Image information metrics from Slanted edges: Equations and Algorithms

\pi w ]nu_x)

Introduction – Summary diagram – Edge variance calculation – Noise method 1 – Noise method 2 
Noise image calculation – NPS – NEQ – SNRi – Object visibility – Noise Autocorrelation 
Meaning of Information capacity – Summary – Links – Binning noise – Calculating Cmax 

This page contains equations and algorithms for the image information metrics, including how information capacity C is calculated from images of slanted-edges, Imatest’s most widely-used test image, and how to signal and noise are calculated from the same location, resulting in a superior measurement of image quality. It is a reference for the calculations in Image information metrics from Slanted edges: Instructions.

Appendix 1 contains a brief introduction to information theory.

The eSFR ISO chart. Automatically detected ROIs, combining MTF, color, tone, and noise measurements

Introduction to the new measurements

This page describes Imatest’s Information capacity-related measurements, which take advantage of recently-discovered properties of slanted edges– Imatest’s most widely used patterns for measuring MTF. Many have been used for medical imaging, but are unfamiliar because they were difficult to perform. We describe a convenient approach to measuring them from slanted edges, taking advantage of two distinct methods for measuring noise in the presence of a signal.

1. The Edge variance calculation measures spatially dependent noise, N(x), and uses it to calculate camera information capacity.
    
2. The Noise image calculation measures frequency-dependent noise, N(f), and uses it to calculate an alternative information capacity as well as several additional image image information metrics including Noise Power Spectrum (NPS), Noise Equivalent Quanta (NEQ), and SNRi.

We can obtain information capacity from images that have very different types of image processing.

• Minimally or uniformly-processed images, converted from raw to TIFF or PNG files from camera raw files. By “minimally-processed”, 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. We can obtain a reasonable approximation of C, but bilateral-filtered images should not be used for the image information metrics (SNRi, etc.) We will show how to identify bilateral-filtered images. 

Summary of the slanted-edge method of calculating MTF, which has been part of the ISO 12233 standard since 2000,

• Find the center each scan line y_l (x) in a slanted-edge Region of Interest (ROI).
• Ft a polynomial curve to the centers.
• Add the line contents to one of four bins, based on the center location.
• Interleave the bins, resulting in a 4× oversampled averaged edge, which has lower noise than the individual scan lines.
• Calculate MTF (Modulation Transfer Function, usually synonymous with Spatial Frequency Response, SFR), by differentiating the averaged edge to obtain the Line Spread Function, LSF, windowing the LSF, then taking its Fourier transform. MTF is the absolute value of the Fourier Transform normalized to 1 (or 100%) at zero frequency.

A: Edge variance calculation (The background color is from the above diagram.)

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 Slanted-Edge Image Quality Measurements: the Edge variance calculation, and in the paper presented at Electronic Imaging 2023, “Measuring Camera Information Capacity with Slanted Edges“.

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_s²(x), which is equivalent to the signal-dependent noise power, N(x), to be calculated. 

\(\displaystyle \sigma_s^2(x) = \frac{1}{L} \sum_{l=0}^{L-1} (y_l(x)-\mu_s(x))^2 = \frac{1}{L}\sum_{l=0}^{L-1} y_l^2(x) \ – \left(\frac{1}{L}\sum_{L=0}^{L-1} y_l(x) \right)^2 \)

Noise power for the Shannon-Hartley Equation  N(x) = σ_s²(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 signal-dependent 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.

1. Uniformly or minimally-processed 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.

2. Bilateral-filtered 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 Signal-to-Noise 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 uniformly-processed 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: Bilateral-filtered in-camera JPEG;  Right Unsharpened TIFF from raw.
The x-axis is the original pixel location of the 4× oversampled signal.

Voltage statistics for the slanted edge

Signal power S for the Shannon-Hartley Equation  The mean signal amplitude for a uniformly-distributed signal of peak-to-peak amplitude V_P-P is \(V_{avg}(f) = V_{P-P} MTF(f) / \sqrt{12}\) — a reasonable number to use for the information capacity calculation using the Shannon-Hartley equation, shown above, which actually uses the average signal power, \(S_{avg}(f) = V^2_{P-P} MTF(f)^2 / 12\).

After removing newly-discovered 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 Shannon-Hartley 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. Frequency-dependence comes from MTF(f).

S_avg(f), N, and W are entered into the Shannon-Hartley 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

C₄ is the direct result of measuring 4:1 contrast ratio slanted-edges. It is calculated from the Shannon Hartley equation, using several assumptions (that the signal is uniformly distributed over the peak-to-peak measurement and the noise power spectral density (NPD) is flat). C₄ 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 (derived in Appendix 2) is a much more stable measurement of the maximum information capacity for the camera starting with C₄ (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 signal-dependent noise σ from
eSFR ISO image converted from raw with minimal processing

B. Noise image calculation (The background color is from the two methods diagram, above.)

Summary: Subtract a low-noise reverse-projected / de-binned 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.

The Noise Image method is the second of two methods for calculating information capacity and related figures of merit. It was developed shortly after the Edge Variance method. It offers a particularly rich set of measurements.

This method involves inverting the ISO 12233 binning procedure. Noting that the 4× oversampled edge was created by interleaving the contents of 4 bins, each of which contains an averaged (noise-reduced) signal derived from the original image, we apply an inverse of the binning algorithm to set the contents of each scan line to its corresponding interleave (Inverse binned… ROI, below). Since the inverse-binned image is a nearly noiseless replica of the original image, we can create a noise image by subtracting the inverse-binned image from the original image. This image is shown, adjusted to make the mean (zero) value middle gray, as the Noise image ROI, on the right below. 

A noise image can be created by subtracting the reverse-projected image from the original image, correct for nonuniformity along the direction of the edge. The three images are shown below. The noise image (below-right), which has a mean value of 0, has been lightened and contrast-boosted for display. The three images are linear: a gamma curve has been applied for display.

Original ROI

Inverse-binned / de-interleaved / reverse-projected ROI

Noise image ROI

These images allow several key image quality parameters to be calculated, including Noise Power Spectrum and Noise Equivalent Quanta, well-known in medical imaging systems, and described in an excellent review paper by Ian Cunningham and Rodney Shaw [4]. These measurements are not well-known 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. An unnormalized 1D Noise Power Spectrum, NPS_U(f), is calculated by dividing the 2D spectrum into several annular regions (the number depends on the size of the ROI; 6-10 are typical), 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 noise power spectrum must be normalized with a normalization constant, c_N : \(NPS(f) = c_N NPS_U(f)\).

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. Assuming that applying Parseval’s theorem,

\(\displaystyle c_N \int_0^{f_{Nyq}} NPS_U(f) df = \int_0^{f_{Nyq}} NPS(f) df = \int \sigma_s^2(x) dx = \int N(x) dx\)   

The 1D Fourier transform described above must be scaled to be consistent with the above equation. 

\(\displaystyle NPS(f) = \frac{NPS_U(f)\ \int N(x) dx}{\displaystyle \int_0^{f_{Nyq}} NPS_U(f) df }\) 

Noise Equivalent Quanta (NEQ)   

is a well-known 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 frequency-dependent Signal-to-Noise (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 \(\mu = V_{P-P}/\sqrt{12}\), is used to calculate information capacity, C_NEQ, from a special case of the Shannon-Hartley 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].

[8] presents the SNRi equation in one dimension, but after considerable effort, we determined that the two-dimensional equation in [9] gives the correct results.

\(\displaystyle SNRi^2 = \int \int \left( \frac{\mu^2 \Delta S^2(\nu_x,\nu_y) MTF^2(\nu_x,\nu_y) }{NPS(\nu_x,\nu_y)} \right) d\nu_xd\nu_y \)

SNRI curves, Micro 4/3 camera, ISO 100

G(ν_x,ν_y)² is the same as μ²ΔS²(ν_x,ν_y). MTF(ν) and NPS(ν) are defined in one dimension, where spatial frequency \(\nu = \sqrt{\nu_x^2+\nu_y^2}\) has units of Cycles/Pixel, and the linearized signal is normalized to a maximum value of 1.

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 the initial analysis) is the peak-to-peak voltage of the slanted edge (shown in the Voltage statistics figure, above), \(\Delta Q = V_{P-P}\) which is typically obtained from a chart with a 4:1 contrast ratio. 

\(\displaystyle \Delta g(x,y) = \Delta Q \cdot \text{rect}(x/w) \cdot \text{rect}(y/kw)\) ,

where    rect(x) = 1   for   -1/2 < x < 1/2 ;  0 otherwise.

G(ν_x,ν_y) is the Fourier transform of the object to be detected, Δg(x,y). It is expressed in two dimensions.

\(\displaystyle G(\nu_x,\nu_y) = kw^2 \Delta Q \frac{\sin(\pi w \nu_x)}{\pi w \nu_x} \ \frac{\sin(\pi  kw \nu_y)}{\pi k w \nu_y} \)

SNRI² is calculated numerically by creating a two dimensional array of frequencies (0 to 0.5 c/p in 51 steps) that has ν_x on the x-axis ν_y on the y-axis, and is filled with \(\nu = \sqrt{\nu_x^2+\nu_y^2}\). These frequencies are used to create a 2D array that can be numerically summed [9]. 

\(\displaystyle SNRi^2 = \Delta \nu_x \Delta \nu_y \sum_{i=1}^{N_x} \sum_{j=1}^{N_y} \frac{MTF^2(i,j)\ V_{P-P}^2}{NPS(i,j)} \Delta S^2(i,j) \)

SNRi is displayed for each color channel for widths w from 1 to 10. The latest values, which emphasize smaller objects, are  w = 1, 1.2, 1.4, 1.7, 2, 2.5, 3, 4, 7, 10. Objects of widths >10 were formerly analyzed, but they provide little insight into system performance.

Object visibility  

The goal of SNRi measurements is to predict object visibility for small, low contrast squares or 4:1 rectangles The SNRi prediction begs for visual confirmation. A simulated image that can do this is shown in Figure 3 of a classic SNRi paper [8].

We have developed a display for Imatest that does this with a real slanted-edge image and a bit of smoke and mirrors. Despite the trickery, the data is directly from the acquired image.

We show two sets of results: one for a relatively low noise image and one high noise image (both from a camera with Micro Four-Thirds sensors, at ISO 100 and 12800, respectively. The sides of the squares are w = 1, 2, 3, 4, 7, 10, 14, and 20 pixels. The original chart has a 4:1 contrast ratio (light/dark = 4), equivalent to a Michelson contrast CMich ((light-dark)/(light+dark)) of 0.6. The outer squares have CMich = 0.6. The middle and inner squares have CMich = 0.3 and 0.15, respectively.

How to use these images — Inconspicuous magenta bars near the margins are designed to help finding the small squares, which are hard to see. The yellow numbers are the square widths in pixels. The SNRi curves (initially, at least) represent the chart contrast — with 4:1 (the ISO 12233 standard) strongly recommended. The outer patches correspond to the SNRi curves, where, according to the Rose model [4], SNRi of 5 (14 dB) should correspond to the threshold of visibility. 

Low noise image, ISO 100

Noisy image, ISO 12800

SNRI curves, Micro 4/3 camera, ISO 12800 Only original pixels were used in these two images, but we used a little smoke and mirrors to make the squares with borders that have the same blur as the device under test. The SNRi curve on the right is for the noisy ISO 12800 image on the right, above. The w = 1 squares are invisible; the w = 2 and 3 squares are marginally visible, and w = 4 squares are clearly visible. In the plot, SNRi at w = 2 is 0-5 dB; it’s 5-10 dB for w =3; close to the expectation that the threshold of visibility is around 14 dB. How the squares were made Expand the image if needed (if the original is less than 170 pixels wide) to make room for all the squares by adding mirrored versions of image to the left and right to the sides of the image. If needed, add a cropped vertical mirrored image to the bottom.  Create a (horizontal) mirror of the full image. This is the “mirror” part. Create a mask with the shares. The background is 0 and the squares are 1. The sides are sharp. Blur the squares with the MATLAB filter2 function. This is the “smoke” part. Determining the blur kernel was challenging. We found that we couldn’t get good results by just taking the 1D Line Spread function (LSF) and using it in 2D. A more complex transformation was required.  Linearize the two images (remove the gamma encoding). Combine the them using the mask, keeping the original image where the mask = 0, using the mirrored image where the mask = 1, and blending them elsewhere. Reapply the gamma encoding.

Noise autocorrelation This plot, which is still in the R&D phase, has been added to examine the hypothesis that the noise power spectrum (and autocorrelation) indicate the amount of electrical crosstalk of image sensor when the effects of demosaicing and fixed-pattern noise are removed 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 white-balanced. The curve is the |inverse Fourier transform| of the noise spectrum, based on the author’s limited understanding of the Wiener-Khinchin theorem.

The image on the right is not White-Balanced. 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. Illumination nonuniformity has been corrected to decrease the (spurious) autocorrelation at large distances.

Additional results, displayed with Information capacity results for convenience

Modulation Transfer Function (MTF) — looks a little different from the standard MTF plot because the y-axis is logarithmic. Used in the calculation of NEQ.

Edge voltage unnormalized — used in the NEQ calculation. Interesting to compare peak-to-peak amplitudes of the different channels.

Meaning of Shannon information capacity

(The Appendix in the white paper, Measuring Information Capacity with Imatest,
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 well-tuned 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 signal-to-noise ratio, S/N. 

The beauty of both the Siemens Star and Slanted-edge 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 half-century to find codes that approached the Shannon capacity). Coding is not an issue with imaging. Rodney Shaw’s paper from 1962 [10] and R. Clark Jones’ paper from 1961 [12] are good (and rare) examples 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 ≅ 2.2 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_bC_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.

Links  (more links in the White Paper)

1. 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.
2. C. E. Shannon, “Communication in the Presence of Noise”, Proceedings of the I.R.E., January 1949, pp. 10-21.
3. 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. 
4. I.A. Cunningham and R. Shaw, “Signal-to-noise optimization of medical imaging systems”, Vol. 16, No. 3/March 1999/pp 621-632/J. Opt. Soc. Am. A 
5. 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.2470-1173.2016.13.IQSP-213.
6. 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. 
7. 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. 
8. 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 27-1 – 27-8,  https://doi.org/10.2352/ISSN.2470-1173.2019.15.AVM-027.
9. 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 41-1 – 41-7, https://doi.org/10.2352/ISSN.2470-1173.2020.16.AVM-041.
10. 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.281-286. 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
11. 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.
12. R. Clark Jones, Information Capacity of Photographic Films, J. Optical Society of America, Vol. 51, No. 11, Nov. 1961, pp. 1159-1171

Appendix 1: Brief introduction to information theory

Claude Shannon

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 Shannon-Hartley 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 = mc² ), 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. And IEEE has an article connecting Shannon with the development of Machine Learning and AI. The 29-minute 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.