# Modulation Transfer Function on Type 18UL with references  Modulation Transfer Function (MTF) values are calculated according to Droege RT, Morin RL. "A practical method to measure the MTF of CT scanners" (Med Phys. 1982 Sep-Oct;9(5):758-60). Authors demonstrate how MTF is related to the standard distribution of pixel values inside a region taken from a bar pattern, with the following relations:

$MTF \left ( f \right ) = \begin{cases} \frac{\pi\sqrt{2}}{4} \cdot \frac{M \left ( f \right )}{M_0}, & \mbox{if } f \ge f_c/3 \\ \frac{\pi\sqrt{2}}{4M_0} \cdot \sqrt{M^2 \left ( f \right ) - \frac{M^2 \left ( 3f \right )}{9} - \frac{M^2 \left ( 5f \right )}{25} - \frac{M^2 \left ( 7f \right )}{49}}, & \mbox{if } f \ge f_c/11 \end{cases}$

where:

$M \left ( f \right ) = \sqrt{\sigma_D^2-N^2}$
$M_0 = |\mu_{bar}-\mu_{bkg}|\sqrt{\frac{n_{bar} \cdot n_{bkg}}{\left ( n_{bar} + n_{bkg}\right )}}$
$N = \frac{\left ( \sigma_{bar}^2 + \sigma_{bkg}^2 \right )}{2}$

and:

• $\sigma_D$: standard deviation of pixel values inside the bar pattern (detail) region
• $\mu_{bar}$: mean pixel value of the bars reference region
• $\mu_{bkg}$: mean pixel value of the background reference region
• $n_{bar}$: number of pixels of the bars reference region
• $n_{bkg}$: number of pixels of the background reference region
• $\sigma_{bar}$: standard deviation of pixel values inside the bars reference region
• $\sigma_{bkg}$: standard deviation of pixel values inside the background reference region

For frequencies outside the range given by the formulas, a linear interpolation is performed.