The Reason Tone Curves
Are 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
Lmax
L
0
pixel value, P
0
Luminance, L
Lmax = 500 lux
I
Iw
Solution #1: The linear camera and monitor
I
P  255
Iw
L max
L  I
Iw
P
L  L max 
255
Lmax
L
0
0
I
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
Tone response of human vision
White 100%
Mid-tone
gray card
E
(Eye Perception
of Brightness) 50%
Black
For the eye adapted to
bright conditions, gb=0.4
 I
E  100% 
 Iw



γb
0
0
2000
4000
6000
8000
10000
Lux Illumination
18% Reflectance
White
Paper
The Camera TTF
White 255
Mid-tone
gray card
P
Pixel Value
128
Black
Set camera contrast
gc=0.4
 I
P  255 
 Iw



γc
0
0
2000
4000
6000
8000
10000
Lux Illumination
18% Reflectance
White
Paper
Invert the process in the monitor
Lmax
400
1
Monitor
Luminance
 P g c
L  L max  

 255
200
0
0
128
P
Pixel Value
255
Solution #2: The gamma-corrected camera and monitor
 I
P  255 
 Iw



γc
L max
L  I
Iw
1
 P g c
L  L max  

 255
Lmax
L
The Same linear relationship, but now
sampled evenly in terms of perception.
0
0
I
Iw
Solution #1: The linear camera and monitor
 I
P  255 
 Iw



γc
L max
L  I
Iw
1
 P g c
L  L max  

 255
Lmax
This is good enough for most ordinary applications.
L
However, if higher quality color reproduction
is required (photographic quality)
then better color management is required. This
typically involves calibrating the monitor to a
0
0
specific tone curve SUCH AS the one shown above.
I
Then modifications of the pixel values are made before sending them to the monitor.
Iw
Iw = 10,000 lux
We assumed our eyes would work the same
way when viewing a monitor and when
viewing the original scene. This often is
not true. L
max
Light, I
L
0
pixel value, P
Luminance, L
0
I
Iw
white
Perception of
Monitor
Brightness
0
0
Perception of
Original
Brightness
white
We assumed the same response under both conditions.
This turns out to be an incorrect assumption.
Original Outdoor Scene
Monitor, office viewing
White100%
Mid-tone
gray card
E
Brightness 50%
Perception
Black 0
White100%
0
Mid-tone
gray card
E
Brightness 50%
Perception
10000
Lux Illumination
White
18% Reflectance Paper
Black 0
0
500
Lux Illumination
White
18% Reflectance Paper
The gamma of the eye decreases as the surrounding light
decreases.
Original Outdoor Scene
Monitor, office viewing
White100%
White 100%
eye g  0.4
E
Brightness 50%
Perception
Black 0
E
Brightness 50%
Perception
Mid-tone
gray card
0
eye g  0.32
10000
Lux Illumination
White
18% Reflectance Paper
Black 0
Mid-tone
gray card
0
500
Lux Illumination
White
11% ReflectancePaper
Iw = 10,000 lux
Light, I
Our original goal is NOT
what we really want.
white
pixel value, P
Luminance, L
Perception of
Monitor
Brightness
0
0
Perception of
Original
Brightness
white
Iw = 10,000 lux
A Gamma correction is required
to adjust for the the adaptation
of vision.
Light, I
white
pixel value, P
Luminance, L
Perception of
Monitor
Brightness
0
0
Perception of
Original
Brightness
white
Iw = 10,000 lux
This Gamma correction is
typically applied in software.
Light, I
pixel value, Pc
white
pixel value, Pm
Perception of
Monitor
Brightness
Luminance, L
0
0
Perception of
Original
Brightness
white
The Gamma correction is
typically applied in software.
Iw = 10,000 lux
255
Pm
Light, I
0
pixel value, Pc
 Pc 
Pm  255 

 255
pixel value, Pm
γ
0
Pc
255
white
Perception of
Monitor
Brightness
Luminance, L
0
0
Perception of
Original
Brightness
white
The Gamma correction is
typically applied in software.
Iw = 10,000 lux
255
Pm
Light, I
pixel value, Pc
 Pc 
Pm  255 

 255
0
γ
0
pixel value, Pm
Luminance, L
255
Pc
Vision adapted to
Outdoor Sun Light
Office Light
Movie Theater
Use g
1.00
1.25
1.50
R.W.G. Hunt, "The Reproduction of Colour",
Fountain Press, England, p. 56, 1987
Summary:
Iw = 10,000 lux
The system requires three
basic tone curves.
Camera
Light, I
pixel value, Pc
 I
Pc  255 
 Iw



γc
P

Processor P  255  c 
m
 255
γ
pixel value, Pm
1
Luminance, L
Monitor
 Pm  g c
L  L max  

 255
Another Parametric Model of the Tone Function
 I
Pc  255 
 Iw
take the log:



γc
Gamma as a power
γc

 I  
LogPc   Log255   

 I w  
 I 
LogPc   γ c  Log   Log255
 Iw 
Gamma as a slope
Other Parametric Models of Tone Functions
 I 
LogPc   γ c  Log   Log255
 Iw 
Gamma as a slope
By analogy, "gamma" is often defined as the slope of the TTF
Output
Variable
y
 dy 
γx    
 dx 
Input Variable, x
For a constant slope:
"gamma" is the Contrast metric,
also called the Window metric.
Output
Variable
y
y
γ

x
Input Variable, x
Slope Called Contrast
Slope Called Window
Gamma, or the slope of the TTF, not only controls
the perception of contrast, it also influences
resolution and noise
Output
Variable
y
y
γ

x
Input Variable, x
Resolution is influenced by contrast
Noise is also influenced by contrast
Summary:
Reasons for Controlling the Tone Transfer Function:
1.
2.
3.
4.
Efficient sampling of an 8 bit gray scale
Color reproduction
Control of resolution
Control of noise