Beam Hardening Correction

• Beam Hardening Correction is a set of techniques designed to mitigate artifacts—such as cupping, shading, and streaks—caused by the nonlinear behavior of polychromatic X-ray beams.
• Methods include empirical calibration using effective attenuation coefficients, spectrum-estimated reprojection, and machine-learning-based approaches to restore linearity in imaging data.
• These strategies apply to various modalities including CT, radiography, and dark-field imaging, improving material quantification and reducing metal or scattering-induced artifacts.

Beam hardening correction (BHC) denotes the set of methods used to compensate the nonlinear transmission behavior produced by polychromatic X-ray sources. In a monochromatic beam, attenuation follows Beer–Lambert’s law with a constant attenuation coefficient, but conventional X-ray tubes emit a broad bremsstrahlung spectrum, and low-energy photons are attenuated more strongly than high-energy photons. The transmitted spectrum therefore shifts to higher energies as it traverses matter, so projection data cease to obey a single-energy exponential law. In radiography and CT this yields cupping, shading, and streak artifacts; in grating-based dark-field imaging it also changes the visibility spectrum and generates an artificial dark-field signal. BHC consequently includes phenomenological calibration, spectrum-estimated reprojection, look-up-table correction, machine-learning-based consistency repair, and acquisition-stage spectral mitigation (Baur et al., 2018, Zhao et al., 2018, Marco et al., 2020).

1. Physical basis and forward models

For photons of a single energy $E$, attenuation is described by the monochromatic Beer–Lambert law,

$I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$

where $\mu(E,Z,\rho)$ depends on photon energy, atomic number, and density. For polychromatic tubes, the measured signal depends on the source spectrum and detector response,

$$I(x) \propto \int N(E)\, \exp\!\big(-\mu(E,Z,\rho)\, x\big)\, S(E)\, \mathrm{d}E,$$

so the effective slope of $-\ln(I/I_0)$ versus $x$ is not constant and decreases with thickness. A standard restoration is to define an $x$-dependent effective attenuation coefficient,

$$I(x) = I_0 \exp\!\big(-\mu_\text{eff}(x)\, x\big), \qquad -\ln\!\frac{I(x)}{I_0} = \mu_\text{eff}(x)\, x,$$

with $\mu_\text{eff}(x)$ depending on material, thickness, spectrum, detector response, geometry, scatter, and filtration (Baur et al., 2018).

In CT, the same nonlinearity appears after log transformation. Under a monochromatic assumption at effective energy $E_0$, one would write a Radon-model line integral, but for a current-integrating system the measured projection is generated by an energy-dependent nonlinear integral model. The consequence is a mismatch between the physical model and the linear integral model used by filtered backprojection, which produces inaccurate quantification of attenuation image and significant beam-hardening artifacts (Cong et al., 2017). In multi-material CT this mismatch is often written as a polychromatic reprojection versus a monoenergetic reprojection of the same segmented template; the difference is then used as an additive correction term (Zhao et al., 2018).

For grating-based dark-field imaging, the attenuation and visibility channels are both polychromatic. Using $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$0 for the effective source spectrum and $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$1 for transmission, the attenuation and dark-field observables can be written as

$I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$2

$I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$3

Even a homogeneous, microstructure-free material changes $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$4 solely because $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$5 hardens the spectrum, inducing an artificial dark-field signal $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$6 that must be removed if dark-field is to reflect ultra-small-angle scattering rather than spectral cross-talk (Lochschmidt et al., 7 Aug 2025).

2. Calibration-driven correction in projection radiography and single-material settings

A direct empirical strategy is to measure $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$7 over a calibration range and then invert the corresponding nonlinear transmission model. In radiographic plate-stack experiments, the proposed phenomenological model is

$I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$8

with $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$9 as an asymptotic effective attenuation and $\mu(E,Z,\rho)$0 capturing how rapidly $\mu(E,Z,\rho)$1 decreases with thickness. Calibration uses known sample thicknesses $\mu(E,Z,\rho)$2 and intensity ratios,

$\mu(E,Z,\rho)$3

followed by nonlinear regression for $\mu(E,Z,\rho)$4. Thickness inversion then solves

$\mu(E,Z,\rho)$5

by Newton’s method, bisection, or a look-up table. The calibration was demonstrated with borosilicate glass, aluminum, and copper plate stacks built from $\mu(E,Z,\rho)$6 mm plates over $\mu(E,Z,\rho)$7–$\mu(E,Z,\rho)$8 mm thicknesses, and the paper reports that fitting three thicknesses spanning the intended range yields nearly identical accuracy to $\mu(E,Z,\rho)$9-point calibration: at $$I(x) \propto \int N(E)\, \exp\!\big(-\mu(E,Z,\rho)\, x\big)\, S(E)\, \mathrm{d}E,$$0 kV, thickness errors $$I(x) \propto \int N(E)\, \exp\!\big(-\mu(E,Z,\rho)\, x\big)\, S(E)\, \mathrm{d}E,$$1 below $$I(x) \propto \int N(E)\, \exp\!\big(-\mu(E,Z,\rho)\, x\big)\, S(E)\, \mathrm{d}E,$$2; at $$I(x) \propto \int N(E)\, \exp\!\big(-\mu(E,Z,\rho)\, x\big)\, S(E)\, \mathrm{d}E,$$3 kV, $$I(x) \propto \int N(E)\, \exp\!\big(-\mu(E,Z,\rho)\, x\big)\, S(E)\, \mathrm{d}E,$$4 below $$I(x) \propto \int N(E)\, \exp\!\big(-\mu(E,Z,\rho)\, x\big)\, S(E)\, \mathrm{d}E,$$5 (Baur et al., 2018).

The same work emphasizes that $$I(x) \propto \int N(E)\, \exp\!\big(-\mu(E,Z,\rho)\, x\big)\, S(E)\, \mathrm{d}E,$$6 is setup-specific rather than a universal material constant. Values can differ by up to a factor of two between two setups even at the same kVp, and any change in tube voltage, filtration, detector type, or geometry requires re-calibration. Filtering narrows the spectrum and reduces beam hardening, but also decreases $$I(x) \propto \int N(E)\, \exp\!\big(-\mu(E,Z,\rho)\, x\big)\, S(E)\, \mathrm{d}E,$$7 significantly: at $$I(x) \propto \int N(E)\, \exp\!\big(-\mu(E,Z,\rho)\, x\big)\, S(E)\, \mathrm{d}E,$$8 kV, $$I(x) \propto \int N(E)\, \exp\!\big(-\mu(E,Z,\rho)\, x\big)\, S(E)\, \mathrm{d}E,$$9 mm Al gives $-\ln(I/I_0)$0 of unfiltered, $-\ln(I/I_0)$1 mm Al gives $-\ln(I/I_0)$2, and $-\ln(I/I_0)$3 mm Cu gives $-\ln(I/I_0)$4 (Baur et al., 2018). This suggests that empirical BHC is inseparable from acquisition protocol control.

A closely related single-material CT formulation replaces the conventional log transform by a calibration-derived monotonic mapping from measured transmission fraction $-\ln(I/I_0)$5 to areal density $-\ln(I/I_0)$6,

$-\ln(I/I_0)$7

The mapping is built from reference sheets of the same material with known density and thickness, and the corrected projections are reconstructed directly as density rather than attenuation. In simulation, cupping was completely corrected without scatter; with scatter, measured densities of $-\ln(I/I_0)$8, $-\ln(I/I_0)$9, and $x$0 g/cm$x$1 versus true $x$2, $x$3, and $x$4 g/cm$x$5 corresponded to $x$6, $x$7, and $x$8 bias. In experimental graphite cylinders, reconstructed average densities agreed within $x$9 of the manufacturer-reported values, with residual bias attributed to scatter and possible composition differences (Deshmukh, 4 Nov 2025).

The phenomenological model in radiography also extends to granular media. For a granular sample of thickness $x$0 with effective grain path length $x$1, the path-averaged volume fraction is

$x$2

The reported example used cuboidal boxes of $x$3, $x$4, and $x$5 cm thickness filled to volume fraction $x$6, and dynamic radiograms of a $x$7 mm$x$8 polycarbonate box filled with sand showed bright stripes where $x$9 is reduced during shaking at $$I(x) = I_0 \exp\!\big(-\mu_\text{eff}(x)\, x\big), \qquad -\ln\!\frac{I(x)}{I_0} = \mu_\text{eff}(x)\, x,$$0 Hz (Baur et al., 2018).

3. Spectrum-estimated and projection-domain correction for multi-material CT

A major class of CT BHC methods constructs a segmented template image, estimates the effective spectrum, and computes a projection-domain correction from the difference between monoenergetic and polychromatic reprojections. In the multi-material spectrum-estimation method, the correction per ray is

$$I(x) = I_0 \exp\!\big(-\mu_\text{eff}(x)\, x\big), \qquad -\ln\!\frac{I(x)}{I_0} = \mu_\text{eff}(x)\, x,$$1

optionally scaled by

$$I(x) = I_0 \exp\!\big(-\mu_\text{eff}(x)\, x\big), \qquad -\ln\!\frac{I(x)}{I_0} = \mu_\text{eff}(x)\, x,$$2

The effective spectrum is modeled as a convex combination of model spectra, the template is obtained by segmenting the uncorrected image into different components using a simple segmentation algorithm, and GPU ray-tracing supplies the per-material path lengths. Numerical simulations, experimental phantom data, and animal data acquired on a Discovery CT750 HD and an Artis Zee showed that the method significantly reduced both cupping and streak artifacts and successfully recovered the Hounsfield Units accuracy (Zhao et al., 2018).

A blind multi-material variant avoids prior material knowledge by first performing a wide sweep of the material based on an experimentally measured look-up table to obtain the closest estimate of the material, then correcting the non-linearity effect of the beam hardening by adding the difference between the estimated monochromatic and the polychromatic simulated projections of the segmented image. The estimated monochromatic projection is simulated by selecting the energy from the polychromatic spectrum which produces the lowest mean square error with the acquired projection, and the polychromatic projection is estimated by minimizing the difference between the acquired projection and the weighted sum of the simulated polychromatic projections using different spectra of different filtration. Extensive experiments on real-world CT data showed that the method can highly reduce the beam hardening artifacts without prior knowledge of the materials, and robustness tests with material misestimation of $$I(x) = I_0 \exp\!\big(-\mu_\text{eff}(x)\, x\big), \qquad -\ln\!\frac{I(x)}{I_0} = \mu_\text{eff}(x)\, x,$$3 left the correction results essentially unchanged (Alsaffar et al., 2022).

Dental cone-beam CT introduces additional difficulties because of offset detector geometry, field-of-view truncation, and low signal-to-noise ratio due to low X-ray irradiation. A two-stage method addresses this by composing sinogram reflection, truncated-data consistency correction, FDK reconstruction, and a deep convolutional neural network,

$$I(x) = I_0 \exp\!\big(-\mu_\text{eff}(x)\, x\big), \qquad -\ln\!\frac{I(x)}{I_0} = \mu_\text{eff}(x)\, x,$$4

The first stage performs a sinogram adjustment in the direction of enhancing data consistency, considering the situation according to the FOV truncation and offset detector; the second stage uses a U-Net because the sinogram correction algorithm significantly reduces beam hardening artifacts caused by high-density materials such as teeth, bones, and metal implants, while tending to amplify special types of noise. Numerous experiments validated the method. For Model 1 at high noise, NRMSD changed from $$I(x) = I_0 \exp\!\big(-\mu_\text{eff}(x)\, x\big), \qquad -\ln\!\frac{I(x)}{I_0} = \mu_\text{eff}(x)\, x,$$5 uncorrected to $$I(x) = I_0 \exp\!\big(-\mu_\text{eff}(x)\, x\big), \qquad -\ln\!\frac{I(x)}{I_0} = \mu_\text{eff}(x)\, x,$$6 after Stage 1 and $$I(x) = I_0 \exp\!\big(-\mu_\text{eff}(x)\, x\big), \qquad -\ln\!\frac{I(x)}{I_0} = \mu_\text{eff}(x)\, x,$$7 after Stage 2, while SSIM changed from $$I(x) = I_0 \exp\!\big(-\mu_\text{eff}(x)\, x\big), \qquad -\ln\!\frac{I(x)}{I_0} = \mu_\text{eff}(x)\, x,$$8 to $$I(x) = I_0 \exp\!\big(-\mu_\text{eff}(x)\, x\big), \qquad -\ln\!\frac{I(x)}{I_0} = \mu_\text{eff}(x)\, x,$$9 and then $\mu_\text{eff}(x)$0; for Model 2 at low noise, NRMSD changed from $\mu_\text{eff}(x)$1 uncorrected to $\mu_\text{eff}(x)$2 after Stage 1 and $\mu_\text{eff}(x)$3 after Stage 2 (Bayaraa et al., 2020).

4. Learning-based BHC and artifact decomposition

Deep learning entered BHC first through sinogram correction for metal-induced beam hardening in CT. A sinogram-consistency learning method was proposed to repair inconsistent sinograms by removing the primary metal-induced beam-hardening factors along the metal trace in the sinogram. Because sufficient training data are difficult to obtain in a medical environment, the learning method was designed to use simulated training data and a patient-type specific learning model was used to simplify the learning process. Its feasibility was investigated using real CT scans of pelvises containing hip prostheses, with different anatomical areas in training and test data to demonstrate that the method extracts the beam hardening features selectively (Park et al., 2017).

A complementary image-domain strategy is implant-specific nonlinear decomposition. In this approach, the generated beam-hardening artifacts can be approximately extracted by subtracting artifacts generated exclusively by metals, and the reconstructed image is modeled as

$\mu_\text{eff}(x)$4

A U-Net–style multi-scale encoder–decoder CNN maps an analytic artifact surrogate $\mu_\text{eff}(x)$5 to the metal-only artifact image $\mu_\text{eff}(x)$6 using the pure $\mu_\text{eff}(x)$7 loss

$\mu_\text{eff}(x)$8

so artifact-free ground-truth CT images are not required for training. On four pelvis cases, NRMSD outside the metal mask changed from $\mu_\text{eff}(x)$9 to $E_0$0, from $E_0$1 to $E_0$2, from $E_0$3 to $E_0$4, and from $E_0$5 to $E_0$6, outperforming the MAC-BC baseline in all four experiments (Park et al., 2017).

Another learning formulation maps conventional current-integrating CT projections to monochromatic-equivalent projections. The method reconstructs a conventional image from raw log projections, samples the pixel values along each path, and uses a 4-layer multilayer perceptron to predict a corrected projection consistent with a target monochromatic energy. Training and testing used a clinical dual-energy dataset from a GE Discovery CT750, and the results show that the proposed method can achieve a high accuracy of the projection correction with a relative error of less than $E_0$7 (Cong et al., 2017).

These methods do not define a single canonical learning architecture for BHC. This suggests that machine learning is being used in several distinct roles: repair of sinogram inconsistency, estimation of metal-only artifact components, and nonlinear projection mapping to monoenergetic equivalents.

5. Beam hardening in dark-field radiography

In dark-field imaging, beam hardening is not confined to attenuation. Because the fringe visibility is energy dependent, beam hardening also changes the visibility spectrum and generates a false dark-field signal even in homogeneous, non-scattering materials. The decomposition

$E_0$8

separates the “true” USAXS dark-field from the beam-hardening-induced component. The physical mechanism is further developed in the visibility-spectrum hardening model, where a dark-field-active object can reduce the effective dark-field signal of downstream objects by modifying the visibility spectrum, and CT simulations show strong “cupping” artifacts if visibility hardening and position-dependent correlation length are not corrected (Lochschmidt et al., 7 Aug 2025, Marco et al., 2020).

A fast empirical correction for clinical dark-field chest radiographs uses per-pixel look-up tables derived from water and aluminum phantoms,

$E_0$9

with an additional bias correction

$I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$00

The method was evaluated qualitatively on three male participants, and applying a correction using a weighted look-up table led to a significant reduction of bone structures in the dark-field image. The weighting of the aluminum component had a substantial impact on the degree to which bone structures remain visible, and the large negative bias dependent on the aluminum weighting was successfully corrected (Lochschmidt et al., 7 Aug 2025).

A more anatomically adaptive variant replaces the global aluminum weighting by deep-learning-based segmentation of ribs and clavicles combined with spectral CT–derived aluminum and water attenuation contribution masks. The material weights satisfy

$I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$01

and pixel-wise weighted LUT interpolation yields the beam-hardening-induced dark-field map, which is subtracted from the raw dark-field image. The clinical evaluation used $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$02 dark-field chest radiographs from $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$03 healthy subjects, $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$04 COPD subjects, and $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$05 COVID-19 subjects. Group-wise normalized dark-field signal decreased after BHC from $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$06 to $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$07 in healthy subjects, from $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$08 to $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$09 in COPD, and from $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$10 to $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$11 in COVID-19; coefficient of variation decreased from $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$12 to $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$13, from $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$14 to $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$15, and from $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$16 to $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$17, respectively; and median overlap between healthy and COPD distributions decreased from $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$18 to $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$19, with both overlap comparisons reported as $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$20 (Kaster et al., 14 Aug 2025).

6. Spectral mitigation, acquisition strategies, and theoretical limits

Some BHC strategies reduce beam hardening at acquisition rather than by post hoc correction. In spectral photon-counting CT with Medipix3RX operated in fully operational charge summing mode, narrow high-energy bins such as $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$21–$I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$22 keV and $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$23–$I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$24 keV reduced cupping and metal streak artifacts relative to wide-band reconstructions. In a titanium scaffold region of interest located in non-metal air, $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$25 changed from $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$26 at $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$27–$I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$28 keV to $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$29 at $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$30–$I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$31 keV, $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$32 at $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$33–$I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$34 keV, and $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$35 at $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$36–$I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$37 keV, with the highest-energy bin limited by photon statistics. The reported strategy is acquisition-stage mitigation rather than post-reconstruction numerical BHC (Rajendran et al., 2013).

Spectral CT can also reconstruct beam-hardening-free CT numbers directly from material-basis images. In a two-basis water-and-bone decomposition, the synthetic effective attenuation is formed voxelwise and converted to synthetic Hounsfield units by an energy-integrating weighting. In simulation, the resulting images showed CT numbers that are consistently accurate for a varying range of tissues and are free of beam hardening artefacts. For ideal photon-counting data, soft-tissue biases were near zero, bone bias was $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$38 HU with RMSE $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$39 HU, and spongiosa bias was $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$40 HU with RMSE $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$41 HU; by comparison, a simulated conventional energy-integrating system with empirical cupping correction retained large residual errors, including bone bias $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$42 HU (Grönberg et al., 2022).

Theoretical work formalizes the nonlinearity behind metal streaks. For strictly convex bodies with

$I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$43

the polynomial nonlinearity is uniquely determined by the single reconstructed image $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$44, provided $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$45 and $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$46 are strictly convex and $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$47. The same analysis proves that $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$48 if and only if $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$49 (Wang, 2023). This suggests that, at least for strictly convex metal regions, the observed artifact geometry contains sufficient information to identify the beam-hardening nonlinearity itself.

Across these diverse formulations, several limitations recur. Setup dependence is explicit in empirical $I(x) = I_0 \exp\big(-\mu(E,Z,\rho)\, x\big),$50 models, spectrum-estimated reprojection depends on accurate material attenuation and effective spectra, dark-field correction depends on visibility-spectrum modeling and bias control, and acquisition-stage spectral narrowing trades beam-hardening reduction against flux or noise. Scatter, absorption edges, and overlapping unknown materials remain the central sources of residual bias or model failure (Baur et al., 2018, Zhao et al., 2018, Marco et al., 2020).