Tag Archives: antialiasing

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:

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

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A Simple Model for Sharpness in Digital Cameras – Aliasing

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 we now turn to the effect of the sampling interval on spatial resolution according to the guiding formula:

(1)   \begin{equation*} MTF_{Sys2D} = \left|(\widehat{ PSF_{lens} }\cdot \widehat{PIX_{ap} })\ast\ast\: \delta\widehat{\delta_{pitch}}\right|_{pu} \end{equation*}

The hats in this case mean the Fourier Transform of the relative component normalized to 1 at the origin (_{pu}), that is the individual MTFs of the perfect lens PSF, the perfect square pixel and the delta grid.

Sampling in the Spatial and Frequency Domains

Sampling is expressed mathematically as a Dirac delta function at the center of each pixel (the red dots below).

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Figure 1. Left, 1a: A highly zoomed (3200%) image of the lens PSF, an Airy pattern, projected onto the imaging plane where the sensor sits. Pixels shown outlined in yellow. A red dot marks the sampling coordinates. Right, 1b: The sampled image zoomed at 16000%, 5x as much, because each pixel’s width is 5 linear units on the side.

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A Simple Model for Sharpness in Digital Cameras – I

The next few posts will describe a linear spatial resolution model that can help a photographer better understand the main variables involved in evaluating the ‘sharpness’ of photographic equipment and related captures. I will show numerically that the combined spectral frequency response (MTF) of a perfect AAless monochrome digital camera and lens in two dimensions can be described as the normalized multiplication of the Fourier Transform (FT) of the lens Point Spread Function by the FT of the (square) pixel footprint, convolved with the FT of a rectangular grid of Dirac delta functions centered at each  pixel, as better described in the article

    \[ MTF_{2D} = \left|(\widehat{ PSF_{lens} }\cdot \widehat{PIX_{ap} })\ast\ast\: \delta\widehat{\delta_{pitch}}\right|_{pu} \]

With a few simplifying assumptions we will see that the effect of the lens and sensor on the spatial resolution of the continuous image on the sensing plane can be broken down into these simple components.  The overall ‘sharpness’ of the captured digital image can then be estimated by combining the ‘sharpness’ of each of them. Continue reading A Simple Model for Sharpness in Digital Cameras – I

The Units of Spatial Resolution

Several sites perform spatial resolution ‘sharpness’ testing of imaging systems for photographers (i.e. ‘lens+digital camera’) and publish results online.  You can also measure your own equipment relatively easily to determine how sharp your hardware is.  However comparing results from site to site and to your own can be difficult and/or misleading, starting from the multiplicity of units used: cycles/pixel, line pairs/mm, line widths/picture height, line pairs/image height, cycles/picture height etc.

This post will address the units involved in spatial resolution measurement using as an example readings from the slanted edge method.

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