In this article we shall find that the effect of a Bayer CFA on the spatial frequencies and hence the ‘sharpness’ captured by a sensor compared to those from a corresponding monochrome imager can go from nothing to halving the potentially unaliased range based on the chrominance content of the image projected on the sensing plane and the direction in which the spatial frequencies are being stressed.
A Little Sampling Theory
We know from Goodman and previous articles that the sampled image ( ) captured in the raw data by a typical current digital camera can be represented mathematically as the continuous image on the sensing plane ( ) multiplied by a rectangular lattice of Dirac delta functions positioned at the center of each pixel:
with the functions representing the two dimensional grid of delta functions, sampling pitch apart horizontally and vertically. To keep things simple the sensing plane is considered here to be the imager’s silicon itself, which sits below microlenses and other filters so the continuous image is assumed to incorporate their as well as pixel aperture’s effects. Continue reading Bayer CFA Effect on Sharpness→
In the last article we saw that the Point Spread Function and the Modulation Transfer Function of a lens could be easily obtained numerically by applying Discrete Fourier Transforms to its generalized exit pupil function twice in sequence.
Obtaining the 2D DFTs is easy: simply feed MxN numbers representing the two dimensional complex image of the pupil function in its space to a fast fourier transform routine and, presto, it produces MxN numbers that represent the amplitude of the PSF on the sensing plane. Figure 1a shows a simple case where pupil function is a uniform disk representing the circular aperture of a perfect lens with MxN = 1024×1024. Figure 1b is the resulting PSF.
Simple and fast. Wonderful. Below is a slice through the center, the 513th row, zoomed in. Hmm…. What are the physical units on the axes of displayed data produced by the DFT?
Less easy – and the subject of this article as seen from a photographic perspective.
Goodman, in his excellent Introduction to Fourier Optics, describes how an image is formed on a camera sensing plane starting from first principles, that is electromagnetic propagation according to Maxwell’s wave equation. If you want the play by play account I highly recommend his math intensive book. But for the budding photographer it is sufficient to know what happens at the exit pupil of the lens because after that the transformations to Point Spread and Modulation Transfer Functions are straightforward, as we will show in this article.
The following diagram exemplifies the last few millimeters of the journey that light from the scene has to travel in order to smash itself against our camera’s sensing medium. Light from the scene in the form of field arrives at the front of the lens. It goes through the lens being partly blocked and distorted by it (we’ll call this blocking/distorting function ) and finally arrives at its back end, the exit pupil. The complex light field at the exit pupil’s two dimensional plane is now as shown below:
This post will continue looking at the spatial frequency response measured by MTF Mapper off slanted edges in DPReview.com raw captures and relative fits by the ‘sharpness’ model discussed in the last few articles. The model takes the physical parameters of the digital camera and lens as inputs and produces theoretical directional system MTF curves comparable to measured data. As we will see the model seems to be able to simulate these systems well – at least within this limited set of parameters.
The following fits refer to the green channel of a number of interchangeable lens digital camera systems with different lenses, pixel sizes and formats – from the current Medium Format 100MP champ to the 1/2.3″ 18MP sensor size also sometimes found in the best smartphones. Here is the roster with the cameras as set up:
The series of articles starting here outlines a model of how the various physical components of a digital camera and lens can affect the ‘sharpness’ – that is the spatial resolution – of the images captured in the raw data. In this one we will pit the model against MTF curves obtained through the slanted edge methodfrom real world raw captures both with and without an anti-aliasing filter.
With a few simplifying assumptions, which include ignoring aliasing and phase, the spatial frequency response (SFR or MTF) of a photographic digital imaging system near the center can be expressed as the product of the Modulation Transfer Function of each component in it. For a current digital camera these would typically be the main ones:
We now know how to calculate the two dimensional Modulation Transfer Function of a perfect lens affected by diffraction, defocus and third order Spherical Aberration – under monochromatic light at the given wavelength and f-number. In digital photography however we almost never deal with light of a single wavelength. So what effect does an illuminant with a wide spectral power distribution, going through the color filter of a typical digital camera CFA before the sensor have on the spatial frequency responses discussed thus far?
Spherical Aberration (SA) is one key component missing from our MTF toolkit for modeling an ideal imaging system’s ‘sharpness’ in the center of the field of view in the frequency domain. In this article formulas will be presented to compute the two dimensional Point Spread and Modulation Transfer Functions of the combination of diffraction, defocus and third order Spherical Aberration for an otherwise perfect lens with a circular aperture.
Spherical Aberrations result because most photographic lenses are designed with quasi spherical surfaces that do not necessarily behave ideally in all situations. For instance, they may focus light on slightly different planes depending on whether the respective ray goes through the exit pupil closer or farther from the optical axis, as shown below:
This series of articles has dealt with modeling an ideal imaging system’s ‘sharpness’ in the frequency domain. We looked at the effects of the hardware on spatial resolution: diffraction, sampling interval, sampling aperture (e.g. a squarish pixel), anti-aliasing OLPAF filters. The next two posts will deal with modeling typical simple imperfections in the system: defocus and spherical aberrations.
Defocus = OOF
Defocus means that the sensing plane is not exactly where it needs to be for image formation in our ideal imaging system: the image is therefore out of focus (OOF). Said another way, light from a distant star would go through the lens but converge either behind or in front of the sensing plane, as shown in the following diagram, for a lens with a circular aperture:
This article is about specifying the units of the Discrete Fourier Transform of an image and the various ways that they can be expressed. This apparently simple task can be fiendishly unintuitive.
The image we will use as an example is the familiar Airy Disk from the last few posts, at f/16 with light of mean 530nm wavelength. Zoomed in to the left in Figure 1; and as it looks in its 1024×1024 sample image to the right:
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:
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.
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).
Now that we know from the introductory article that the spatial frequency response of a typical perfect digital camera and lens can be modeled simply as the product of the Modulation Transfer Function of the lens and pixel area, convolved with a Dirac delta grid at cycles-per-pixel spacing
While perusing Jim Kasson’s excellent Longitudinal Chromatic Aberration tests I was impressed by the quantity and quality of the information the resulting data provides. Longitudinal, or Axial, CA is a form of defocus and as such it cannot be effectively corrected during raw conversion, so having a lens well compensated for it will provide a real and tangible improvement in the sharpness of final images. How much of an improvement?
This is a vast and complex subject for which I do not have formal training. In this and the previous article I present my thoughts on how MTF50 results obtained from raw data of the four Bayer CFA channels off a uniformly illuminated neutral target captured with a typical digital camera through the slanted edge method can be combined to provide a meaningful composite MTF50 for the imaging system as a whole1. Corrections, suggestions and challenges are welcome. Continue reading COMBINING BAYER CFA MTF Curves – II→
In fact the question is more generic than that. Smaller format lens designers try to compensate for their imaging system geometric resolution penalty (compared to a larger format when viewing final images at the same size) by designing ‘sharper’ lenses specifically for it, rather than recycling larger formats’ designs (feeling guilty APS-C?) – sometimes with excellent effect. Are they succeeding? I will use mFT only as an example here, but input is welcome for all formats, from phones to large format.
Over the last couple of years I’ve been using Frans van den Bergh‘s excellent open source MTF Mapper to measure the Modulation Transfer Function of imaging systems off a slanted edge target, as you may have seen in these pages. As long as one understands how to get the most out of it I find it a solid product that gives reliable results, with MTF50 typically well within 2% of actual in less than ideal real-world situations (see below). I had little to compare it to other than to tests published by gear testing sites: they apparently mostly use a commercial package called Imatest for their slanted edge readings – and it seemed to correlate well with those.
Then recently Jim Kasson pointed out sfrmat3, the matlab program written by Peter Burns who is a slanted edge method expert who worked at Kodak and was a member of the committee responsible for ISO12233, the resolution and spatial frequency response standard for photography. sfrmat3 is considered to be a solid implementation of the standard and many, including Imatest, benchmark against it – so I was curious to see how MTF Mapper 0.4.1.6 would compare. It did well.
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.
You have obtained a raw file containing the image of a slanted edge captured with good technique. How do you get the MTF curve of the camera and lens combination that took it? Download and feast your eyes on open source MTF Mapper by Frans van den Bergh. No installation required, simply store it in its own folder.
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 immensely 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 Point Spread Function of the camera/lens system in the direction perpendicular to the edge.
Much of what there is to know about a system’s spatial resolution performance can be deduced by analyzing such a curve, starting from the perceptually relevant MTF50 metric, discussed a while back. And all of this simply from capturing the image of a black and white slanted edge, which one can easily produce and print at home.
Why Raw? The question is whether one is interested in measuring the objective, quantitative spatial resolution capabilities of the hardware or whether instead one would prefer to measure the arbitrary, qualitatively perceived sharpening prowess of (in-camera or in-computer) processing software as it turns the capture into a pleasing final image. Either is of course fine.
My take on this is that the better the IQ captured the better the final image will be after post processing. In other words I am typically more interested in measuring the spatial resolution information produced by the hardware comfortable in the knowledge that if I’ve got good quality data to start with its appearance will only be improved in post by the judicious use of software. By IQ here I mean objective, reproducible, measurable physical quantities representing the quality of the information captured by the hardware, ideally in scientific units.
You want to measure how sharp your camera/lens combination is to make sure it lives up to its specs. Or perhaps you’d like to compare how well one lens captures spatial resolution compared to another you own. Or perhaps again you are in the market for new equipment and would like to know what could be expected from the shortlist. Or an old faithful is not looking right and you’d like to check it out. So you decide to do some testing. Where to start? Continue reading How Sharp are my Camera and Lens?→
Is MTF50 a good proxy for perceived sharpness? It turns out that the spatial frequencies that are most closely related to our perception of sharpness vary with the size and viewing distance of the displayed image.
For instance if an image captured by a Full Frame camera is viewed at ‘standard’ distance (that is a distance equal to its diagonal) the portion of the MTF curve most representative of perceived sharpness appears to be around MTF90. Continue reading MTF50 and Perceived Sharpness→