Chapter 5: Fan-Beam Geometry and Helical CT

Parallel-beam projection as used so far assumes that source and detector are translated for every single ray, and that this sweep is repeated at every angle (first-generation CT). Imaging one slice this way takes minutes. With a beam that spreads from the source as a fan and a multi-element detector, one source position captures an entire projection at once, and rotating source and detector is all it takes to collect the full dataset (third-generation CT). Modern clinical scanners are based on this scheme, completing a rotation in under a second.

Let us define the geometry first. The source travels on a circle of radius $D_{\mathrm{SO}}$ around the isocenter (origin), parameterized by the source angle $\beta$, so $S = D_{\mathrm{SO}}(\cos\beta,\ \sin\beta)$. The physical detector sits on the far side of the source, but for computation it is convenient to use a virtual flat detector scaled down to pass through the isocenter. The angle between the ray toward detector position $u$ and the central ray (from the source to the origin) is the fan angle $\gamma$.

$$S = D_{\mathrm{SO}}(\cos\beta,\ \sin\beta), \qquad \gamma = \arctan\!\frac{u}{D_{\mathrm{SO}}}$$

Each individual fan-beam ray is, viewed differently, the same straight line as some parallel-beam ray $(\theta, s)$. With the conventions of this course the correspondence is $\theta = \beta + \pi/2 - \gamma$ and $s = D_{\mathrm{SO}}\sin\gamma$. Re-sorting (interpolating) the fan sinogram $(\beta, u)$ onto the parallel grid $(\theta, s)$ therefore lets us reuse the parallel-beam FBP of the previous chapters unchanged. This re-sorting is called rebinning.

$$\theta = \beta + \frac{\pi}{2} - \gamma, \qquad s = D_{\mathrm{SO}}\sin\gamma$$

An FBP that reconstructs directly from the fan sinogram, without rebinning, can also be derived. It has three steps. (1) Weight each detector value by the cosine factor $D_{\mathrm{SO}}/\sqrt{D_{\mathrm{SO}}^2 + u^2}$. (2) Convolve each row with the ramp filter. (3) Backproject with a per-pixel distance weight $1/U^2$. Here $U$ is the pixel's depth from the source along the central ray, normalized by $D_{\mathrm{SO}}$, and $u^*$ is where the ray through the pixel hits the virtual detector. In a full $2\pi$ scan every ray is measured twice, hence the factor $1/2$.

$$f(x,y) = \frac{1}{2}\int_0^{2\pi} \frac{1}{U^2}\; q\bigl(\beta,\, u^*\bigr)\, d\beta, \qquad q(\beta,\cdot) = \left[\frac{D_{\mathrm{SO}}}{\sqrt{D_{\mathrm{SO}}^2 + u^2}}\; p(\beta, u)\right] * h$$ $$U = \frac{D_{\mathrm{SO}} - x\cos\beta - y\sin\beta}{D_{\mathrm{SO}}}, \qquad u^* = \frac{D_{\mathrm{SO}}\,(-x\sin\beta + y\cos\beta)}{D_{\mathrm{SO}} - x\cos\beta - y\sin\beta}$$

Simulation: fan-beam geometry

The ray bundle from the source (orange) on its circular orbit to the virtual detector (purple) is shown. Rotate the source with the $\beta$ slider or the play button, and move it closer or farther with the $D_{\mathrm{SO}}$ slider. Decreasing $D_{\mathrm{SO}}$ enlarges the fan half-angle $\gamma_{\max}$ for the same detector width, making it easier to cover the FOV (blue circle). In exchange, the beam divergence (geometric distortion) grows stronger.

Simulation: fan sinogram and rebinning

Three images are shown side by side: the fan-beam sinogram of the selected phantom ($\beta \times u$, 360°), its rebinned version on the parallel grid ($\theta \times s$, 180°), and the true parallel-beam sinogram computed directly. Confirm that the rebinned result matches the true parallel sinogram almost exactly. With the point phantom you can also observe the trace in the fan sinogram deviating slightly from a pure sinusoid (the nonlinearity of $\gamma$).

Simulation: direct fan-beam FBP

The image is reconstructed directly from the fan sinogram with FBP (Ram-Lak). How does the image quality respond as you vary the number of projections ($\beta$ positions) and $D_{\mathrm{SO}}$? A parallel-beam FBP (180 projections) is shown alongside for comparison. A full $2\pi$ fan scan measures every ray twice, so 360 fan projections carry roughly the same information as 180 parallel ones. RMSE is evaluated against the true phantom inside the central 90% circle of the FOV.

Helical CT and pitch

Clinical CT usually rotates the source while moving the patient table continuously. The source follows a helix around the patient, allowing a large volume to be covered in one breath-hold. Pitch is the table travel per rotation divided by the nominal width of all active detector rows:

$$\text{pitch} = \frac{\text{table travel per rotation}}{\text{total nominal detector width}}$$

A larger pitch covers more anatomy per unit time but samples the longitudinal direction less densely. At a fixed tube current it also generally reduces dose. The scanner interpolates measurements from the helix to reconstruct images at chosen positions.

The key steps in fan-beam reconstruction

The fan beam is the practical geometry that collects all projections in a single rotation. Each of its rays corresponds one-to-one to a parallel ray $(\theta, s)$ through the fan angle $\gamma = \arctan(u/D_{\mathrm{SO}})$, so rebinning lets parallel-beam FBP be reused as is. The direct fan FBP, built from cosine weighting, ramp filtering, and distance-weighted backprojection, yields an equivalent image. In the next chapter the detector grows to two dimensions. What happens, then, when the fan becomes a cone?

References

• Kak AC, Slaney M. Principles of Computerized Tomographic Imaging, Chapter 3. IEEE Press (1988).
• Crawford CR, King KF. Computed tomography scanning with simultaneous patient translation. Medical Physics 17, 967–982 (1990).
• Kalender WA et al. Spiral volumetric CT with single-breath-hold technique, continuous transport, and continuous scanner rotation. Radiology 176, 181–183 (1990).