The Reason Tone CurvesAre The Way They Are

Tone Curves in a common imaging chain.

An Example Goal: Make the Monitor luminance, L, directly proportional to original scene intensity, I.

Light, I

pixel value, P

Luminance, L

Lmax = 500 lux

L

I00

Lmax

Iw

Solution #1: The linear camera and monitor

wI

I255P ⋅=

255

PLL max⋅=

w

max

I

LIL ⋅=

L

I00

Lmax

Iw

Problems: 1. Half the amount of light does not LOOK like half the light. 2. Non-uniform in PERCEPTUAL Sampling of the gray scale. (An 8 bit gray scale allows only 256 samples.)

Original Linear Transformation

M M2

0 2000 4000 6000 80000

50%

100%

E(Eye Perceptionof Brightness)

10000

Lux Illumination

WhitePaper

Mid-tonegray card

18% Reflectance

White

Black

bã

wI

I100%E ⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅=

For the eye adapted tobright conditions, b=0.4

Tone response of human vision

0 2000 4000 6000 80000

128

255

PPixel Value

10000

Lux Illumination

WhitePaper

Mid-tonegray card

18% Reflectance

White

Black

cã

wI

I255P ⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅=

Set camera contrastc=0.4

The Camera TTF

0 128 2550

200

400

Invert the process in the monitor

PPixel Value

MonitorLuminance

Lmax

c1

max 255P

LL ⎟⎠

⎞⎜⎝

⎛⋅=

Solution #2: The gamma-corrected camera and monitor

w

max

I

LIL ⋅=

L

I00

Lmax

Iw

c1

max 255P

LL ⎟⎠

⎞⎜⎝

⎛⋅=

cã

wI

I255P ⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅=

The Same linear relationship, but nowsampled evenly in terms of perception.

Solution #1: The linear camera and monitor

w

max

I

LIL ⋅=

L

I00

Lmax

Iw

c1

max 255P

LL ⎟⎠

⎞⎜⎝

⎛⋅=

cã

wI

I255P ⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅=

This is good enough for most ordinary applications.However, if higher quality color reproduction is required (photographic quality)then better color management is required. Thistypically involves calibrating the monitor to a specific tone curve SUCH AS the one shown above. Then modifications of the pixel values are made before sending them to the monitor.

We assumed our eyes would work the sameway when viewing a monitor and whenviewing the original scene. This often is not true.

Light, I

pixel value, P

Luminance, L

Iw = 10,000 lux

L

I00

Lmax

Iw

00

Perception ofMonitorBrightness

Perception ofOriginalBrightness

white

white

0

50%

100%

EBrightnessPerception

10000

Lux IlluminationWhitePaper

0

Mid-tonegray card

18% Reflectance

White

Black0

50%

100%

EBrightnessPerception

500

Lux IlluminationWhitePaper

0

Mid-tonegray card

18% Reflectance

White

Black

We assumed the same response under both conditions.This turns out to be an incorrect assumption.

Original Outdoor Scene Monitor, office viewing

0

50%

100%

EBrightnessPerception

10000

Lux IlluminationWhitePaper

0

Mid-tonegray card

18% Reflectance

White

Black

Original Outdoor Scene

The gamma of the eye decreases as the surrounding light decreases.

EBrightnessPerception

Lux IlluminationWhitePaper11% Reflectance

White

Black

Monitor, office viewing

0

50%

100%

5000

Mid-tonegray card

eye 0.4 eye 0.32

Our original goal is NOTwhat we really want.

Light, I

Iw = 10,000 lux

pixel value, P

Luminance, L0

0

Perception ofMonitorBrightness

Perception ofOriginalBrightness

white

white

A Gamma correction is requiredto adjust for the the adaptationof vision.Light, I

Iw = 10,000 lux

pixel value, P

Luminance, L0

0

Perception ofMonitorBrightness

Perception ofOriginalBrightness

white

white

Light, I

Iw = 10,000 lux

pixel value, Pc

Luminance, L

pixel value, Pm

00

Perception ofMonitorBrightness

Perception ofOriginalBrightness

white

white

This Gamma correction is typically applied in software.

The Gamma correction is typically applied in software.

00

Perception ofMonitorBrightness

Perception ofOriginalBrightness

white

white

00

Pm

Pc

255

255ã

cm 255

P255P ⎟

⎠

⎞⎜⎝

⎛⋅=

Light, I

Iw = 10,000 lux

pixel value, Pc

Luminance, L

pixel value, Pm

The Gamma correction is typically applied in software.

00

Pm

Pc

255

255

Vision adapted toOutdoor Sun LightOffice LightMovie Theater

Use 1.001.251.50

R.W.G. Hunt, "The Reproduction of Colour", Fountain Press, England, p. 56, 1987

ã

cm 255

P255P ⎟

⎠

⎞⎜⎝

⎛⋅=

Light, I

Iw = 10,000 lux

pixel value, Pc

Luminance, L

pixel value, Pm

ã

cm 255

P255P ⎟

⎠

⎞⎜⎝

⎛⋅=

Summary:

The system requires threebasic tone curves.

c1

mmax 255

PLL ⎟

⎠

⎞⎜⎝

⎛⋅=

cã

wc I

I255P ⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅=Camera

Monitor

Processor

Light, I

Iw = 10,000 lux

pixel value, Pc

Luminance, L

pixel value, Pm

cã

wc I

I255P ⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅=

Another Parametric Model of the Tone Function

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⋅=

cã

wc I

I255LogPLog

( ) ( )255LogI

ILogãPLog

wcc +⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅=

Gamma as a power

Gamma as a slope

take the log:

Other Parametric Models of Tone Functions

( ) ( )255LogI

ILogãPLog

wcc +⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅=

Gamma as a slope

By analogy, "gamma" is often defined as the slope of the TTF

( ) ⎟⎠

⎞⎜⎝

⎛=dx

dyxã

Input Variable, x

OutputVariable

y

For a constant slope:

"gamma" is the Contrast metric,also called the Window metric.

⎟⎠

⎞⎜⎝

⎛ΔΔ

=x

yã

Input Variable, x

OutputVariable

y

Slope Called Window

Slope Called Window

Gamma, or the slope of the TTF, not only controls the perception of contrast, it also influencesresolution and noise

⎟⎠

⎞⎜⎝

⎛ΔΔ

=x

yã

Input Variable, x

OutputVariable

y

Resolution is influenced by contrast

Noise is also influenced by contrast

Summary:

Reasons for Controlling the Tone Transfer Function:

1. Efficient sampling of an 8 bit gray scale

2. Color reproduction

3. Control of resolution

4. Control of noise