Chapter 2: Forward Projection and the Sinogram

The previous chapter dealt with the line integral $p$ of a single ray. Keeping the angle $\theta$ fixed and sweeping the detector position $s$ uniformly yields a one-dimensional projection profile $p(\theta,\cdot)$ as a function of $s$. Because all rays are parallel, this acquisition scheme is called parallel-beam projection. The transform that maps an object $\mu(x,y)$ to its line integrals over all angles and positions is the Radon transform. In the equation below, the $\delta$ function picks out the integral along the line $x\cos\theta + y\sin\theta = s$.

$$p(\theta, s) = \iint \mu(x, y)\, \delta(x\cos\theta + y\sin\theta - s)\, dx\, dy$$

Collecting projections while rotating $\theta$ from 0° to 180° and stacking them as a 2D image, with $s$ horizontal and $\theta$ vertical, gives the sinogram. For parallel beams $p(\theta+180^\circ, s) = p(\theta, -s)$, so 180° already contains a full revolution of information. The sinogram is the raw data a CT scanner actually measures, and reconstruction means recovering $\mu(x,y)$ from it.

$$s(\theta) = x_0\cos\theta + y_0\sin\theta = r\cos(\theta - \varphi), \quad r = \sqrt{x_0^2 + y_0^2}$$

Simulation: scanning and the growing sinogram

Moving the angle slider updates the ray bundle over the phantom (the current projection direction) together with the projection profile $p(\theta,\cdot)$ at the top right. Press "Scan" to advance the angle automatically and watch the sinogram at the bottom right fill in row by row from the top. Start with the point phantom and let the scan run: exactly one sine curve appears in the sinogram. Switching to Shepp-Logan reveals many sinusoids overlapping.

From forward projection to the inverse problem

A sinogram stacks the projection profiles from every angle into a 2D dataset. A single point in the object traces a sine curve within it. So far we have mapped an object to its measurements. The next chapter reverses the process with simple backprojection, which returns each measurement along its ray but produces a blurred image.

References

• Radon J. On the determination of functions from their integral values along certain manifolds. IEEE Transactions on Medical Imaging 5, 170–176 (1986; English translation of the 1917 paper).
• Kak AC, Slaney M. Principles of Computerized Tomographic Imaging. IEEE Press (1988).