Chapter 3: Simple Backprojection (SBP)

Let us start with the most naive way back from the sinogram to an image. A measurement $p(\theta,s)$ tells us only how much attenuation the ray accumulated along its path, not where along the path it happened. So we smear each measured value back uniformly over the entire ray path. A single direction produces nothing but streaks. Summing over all directions, however, locations where the object actually was receive contributions from many rays and grow bright. This is simple backprojection (SBP). In symbols, the SBP image $b(x,y)$ is the sum of the measurements of all rays passing through the pixel $(x,y)$.

$$b(x, y) = \int_0^{\pi} p\bigl(\theta,\; x\cos\theta + y\sin\theta\bigr)\, d\theta$$

The SBP image $b(x,y)$ does not equal the original $\mu(x,y)$, though. Consider an object consisting of a single point. Each angle backprojects one straight line through the point, so with few angles the result is a star-shaped (radial) artifact. As the number of angles grows toward the continuous limit, the streaks merge smoothly, but the point keeps a tail that decays as $1/r$ from the center. In other words, SBP yields the true image convolved with a point spread function (PSF) of $1/r$.

$$b(x,y) = \mu(x,y) \;{*}{*}\; \frac{1}{r}, \qquad r = \sqrt{x^2 + y^2}$$

The 1/r point spread function of SBP. A systematic blur that no number of projections removes.

Simulation: number of projections vs. the SBP image

Use the projection-count slider to inspect the SBP image built from $k$ projections spread evenly over 0–180°. Start with the point phantom and increase the count 1 → 2 → 8. At first you see a star artifact of straight lines through the point. At 180 the star disappears, yet the point never sharpens back: it stays a blurred disk with a $1/r$ tail. With Shepp-Logan you can confirm that the result remains hazy and low in contrast. The "Scan" button plays an animation stacking up the projections one by one.

The blur left by simple backprojection

Simple backprojection just smears each measurement back along its ray and sums. The result is the true image convolved with $1/r$: the star artifacts vanish as projections are added, but the $1/r$ blur persists no matter how many rays are collected. The cause was the over-weighting of low spatial frequencies. The cure is to correct the projection profiles with a high-pass-boosting filter before backprojecting. The next chapter derives this filtered backprojection (FBP).

References

• Bracewell RN, Riddle AC. Inversion of fan-beam scans in radio astronomy. Astrophysical Journal 150, 427–434 (1967).
• Kak AC, Slaney M. Principles of Computerized Tomographic Imaging, Chapter 3. IEEE Press (1988).