$4 phantom, they were FF 170.27, STC 46.70, ATC 30.50, PMDC 37.16, and STEPC 21.75. For CaCl$\Delta R_j = R_j - R_j^{\text{ideal}}$5 inserts, they were FF 196.21, STC 36.02, ATC 34.74, PMDC 33.08, and STEPC 31.12. For a 50 mg/mL iodixanol phantom, they were FF 167.53, STC 56.71, ATC 23.78, PMDC 218.23, and STEPC 21.99. For CaCl$\Delta R_j = R_j - R_j^{\text{ideal}}$6 plus iodixanol inserts, they were FF 193.17, STC 30.19, ATC 27.03, PMDC 56.46, and STEPC 24.49. In mouse data, the head scan yielded FF 339.10, STC 94.25, ATC 56.88, PMDC 56.45, and STEPC 40.98; the kidney scan yielded FF 429.04, STC 83.26, ATC 56.13, PMDC 162.80, and STEPC 54.91 (Zhou et al., 20 Jul 2025).

Qualitatively, FF showed severe rings in phantoms and mouse scans. STC performed well for PMMA but left ring artifacts in CaCl$\Delta R_j = R_j - R_j^{\text{ideal}}$7, iodixanol, and mixed phantoms. ATC was described as decent but still showed visible rings, especially in low-energy images and between inserts. PMDC performed well for PMMA and CaCl$\Delta R_j = R_j - R_j^{\text{ideal}}$8 phantoms but failed dramatically in iodine-rich scenarios, with strong rings and streaks. STEPC was reported to suppress ring artifacts consistently across phantom and in vivo datasets, with minimal residual rings (Zhou et al., 20 Jul 2025).

6. Robustness, limitations, and broader significance

The study reports robustness across materials, thicknesses, energy bins, and object types. STEPC maintained low MLSD across PMMA-plus-aluminum slab combinations, including thick slabs, and across PMMA-plus-aluminum-plus-iodixanol combinations. It was reported to be particularly robust at low energies, where nonlinear effects are strongest and where STC and ATC often degrade. The method was also reported to work consistently on homogeneous phantoms, multi-material insert phantoms, and in vivo mouse head and kidney scans (Zhou et al., 20 Jul 2025).

Sensitivity analysis examined polynomial order and calibration material set. With PMMA plus aluminum calibration only, moving from first-order to second-order polynomial reduced RAD significantly for PMMA and CaCl$\Delta R_j = R_j - R_j^{\text{ideal}}$9, while moving to third order yielded little benefit or even worse RAD in some cases, especially for iodixanol phantoms when iodine was absent from calibration. When iodixanol was included in calibration, a third-order polynomial further improved RAD for iodixanol phantoms significantly, with only a modest increase in RAD for PMMA. The stated conclusion is that second-order or third-order polynomial models are sufficient, with third order used in the main experiments and second order often sufficient with lower risk of overfitting (Zhou et al., 20 Jul 2025).

The same sensitivity study also showed the importance of the calibration material set. PMMA-plus-aluminum calibration was very good for PMMA and CaCl$S(E)e^{-\int_l \mu(E,\vec{x})\,dl}$0 but not optimal for iodixanol. Adding iodixanol substantially improved correction for iodixanol objects and contrast-enhanced scans with minimal impact on PMMA and CaCl$S(E)e^{-\int_l \mu(E,\vec{x})\,dl}$1. The recommended practice was to use PMMA plus aluminum calibration for non-contrast applications and to add iodinated contrast to calibration for contrast-enhanced applications (Zhou et al., 20 Jul 2025).

The paper identifies several limitations. Calibration currently uses many slab combinations and 600 views per combination, which is time-consuming. The study was conducted on a dual-threshold detector with three bins, so extension to more thresholds remains future work. The polynomial model is interpretable and efficient but may have limited capacity in very complex cases; the authors suggest that neural network models could be explored while preserving the same calibration concept. The method also assumes detector-response nonuniformity is stable over time, so long-term drifts or temperature-dependent effects may require periodic recalibration. Finally, the framework was demonstrated on a specific Micro-PCCT system with a tiled detector; the paper states that the concept should apply to other PCD-based CT systems, though practical performance may depend on detector architecture such as CdTe versus CZT and tile boundaries (Zhou et al., 20 Jul 2025).

Within the scope of the reported experiments, STEPC is presented as a general-purpose calibration framework because it works with limited thresholds yet supports more than two effective materials, requires only standard flat-field correction and log transformation, operates in the projection domain before reconstruction, and does not depend on precise geometric measurements such as phantom thickness or alignment. This suggests a methodological shift away from explicit material-thickness inversion toward direct calibration of spectrum-conditioned detector error, which is the central conceptual contribution of STEPC.