# Measuring the MTF

Determining the MTF isn't completely straightforward. The MTF is calculated by performing a Fourier transform of the line spread function, and the line spread function is calculated by differentiating the edge response function, which is what we measure on an image.

Edge Response Function (ERF) (measured) ....*differentiate*:

→ Line Spread Function (LSF) ....*Fourier transform*:

→ Modulation Transfer Function (MTF).

The ERF is dead simple. It's just a plot of the image intensity across the edge of an angled block in an image. The differentiation of this ERF shape is obvious (or should be, to a physicist!). Why a Fourier transform is the thing to do to then get MTF vs *ν*, should also be obvious to you if you have looked at Fourier transforms.

Resources that describe MTF measurement in optics often quote a simple equation that looks like this:

modulation = (L_{max} - L_{min} ) / (L_{max} + L_{min})

where L_{max} is the maximum luminance (white lines) and L_{min} is the minimum (dark lines). This is then calculated for the image and the object an equation is used. So why do we bother with ERFs, differentiation, LSFs and Fourier transforms? Using the *modulation* equation on this page would be fine to calculate a MTF for a specific spatial frequency (which would be determined by the lines used). However we want to know the MTF across a *range* of spatial frequencies. It is possible to make images for a new set of lines for every point on an MTF vs *ν* graph, but this can be very time consuming.

Further reading on this topic:

Books: MRI From Picture to Proton p209-212, IPEM Report 80 p20-24