Using InfoDR, Part 2: Chart analysis

1.   Edge and SFR (MTF)
17. Edge & Info Capacity noise

Since the Edge and SFR (MTF) plot (1) is highly familiar — it has been a part of Imatest since the very beginning — only the Edge & Info Capacity (spatially dependent) noise plot is shown. These plots are available for all slanted-edge modules.

The upper (Edge) plot shows the C₄ and C_max information capacities measured the 4:1 contrast edge. 

the lower plot shows the spatially-dependent noise. A large peak near the edge indicates bilateral (nonuniform) filtering, common in JPEG files from cameras. Several statistics (mean signal and noise on either side of the edge, etc.) are displayed.

Edge (top) and spatially dependent noise
for ROI 2, where C₄ = 2.29 bits/pixel

In this plot, Show large ROI and noise has been checked. This displays a large image of the ROI, lightened if it’s dark. This is for a very dark region, where performance is significantly worse than for the light regions.

Enlarged and lightened image of ROI 17,
where C₄ = 0.182 bits/pixel

Edge and spatially dependent noise
on bottom Left & Right

18. Info-related: NPS, NEQ, SNRi… 

Image information metrics (NPS, NEQ, SNRi, Error Probability, and more) as well as several traditional metrics (MTF, LSF, …) for individual regions are displayed in the Information-related: NEQ, SNRi, … display.

A key feature of the Information-related… plot is that it displays two results: one at the top and one at the bottom, making it easy to compare and correlate different results. This feature has been used extensively during the development of the new metrics.

The Log x-axis checkbox between the two dropdown menus (shown inside a red oval on the right) lets you select a linear or logarithmic x-axis for frequency plots, where appropriate.

A large selection of results is available in the Information-related… dropdown menus for the upper and lower plots, shown inside the red ovals in the Rescharts  Display  area, above right.

The Information-related… dropdown menu is something of a monster. It contains more entries than ideal (you have to scroll to see them all; not all are shown on the right).

More plot examples are shown in Image information metrics from Slanted edges: Instructions — Results

The upper plot displays the selected variable, from the list below. 1., 3., and 4. 

1. Info Capacity C_4 — The information capacity for a 4:1 contrast edge or object, C₄ (or C_4). MTF values ordinarily don’t vary much with exposure (though they can be affected by image processing).
2. Info Capacity C_max
3. SNRi (nx4n pixels) 
4. log10(Bit Error Rate from SNRi) Contrast ratios from 10:1 to 1.2:1 can bd selected (also for 5. and 6.).
5. Edge Location sigma (nx4n pixels)
6. Gamma (from edge)
7. Mean ROI level
8. Edge contrast ratio
9. MTF50
10. MTF50P
11. MTF Area Peak Norml
12. MTF20
13. (Secondary readout 1)
14. (Secondary readout 2)
15. Mean noise
16. Mean SNR
17. Delta Edge Density
18. Edge contrast from Density Diff.
19. Others, added as needed.

X-axis units can be set to 

• Log_10 exposure (-Optical Density  (-OD)), Log₁₀ units, preferred by optical scientists,
• Exposure dB, 20×log₁₀ units, preferred by electrical engineers,
• F-stops (EV), Log₂ units, preferred by photographers,
• Patch number, mostly for debugging and diagnostics,
• Luminance cd/m^2, the patch luminance: recommended for cameras (often with fixed aperture) that need to perform well in low light. Note that for reflective charts (not recommend for measuring dynamic range),
  Lightbox luminance (cd/m²) = L = E/π  for Chart illuminance E (Lux), i.e.,
  Chart illuminance  E (Lux) = π Lightbox Luminance L (cd/m²)
• Sensor Exposure H in Lux-Seconds, the same units used for film characteristic curves, or
• log₁₀(Sensor Exposure H in Lux-Seconds).

The lower plot displays the density response (log₁₀(signal)). 

InfoDR control area

Information capacity units (C-units)

• Bits/pixel  (preferred most of the time)
• Megabits/mm²  (pixel pitch must be entered)
• Megabits/image  (involves extrapolation)

The first three x-axis units are relative units based on logarithms: OD (log₁₀), dB (20×log₁₀), and EV (log₂).

The last three are absolute units, based on illuminance or luminance measurements described in Part 1 of the InfoDR instructions.

The last two entries, for Sensor exposure H (the total illumination reaching the sensor in Lux-seconds), are derived from an approximation in ISO 12233:2019, Annex B.

Appendix. SNRi and Error Probability

We use results from detection theory to calculate the total error probability, which is the sum of the probabilities of false positives and false negatives, as illustrated in the Receiver Operating Curve (ROC), below. For the gaussians with means μ₁ and μ₂ and identical standard deviations σ = σ₁= σ₂, the minimum error probability occurs when the decision threshold, = ( + )⁄. Note that the system is symmetrical around x = 0.5. More details on detection theory can be found in Wikipedia and in reference papers.

Receiver Operating Curve (ROC)

For the gaussian curves,

The key assumption is that SNRi can be treated as a standard Signal-to-Noise Ratio (SNR) with signal = Δμ = μ2 – μ1 and noise = σ, illustrated above.

Thanks to the symmetry of the ROC, we can use the left gaussian to calculate the error rate because its mean, μ1 = 0, makes the calculation simpler. The error probability, which we also call Bit Error Rate, BER, is

For the gaussian curve on the left, the decision threshold is \(x_0=(\mu_1 + \mu_2)/2 = 1/2\).

     Let \(u^2 = (x-\mu)^2/(2 \sigma^2). \text{  Then } u = (x-\mu)/(\sigma \sqrt{2}) , \ du = dx/(\sigma/\sqrt{2}) , \ dx = \sigma\sqrt{2}\ du\)

     and \( u_0 = 0.5/(\sigma\sqrt{2}) = 1/(2\sqrt{2}\ \sigma) = SNRi/(2\sqrt{2}) \) .

From the MATLAB documentation for erfc, the MIT class notes, and other sources,

BPSK (Binary Phase Shift Keying) equation (for checking results) — BPSK is symmetrical around V = 0 with μ = ±0.5, i.e., \(V_0=-\sqrt{E_b}\) and \(V_1=\sqrt{E_b} = 1/2\) in the above diagram. The standard equation for BPSK is,

\(\displaystyle P_b=Pr(\text{error})=\frac{1}{2}\text{erfc}\left(\sqrt{\frac{E_b}{N_0}}\right)\)

There was a discrepancy when we assumed that \(N_0=\sigma^2 ;  \sqrt{N_0}=\sigma\), which is explained in the MIT class notes. 

“We will denote 2σ² by N_o. It has already been mentioned that σ² is a measure of the expected power in the under­lying AWGN process. However, the quantity N_o is also often referred to as the noise power, and we shall use this term for N_o too.”

Based on this statement, it appears that for BPSK, N_o , rather than σ , is what is actually measured, so that \(N_0=2\sigma^2 \text{ , and } \sqrt{N_0}=\sigma/2\). Using \(SNRi = \Delta\mu/\sigma\) and assuming \(\sqrt{E_b} = \Delta\mu/2\),

\(\displaystyle P_b=Pr(\text{error}) = \frac{1}{2}\text{erfc}\left(\sqrt{\frac{E_b}{N_0}}\right) = \frac{1}{2}\text{erfc}\left(\frac{\Delta\mu}{2\sqrt{2}\ \sigma}\right) = \frac{1}{2}\text{erfc}\left(\frac{SNRi}{2 \sqrt{2}}\right)\)

This resolved the discrepancy we struggled with when we assumed N₀ = σ².