Chapter 11: Artifacts and Their Reduction

Everything we have reconstructed so far rested on assumptions we never spelled out: the X-rays are monochromatic (every photon has the same energy), the projection data are consistent and error-free, the object holds still for the whole scan, the detector responds perfectly uniformly, and only primary photons traveling in straight lines reach it. In a real scanner, none of these holds exactly. When an assumption breaks, the reconstruction acquires a systematic false structure (an artifact) that is qualitatively different from noise. The few-projection streaks of Chapter 4 were already an example: the assumption of sufficient angular sampling had broken. This chapter walks through five representative artifacts, each in the same order: cause, signature in the sinogram, and remedy.

Beam hardening

A real X-ray tube emits not a single energy but a continuous spectrum $S(E)$, and the attenuation coefficient depends on energy ($\mu = \mu(x,y;E)$). The detected intensity is therefore a spectrum-weighted average of Beer-Lambert transmissions.

$$I = \int S(E)\, \exp\!\left(-\int_L \mu(x,y;E)\, dl\right) dE$$

Taking the log as in Chapter 1, $p_{\mathrm{poly}} = -\ln(I/I_0)$, no longer yields a line integral of $\mu$. Discretizing the spectrum into energy bins shows that the polychromatic measurement is a concave function of the monochromatic line integral $p$:

$$p_{\mathrm{poly}} = -\ln\!\left(\sum_k w_k\, e^{-r_k p}\right), \qquad \sum_k w_k = 1,\quad \sum_k w_k r_k = 1$$

where $w_k$ are the bin weights and $r_k$ the attenuation ratios relative to a reference energy. Low-energy photons are absorbed preferentially ($r_k$ large), so the beam's effective energy rises as it penetrates; the beam "hardens". The effect grows with path length: $p_{\mathrm{poly}} \le p$, with the deficit growing as $p$ grows. Long central paths are underestimated the most, so the FBP image sags toward the center in a dish shape. This is cupping.

The standard remedy is water correction. If the object can be treated as a single material (the human body is mostly water), the hardening curve $p_{\mathrm{poly}} = f(p)$ is monotonic, so applying the inverse map $f^{-1}$ to the measurements recovers monochromatic-equivalent line integrals. Real scanners ship with this correction built into their calibration.

Simulation: cupping and water correction

Increase the spectral width $\beta$ and watch the center of the uncorrected reconstruction sink. The center-row profile shows the interior (flat in the ground truth) becoming dish-shaped, and the water correction restoring it almost exactly. $\beta = 0$ corresponds to the monochromatic world of the chapters so far.

Scattered radiation

X-rays are not only absorbed inside the object; many survive Compton scattering with a new direction. A scattered photon reaches the detector having forgotten which ray it traveled, so the measured intensity carries, on top of the primary beam, a scatter component $S$ spread smoothly across the detector:

$$I = I_0\, e^{-p} + S, \qquad p' = -\ln(I/I_0) \le p$$

The extra light means attenuation is always underestimated, and the relative impact is largest where the primary signal is weakest, along strongly attenuating paths. The result has the same shape as beam hardening: cupping toward the center, plus dark bands and contrast loss between dense structures. Scatter grows with irradiated volume, so it is mild for a thin fan beam but severe for the cone-beam flat-panel systems of Chapter 6, where the scatter-to-primary ratio (SPR) can exceed 1.

Countermeasures come in two layers: an anti-scatter grid in front of the detector physically blocks obliquely incident photons, and the remainder is estimated and subtracted in software by exploiting the smoothness of scatter (kernel methods, Monte-Carlo model estimation). Scatter does not disappear with photon-counting detectors (Chapter 13), so scatter correction remains an active engineering topic.

Simulation: scatter and kernel correction

Raise the SPR and watch the uncorrected reconstruction develop a central sag and dark bands between the cylinders, with the profile dipping into a dish shape. A simple kernel correction that relies only on the smoothness of scatter restores the image almost to truth. Note how similar this cupping looks to beam hardening's, which is one reason the two are hard to disentangle in real data.

Metal artifacts

Implants and dental fillings have attenuation coefficients orders of magnitude above soft tissue. Two things happen at once along rays through metal. First, beam hardening becomes so extreme that single-material water correction cannot fix it. Second, almost no photons reach the detector ($I \to 0$), so the log transform $-\ln(I/I_0)$ amplifies noise explosively (photon starvation). The result is dark bands connecting the metal objects and bright/dark streaks radiating from them.

If the rays through metal cannot be trusted, why not replace them with an estimate? That is the idea of metal artifact reduction (MAR), phrased as sinogram inpainting:

The simplest interpolation is linear (LI-MAR): in each angular row, connect the two edges of the trace with a straight line. But LI also erases the genuine structure (bone, for instance) that lay under the trace, which makes the filled region inconsistent with its surroundings and introduces secondary artifacts.

NMAR (normalized MAR) fixes this with a prior image. Classify the uncorrected reconstruction into air, soft tissue, and bone; forward-project that prior; divide the sinogram by the prior's projection before interpolating; and multiply it back afterwards:

$$\tilde{p} = \frac{p}{p_{\mathrm{prior}}}, \qquad p_{\mathrm{corr}} = \tilde{p}_{\mathrm{interp}} \cdot p_{\mathrm{prior}}$$

After normalization, $\tilde{p}$ is nearly flat, since the anatomical structure has been divided out, so even simple linear interpolation makes little error. Remove the structure before interpolating, put it back after. This is the core of NMAR.

Simulation: metal artifacts and MAR

Switch between phantoms and MAR methods. With method "None", the metal trace runs through the sinogram as a bright band and the reconstruction shows dark bands and streaks. LI fills the trace smoothly and reduces the streaks; NMAR preserves the surrounding structure better still. RMSE is computed inside the FOV circle excluding metal pixels. Lowering $I_0$ adds photon-starvation streaks on top.

Ring artifacts

If one detector element's sensitivity is slightly off (gain $g_d \ne 1$), its measurements shift by a constant after the log transform:

$$p'(\theta, s_d) = p(\theta, s_d) - \ln g_d$$

The shift lands on the same column $s_d$ at every angle $\theta$, so it appears as a vertical stripe in the sinogram. Under backprojection, the contribution at detector position $s_d$ is smeared along the line at distance $|s_d|$ from the rotation center, at every angle. Sweeping the angle, the envelope of this family of lines is a circle of radius $|s_d|$, a ring around the rotation center. One vertical line in the sinogram maps to one ring in the image.

Because the artifact is so well-behaved, so is the fix. Take each detector column's mean, smooth the means along the detector axis, and subtract the residual (the column-specific offset component). This removes the gain error while preserving the smooth column-mean structure of the true signal. Real scanners also calibrate continuously with air scans.

Simulation: ring artifacts and detector correction

Increase the gain error $\sigma$ and watch the sinogram stripes and the image rings darken together. Verify the correspondence "vertical lines in the sinogram ⇔ concentric rings in the image", and the effect of the column normalization.

Motion artifacts

Projections are not acquired all at once; each angle is a snapshot taken at a different time. If the object moves during the scan, each projection sees a slightly different object, and the sinogram is no longer consistent with any single object. Reconstruction backprojects as if all projections saw the same thing, so the inconsistency surfaces as double contours and streaks.

The signature depends on how the object moves. A single mid-scan shift (step) creates a discontinuity between sinogram rows; a slow one-directional drift distorts the sinusoidal trajectories; periodic motion such as breathing makes the sinogram wobble.

Motion is the hardest artifact to fix after the fact, so practice focuses on avoiding it: fast gantry rotation and half-scans (180° plus the fan angle) shorten the acquisition itself; cardiac CT uses ECG gating (using only the quiet phase of the R-R interval); respiratory gating handles breathing. Estimating the motion from the data and folding it into reconstruction is an active research topic, one that also pairs naturally with the learning-based methods of the next chapter.

Rotation time and temporal resolution are not identical. A full-scan reconstruction uses nearly one rotation of data, while fan-beam redundancy permits a half-scan reconstruction from roughly half a rotation. Dual-source CT uses two source–detector pairs at different angles to improve cardiac temporal resolution further. Prospective ECG triggering exposes the patient only near selected cardiac phases and generally reduces dose. Retrospective gating scans throughout the cardiac cycle and allows the reconstruction phase to be chosen later, usually at a higher dose.

Simulation: motion artifacts

Vary the motion type and amplitude, and observe the sinogram inconsistency (the step discontinuity is the easiest to spot) and how the reconstruction degrades. The same amplitude breaks the image differently depending on the motion type.

Summary: cause, signature, remedy

These remedies fall into three groups: correct a known physical effect, as in water correction; estimate missing or corrupted data, as in MAR; or prevent the inconsistency during acquisition, as in gating. Different causes can produce similar-looking artifacts, so the image, sinogram, and acquisition conditions all matter when diagnosing them. The next chapter considers priors learned from data.

References

• Herman GT. Correction for spectral artifacts in computerized tomography. Physics in Medicine & Biology 21, 390–398 (1976).
• Meyer E et al. Normalized metal artifact reduction (NMAR) in computed tomography. Medical Physics 37, 5482–5493 (2010).
• Boas FE, Fleischmann D. CT artifacts: causes and reduction techniques. Imaging in Medicine 4, 229–240 (2012).