News: Imatest 23.1 (March 2023) (available in the Imatest Pilot program). New methods for calculating camera information capacity, Noise Power Spectrum (NPS), Noise Equivalent Quanta (NEQ), and Ideal observer SNR (SNRi) from slantededge patterns are now available. 
The basic premise of this work is that Information capacity is a superior Shannon information capacity can also be calculated from images of the Siemens star, introduced in 2020.
The Siemens star method was presented at the Electronic Imaging 2020 conference, and published in the paper, “Measuring camera Shannon information capacity from a Siemens star image” and announced in the Imatest News Post: Measuring camera Shannon information capacity with a Siemens star image. The 2020 white paper, Camera information capacity: a key performance indicator for Machine Vision and Artificial Intelligence systems, is a more readable introduction to the Siemens Star measurement. This page presents new calculations of information capacity and additional measurements from slanted edges. 
Introduction – Edge variance calculation – Noise image calculation – NPS – NEQ – SNRi – Object visibility – Noise Autocorrelation
Meaning of Information capacity – Summary – Links – Binning noise
Instructions page: Acquiring and framing – Running the MTF module – Results – Edge/MTF plot –
Edge/noise plot – 3D plot and Total information capacity –
Nothing like a challenge! There is such a metric for electronic communication channels— one that quantifies the maximum amount of information that can be transmitted through a channel without error. The metric includes sharpness and noise (grain in film). And a camera— or any digital imaging system— is such a channel.
The metric, first published in 1948 by Claude Shannon* of Bell Labs [1,2], has become the basis of the electronic communication industry. It’s called the Shannon channel capacity or Shannon information capacity C, and has a deceptively simple equation [3]. (See the Wikipedia page on the ShannonHartley theorem for more detail.)
\(\displaystyle C = W \log_2 \left(1+\frac{S}{N}\right) = W \log_2 \left(\frac{S+N}{N}\right) = \int_0^W \log_2 \left( 1 + \frac{S(f)}{N(f)} \right) df\)
W is the channel bandwidth, S(f) is the average signal energy (the square of signal voltage; proportional to MTF(f)^{2}), and N(f) is the average noise energy (the square of the RMS noise voltage), which corresponds to grain in film. It looks simple enough (only a little more complex than E = mc^{2 }), but it’s not easy to apply.
*Claude Shannon was a genuine genius. The article, 10,000 Hours With Claude Shannon: How A Genius Thinks, Works, and Lives, is a great read. There are also a nice articles in The New Yorker and Scientific American. The 29minute video “Claude Shannon – Father of the Information Age” is of particular interest to me it was produced by the UCSD Center for Memory and Recording Research. which I frequently visited in my previous career.
This page describes how to calculate information capacity C from images of slantededges, Imatest’s most widelyused test image, which (thanks to a recent discovery) allows signal and noise to be calculated from the same location, resulting in a superior measurement of image quality. The earlier (2020) Siemens star method is described in Shannon information capacity from Siemens stars.
Introduction to the new measurements
This page describes several Information capacityrelated measurements introduced in Imatest 23.1, released in March 2023. These measurements take advantage of newlydiscovered properties of slanted edges– Imatest’s most widely used patterns for measuring MTF. Many have been used for medical imaging, but are unfamiliar elsewhere, in part because they were difficult to perform. We describe two convenient new measurement methodologies, both of which use the slanted edge.
 The Edge variance calculation conveniently calculates camera information capacity.
 The Noise image calculation, which uses a different approach to calculate information capacity and several additional image quality factors (NPS, NEQ, SNRi, and more).
This page introduces the new calculations and presents detailed equations and algorithms. The Instructions page introduces the new calculations, shows how to obtain them, then presents Key Results. 
Motivation — We need to obtain information capacity from images that have very different types of image processing.
 Minimally or uniformlyprocessed images, converted from raw to TIFF or PNG files. By “minimallyprocessed”, we mean no sharpening, no noise reduction, at most a simple gamma curve (no complex tonal response curves or local tone mapping). A color matrix may be applied (it affects noise and SNR, but not MTF). When available, these images give the most reliable information capacity measurements.
 JPEG files from cameras, which usually have bilateral filters — filters that sharpen images near contrasty features like edges but blur them to reduce noise elsewhere — making it appear that the image contains more information than it actually has. This improves conventional SNR measurements (made from flat patches) while actually removing information.
In late October 2022, we discovered how to extract signaldependent noise from slantededges using an overlooked capability of the ISO 12233 slantededge algorithm, described briefly below and in more detail in the white paper, Measuring Camera Information Capacity from Slantededges.
In March 2023, we discovered a second overlooked capability that allows Imatest to calculate a camera’s Noise Power Spectrum (NPS), Noise Equivalent Quanta (NEQ), and Ideal Observer SNR (SNRi), but only works with minimally / uniformlyprocessed images.
The slantededge method of calculating MTF, which has been part of the ISO 12233 standard since 2000,
 takes each scan line y_{l }(x) in a slantededge Region of Interest (ROI),
 finds its center,
 fits a polynomial curve to the centers, then
 depending on the relation between the line center and the curve, adds the line contents to one of four bins.
 The bins are then interleaved, resulting in a 4× oversampled averaged edge, which has lower noise than the individual scan lines.
 Calculates MTF (Modulation Transfer Function, usually synonymous with Spatial Frequency Response), by differentiating the averaged edge to obtain the Line Spread Function, LSF, windowing the LSF, then Fouriertransforming it. MTF is the absolute value of the Fourier Transform normalized to 1 (or 100%) at zero frequency.
The new information capacity measurements take advantage of overlooked capabilities of the slantededge method.
Why two calculations (Edge variance and Noise image)?The Edge variance was developed first, starting in late October 2022. It was presented at the Electronic Imaging 2022 conference. The Noise image was developed in February 2023, about a month after the conference. It measures more image quality parameters than the Edge variance method. We currently recommend the Noise image method because we believe the camera information capacity measurement is slightly more accurate. The reason: it calculates the Noise Power Spectrum, while the Edge variance assumes the NPS is flat (white). On the other hand, The Edge variance method can provide useful (if imperfect) results for bilateralfiltered images (most JPEGs from consumer cameras), while the Noise image method is only recommended for uniformly or minimallyprocessed images. 
A: Edge variance calculation
Summary: Sum the squares of the scan lines to obtain the edge variance, then use it to calculate information capacity.
The Edge variance Information capacity calculation is described in detail in the white paper, New SlantedEdge Image Quality Measurements: the Edge variance calculation, and in the paper presented at Electronic Imaging 2023, “Measuring Camera Information Capacity with Slanted Edges“. The concise descriptions on this page omit many of the details in the two linked documents.
The calculation starts with images of slanted edges, (Original ROI, below), typically made from a 4:1 contrast chart (chart contrasts between 2:1 and 10:1 are acceptable). In addition to the binning/summing described above, the squares of the scan lines are summed. This allows the variance of the edge, σ_{s}^{2}(x), which is equivalent to the signaldependent noise power, N(x), to be calculated.
\(\displaystyle \sigma_s^2(x) = \frac{1}{L} \sum_{l=0}^{L1} (y_l(x)\mu_s(x))^2 = \frac{1}{L}\sum_{l=0}^{L1} y_l^2(x) \ – \left(\frac{1}{L}\sum_{L=0}^{L1} y_l(x) \right)^2 \)
Noise power for the ShannonHartley Equation N(x) = σ_{s}^{2}(x) and voltage σ_{s}(x) are important because many images— including most JPEGs from consumer cameras— have bilateral filters, which sharpen the image (boosting noise) near sharp areas like edges, and blurs it (to reduce visible noise) elsewhere. This obscures the noise at edges, which is critical to the performance — and information capacity — of the system. The new technique makes signaldependent noise near the edge visible so that it can be used in the information capacity calculation. It is also highly convenient.
The selection of N depends on the image processing. Two major classes have been identified.
 Uniformly or minimallyprocessed images, often TIFFs converted from raw files (raw→TIFF) without bilateral filtering, i.e., they either have no or uniform sharpening or noise reduction. Most cameras intended for Machine Vision/Artificial Intelligence fall into this category.
Since noise can be a very rough function of x, a large region size is required for a stable value of N. We average over all values of x in the ROI.
\(N_{uniform} = \text{mean}(\sigma_s^2(x))\) for all values of x in the ROI.  Bilateralfiltered images include most JPEG images from consumer cameras.
Bilateral filters sharpen images near contrasty features such as edges, but blur them (to reduce noise) elsewhere. This causes a noise peak near the edge (below, left). The blurring improves SignaltoNoise Ratio (SNR), but it removes information. Because of this, noise near the edge can dominate camera performance, and should be strongly weighted in calculating N. We have long known about the noise peak, but until the present method was developed, there was no easy way to observe or measure it (or detect bilateral filtering).
For calculating information capacity C, we use the square of the voltage, σ, at the peak, smoothed (with a rectangular kernel of length PW20/2) to remove jaggedness. \(N_{bilateral} = \sigma^2_{peak}\). This is a somewhat arbitrary choice that produces reasonably consistent results. This method also works with uniformlyprocessed images, but results are less consistent.
Imatest lets you select the calculation of noise N: it can be N_{uniform}, N_{bilateral} , or automatically detected depending on the presence of a peak.
Edge noise voltage for compact camera @ ISO 100. Left: Bilateralfiltered incamera JPEG; Right Unsharpened TIFF from raw.
The xaxis is the original pixel location of the 4× oversampled signal.
Signal power for the ShannonHartley Equation The mean signal amplitude for a uniformlydistributed signal of peaktopeak amplitude V_{PP} is \(V_{avg}(f) = V_{PP} MTF(f) / \sqrt{12}\) — a reasonable number to use for the information capacity calculation using the ShannonHartley equation, shown above, which actually uses the average signal power, \(S_{avg}(f) = V^2_{PP} MTF(f)^2 / 12\).
After removing newlydiscovered binning noise, selecting noise power calculation, and adjusting the signal level from the edge, which is a square wave, to be more representative of an “average’ signal, numbers are entered into the ShannonHartley equation (above) to calculate the information capacity, for the chart contrast.
Bandwidth W is always 0.5 cycles/pixel (the Nyquist frequency). Signals above Nyquist do not contribute to the information content; they can reduce it by causing aliasing — spurious low frequency signals like Moiré that can interfere with the true image. Frequencydependence comes from MTF(f).
S_{avg}(f), N, and W are entered into the ShannonHartley equation to obtain information capacity C.
\(\displaystyle C = \int^{0.5}_0{\log_2\left(1+\frac{S_{avg}(f)}{N}\right)}df \ \approx \sum_{i=0}^{0.5/\Delta f} {\log_2\left(1+\frac{S_{avg}(i\Delta f)}{N}\right)} \Delta f \)
The key results are
C_{4} is the direct result of measuring 4:1 contrast ratio slantededges. It is calculated from the Shannon Hartley equation, using several assumptions (that the signal is uniformly distributed over the peaktopeak measurement and the noise power spectral density (NPD) is flat). C_{4} is a special case of C_{n}, for a n:1 contrast ratio (with ISO standard 4:1 strongly recommended) C_{n} is sensitive to chart contrast ratio and exposure, making it interesting for measuring performance as a function of exposure but less robust than ideal for calculating a camera’s maximum information capacity.
C_{max} is a much more stable measurement of the maximum information capacity for the camera starting with C_{4} (for the 4:1 contrast chart). It is also insensitive to exposure, at least for linear sensors, where noise is a known function of signal voltage.
Here are some key results of the Edge variance method.
Line Spread Function (LSF) and signaldependent noise σ from
eSFR ISO image converted from raw with minimal processing
B. Noise image calculation
Summary: Subtract a lownoise reverseprojected / debinned ROI image from the original image to obtain a noise image, which can be used to calculate Noise Power Spectrum (NPS) and several additional measurements.
Measurement  Description 
Noise Power Spectrum, NPS(f)  NPS was implicitly assumed to be constant (white noise) in the Edge variance method. 
Noise Equivalent Quanta, NEQ(f) and NEQ_{info}(f) 
measures of frequencydependent signaltonoise ratio (SNR). \(NEQ(f) = \mu^2\ MTF^2(f) / NPS(f)\text{, where }\mu = V_{mean}\) has been used for quantifying medical image quality, but are much less familiar in general imaging. NEQ(f) is equivalent to the number of quanta detected by the sensor when photon shot noise is dominant. It is appropriate for calculating Digital Quantum Efficiency (DQE), when the density of quanta reaching the image sensor is known. NEQ_{info}(f) is derived from \(\mu = V_{PP} / \sqrt{12}\), making it wellsuited for calculating information capacity C_{NEQ}. 
Information capacities C_{4NEQ} and C_{maxNEQ} 
correspond to C_{4} and C_{max} from the Edge variance method, but are derived from NEQ_{info}(f). They are close, but not identical. 
Ideal observer SignaltoNoise Ratio, SNRi  From Skorka and Kane [9], “The Ideal Observer is a Bayesian decision maker that maximizes the statistical precision of a hypothesis test with two possible outcomes.” SNRi as we present is, is a metric of the detectability of small objects (squares or rectangles), typically of low contrast. 
Noise autocorrelation  The inverse Fourier transform of the Noise Voltage Spectrum. Related to sensor electrical crosstalk. 
This method involves inverting the ISO 12233 binning procedure. Noting that the 4× oversampled edge was created by interleaving the contents of 4 bins, we apply an inverse of the binning algorithm to set the contents of each scan line to its corresponding bin (Inverse binned… ROI, below). Since the inversebinned image is nearly noiseless, we can create a noise image by subtracting the inversebinned image from the original image. This image is shown, adjusted to make the mean (zero) value middle gray, as the Noise image ROI, below.
The 4× oversampled averaged edge, described above, was created by adding each scan line in the original ROI image (below, left) to one of four interleaved bins, each of which contains an averaged (noisereduced) signal. It can be deinterleaved (debinned or reverseprojected; the nomenclature isn’t final) by filling each line in a new image with the averaged signal of the corresponding interleave. This creates a lownoise replica of the original image (belowmiddle).
A noise image can be created by subtracting the reverseprojected image from the original image, correct for nonuniformity along the direction of the edge. The three images are shown below. The noise image (belowright), which has a mean value of 0, has been lightened and contrastboosted for display. The three images are linear: a gamma curve has been applied for display.
Original ROI  Inversebinned / deinterleaved / reverseprojected ROI 
Noise image ROI 
These images allow several key image quality parameters to be calculated, including Noise Power Spectrum and Noise Equivalent Quanta, wellknown in medical imaging systems, and described in an excellent review paper by Ian Cunningham and Rodney Shaw [4]. These measurements are not wellknown outside of medical imaging, largely because they have been difficult to measure.
Noise Power Spectrum (NPS)
(the square of the Noise (voltage) spectrum) is calculated by taking the 2D Fourier transform of the noise ROI (Region of Interest) and noting that the initial 2D spectrum has zero frequency at the center of the image. A 1D Noise Power Spectrum, NPS_{1}, is calculated by dividing the 2D spectrum into several annular regions (the number depends on the size of the ROI), then taking the average noise power of each region. This transformation allows calculations to be performed in one dimension (rather than two) under the assumption that the vertical and horizontal MTFs are close.
The relationship between NPS and the variance of the noise image is given in equations (3) and (8) of Cunningham and Shaw [4], which we have reduced to one dimension and with the integration limits changed from {∞,∞} to {0, f_{Nyq}}, where f_{Nyq} = Nyquist frequency = 0.5 cycles/pixel.
\(\displaystyle \sigma^2 = \int_0^{f_{Nyq}} NPS(f) df \)
The 1D Fourier transform described above must be scaled to be consistent with the above equation.
\(\displaystyle NPS(f) = \frac{NPS_1(f)\ \sigma^2}{\displaystyle \int_0^{f_{Nyq}} NPS_1(f) df }\)
Noise Equivalent Quanta (NEQ)
is a wellknown figure of merit in medical imaging, but is unfamiliar in general imaging. It is described in a 2016 paper by Brian Keelan [5]. Essentially, it is a frequencydependent SignaltoNoise (power) Ratio. Units are the equivalent number of quanta that would generate the measured SNR when photon shot noise is dominant.
\(\displaystyle NEQ(f) = \frac{\mu^2 MTF^2(f)}{NPS(f)}\)
where the mean linear signal, μ, can be defined in either of two ways, depending on how NEQ is to be interpreted.
If NEQ is to be used for calculating DQE (Digital Quantum Efficiency), where \(DQE(f) = NEQ(f) / \overline{q}\), then μ should be the mean value of the linearized signal voltage in the original image. Measuring DQE requires a separate measurement of the mean number of quanta reaching each pixel. We may add this in the future.
Getting familiar with the meaning and use of NEQ will take some time. Characterization of imaging performance in differential phase contrast CT compared with the conventional CT: Spectrum of noise equivalent quanta NEQ(k) by Tang et. al. is an excellent example of how NEQ is used in medical imaging: it has real technical depth.
Information capacity from NEQ:
A special form of NEQ, NEQ_{info}(f), calculated using \(\mu = V_{PP}/\sqrt{12}\), is used to calculate information capacity, C_{NEQ,} from a special case of the ShannonHartley equation. NEQ_{info} is not plotted.
\(\displaystyle C_{NEQ} = \int_0^W \log_2 \left( 1 + NEQ_{info}(f)\right) df\)
where bandwidth W is the camera’s Nyquist frequency, \(W = f_{Nyq} = 0.5 \text{ Cycles/Pixel}\). [Author’s note: I thought I’d discovered this connection, but it’s in papers on PET scanners and Digital Mammography by Christos Michail et. al. [6,7] Not papers anybody outside medical imaging is like to chance upon.]
Ideal Observer SNR (SNRi)
is a measure of the detectability of small objects. It is described in papers by Paul Kane [8] and Orit Skorka and Paul Kane [9]. There is a problem with the equations for SNRi in these two papers. They are presented in both one and two dimensions, even though the Fourier transform of the object to be detected is expressed in two dimensions.
In [8], the equation is presented in one dimension,
\(\displaystyle SNRi^2 = \int_0^{f_{Nyq}}{\frac{G(\nu)^2 MTF^2(\nu)}{NPS(\nu)} }d\nu \)
where spatial frequency ν has units of Cycles/Pixel, and the linearized signal is normalized to have a maximum value of 1.
In [9], it is presented in two dimensions.
\(\displaystyle SNRi^2 = \int \int \left( \frac{\mu^2 \Delta S^2(\nu_x,\nu_y) MTF^2(\nu_x,\nu_y) }{NPS(\nu_x,\nu_y)} \right) d\nu_xd\nu_y \)
We assume that G(ν)^{2} is the same as μ^{2}ΔS^{2}(ν_{x},v_{y}) and that \(\nu_x = \nu_y = \nu\), which makes MTF and NPS essentially onedimensional.
The object to be detected is typically a rectangle of dimensions w × kw, where k = 1 (for a square) or 4 for a 1×4 aspect ratio rectangle. Its amplitude (for now) is the peaktopeak voltage of the slanted edge (shown in the Voltage statistics figure, above), \(\Delta Q = V_{PP}\) which typically has a 4:1 contrast ratio.
\(\displaystyle \Delta g(x,y) = \Delta Q \cdot \text{rect}(x/w) \cdot \text{rect}(y/kw) \ , \ \text{ where rect}(x) = 1 \text{ for } 1/2 < x < 1/2 \text{ ; 0 otherwise.}\)
G(υ) is the Fourier transform of the object to be detected, Δg(x,y). It can be expressed in one or two dimensions.
In two dimensions, \(\displaystyle G_{2D}(\nu_x,\nu_y) = kw^2 \Delta Q \frac{\sin(\pi w \nu_x)}{\pi w \nu_x} \frac{\sin(\pi kw \nu_y)}{\pi k w \nu_y}\) , where, as we noted, \(\nu_x = \nu_y = \nu\)
In one dimension, \(\displaystyle G_{1D}(k \nu) = kw \Delta Q \frac{\sin(\pi kw \nu)}{\pi k w \nu}\)
To deal with ambiguity about how to evaluate the integral, we assume that the correct equation is the double integral, where the inner part is evaluated first, then the outer part.
\(\displaystyle SNRi^2 = \int \left[ \int \left( \frac{G_{1D}(\nu)^2 MTF^2(\nu) }{NPS(\nu)} \right) d\nu \right] \left( \frac{G_{1D}(k \nu)^2 MTF^2(\nu) }{NPS(\nu)} \right) d \nu \)
We choose this in preference to single integral we previously tried. \(\displaystyle SNRi^2 = \int \left( \frac{G_{2D}(\nu)^2 MTF^2(\nu) }{NPS(\nu)} \right) d\nu \)
1D analysis — Because of the conflicts and ambiguities of 2D analysis, we have also considered the simpler 1D analysis.
\(\displaystyle SNRi^2 = \int \left( \frac{G_{1D}(\nu)^2 MTF^2(\nu) }{NPS(\nu)} \right) d\nu \)
As of March 2023, we are working to resolve the question of which equation is best. The good news is that they correlate well with each other. They differ mostly in absolute magnitude.
SNRi is displayed for each color channel for w from 1 to 40 in increments of approximately the square root of w (1, 1.4, 2, …). The images below are for squares with widths w = 1, 2, 3, 4, 7, 10, 14, 20.
Object visibility
Noise autocorrelationThis plot is still in the R&D phase. The author (NLK) added it to examine his hypothesis that the noise power spectrum (and autocorrelation) indicate the amount of electrical crosstalk of image sensor when the effects of demosaicing and fixedpattern noise are removed (not the case for the image on the right) and the primary noise source is photon shot noise. The idea behind the hypothesis is that light incident on the sensor is entirely uncorrelated, so that if there were no crosstalk the noise would be white. This image on the right was whitebalanced. The curve is the inverse Fourier transform of the noise spectrum, based on the author’s limited understanding of the WienerKhinchin theorem. 

The image on the right is not WhiteBalanced. The red channel has a larger autocorrelation distance than the other channels, as we would expect. Click on the image to enlarge it. A similar autocorrelation plot can also be obtained from a flat field image in the Image Statics module. The relatively large autocorrelation (>1.3) at large distances (>4 pixels) is a definite concern. 
Meaning of Shannon information capacity
(The Appendix in the white paper, Measuring Camera Information Capacity with slantededges,
has a concise definition of information.)
In electronic communication channels the information capacity is the maximum amount of information that can pass through a channel without error, i.e., it is a measure of channel “goodness.” The actual amount of information depends on the code— how information is represented. But although coding is integral to data compression (how an image is stored in a file), it is not relevant to digital cameras. What is important is the following hypothesis:
Hypothesis: Perceived image quality (assuming a welltuned image processing pipeline) and also the performance of machine vision and Artificial Intelligence (AI) systems, is proportional to a camera’s information capacity, which is a function of MTF (sharpness), noise, and artifacts arising from demosaicing, clipping (if present), and data compression.
I stress that this statement is a hypothesis— a fancy mathematical term for a conjecture. It agrees with my experience and with numerous measurements, but it needs more testing and verification. Now that information capacity can be conveniently calculated with Imatest, we have an opportunity to learn more about it.
The information capacity, as we mentioned, is a function of both bandwidth W and signaltonoise ratio, S/N.
In texts that introduce the Shannon capacity, bandwidth W is often assumed to be the halfpower frequency, which is closely related to MTF50. Strictly speaking, W log_{2}(1+S/N) is only correct for white noise (which has a flat spectrum) and a simple low pass filter (LPF). But digital cameras have varying amounts of sharpening, which can result in response curves with response that deviate substantially from simple LPF response. For this reason we use the integral form of the ShannonHartley equation: \(\displaystyle C = \int_0^W \log_2 \left( 1 + \frac{S(f)}{N(f)} \right) df = \int_0^W \log_2 \left(\frac{S(f)+N(f)}{N(f)} \right) df \) S and N are mean values of signal and noise power; they are not directly tied to the camera’s dynamic range (the maximum available signal). For this reason, we reference calculations of C to the contrast ratio of the chart used for the measurement, most frequently C_{4} for 4:1 contrast charts that conform to the ISO 12233 standard. For Siemens star analysis, we this equation was altered to account for the twodimensional nature of pixels by converting it to a double integral, then to polar form, than back to one dimension. But this wasn’t necessary for slantededges, which are already one dimensional. 
The beauty of both the Siemens Star and Slantededge methods is that signal power S and noise power N are calculated from the same location: important because noise is not generally constant over the image.
Summary
 Shannon information capacity C has long been used as a measure of the goodness of electronic communication channels. It specifies the maximum rate at which data can be transmitted without error if an appropriate code is used (it took nearly a halfcentury to find codes that approached the Shannon capacity). Coding is not an issue with imaging. Rodney Shaw’s paper from 1962 [10] is a particularly good example of measuring C for photographic film— it wasn’t easy back then.
 C is ordinarily measured in bits per pixel. The total capacity is \( C_{total} = C \times \text{number of pixels}\).
 The channel must be linearized before C is calculated, i.e., an appropriate gamma correction (signal = pixel level gamma, where gamma ~= 2 for images in standard color spaces such as sRGB or Adobe RGB) must be applied to obtain correct values of S and N. The value of gamma (close to 2) can be determined from runs of any of the Imatest modules that analyze grayscale step charts: Stepchart, Colorcheck., Color/Tone, Multitest, SFRplus, or eSFR ISO. But in most cases it can be determined from the edge image if the chart contrast is entered and Use for MTF is checked.
 We hypothesize that C can be used as a figure of merit for evaluating camera quality, especially for machine vision and Artificial Intelligence cameras. (It doesn’t directly translate to consumer camera appearance because they have to be carefully tuned to reach their potential, i.e., to make pleasing images). It provides a fair basis for comparing cameras, especially when used with images converted from raw with minimal processing.
 Imatest calculates the Shannon capacity C for the Y (luminance; 0.212*R + 0.716*G + 0.072*B) channel of digital images, which approximates the eye’s sensitivity. It also calculates C for the individual R, G, and B channels as well as the C_{b} and C_{r} chroma channels (from YC_{b}C_{r}).
 Shannon capacity has not been used to characterize photographic images because it was difficult to calculate and interpret. But now it can be calculated easily, its relationship to photographic image quality is open for study.
 Since C is a new measurement, we are interested in working with companies or academic institutions who can verify its suitability for Artificial Intelligence systems.
Note: A slantededge information capacity measurement used prior to Imatest 2020, used primarily to obtain total information capacity from Siemens star measurements, has been deprecated completely because it was not sufficiently accurate. 
Links (more links in the White Paper)
 C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J., vol. 27, pp. 379–423, July 1948; vol. 27, pp.
623–656, Oct. 1948.  C. E. Shannon, “Communication in the Presence of Noise”, Proceedings of the I.R.E., January 1949, pp. 1021.
 Wikipedia – Shannon Hartley theorem has a frequency dependent integral form of Shannon’s equation that is applied to both Imatest’s sine pattern and slanted edge Shannon information capacity calculation.
 I.A. Cunningham and R. Shaw, “Signaltonoise optimization of medical imaging systems”, Vol. 16, No. 3/March 1999/pp 621632/J. Opt. Soc. Am. A
 Brian W. Keelan, “Imaging Applications of Noise Equivalent Quanta” in Proc. IS&T Int’l. Symp. on Electronic Imaging: Image Quality and System Performance XIII, 2016, https://doi.org/10.2352/ISSN.24701173.2016.13.IQSP213.
 Michail C, Karpetas G, Kalyvas N, Valais I, Kandarakis I, Agavanakis K, Panayiotakis G, Fountos G., Information Capacity of Positron Emission Tomography Scanners. Crystals. 2018; 8(12):459. https://doi.org/10.3390/cryst8120459.
 Christos M. Michail, Nektarios E. Kalyvas, Ioannis G. Valais, Ioannis P. Fudos, George P. Fountos, Nikos Dimitropoulos, Grigorios Koulouras, Dionisis Kandris, Maria Samarakou, Ioannis S. Kandarakis, “Figure of Image Quality and Information Capacity in Digital Mammography”, BioMed Research International, vol. 2014, Article ID 634856, 11 pages, 2014. https://doi.org/10.1155/2014/634856.
 Paul J. Kane, “Signal detection theory and automotive imaging”, Proc. IS&T Int’l. Symp. on Electronic Imaging: Autonomous Vehicles and Machines Conference, 2019, pp 271 – 278, https://doi.org/10.2352/ISSN.24701173.2019.15.AVM027.
 Orit Skorka, Paul J. Kane, “Object Detection Using an Ideal Observer Model”, IS&T Int’l. Symp. on Electronic Imaging: Autonomous Vehicles and Machines, 2020, pp 411 – 417, https://doi.org/10.2352/ISSN.24701173.2020.16.AVM041.
 R. Shaw, “The Application of Fourier Techniques and Information Theory to the Assessment of Photographic Image Quality”, Photographic Science and Engineering, Vol. 6, No. 5, Sept.Oct. 1962, pp.281286. Reprinted in “Selected Readings in Image Evaluation,” edited by Rodney Shaw, SPSE (now SPIE), 1976. A fascinating and difficult calculation of information capacity of photographic film. Available for download
 X. Tang, Y. Yang, S. Tang, Characterization of imaging performance in differential phase contrast CT compared with the conventional CT: Spectrum of noise equivalent quanta NEQ(k), Med Phys. 2012 Jul; 39(7): 4467–4482. Published online 2012 Jun 29. doi: 10.1118/1.4730287.
Appendix 1. Binning noise
Binning noise, which has identical statistics to quantization noise, is a recentlydiscovered artifact of the ISO 12233 binning algorithm. It is largest near the image transition — where the Line Spread Function \(LSF(x) = d\mu_s(x)/dx\) is maximum, and it can affect information capacity measurements. It appears because the individual scan lines are added to one of four bins, based on a polynomial fit to the center locations of the scan lines, which is a continuous function.
Assume that n identical signals μ_{s}(x) are binned over an interval {Δ/2, Δ/2}, where Δ = 1 in the 4× oversampled output of the binning algorithm (noting that Δ = (original pixel spacing)/4). If there were no binning noise, we would expect the binning noise power σ_{Bnoise}^{2} to be zero. However, the values of μ_{s}(x_{k}) are summed at uniformlydistributed locations x_{k} over the interval Δ, so they take on values
\(\displaystyle \mu_k = \mu_s(x_k) = \mu_s(x_0+\delta) = \mu_s(x_0) + \delta\ \frac{d\mu(x)}{dx} = \mu_s(x_0) + \delta\ LSF(x)\)
for Line Spread Function LSF(x). Noting that δ is uniformly distributed over {1/2, 1/2} we apply the equation for the variance of a uniform distribution (similar to quantization noise) to get
\(\sigma_{Bnoise}^2(x) = LSF^2(x)\ \sigma^2_{Uniform} = LSF^2(x)/12 \ \ \ \ \text{ or }\ \ \ \ \sigma_{Bnoise} = LSF(x)/\sqrt{12}\).
Although this equation involves some approximations, we have had good success calculating the corrected noise, \(\sigma_s^2(\text{corrected}) = \sigma_s^2 – \sigma^2_{Bnoise}\). Binning noise has no effect on conventional MTF calculations.
Edge with binning noise Binning noise removed
from a raw image from a Micro FourThirds camera, ISO 100, converted to TIFF with minimal processing
Binning noise also affects JPEG files with bilateral filtering (nonuniform sharpening). Removing it is important for robust calculations.
Appendix 2. SNRi information capacity —

SNRi 1D 
Note that C_{SNi } appears to approach a constant value for large w, and the equation for C_{SNi }uses G(ν)^{2}. The absolute value bars (…) are redundant.
\(\displaystyle G(\nu)^2 =\frac{ \Delta Q^2 \sin^2(\pi w \nu) }{(\pi \nu)^2}\) where \(\sin^2(x) = (1\cos(2x))/2\).
\(\displaystyle G(\nu)^2 =\frac{ \Delta Q^2 (1\cos(2\pi w \nu)) }{2 (\pi \nu)^2} \)
Note that in the limit as w → ∞, the cosine term in the above equation go through multiple cycles, and averages to 0 inside the integral. This enables G_{∞}^{2} to be substituted for G(ν)^{2} inside the integral, where
\(\displaystyle G^2_\infty = \lim_{w\to \infty}G(\nu)^2 = \frac {\Delta Q^2}{2(\pi \nu)^2} = \frac {V^2_{PP}}{2 (\pi \nu)^2} \)
Unfortunately, the resulting integral does not appear to have an exact solution.
The result is certainly interesting, but is it a valid measurement? I’ll be happy to talk about it.