# What is the point spread function and optical transfer function and what uses are they in image processing

So I seem to really be struggling with the concepts of the point spread function (PSF) and optical transfer function (OTF) and what they are used for in image processing.

Everything I seem to google on the concepts seems to come back with telescopes and microscopes or I seem to get information about Delta functions and impulse response of a system and how this relates to convolution, but nothing seems to tell me about what PSF or OTF actually do or measure.

From what I have gathered so far is the following.

1. You start with a linear time invariant (LTI) system you then apply a signal which can be broken down into Diracdelta functions ($$\delta(x)$$), aka the impulse. The (LTI) then outputs a response aka the impulse response ($$h(x)$$) and according to some web sites this is the PSF. Now I can follow the maths to a degree and see how the convolution between the impulse and the impulse response, but what I can wrap my head around is what the PSF is actually useful for.

2. Why is it useful for imaging systems? From a video that I glimpsed at on you tube the PSF spreads the Dirac delta function so in terms of image processing is this effectively telling us how much blur will be in the image itself? Is the the only thing the PSF tells us?

3. Moving on to the OTF, so this is the Fourier transform of the PSF and that as far as I have gotten, as I don't fully understand what the PSF dose for an imaging system. I really cant figure what the OTF does for an imaging system.

4. What's more confusing is now, I keep thinking of the OTF as a kinda of filter function when I see the convolution equation in the frequency domain, dose the OTF itself play a role in frequency filtering if so how?

Fundamentally could someone explain the concepts of PSF and OTF and what they actually tell me about the system in terms of image processing.

PSF (point spread function) is the spatial-domain impulse response of the optical imaging system assuming it's modeled as an LSI (linear shift invariant) system.

In principle, PSF can be obtained by inputting an impulse (small bright point) to the system, and measuring the response where the point is typically spread into a blurry circle; hence the name point-spread function.

Also for CRT monitors, PSF can be obtained by sending a single dot to the display, and observing the brightness spread on the screen.

Once you obtain the impulse response (or PSF) of an LSI system, then you can mathematically compute the response of the system to any input by convolving the PSF with the input signal. This concept lies at the heart of the signal processing, and that's why PSF is a very important characterisation of optical systems.

OTF is the Fourier transform of the PSF, and therefore carries the same information, but, stated in frequency-domain instead.

Three main points for your reference.

• PSF does NOT change the density (LPMM) of input image. Therefore, PSF is LSI.
• Modulation is powered by multiplication which could change frequency, so it is NOT LTI.
• PSF is powered by summation in a logical way which does NOT change frequency, so it is LTI/LSI.

OTF supports us to make up "Modulation Transfer Function" which is a great equation to verify the resolution. I believe it is a great start to figure out "blur" and "deblur".