Chapter 20: Tomosynthesis
When you cannot rotate all the way around: depth blur from a limited angular range, and in-plane vs through-plane resolution, in simulation.
Welcome to the frontiers part. The CT chapters rotated 180° or more around the object to complete the data. In reality, though, there are many ways of imaging that cannot go all the way around. Compressing a breast into a thin layer, moving only the source over a chest on the table, tracing a dental arch from outside the mouth — each of these can move the source over only a narrow range of angles. When the angle is not enough, what happens to the reconstruction, what do you give up, and what do you gain? That is this chapter's theme.
You cannot rotate all the way
Tomosynthesis fixes the detector and moves the source over a narrow arc (typically ±15–25°). It is used in breast imaging (DBT, digital breast tomosynthesis), chest, and dental panoramic imaging. A single radiograph is one angle, with no depth information at all. Tomosynthesis adds just a little angle to obtain limited tomographic information — an intermediate between the plain radiograph and full CT.
Full CT (left) rotates essentially all the way around the object to complete the data. Tomosynthesis (right) fixes the detector and moves the source over a narrow arc (about ±25°). With too little angle, the in-plane structure is solvable but depth (through-plane) cannot be separated.
Why not rotate all the way? The reasons differ by application, but they share a "cannot or would rather not rotate" character: the breast is fixed between compression plates, you want to limit dose, keep the machine small and cheap, and scan quickly. If a limited angle can stand in for full CT, the payoff is large. The catch is that the image pays for it.
What the limited angle takes away
The reconstruction algorithm can stay exactly as in the CT chapters: forward-project over the limited angle set and back out with FBP. What changes is the quality — limited angle brings its own artifact. In-plane resolution (parallel to the detector) is largely preserved, while through-plane (depth) blurs. A point-like structure stretches into a tail along the depth direction. This is depth blur.
Why is only the through-plane sacrificed? Recall the Fourier slice theorem of Chapter 4. Measuring the projection at angle measures the object's 2D spectrum along one line through the origin. Rotating all the way fills the whole spectrum; a limited angle fills only the range corresponding to the measured projection angles. The unmeasured angles remain as a fan-shaped blank in Fourier space — the missing wedge.
The two faces of limited angle. Left (image space): the point spreads through-plane (depth blur). Right (Fourier space): by the Fourier slice theorem, only the range of measured projection angles fills the spectrum, leaving the unmeasured angles as a missing wedge. Two sides of the same gap.
Depth blur and the missing wedge are the same thing
"The point stretches into depth" in image space and "a wedge is missing" in Fourier space are two sides of the same phenomenon. The direction of the missing wedge corresponds to the unmeasured projection angles; the high frequencies in that direction are lost, so edges in that direction blur. The depth blur of tomosynthesis is nothing but the image-space appearance of this gap. In Chapter 22, electron tomography, exactly the same missing wedge stands in the way, now in three dimensions.
Simulation: limited angle and depth blur
A grid of disks is imaged both with full 180° and with a limited ±half-span, then reconstructed by FBP. Lower the half-span slider and watch the disks on the limited-angle side stretch vertically (through-plane). The elongation ratio of the center disk (through/in-plane) is shown, so you can confirm numerically that narrowing the angle increases the elongation. Note too that the horizontal (in-plane) direction barely changes — the angle deficit acts in a chosen direction.
True object
Full 180°
Computing…
Limited angle
Computing…
The same grid of disks is imaged with full 180° and with a limited ±half-span, then reconstructed by FBP. Narrowing the sweep stretches each disk through-plane (vertically) into depth blur. In-plane (horizontal) resolution is largely preserved while the through-plane depends strongly on the angle — the essence of limited-angle tomography, and the elongation ratio grows as the angle shrinks.
Why use it anyway
Despite all these drawbacks, tomosynthesis is widely used clinically, because the comparison is not "full CT" but "a single radiograph." In mammography, overlapping breast tissue hiding a lesion (tissue superposition) was the biggest problem. Tomosynthesis separates the volume into thin slices along depth, peeling apart the overlap and making lesions easier to find. The through-plane resolution may be coarse, but it is enough to reduce superposition — and the dose is far lower than full CT.
There is also active work on filling the limited-angle gap with prior knowledge to improve image quality. The compressed sensing of Chapter 10 (filling the gap with sparsity) and the deep learning of Chapter 12 (learning how to fill it from data) apply directly as tools for filling the missing wedge. Limited-angle tomography is a place where the techniques of this textbook for recovering images from incomplete data are brought together.
Key points
Tomosynthesis is limited-angle tomography: the detector is fixed and the source moves over a narrow arc. The reconstruction algorithm is the same as CT, but the missing angle leaves a missing wedge in Fourier space, blurring resolution in its direction — effectively the depth (through-plane) direction. Yet it yields far more information than a single radiograph and peels apart tissue superposition, serving the clinic at low dose. It is also a target for the gap-filling of compressed sensing and deep learning. The next chapter moves to photoacoustic tomography, where the source is inside the body rather than outside, and the very geometry of reconstruction changes.
References
- Dobbins JT, Godfrey DJ. Digital x-ray tomosynthesis: current state of the art and clinical potential. Physics in Medicine & Biology 48, R65–R106 (2003).
- Sechopoulos I. A review of breast tomosynthesis. Part I. The image acquisition process. Medical Physics 40, 014301 (2013).
- Frikel J, Quinto ET. Characterization and reduction of artifacts in limited angle tomography. Inverse Problems 29, 125007 (2013).