# Wavefront to PSF to MTF: Physical Units

In the last article we saw that the Point Spread Function and the Modulation Transfer Function of a lens could be easily obtained numerically by applying Discrete Fourier Transforms to its generalized exit pupil function twice in sequence.[1]

Obtaining the 2D DFTs is easy: simply feed MxN numbers representing the two dimensional complex image of the pupil function in its space to a fast fourier transform routine and, presto, it produces MxN numbers that represent the amplitude of the PSF on the sensing plane, as shown below for the pupil function of a perfect lens with a circular aperture and MxN = 1024×1024.

Simple and fast.  Wonderful.  Below is a slice through the center, the 513th row, zoomed in.  Hmm….  What are the physical units on the axes of displayed data produced by the DFT?

Less easy – and the subject of this article as seen from a photographic perspective.

# Aberrated Wave to Image Intensity to MTF

Goodman, in his excellent Introduction to Fourier Optics[1], describes how an image is formed on a camera sensing plane starting from first principles, that is electromagnetic propagation according to Maxwell’s wave equation.  If you want the play by play account I highly recommend his math intensive book.  But for the budding photographer it is sufficient to know what happens at the exit pupil of the lens because after that the transformations to Point Spread and Modulation Transfer Functions are straightforward, as we will show in this article.

The following diagram exemplifies the last few millimeters of the journey that light from the scene has to travel in order to smash itself against our camera’s sensing medium.  Light from the scene in the form of  field   arrives at the front of the lens.  It goes through the lens being partly blocked and distorted by it (we’ll call this blocking/distorting function ) and finally arrives at its back end, the exit pupil.   The complex light field at the exit pupil’s two dimensional plane is now   as shown below:

# Equivalence in Pictures: Focal Length, f-number, diffraction

Equivalence – as we’ve discussed one of the fairest ways to compare the performance of two cameras of different physical formats, characteristics and specifications – essentially boils down to two simple realizations for digital photographers:

1. metrics need to be expressed in units of picture height (or diagonal where the aspect ratio is significantly different) in order to easily compare performance with images displayed at the same size; and
2. focal length changes proportionally to sensor size in order to capture identical scene content on a given sensor, all other things being equal.

The first realization should be intuitive (future post).  The second one is the subject of this post: I will deal with it through a couple of geometrical diagrams.

# 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? Continue reading Deconvolution PSF Changes with Aperture

# What Is Exposure

When capturing a typical photograph light from one or more sources is reflected from the scene, reaches the lens, goes through it and eventually hits the sensing plane. Continue reading What Is Exposure