Computed tomography as an inverse problem

Example

Let $\mu \colon \R^2 \to [0, \infty)$ be the X-ray attenuation coefficient of a slice of a body, a function with compact support. An X-ray beam that crosses the slice along a line $L$ enters with intensity $I_0$ and leaves with intensity $I$. By the Beer–Lambert law of absorption, $I = I_0 \exp\left(-\int_L \mu \, ds\right)$, where $ds$ is arc length on $L$. So each measurement gives the number $$-\log \frac{I}{I_0} = \int_L \mu \, ds = (R\mu)(L),$$ the value of the Radon transform $R\mu$ at the line $L$. A scanner measures $R\mu(L)$ for many lines $L$, and the task is to recover $\mu$ from these values.

This is an inverse problem: the forward map $\mu \mapsto R\mu$ is known and linear, and one must invert it from data. When $R\mu(L)$ is known for all lines, $\mu$ is determined, and the Radon inversion formula reconstructs it. The inversion is unstable, because it involves a derivative, a "ramp filter" $\abs{\omega}$ in the Fourier variable. So small errors at high frequencies in the data become large errors in $\mu$. With data from only a limited range of directions the instability is much worse.