# Deconvolution PSF Changes with Aperture

We have  seen in the previous post how the radius for deconvolution capture sharpening by a Gaussian PSF can be estimated for a given setup in well behaved and characterized camera systems.  Some parameters like pixel aperture and AA strength should remain stable for a camera/prime lens combination as f-numbers are increased (aperture is decreased) from about f/5.6 on up – the f/stops dear to Full Frame landscape photographers.  But how should the radius for generic Gaussian deconvolution  change as the f-number increases from there?

We will use as an example the D4+85mm:1.8G which was shown at f/5.6. in the last post.  In order to maximize curve fit to the measured MTF curve we chose the radius of Gaussian PSF  deconvolution so that the two curves would intersect in the middle at MTF50.  Below you can see the modeled Total System MTF as a black line and the MTF for a Gaussian PSF as the dotted yellow line.

At f/5.6, for the two curves to intersect in the middle (the MTF50 of the measured MTF, 0.29 cycles per pixel) the Gaussian PSF needs to have a radius/standard deviation of 0.646 pixels, as determined by the formula in the previous post.

Let’s now model what Gaussian PSF radius would be necessary to achieve our MTF50 intersect criterion at increasing f-numbers, assuming that other parameters remain unchanged from their values at f/5.6 (a stretch as far as lens blur is concerned, I know):

The  radius (standard deviation) of the Gaussian PSF increases fairly linearly with the f-number at these apertures in our simple theoretical model.

There is only one problem: as the f-number is increased the shape of the modeled Total MTF curve starts to look less and less like a Gaussian because the increasingly strong diffraction component becomes more and more dominant in shaping the overall curve.  Diffraction does not look like a Gaussian MTF.  This makes a Gaussian PSF alone less and less suitable for deconvolution.  Depicted below is f/16.

Recall that when we divide (deconvolve) dissimilar curves in the frequency domain, spatial frequencies in our images get amplified or attenuated in a way totally disconnected from reality:

It looks like deconvolution by a Gaussian PSF alone is no longer suitable at f/16 (and earlier).  Perhaps by deconvolving first with an Airy PSF to cover diffraction followed by a second deconvolution with a weaker Gaussian PSF would be better.  The yellow dotted line below represents an Airy PSF for f/16 combined with a Gaussian PSF of radius 0.85 pixels:

A bit better, but that area above 0.4 cycles per pixels fast approaching zero in the denominator does not bode well, certainly requiring further filtering  to even things out.

So we see that at least in theory for the D4+85mm:1.8G a Gaussian PSF alone may not be the best solution for Capture Sharpening at all camera/lens settings.  Introducing an Airy PSF helped.  As would applying a low pass filter.

Many programs/plug-ins that do deconvolution have the ability to specify different PSFs and apply low pass filters.  For instance Photoshop Smart Sharpen (more accurate ticked) has settings for Gaussian or Lens (aka Airy) Blur.  So in the f/16 example above  one could use a two pass Smart Sharpen strategy: once to ‘Remove’ Lens Blur and once Gaussian Blur, with appropriately different radii.  But each application of Smart Sharpen is so noisy as to discourage multi-pass usage.  Wouldn’t it be better to combine the PSFs and Remove them once only, applying appropriate filtering?

In deconvolution plug-ins Low Pass functions are often controlled by settings called Softness, Suppress, Damping .   Or they can be applied separately in a different layer or program.  However I can’t help but think that deconvolution controls in plug-ins aimed at photographic capture sharpening need to get much more explicit,  flexible and sophisticated to exit a niche and  become truly useful.

For instance, what is one to do with AA-less sensors, which do not show gaussian MTFs from images captured at their sharpest apertures with good technique?  Future article.