Chapter 7: Industrial CT

Every CT so far has been aimed at a human body. But the cone beam and FDK we assembled in the previous chapter run just as well in another world: industrial CT, which hunts for voids inside castings, peers at solder joints on circuit boards, and studies fossils without breaking them. Same physics, same mathematics. And yet, the moment the subject stops being a patient, the design constraints swap out wholesale. This chapter is about what that swap makes possible.

The constraints swap out

The constraint that shaped medical CT above all was dose. The patient is irradiated, so scans must be fast and photons few. In industrial CT the subject is an aluminium casting or a connector. Nobody is being irradiated, so the constraint lifts.

• You may take your time: a scan can run for tens of minutes or hours, where medical CT must finish in under a second.
• You may spend photons freely: long exposures beat the noise down. The $\sigma \propto 1/\sqrt{I_0}$ of Chapter 9 applies with no ceiling on dose.
• You may take as many projections as you like: thousands are practical, so the sparse-view problem of Chapter 10 rarely bites.
• The subject holds still: parts do not breathe. The motion artifacts of Chapter 11 simply do not arise.

New severities arrive in exchange. The subjects are metal, so beam hardening and photon starvation bare their teeth daily (Chapter 11). And the biggest difference of all is the machine's layout.

Rotate the part

Medical CT lays the patient down and spins the source and detector around them on a gantry, because the patient cannot be spun. Industrial CT does the opposite: source and detector are bolted to the floor, and the part rides a turntable. The part does not complain, so it is the part that turns.

That single move is decisive. Being free to place the subject anywhere means you can move it as close to the source as you like.

Call the source-object distance $\mathrm{SOD}$ and the source-detector distance $\mathrm{SDD}$. The part's shadow lands on the detector magnified, and this geometric magnification $M$ is the protagonist of industrial CT:

$$M = \frac{\mathrm{SDD}}{\mathrm{SOD}}$$

The closer the part sits to the source (the smaller $\mathrm{SOD}$), the bigger the shadow. The detector pixel pitch $p_{\mathrm{det}}$, referred back to the scale of the object, shrinks in inverse proportion:

$$p_{\mathrm{voxel}} = \frac{p_{\mathrm{det}}}{M}$$

A detector with a 100 µm pitch yields 10 µm voxels at $M = 10$. Where medical CT is pinned near $M \approx 1$ and can never beat its pixel pitch, industrial CT buys resolution with geometry alone. That is how micro-CT reaches below a micrometre.

Except the focal spot gets in the way

So shrink $\mathrm{SOD}$ without limit and get unlimited resolution? No. We have been treating the source as a point, but a real X-ray tube's focal spot has size.

Given a focal spot of size $f$, the edge of a shadow blurs: light from the top of the spot and light from the bottom draw the edge at different places on the detector. This penumbra has width

$$u_{\mathrm{det}} = f \cdot (M - 1)$$

and it grows as you magnify.

Combining voxel and penumbra, the total unsharpness is estimated as the root sum of squares:

$$u_{\mathrm{total}} = \sqrt{u_{\mathrm{obj}}^2 + p_{\mathrm{voxel}}^2}$$

Simulation: magnification and voxel size

Shrink $\mathrm{SOD}$ and the magnification rises: the effective voxel (the falling straight line) keeps getting finer. But the focal-spot penumbra flattens out at the focal-spot size, and the total unsharpness stops there with it. Raise the focal spot to 50 µm and confirm that no amount of magnification rescues you. Note too that the field of view shrinks as $1/M$ — resolution and field of view are a trade-off wired together by geometry.

What is it all for?

Industrial CT splits broadly into two jobs.

Non-destructive testing (NDT) is the business of finding flaws: porosity in castings, cracks in welds, delamination in composites, voids in solder joints. The whole value is seeing inside without breaking anything, so everything comes down to whether the flaw is visible — and visibility is governed by the total unsharpness we just derived.

Dimensional metrology is the business of measuring. CT can measure internal geometry that no caliper or touch probe can reach, without cutting the part open. But measuring raises the bar sharply. Detecting a flaw tolerates some blur as long as you can tell it is there; measuring a dimension requires locating a surface to within micrometres. If beam hardening shifts an edge, that shift becomes measurement error outright. The traceability standards for CT metrology (VDI/VDE 2630 and others) exist because of exactly this difficulty.

Simulation: NDT and resolution

A cast-part phantom (porosity and a fine crack in solid material) is scanned and thresholded to flag defects (red). Increase $\mathrm{SOD}$ to lower the magnification, or raise the focal-spot size, and the small pores and the fine crack are the first to vanish as the detection rate falls. The crucial part: raising the incident photon count $I_0$ to crush the noise does not bring the detection rate back. Noise can be bought with photons; unsharpness cannot. It is precisely because industrial CT has no dose limit that resolution design is what matters.

Key points

Industrial CT takes the physics and reconstruction mathematics of medical CT into a world with different constraints. Because the subject is not a patient, time, photons, and projection count are all free — and the subject itself can be rotated and brought close to the source. That geometric magnification $M = \mathrm{SDD}/\mathrm{SOD}$ divides the detector pixel pitch by $M$ and delivers micro-CT resolution. The focal spot casts a penumbra, though, so resolution bottoms out at the spot size, and the field of view shrinks as $1/M$. The part's size sets the magnification, and the magnification sets the voxel: that is the skeleton of a scan plan. The next chapter returns to reconstruction algorithms, solving the equations by iteration. Metal artifacts, the daily enemy of industrial CT, come back in Chapter 11.

References

• Feldkamp LA, Davis LC, Kress JW. Practical cone-beam algorithm. Journal of the Optical Society of America A 1, 612–619 (1984) — the de facto standard reconstruction for industrial CT as well (shared with Chapter 6).
• De Chiffre L et al. Industrial applications of computed tomography. CIRP Annals 63, 655–677 (2014).
• Kruth JP et al. Computed tomography for dimensional metrology. CIRP Annals 60, 821–842 (2011).
• VDI/VDE 2630 Part 1.2: Computed tomography in dimensional measurement — Influencing variables on measurement results and recommendations for computed tomography dimensional measurements (2018).
• Stock SR. MicroComputed Tomography: Methodology and Applications, 2nd ed. CRC Press (2019).