The plot in Figure P6.5(a) shows the input energy spectrum of a polychromatic x-ray sour...

The plot in Figure P6.5(a) shows the input energy spectrum of a polychromatic x-ray source and the linear attenuation coefficient of a material as a function of photon energy.

a) Sketch the energy spectrum of the x-ray beam after it passes through the material. Explain your sketch.

(b) Suppose that an object is imaged using a 1G CT scanner. In Figure P6.5(b) there are two different positions of the source and detector while imaging this same object. Explain why the two measurements should ideally be the same but are actually different in practice.

(c) The fact that the two measurements are different causes artifacts in CT reconstruction. One way to reduce these artifacts is to acquire two measurements with two different input x-ray spectra, one having

higher mean energy than the other. Describe how you might obtain these two measurements using the same x-ray tube.

(d) This so-called dual-energy CT setup requires calibration. Four measurements are made on two test objects (Figure P6.6) using the two input spectra, yielding measurements given in the table below. The

values in the table are line integrals (not x-ray intensities or numbers of photons). Explain why

(f) For calibration, coefficients are to be determined so that measurements using the high and low spectra can be combined in a linear equation to yield what is expected from a monoenergetic 75 keV x-ray source. Given your result from part (e), find the coefficients such that

(g) For an arbitrary object, your dual-energy CT scanner will obtain two sinograms Write a mathematical expression for the reconstructed object, ??(object, 75 keV), given these sinograms.