Gamma, Tonal Response Curve, and related concepts

Introduction

In this post we discuss a number of concepts related to tonal response and gamma that are scattered around the Imatest website, making them hard to find. They are assembled here in anticipation of questions about gamma and related concepts, which keep arising.

Gamma (γ) is the average slope of the function that relates the logarithm of pixel levels (in an image file) to the logarithm of exposure (in the scene).

This relationship is called the Tonal Response Curve, (It’s also called the OECF = Opto-Electrical Conversion Function). The average is typically taken over a range of pixel levels from light to dark gray.

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Tonal Response Curve for Fujichrome Provia 100F film. The y-axis is reversed from the digital plots.

Tonal Response Curves (TRCs) have been around since the nineteenth century, when they were developed by Hurter and Driffield— a brilliant accomplishment because they lacked modern equipment. They are widely used for film. The Tonal Response Curve for Fujichrome Provia 100F film is shown on the right. Note that the y-axis is reversed from the digital plot shown below, where log(normalized pixel level) corresponds to

Equivalently, gamma can be thought of as the exponent of the curve that relates pixel level to scene luminance.

There are actually two gammas:

1. encoding gamma, which relates scene luminance to image file pixel levels, and

2. display gamma, which relates image file pixel levels to display luminance. It is typically found in equations of the form,

The equation, (above and in most references to gamma on this page), refers to encoding gamma. Exceptions are for characterizing color spaces (such as sRGB, where display gamma is approximately 2.2) and when display gamma is explicitly referenced, as in the Appendix on Monitor gamma.

The overall system contrast is the product of the encoding and decoding gammas. More generally, we think of gamma as contrast.

Encoding gamma is applied in the image processing pipeline because the output of image sensors, which is linear for most standard (non-HDR) image sensors, is never gamma-encoded. Encoding gamma is typically measured from the tonal response curve, which can be obtained photographing a grayscale test chart and running Imatest’s Color/Tone module (or the legacy Stepchart and Colorcheck modules).

Display gamma is typically specified by the color space of the file. For the most common color space in Windows and the internet, sRGB, display gamma is approximately 2.2. (It actually consists of a linear segment followed by a gamma = 2.4 segment, which together approximate gamma = 2.2.) For virtually all computer systems, display gamma is set to correctly display images encoded in standard (gamma = 2.2) color spaces. 

Here is an example of a tonal response curve for an in-camera JPEG for a high quality consumer camera, measured with Imatest Color/Tone.

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Tonal response curve measured by the Imatest Color/Tone module
from the bottom row of the 24-patch Colorchecker,

Note that this curve is not a straight line. It’s slope is reduced on the right, for the brightest scene luminance. This area of reduced slope is called the “shoulder”. It improves the perceived quality of pictorial images (family snapshots, etc.) by reducing saturation or clipping (“burnout”) of highlights, thus making the response more “film-like”. A shoulder is plainly visible in the lower-right of the Fujichorme Provia 100F curve, above. Shoulders are almost universally applied in consumer cameras; they’re less common in medical or machine vision cameras. [A region of reduced slope in dark regions is called a “toe”. It is common in film, but relatively rare in digital images.]

Because the tonal response is not a straight line, gamma has to be derived from the average (mean) value of a portion of the tonal response curve. 

Why use logarithms?

Logarithmic curves have been used to express the relationship between illumination and response since the nineteenth century because the eye’s response to light is logarithmic. This is a result of the Weber-Fechner law, which states that the perceived change dp to a change dS in an initial stimulus S is

Applying a little math to this curve, we arrive at  where ln is the natural logarithm (log_e).

From G. Wyszecki & W. S. Stiles, “Color Science,” Wiley, 1982, pp. 567-570, the minimum light difference ΔL that can be perceived by the human eye is approximately

The brighter areas take up far too much space in linearly-displayed tonal response curves, making it difficult to see the response in darker areas.

How gamma encoding increases Dynamic Range

In photography, we often talk about zones (derived from Ansel Adams’ zone system). A zone is a range of illumination L that varies by a factor of two, i.e., if the lower and upper boundaries of a zone are L₁ and L₂, then L₂ /L₁ = 2 or log₂(L₂) – log₂(L₁) = 1.

The steps between each zone has equal visual weight, except for very dark zones where the differences are hard to see.

For a set of zones z = {1, 2, 3, ..} the relative illumination of zone n is 2^-(z-1) = {1, 0.5, 0.25, 0.125, …}. The illumination difference between zones is {0.5, 0.25, 0.125, …}. The scene illumination ratio between zones is always 2:1.

What gamma (and Tonal Response Curve) should I expect?
And what is good?

JPEG images from consumer cameras typically have complex tonal response curves (with shoulders), with gamma (average slope) in the range of 0.45 to 0.6. This varies considerably for different manufacturers and models. The shoulder on the tonal response curve allows the the slope in the middle tones to be increased without worsening highlight saturation. This increases the apparent visual contrast, resulting in “snappier” more pleasing images.

RAW images from consumer cameras have to be decoded using LibRaw/dcraw. Their gamma depends on the Output gamma and Output color space settings (display gamma is shown in the settings window). Typical results are

• Encoding gamma ≅ 0.4545 with a straight line TRC (no shoulder) if conversion to a color space (usually sRGB or Adobe RGB) is selected;

• Encoding gamma ≅ 1.0 if a minimally processed image is selected.

RAW images from binary files (usually from development systems) have straight line gamma of 1.0, unless the Read Raw Gamma setting (which defaults to 1) is set to a different value. 

Flare light can reduce measured gamma by fogging shadows, flattening the Tonal Response Curve for dark regions. Care should be taken to minimize flare light when measuring gamma.

That said, we often see values of gamma that differ significantly from the expected values of ≅0.45-0.6 (for color space images) or 1.0 (for raw images without gamma-encoding). It’s difficult to know why without a hands-on examination of the system. Perhaps the images are intended for special proprietary purposes (for example, for making low contrast documents more legible by increasing gamma); perhaps there is a software bug. 

What effect does entering gamma have on Imatest measurements?

Gamma can be entered into imatest in several places. Here is what it does, and equally important, what it does not do.

Gamma can be measured from grayscale test charts. Entered values of gamma do not affect grayscale measurements.

• Gamma is used to linearize images for MTF calculations. The entered value of gamma can be overridden for slanted-edge patterns if the pattern contrast ratio (often 4) has been entered and Use for MTF has been checked. 

• In Flatfield, gamma is used to linearize ISO 18844 flare light measurements and also to calculate F-stop contour plots (not one of Imatest’s major displays; luminance (pixel) contour plots are usually preferred.)

• In Color/Tone, gamma is used to linearize the image for calculating the Color Correction Matrix (important) and for correcting illumination nonuniformity (an infrequently-used feature that requires a second flat image). 

Gamma and MTF measurement

MTF (Modulation Transfer Function, which is equivalent to Spatial Frequency Response), which is used to quantify image sharpness, is calculated assuming that the signal is linear. For this reason, gamma-encoded files must be linearized, i.e., the gamma encoding must be removed. The linearization doesn’t have to be perfect, i.e., it doesn’t have be the exact inverse of the tonal response curve. For most images (especially where the chart contrast is not too high), a reasonable estimate of gamma is sufficient for stable, reliable MTF measurements. The settings window for most MTF calculation has a box for entering gamma (or setting gamma to be calculated from the chart contrast).

Gamma is entered in the Setup or More settings window for each MTF module. They are described in the documentation for individual Rescharts modules. For Slanted-edge modules, they appear in the Setup window (crop shown on the right) and in the More settings window (crop shown below).

Gamma (input) defaults to 0.5, which is a reasonable approximation for color space files (sRGB, etc.), but is incorrect for raw files, where gamma ≅ 1. Where possible we recommend entering a measurement-based estimate of gamma.

Determine gamma from a grayscale pattern

Gamma is often measured from grayscale (OECF) patterns, which can be in separate charts or included in any of several sharpness charts— SFRplus, eSFR ISO, Star, Log F-Contrast, and Random (but not Checkerboard). The grayscale pattern from the eSFR ISO and SFRplus chart is particularly interesting because it shows the levels of the light and dark portions of the slanted-edge patters used to measure MTF.

Gamma = 0.588 here: close to the value (shown above) measured from the X-Rite Colorchecker for the same camera. The interesting thing about this plot is the pale horizontal bars, which represent the pixel levels of the light and dark portions of the selected slanted-edge ROIs (Regions of Interest). This lines let you see if the slanted-edge regions are saturating or clipping. This image shows that there will be no issue.

Determine gamma from each individual slanted-edge

Enter chart contrast and check Use for MTF. Only available for slanted-edge MTF modules. When Use for MTF is checked, the gamma (input) box is disabled.

This setting uses the measured of contrast of flat areas P₁ and P₂ (light and dark portions of each individual slanted-edge Region of Interest (ROI), away from the edge itself) to calculate the gamma for each edge. It is easy to use and quite robust. The printed chart contrast ratio (<=10) must be entered and the OECF curve should be close to .

In many cases this is more accurate than using results from grayscale (OECF) patterns because pixel levels above P₁ or below P₂ are not relevant to the ROI edge linearization.

How strongly does gamma affect MTF?  i.e., if the estimate of gamma is incorrect, how are far off is the MTF?

We ran an image from a compact digital camera, converted from raw with dcraw, with a number of different gamma settings to see how the settings affected MTF rsults.

The chart and images are several years old. The chart may have been printed with lower contrast than intended, resulting in a lower gamma measurement. This has no effect on the relative measurements below

Note that errors in estimating gamma (differences from 0.322) have very little effect on MTF measurements. MTF50 and MTF Area change very slightly, even with relatively large gamma errors. This is a good place discuss Imatest’s ISO 12233 compliance. Imatest uses a simple first-order inverse of the OECF curve (tonal response) measurement to linearize the image. It does not use a higher-order inverse. But because MTF is relatively insensitive to gamma (the slope of the tonal response curve), the simple fit results in nearly identical MTF measurements (except the the exposure is close to saturation, which where it’s hard to avoid dodgy results). 

Which patches are used to calculate gamma in Imatest?

This is an important question because gamma is not a “hard” measurement. Unless the Tonal Response Curve says pretty close to a straight line log pixel level vs. log exposure curve, its measured value depends on which measurement patches are chosen. Also, there have been infrequent minor adjustments to the patches used for Imatest gamma calculations— enough so customers occasionally ask about tiny discrepancies they find.

Light and dark patches where the tonal response curve flattens out are generally omitted from gamma calculations.

For the 24-patch Colorchecker, patches 2-5 in the bottom row are used (patches 20-23 in the chart as a whole).

For all other charts analyzed by Color/Tone or Stepchart, the luminance Y_i for each patch i (typically 0.2125×R + 0.7154×G + 0.0721×B) is found, and the minimum value Y_min, maximum value Y_max, and range Y_range = Y_max – Y_min is calculated.

Gamma is the slope of the straight line fit (linear regression) to patches Y_i , where Y_min + 0.2×Y_range  < Y_i  < Y_max – 0.1×Y_range . This ensures that light through dark gray patches are included and that saturated patches are excluded.

History: where did gamma ≅ 2.2 come from?

It came from the Cathode Ray Tubes (CRTs) that were universally used for television and video display before modern flat screens became universal. In CRTs, screen brightness was proportional to the control grid voltage raised to the 2 to 2.5 power. For this reason, signals had to be encoded with the approximate inverse of this value, and this encoding stuck. As we describe above, in “What is gained by applying a gamma curve?”, there is a real advantage to gamma encoding in image files with a limited bit depth, especially 8-bit files, which only have 256 possible pixel levels (0-255).

Why are images converted from raw files often darker
than JPEGs from the same capture?

Many cameras have the option of producing both interchangeable image files (usually JPEG) and raw files from the same image capture. The raw-converted files are frequently darker— sometimes much darker— than the corresponding JPEGs. This is not an accident. 

In Imatest, raw files are typically converted to interchangeable files (usually TIFF) using LibRaw, which has several presets and additional options. The most commonly-used presets are Color 24-bit sRGB (gamma ≅ 2.2) and Color 48-bit Adobe RGB (gamma = 2.2). These settings use EXIF metadata in the raw file to perform white balance and color correction (for the selected color space) and apply an appropriate straight-line gamma curve, . The tonal response curve, displayed logarithmically, is a straight line with no “shoulder” or other curvature.

JPEGs, on the other hand, frequently have a “shoulder”— a region where gamma (contrast) is gradually reduced— in the highlights. The shoulder improves the perceived pictorial quality of the image by reducing saturation (or “burnout”) of the highlights. It was a part of film response curves, long taken for granted. In order to achieve this shoulder, the exposure has to be reduced below the optimum level for a straight gamma curve to allow some “headroom” for the shoulder (before highlights are saturated). Here is an example:  in-camera JPEG and LibRaw-converted raw images of the Colorchecker charts and doll, whose tonal response curve are shown above or the in-camera JPEG and below for the LibRaw-converted TIFF.

Here is the (nearly) straight-line density response for the converted CR2 raw image on the right. Though this image is typical, we have seen much larger differences between JPEG and converted raw files.

Tone mapping

Tone mapping  is a form of nonuniform image processing that lightens large dark areas of images to make features more visible. It reduces global contrast (measured over large areas) while maintaining local contrast (measured over small areas) so that High Dynamic Range (HDR) images can be rendered on displays with limited dynamic range. it can usually be recognized by extremely low values of gamma (<0.25; well under the typical values around 0.45 for color space images), measured with standard grayscale charts. 

Tone mapping can seriously spoil gamma and Dynamic Range measurements, especially with standard grayscale charts. The Contrast Resolution chart was designed to give good results (for the visibility of small objects) in the presence of tone mapping, which is becoming increasingly popular with HDR images.

Contrast definitions

Several definitions of contrast are used in imaging. For two luminances (or reflectances) L_max and L_min, which could, for example, represent two patches on a test chart,

Ratio

The most common contrast metric in Imatest documentation is Contrast ratio, 

Weber

The Weber contrast metric is based on the Weber-Fechner law on human perception. The contrast measures the difference between the high and low luminances (L) divided by the lower luminance:

Michelson

Michelson contrast is more common in the optics field. It measures the relationship between the spread and the sum of the two luminances:

Michelson contrast is sometimes called “modulation”, and is the basis of Modulation Transfer Function (MTF) measurements. In the SPIE Optipedia page on Modulation Transfer Function, M in equation (1.12) is none other than Michelson contrast. MTF is defined as Michelson contrast at spatial frequency f, normalized to 1 at f = 0. A_max and A_min are derived from sine waves. If sine waves are unavailable (e.g., for slanted-edges), f can be calculated from a Fourier transform (it’s 1/f for a one-dimensional edge, which has the lovely property of spatial invariance). 

Michelson contrast is named after Albert Michelson, whose work on the famous Michelson-Morley experiment was a critical factor leading to Einstein’s theory of special relativity. Michelson’s life story is quite fascinating. He grew up in the wild-west mining towns of Murphy’s Camp, California and Virginia City, Nevada (there is a fictionalized Bonanza episode about his youth), and he went on to become the first American to win the Nobel prize in physics.

Logarithmic encoding

Logarithmic encoding, which has a similar intent to gamma encoding, is widely used in cinema cameras. According to Wikipedia’s Log Profile page, every camera manufacturer has its own flavor of logarithmic color space. Since logarighmic encoding is rarely, if ever, used for still cameras, Imatest does little with it apart from calculating the log slope (n₁, below). A detailed encoding equation from renderstory.com/log-color-in-depth is .

Comparison of linear, gamma (0.5) and logarithmic response curves

From context, this can be expressed as 

Since log(0) = -∞, exposure must be greater than 1/n₃ for this equation to be valid, i.e., there is a minimum exposure value (that defines a maximum dynamic range). 

By comparison, the comparable (simplified) equation for gamma encoding is

The primary difference is that pixel level instead of log(pixel level) is used the equation. The only thing Imatest currently does with logarithmic encoding is to display the log slope n₁, shown in some tonal response plots. Imatest could do more on customer request.

Logarithmic encoding has a very different appearance from gamma encoding when viewed in log-log density plots, as shown on the right. These curves are useful for identifying logarithmic encoding. The plot below is a density response curve, made with an Imatest dynamic range chart, for a camera that produces still and video output. At this time we’re not sure it’s intentionally logarithmic, but it certainly looks it. The flattened response and steps on the left are caused by flare light.

Dynamic range limitation —  An issue with logarithmic spaces is that their dynamic range is strictly limited by the math since log(0) = -∞. Using the above equation and assuming that log represents log₁₀ and n₂ = 1, 

If we assume that exp is normalized to a maximum value of exp_max = 1,  p(exp_max) = p(1) = 1. The darkest level that can be reproduced is p(exp_min) = γ log₁₀(exp_min)+1 = 0. 

For gamma = 0.5 (shown in the plot, above right), exp_min = 10^-2 = 0.01, equivalent to 

Logarithmic response example

The truth about v-log has detailed equations for Panasonic V-LOG encoding. We programmed them because

1. it’s a real example and

2. it was easy.

The curve is a hybrid curve with a logarithmic section for illumination above cut1 = 0.01 and a linear section below cut1. In this respect it is somewhat similar to sRGB, which has linear and gamma sections that average to approximately gamma = 2.2. 

The encoding is intended to improve dynamic range, but we have some doubts because the slope of the log-log plot is very low for the darkest regions (x < 10^-3).

Appendix I: Monitor gamma

Monitors are not linear devices. They are designed so that Brightness is proportional to pixel level raised to the gamma power, i.e., .

For most monitors, gamma should be close to 2.2, which is the display gamma of the most common color spaces, sRGB (the Windows and internet standard) and Adobe RGB.

To view the gamma chart (on the right) correctly, right-click on it, copy it, then paste it into Fast Stone Image Viewer.

This worked well for me on my fussy system where the main laptop monitor and my ASUS monitor (used for these tests) have different scale factors (125% and 100%, respectively).

The gamma will be the value on the scale where the gray area of the chart has an even visual density. For the (blurred) example on the left, gamma = 2.

Although a full monitor calibration (which requires a spectrophotometer) is recommended for serious imaging work, good results can be obtained by adjusting the monitor gamma to the correct value. We won’t discuss the process in detail, except to note that we have had good luck with Windows systems using QuickGamma. 

Appendix II: Tonal response, gamma, and related quantities

For completeness, we’ve updated and kept this table from elsewhere on the (challenging to navigate) Imatest website.

Appendix III: New linearization methods introduced in Imatest 23.2

Feature summary

Imatest 23.2 introduced new options for “linearization” of slanted edge ROI image data for edge SFR analyses. The new options take the form of a dropdown on both the “Setup” (ROI selection) window and the “More Settings” window. Users can now select a “linearization method” with the following options for eSFR ISO and SFRplus analyses:

1. Input gamma value (legacy, default) – The inverse of the value entered into the “Gamma (input)” field is used to linearize the data.

   1. Note: The input gamma value represents the “forward” (encoding) gamma value (typically around 0.46 = 1/2.2 for sRGB or Adobe RGB color spaces); the inverse (display gamma) is used for linearization.

2. Gamma calculated from chart contrast (legacy) – The value selected for the “Chart contrast” setting is used, in combination with the contrast measured from the edge ROI region, to calculate a value for gamma on a per edge ROI basis. The inverse of this gamma is used to linearize the data.

   1. The gamma value is estimated from the ratio of the light and dark pixel levels P₁ and P₂ in the slanted-edge region (away from the edge) if the chart contrast ratio (light/dark reflectivity) has been entered (and is 10:1 or less). Starting with ,

   2. Note: The calculated gamma value(s) are included in the JSON output as the field “gamma_from_chart”, on a per edge ROI basis (this is not a new output)

3. No linearization (new) – The data is not linearized.

   1. Note: “Input gamma value” of 1 is equivalent to “No linearization”.

4. Linearly interpolated LUT from step chart (new, requires step chart analysis to be checked) – A linearly interpolated look-up-table (LUT), representing the inverse of the measured OECF, is calculated from the mean of the grayscale patch (step chart) ROIs and the nominal patch densities (or those from a grayscale reference file). The LUT is used to linearize the data.

   1. Note: A grayscale reference file contains measured patch densities and can be loaded from the Rescharts more settings window

5. Gamma calculated from step chart (new, requires step chart analysis to be checked) – The mean of the grayscale patch (step chart) ROIs and the nominal patch densities (or those from a grayscale reference file) are used to calculate a value for gamma. The calculated gamma is the slope of a linear fit of the patch densities and log(grayscale patch means). The inverse of this gamma is used to linearize the data.

   1. Note: A grayscale reference file contains measured patch densities and can be loaded from the Rescharts more settings window

   2. Note: The calculated gamma value is the same as that which appears on the Rescharts Tonal Response plot, also corresponding to the gray dotted line on the plot (the gamma value is the slope of this line). Note that there is also a gray dashed line that represents a 2nd order polynomial fit.

      1.  

   3. Note: The calculated gamma value is included in the JSON output as the field “gamma_from_stepchart” (this is not a new output)

   4. Note: The range of patches is limited for the calculation of gamma, specifically using the patches that have a mean within the central 80% of the total range of patch means. Therefore, some darker or lighter patches may not be used for calculation of this gamma.

For edge SFR analyses that do not contain a grayscale step chart pattern, the “linearization method” options are limited to:

1. Input gamma value (legacy, default)

2. Gamma calculated from chart contrast (legacy)

3. No linearization (new, equivalent to input gamma value of 1)

Note: The new dropdown replaces the “Use for MTF” checkbox on all relevant GUIs, which now corresponds with the “Gamma calculated from chart contrast” option in the dropdown.

INI settings

A new INI field has been added for control of the new “Linearization method” setting on a per INI section (module/analysis) basis:

Valid INI values are 1-5, corresponding with valid LinearizationMethod enum values.

• 1 = Input gamma value

• 2 = Gamma calculated from chart contrast

• 3 = No linearization

• 4 = Linearly interpolated LUT from step chart

• 5 = Gamma calculated from step chart

The backend logic preserves legacy setting behavior as such:

Outputs

The user-selected linearization method is now included as a new field in the CSV and JSON outputs, as a way of documenting the input method. The new CSV field is “Linearization method”. The new JSON field is “linearization_method”.