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.

Sampling in the Spatial Domain

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.

In typical photographic sensors pixels are square and laid out in a rectangular grid as shown above.  Therefore the Dirac delta functions at the center of each pixel follow the same grid pattern, also known as a lattice or a two dimensional comb function.  Assuming a very large number of pixels, the two dimensional Dirac delta comb can be assumed to be infinitely long.

Sampling in the Frequency Domain

The 2D delta sampling grid can be expressed as follows in the spatial (left) and frequency domains (right):

(2)  

The Fourier transform of a 2D comb is a 2D comb.  In our case  a and b represent pixel pitch () and are the same so the deltas are in a rectangular grid  at cycle/pixel-pitch spacing in the frequency domain (c/p or cycle per pixel in short).

Convolutions in the spatial domain become multiplications in the frequency domain and vice versa.  So applying the spatial domain operations above in the frequency domain we obtain Equation (1).  The convolution of the perfect lens MTF and square pixel aperture MTF discussed in the previous article is the product of their individual MTFs:

Then sampling is accomplished by convolving in two dimensions the rectangular lattice of deltas one cycle per pixel pitch apart with the tent-looking combined lens+pixel area MTF in Figure 2a.  The result is many tent-like MTFs,  each centered on a delta function in the grid:

We are now in the discrete domain.  Sampling the continuous function may have introduced aliasing.

Modeling Aliasing

It is clear that if the reference lens+pixel 2D MTF has some energy above 0.5 c/p (the Nyquist-Shannon frequency) it will start interfering with its neighbors once convolved with the delta sampling grid: the center of each solid is only 1 c/p away.  This interference is called aliasing and it becomes more obvious when viewing Figure 3 in profile, projected against the X-Z plane:

We are normally used to seeing this information in the 0-1 c/p range only, as shown in Figure 2b above.  Note how energy at spatial frequencies above 0.5 c/p intermingle with those of their neighbours.  The result is that they are able to sneak back below Nyquist under an ‘alias’, masquerading as lower frequencies.  Ignoring phase in our simple model and because the convolved MTFs look like mirror images of each other, frequencies higher than Nyquist can be thought of as ‘folding’ back around 0.5 c/p.  Once this happens it is impossible to tell the real low frequencies from the folded aliased ones, which are then free to produce undesirable artifacts like stair stepping, false color and moiré in the final photograph.

Aliasing is the reason why we see that uptick near 0.5 c/p in the system MTF horizontal radial slice shown in Figure 2b.  If we ignore phase, it can be modeled by folding aliased frequencies above Nyquist back towards the origin and adding them to the unaliased model, as shown in Figure 5.

Anti Aliasing

The negative effect of aliasing can be controlled by filtering the original signal before sampling to limit energy captured above the Nyquist frequency.  This is the job of the Anti Aliasing filter that we will discussed in a future article.  There is always a trade-off because no filter is  perfect, so reducing the impact of frequencies above 0.5 c/p means necessarily also lowering some good frequencies below that – and perceived ‘sharpness’ with them.

Another way to reduce aliasing all other things being equal is to increase the sampling rate.  One of the properties of spatial-frequency duality is that narrower features in one domain become larger ones in the other (the a and b factors of equation 2 are in the numerator in one and in the denominator in the other).  Sampling at a smaller pitch spreads apart the 2D Dirac deltas in the frequency domain grid, therefore pushing the convolved MTF ‘tents’ further apart, reducing the chance of overlap.

Figure 4  also shows clearly why in order to be able to recover a signal perfectly it is necessary for contiguous MTF solids not to overlap, which means filtering away frequencies above the maximum desirable spatial frequency and sampling at least at twice that rate.  This is known as the Whittaker-Shannon sampling theorem.

Directional Slices of 2D MTFs

We know that the two dimensional system MTF of the sampled image is not rotationally symmetric because in typical photographic sensors the sampling grid has a rectangular (or square) layout.

Neighbours above and below or to the left and right are normally one cycle per pixel pitch apart.  Those diagonally across are however further away, c/p apart.  This suggests that in a typical photographic sensor there is less chance of aliasing when the detail being evaluated is in the 45 degree direction with respect to its origin, as can be gleaned from this central cutout of the 2D MTF in Figure 3:

In our f/16 example it can be seen that there is aliasing overlap in the horizontal (x-axis, c/p) or vertical (y-axis, c/p) directions, say going from (0,0) to (1,0) but that there is no overlap diagonally, say from (0,0) to (1,1).  This is quite obvious when plotting the two respective directional MTF radial slices on the same graph:

Therefore in this simple monochrome example there is aliasing in the vertical and horizontal direction, but not diagonally.

The Simplified Perfect Model

This concludes numerical verification of our simple 2D MTF model. Ignoring phase, it works as advertised in its current form based on a perfect lens and monochrome sensor with perfect square pixels of 100% fill factor:

(3)  

with the linear spatial frequency normalized for extinction: ; and the horizontal and vertical spatial frequency components so that ; the effective linear size of a perfect square pixel on the side directly related to linear fill factor; pixel pitch, the horizontal and vertical spacing of the centers of pixels as laid out on a rectangular grid in the sensor; and two dimensional convolution.

We can use this simplified model to start answering questions about the effects of diffraction and pixel size on the spatial resolution performance of our photographic equipment.  Next we will add antialiasing filters and simple aberrations to the model, then test it against real captures.

 

*The units of spatial frequency are described in detail in this article.