X-photons/matter Interactions (1/2)

• X-photons cross matter;
• During their path into any material, they can interact with matter.

Illustration of X-ray photon/matter interaction

1. Directly transmitted photons (no interaction);
2. Absorbed photons;
3. Scattered photons;
4. Absorbed scattered photons.

X-photons/matter Interactions (2/2)

For most X-rays imaging modalities, only directly transmitted photons are essential:

• Scattered photons decrease the image quality;
• Absorbed photons do not reach the detector;
• Scattered photons may be ignored (but not necessarily).

Illustration of X-ray photon/matter interaction

1. Directly transmitted photons (no interaction);
2. Absorbed photons;
3. Scattered photons;
4. Absorbed scattered photons.

Beer-Lambert Law (Attenuation Law)

Illustration of the Beer-Lambert law

• N_out(E) = N_in(E) e^{(-Σ_i μ_i(E,ρ,Z) L_p(i))}
  • N_in(E) the number of incident photons at energy E;
  • N_out(E) the number of transmitted photons of energy E;
  • μ_i the linear attenuation coefficient (in cm^-1) of the i^th object. It depends on:
    • E the energy of incident photons;
    • ρ the material density of the object;
    • Z the atomic number of the object material.
  • L_p(i) the path length of the ray in the i^th object.
• E_out = N_out(E) x E
  • E_out the energy received by the pixel, i.e. as recorded in the X-ray image.

Example (monochromatic case)

http://gvirtualxray.sourceforge.net/validation/validation_03/beer_lambert_law_monochromatic.php

Beer-Lambert Law in the polychromatic case

• There are more than one energy in the incident beam spectrum
• Just iterate over the energy channels:

E_out = Σ_j E_j x N_out(E_j)

E_out = Σ_j E_j x N_in(E_j) e^{(-Σ_i μ_i(E_j,ρ,Z) L_p(i))}

with j the j-th energy channel

Example (polychromatic case)

http://gvirtualxray.sourceforge.net/validation/validation_05/beer_lambert_law_polychromatic.php

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