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We understand from the previous article that the process of digitizing an optical image with a photographic sensor can be thought of as two subsequent operations:

filtering (convolution) of the optical image on the sensing plane by the pixel’s finite effective active area (aka pixel aperture);
point sampling the convolved image at a given fixed rate and position, often corresponding to the center of each pixel.

Both affect resolution in different ways: the former can be thought of as modifying continuously the analog optical image, as seen below right; the latter as possibly introducing interference (aliasing) into the result.

Figure 1. Digitizing an optical image corresponds to convolution with pixel aperture followed by Dirac delta sampling at the center of each pixel (red dots). Highly magnified images of two simulated stars separated by the Rayleigh limit: the stars are resolved after just the optics to the left; and unresolved after smoothing by an ideal square pixel with 100% Fill Factor to the right.

In this page I will explore how the act of digitizing that image – the process of sampling – fundamentally alters what we can resolve. In the next one we will discuss the impact on resolution of pixel-shift modes available in current mirrorless cameras. Continue reading The Effect of Sampling on Image Resolution →

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

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; $**$ 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.

Footprint-PSF3 — 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 in this example each pixel’s width is 5 linear units on the side.

Continue reading A Simple Model for Sharpness in Digital Cameras – Sampling & Aliasing →

Musings about Photography