# How to derive the full width half maximum of a point spread function from a Modulation Transfer Function (MTF) assessment at Nyquist frequency?

I was given a table with Modulation Transfer Function (MTF) assessments at Nyquist frequency, and was tasked to apply the corresponding Point Spread Function (PSF) to an image to simulate a sensor.

Starting from the PSF, it is easy to derive the Full Width Half Maximum to parametrize a Gaussian kernel. Is there a way to derive the width of the kernel directly from the MTF assessment at Nyquist frequency?

• Can you define all the initialisms/acronyms in your question to help the reader? – honeste_vivere Oct 31 '16 at 17:22

Because the modulation transfer function and point spread function are related by a Fourier transform their widths, as defined by variances, will obey a Heisenberg uncertainty principle type relation: $$\sigma_{\mathrm{PSF}} \sigma_{\mathrm{MTF}} \ge \frac{1}{2},$$ where the lower bound is obtained for Gaussian functions.
If the width needed is not the standard deviation, but the functions are Gaussian, then it is possible to adapt the inequality. If the width is related to the standard deviation as $w = c \sigma$, for some numerical constant $c$, then the equality becomes: $$w_{\mathrm{PSF}} w_{\mathrm{MTF}} \ge \frac{c^2}{2},$$ so you'll need to figure out the constant of proportionality that relates the desired width to the standard deviation.