MedicalVisualization

## Line Integral

With computer tomography an object is exposed to X-rays from varying directions to reconstruct the density of the corresponding volume.

The object’s density function is defined by $f(x,y,z)$ with the function values being interpreted as attenuation coefficients.

Then the intensity of a pixel on the radiograph is determined by the accumulated attenuation of the radiated energy on the corresponding ray segment through the volume.

The accumulated attenuation is equal to the line integral of the attenuations on each ray segment:

$I = I_0 e^{−\int f_{\mu}(s) \cdot ds}$

Discrete formulation of the line integral:

$I = I_0 e^{−\sum f_{\mu}(s) \cdot \Delta s}$

Observation: The line integral is view independent!

X-ray images from angles of 0 to 180 degrees are sufficient to reconstruct the density function!