A Simple Model for Sharpness in Digital Cameras – AA

This article will discuss a simple frequency domain model for an AntiAliasing (or Optical Low Pass) Filter, a hardware component sometimes found in a digital imaging system[1].  The filter typically sits right on top of the sensing plane and its objective is to block as much of the aliasing and moiré creating energy above the Nyquist spatial frequency while letting through as much as possible of the real image forming energy below that, hence the low-pass designation.

Downsizing Box 4X
Figure 1. The blue line indicates the pass through performance of an ideal anti-aliasing filter presented with an Airy PSF (Original): pass all spatial frequencies below Nyquist (0.5 c/p) and none above that. No filter has such ideal characteristics and if it did its hard edges would result in undesirable ringing in the image.

In consumer digital cameras it is often implemented  by introducing one or two birefringent plates in the sensor’s filter stack.  This is how Nikon shows it for one of its DSLRs:

d800-aa1
Figure 2. Typical Optical Low Pass Filter implementation  in a current Digital Camera, courtesy of Nikon USA (yellow displacement ‘d’ added).

Birefringent plates are typically made of a material such as Lithium Niobate.  They split normally incident beams of light into two, the new beam displaced by a known distance (d).  In some implementations a single plate is used, providing antialiasing action in one direction only (e.g. D610, a7II, XA1, etc.).  In others two plates are used, one for the horizontal and one for the vertical direction as shown in Figure 2 (e.g. D800, 5DS, K5, etc.).  As pixel pitch decreases below 5 microns sometimes no AA filter  is used at all.  In this case the manufacturer is hoping/expecting that the low pass function will be provided by physical constraints placed on the increasingly smaller sensing elements (e.g. lens blur, diffraction, defocus, motion/vibration blur etc.).

2 and 4 Dot Beam Splitting AAs

In the more generic case of Figure 2 a single spot of light (an impulse) reaching the filter stack is first split into two horizontally by the H-aligned birefringent plate.  The resulting two ‘dots’ are then split vertically by a second V-aligned birefringent plate, presumably by the same displacement (d).  This arrangement is called a four-dot beam splitter:

4-dot-beam-splitter
Figure 3. Effect of a 4-dot beam splitter on the sensing plane in two dimensions; and just the x-axis profile.

Assuming linearity and space invariance we can use transfer function theory to model this anti-aliasing filter configuration in isolation.  The birefringent plates act on one axis at the time so we can represent what happens to impulse \delta(x,y) as it is split horizontally with displacement (d) along the x axis separately and in one dimension.  The ideal transfer function is simply the sum of two delta functions separated by distance (d), with half the energy of the original impulse[2]:

    \[ H = \frac{1}{2}[\delta(x) + \delta(x-d)] \]

The relative spectral frequency response in the horizontal direction (x) of such a filter is the normalized magnitude of its Fourier transform[3].  It represents the MTF of an ideal 2-dot beam splitter in the direction of the split, with spatial frequency f in cycles per the same units as displacement d (e.g. microns or pixel pitch):

(1)   \begin{equation*} MTF_{AA_{2dot}} = |cos(2\pi \frac{d}{2} f)| \end{equation*}

For a 4-dot beam splitter the horizontal split along the x axis is followed by the same function rotated 90 degrees in the vertical direction along the y axis. The transfer function of the ideal anti-aliasing optical low pass filter in two dimensions is then

(2)   \begin{equation*} MTF_{AA_{4dot}} =  |cos(2\pi \frac{d_x}{2} f_x)| |cos(2\pi \frac{d_y}{2} f_y)| \end{equation*}

with spatial frequencies f_{x} and f_{y} in cycles per the same units as displacement d (e.g. microns or pixels).

Spatial Frequency Properties of Birefringent AA

The shape of the relative MTF in one dimension produced by formula (1) is obviously a cosine, starting with a value of 1 at zero frequency and decreasing monotonically until hitting a first zero at

(3)   \begin{equation*} f_{MTF0_1} = \frac{1}{2d} \end{equation*}

in units of cycles/pixel if d is expressed in pixels.  Because the MTF represents the magnitude of the cosine, the curve then appears rectified and bounces back, producing a mirror image of its trajectory up to the zero.  Here it is in isolation for a displacement of 0.7 pixels (or 0.35×2 half displacements as also sometimes seen in the literature)

birefringent-aa
Figure 4. Birefringent AA MTF cosine for a displacement of 0.7 pixels, with a zero at 1/2d = 0.714 cycles/pixel

Note the strong attenuation past the Nyquist frequency (0.5 c/p), the expected zero at f = \frac{1}{2\cdot0.7} = 0.7143 c/p and the rectified bounce beyond that.  Approaching 1 cycle/pixel there is usually very little energy left  in the signal of a real image – so less need for low pass filtering – because of the combination of the effects of diffraction and pixel aperture discussed in previous articles.

Note also the undesired attenuation in the desirable frequencies below Nyquist, with this fairly typical optical low pass filter reducing contrast of frequencies near 0.5 c/p by more than half.  This is the reason why many current digital cameras are not equipped with an antialiasing filter, favoring real and imaginary ‘sharpness’ at the cost of the possibility of more visible aliasing and moiré.  Sharper filters could be implemented but probably at the expense of cost and complexity.

Here is what the spatial frequency response of an ideal 4-dot beam splitter looks like in two dimensions, with the same displacement d = 0.7 pixels in both the horizontal and vertical directions

aa-2d-solid
Figure 5. Two dimensional MTF of 4-dot beam splitter anti-aliasing filter, for displacement d = 0.7 pixels in both the x and y directions, frequency units of cycles/pixel

Below the same anti-aliasing filter is shown directly from above. Note that it makes little difference to MTF values when reading off a directional radial slice through the origin just a few degrees from the horizontal or vertical axis – but as the angle increases things change.  For instance in the case of d = 0.7 pixels the zero for a 9 degree slice through the origin is at 0.72 c/p vs 0.71 c/p for the zero on the horizontal axis.  But the zero for a 45 degree radial slice occurs at 1.01 c/p.

aa-2d
Figure 6. Same as Figure 5 but seen from above

Can the Strength of an AA Filter Be Measured?

Indeed it can, as originally shown by Frans van den Bergh[1].  Here for instance is an example based on two Canon DSLRs, identical in setup other than the fact that the 5DS has an anti-aliasing filter while the 5DSR does not.  The aggregate MTFs shown were measured from the green channels of slanted edges in DPR’s studio scene raw captures using MTF Mapper.   The edges were slanted about 9 degrees off the horizontal sensing plane.

mtf-5ds-and-5dsr-g-hcrop
Figure 7. Aggregate MTFs measured on same sensor, same lens, same setup: one camera with AA, one without.

The drop in response (0.25 vs 0.3 c/p at MTF50, 0.125 vs 0.25 c/p at Nyquist) and the zero around 0.73 c/p caused by the anti-aliasing filter are quite obvious in the 5DS.  Every digital camera with AA I have measured over the last few years has shown zeros in the range of 0.62-0.78 cycles/pixel (0.8-0.64 pixel displacement d).

On the other hand some cameras like the D610 appear to have an AA in one direction only.  Here for instance is the ratio of the MTF curves derived from the vertical and horizontal edges  in the same DPR studio scene raw capture.  Assuming the same aberrations and rotational symmetry the ratio should pretty well be one and the same.  Instead it looks like a cosine, the MTF of a birefringent AA.  So the D610 and its siblings apparently only filter the signal in the vertical direction (ignore values in frequencies above the first null, things become unreliable there because we are dividing two very small noisy numbers together).

d610-aa
Figure 8. ‘Measured’ MTF of the D610’s AA 2-dot Beam Splitter: it apparently only filters the signal in the vertical direction.

In case you are curious, here is instead the Sony a6300 which definitely does not have an AA in either direction, the ratio between the MTF curves off the horizontal and vertical edges being pretty well unity throughout (ignoring a little noise).

a6300-g-ch-dpr-100-200-iso

Adding to the Model

In conclusion we have one more component that, ignoring phase, can be multiplied into our simplified model: a birefringent anti-aliasing filter per equation (2) above

    \[ MTF_{AA_{4dot}} =  \left|cos(2\pi \frac{d_x}{2} f_x)\right| \left|cos(2\pi \frac{d_y}{2} f_y)\right| \]

with spatial frequencies f_{x} and f_{y} in cycles per the same units as displacement d (e.g. microns or pixels).

Defocus is next.

Notes and References

1. Frans van den Bergh introduced me to this subject a few years ago through this excellent blog article.
2. There is a good discussion of this subject in Modulation Transfer Function in Optical and Electro-Optical Systems. Glenn D Boreman. SPIE Press 2001. p. 40.
3. See for instance here for why two impulses in the spatial domain Fourier Transform into a cosine in the frequency domain

 

2 thoughts on “A Simple Model for Sharpness in Digital Cameras – AA”

  1. So is A6300’s AA filter so weak that it is as if without OLPF?

    If that is the case, is not this the best of both worlds, where the AA filter prevents moirè but does not compromise sharpness?

    By the way, do you have an idea if A6500 the same or any different?

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