Imaging Physics Laboratory

In x-ray imaging, pixel values represent effective attenuation measurements of an object in the beam and can be expressed by the following equation: 

N = N₀e^-ut

Where N is the number of detected photons, N0 is the number of photons incident on the object, u is the attenuation coefficient and is material specific and a function of energy,  tis the object thickness. When imaging two distinct materials, the equation becomes: 

N = N₀e^-(u_j^t_j^+u_a^t_a⁾

Only in limited situations with specific a priori knowledge can information about the thickness of both materials be discerned from a single measurement. 

        One alternative to recover additional information regarding the two materials is to make use of “dual energy” methods in x-ray imaging. Dual energy imaging exploits the differential energy dependence of two materials’ attenuation coefficients. Each pixel can be decomposed into two separate values for thickness or mass, one for each material under consideration.
A second x-ray exposure at a different energy yields a system of two equations and two unknowns: 

                                                                N₁ = N₁₀e^-(u_j^(E₁^)t_j^+u_a^(E₁^)t_a⁾

                                                                N₂ = N₂₀e^-(u_j^E₂^)t_j^+u_a^(E₂^)t_a⁾

        In principle, the thicknesses of each material can be determined by solving the system of equations.
An example of dual energy imaging in chest radiography is shown in the figure below:

 Defining the log signal as the negative natural logarithm of the ratio of the transmitted and incident number of photons, the system can be rewritten in matrix form as:

And inverted to solve for values t_f and t_a: