Now that we have the lens Point Spread Function it should be straight forward to reverse its blurring effect out of the image, right?
Deconvolution by Naive Division
Tag Archives: frequency domain
A Simple Model for Sharpness in Digital Cameras – Sampling & 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 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.
Continue reading A Simple Model for Sharpness in Digital Cameras – Sampling & Aliasing
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 magnitude of the normalized product of the Fourier Transform (FT) of the lens Point Spread Function by the FT of the pixel footprint (aperture), convolved with the FT of a rectangular grid of Dirac delta functions centered at each pixel:
![\[ MTF_{2D} = \left|\widehat{ PSF_{lens} }\cdot \widehat{PIX_{ap} }\right|_{pu}\ast\ast\: \delta\widehat{\delta_{pitch}} \]](https://i0.wp.com/www.strollswithmydog.com/wordpress/wp-content/ql-cache/quicklatex.com-3409f065a49bb47fd4618ed595be72fc_l3.png?resize=320%2C37&ssl=1 "Rendered by QuickLaTeX.com")
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.
The stage will be set in this first installment with a little background and perfect components. Later additional detail will be provided to take into account pixel aperture and Anti-Aliasing filters. Then we will look at simple aberrations. Next we will learn how to measure MTF curves for our equipment, and look at numerical methods to model PSFs and MTFs from the wavefront at the aperture. Continue reading A Simple Model for Sharpness in Digital Cameras – I
Linearity in the Frequency Domain
For the purposes of ‘sharpness’ spatial resolution measurement in photography cameras can be considered shift-invariant, linear systems when capturing scene detail of random size and direction such as one often finds in landscapes.
Shift invariant means that the imaging system should respond exactly the same way no matter where light from the scene falls on the sensing medium . We know that in a strict sense this is not true because for instance pixels tend to have squarish active areas so their response cannot be isotropic by definition. However when using the slanted edge method of linear spatial resolution measurement we can effectively make it shift invariant by careful preparation of the testing setup. For example the edges should be slanted no more than this and no less than that. Continue reading Linearity in the Frequency Domain
The Units of Spatial Resolution
Several sites for photographers perform spatial resolution ‘sharpness’ testing of a specific lens and digital camera set up by capturing a target. 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 popular slanted edge method, although their applicability is generic.
The Slanted Edge Method
My preferred method for measuring the spatial resolution performance of photographic equipment these days is the slanted edge method. It requires a minimum amount of additional effort compared to capturing and simply eye-balling a pinch, Siemens or other chart but it gives more, useful, accurate, quantitative information in the language and units that have been used to characterize optical systems for over a century: it produces a good approximation to the Modulation Transfer Function of the two dimensional camera/lens system impulse response – at the location of the edge in the direction perpendicular to it.
Much of what there is to know about an imaging system’s spatial resolution performance can be deduced by analyzing its MTF curve, which represents the system’s ability to capture increasingly fine detail from the scene, starting from perceptually relevant metrics like MTF50, discussed a while back.
In fact the area under the curve weighted by some approximation of the Contrast Sensitivity Function of the Human Visual System is the basis for many other, better accepted single figure ‘sharpness‘ metrics with names like Subjective Quality Factor (SQF), Square Root Integral (SQRI), CMT Acutance, etc. And all this simply from capturing the image of a slanted edge, which one can actually and somewhat easily do at home, as presented in the next article.