This article is about specifying the units of the Discrete Fourier Transform of an image and the various ways that they can be expressed. This apparently simple task can be fiendishly unintuitive.
The image we will use as an example is the familiar Airy Pattern from the last few posts, at f/16 with light of mean 530nm wavelength. Zoomed in to the left in Figure 1; and as it looks in its 1024×1024 sample image to the right:
Having shown that our simple two dimensional MTF model is able to predict the performance of the combination of a perfect lens and square monochrome pixel with 100% Fill Factor we now turn to the effect of the sampling interval on spatial resolution according to the guiding formula:
(1) 
The hats in this case mean the Fourier Transform of the relative component normalized to 1 at the origin (
), that is the individual MTFs of the perfect lens PSF, the perfect square pixel and the delta grid; 
represents two dimensional convolution.
While exposed a pixel sees the scene through its aperture and accumulates energy as photons arrive. Below left is the representation of, say, the intensity that a star projects on the sensing plane, in this case resulting in an Airy pattern since we said that the lens is perfect. During exposure each pixel integrates (counts) the arriving photons, an operation that mathematically can be expressed as the convolution of the shown Airy pattern with a square, the size of effective pixel aperture, here assumed to have 100% Fill Factor. It is the convolution in the continuous spatial domain of lens PSF with pixel aperture PSF shown in Equation (2) of the first article in the series.
Sampling is then the product of an infinitesimally small Dirac delta function at the center of each pixel, the red dots below left, by the result of the convolution, producing the sampled image below right.
Continue reading A Simple Model for Sharpness in Digital Cameras – Sampling & Aliasing
Is MTF50 a good proxy for perceived sharpness? In this article and those that follow MTF50 indicates the spatial frequency at which the Modulation Transfer Function of an imaging system is half (50%) of what it would be if the system did not degrade detail in the image painted by incoming light.
It makes intuitive sense that the spatial frequencies that are most closely related to our perception of sharpness vary with the size and viewing distance of the displayed image.
For instance if an image captured by a Full Frame camera is viewed at ‘standard’ distance (that is a distance equal to its diagonal), it turns out that the portion of the MTF curve most representative of perceived sharpness appears to be around MTF90. On the other hand, when pixel peeping the spatial frequencies around MTF50 look to be a decent, simple to calculate indicator of it, assuming a well set up imaging system in good working conditions. Continue reading MTF50 and Perceived Sharpness