# Image Histograms by zH37CPH8

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```									                 The Properties of Images
and Imaging Devices

Group II of the Imaging Chain

Imaging Science Fundamentals     Chester F. Carlson Center for Imaging Science
Think of an image mathematically.

A simple black & white image is a volume of information!

Two dimensions are spatial dimensions (cm for example)

The other dimension is lightness/darkness
y                      gray level

y

Imaging Science Fundamentals   x                x Chester F. Carlson Center for Imaging Science
Three dimensional graphs
are hard to understand.

y                        gray level

y

x                x

Imaging Science Fundamentals                        Chester F. Carlson Center for Imaging Science
Contour Plot:
Another way to represent 3 dimensional information
y

Lines are regions
x
of constant gray level
Imaging Science Fundamentals                   Chester F. Carlson Center for Imaging Science
The 3D graph and the contour plot
are not used much in imaging science.

Other graphical representations
are more often used.

One that is commonly
used is the "Histogram"

Imaging Science Fundamentals                 Chester F. Carlson Center for Imaging Science
The Histogram
A statistical way of looking at an image
Step #1: Sample the image
Divide it up into "pixels"

Imaging Science Fundamentals                       Chester F. Carlson Center for Imaging Science
Step #2: Look at the gray value of each "pixel"

Gray Scale
R = 1 is white

R = 0 is black

Gray Value: R = 0.653

Imaging Science Fundamentals             Chester F. Carlson Center for Imaging Science
Step #3: Take the gray pixels out of the image.

We have cut the image into lots of pieces, called "pixels".
We measure the gray value, R, for each patch.

Imaging Science Fundamentals                  Chester F. Carlson Center for Imaging Science
Step #4: Sort the pixels from darkest to lightest.

There are many pixels at this gray level.

Only a few at these gray levels.

Imaging Science Fundamentals                 Chester F. Carlson Center for Imaging Science
Each pixel in the image has its
place in the stacks of rearranged pixels.

Number
at a
given R

R=0                                         R=1

Imaging Science Fundamentals                       Chester F. Carlson Center for Imaging Science
Step #5: Form a graph to
represent the sorted pixels.

Number
of pixels
at a
given R

0                                              1
R

Each pixel in the image has its place in the stacks of rearranged pixels.
Imaging Science Fundamentals                              Chester F. Carlson Center for Imaging Science
If you use very small pixels,
the graph becomes sharper.

Number
of pixels
at a
given R

0                                    1
R

This is called the image Histogram.

Imaging Science Fundamentals                   Chester F. Carlson Center for Imaging Science
The Image Histogram Has Many Uses

Use #1:

A way to describe Properties of an Image

Imaging Science Fundamentals             Chester F. Carlson Center for Imaging Science
The Image Histogram Has Many Uses
Use #1: A way to describe Image "Brightness"

N                 N                  N

R               R                     R

Average R = 0.2           Average R = 0.47    Average R = 0.78

Imaging Science Fundamentals                      Chester F. Carlson Center for Imaging Science
N

R

Range = 1
Same Average
N
R=0.47

R

Range = 0.7
Different Ranges
N

R

Range = 0.2

Imaging Science Fundamentals                        Chester F. Carlson Center for Imaging Science
Vary Both Brightness AND Contrast
A graph of graphs!!

N                 N                  N

Contrast                                  C
Range                                    o          R                    R                   R

of R                                    n
t   N                 N                  N
r
a
R                    R                   R
s
t
N                N                   N

R                   R                    R

Brightness (Average R)                              Brightness

Imaging Science Fundamentals                             Chester F. Carlson Center for Imaging Science
A Summary of Brightness & Contrast

1. These are properties of the image.

2. The Histogram is a way of
quantifying Brightness and Contrast.

Number
of pixels
at a
given R

0                                             1
R

Imaging Science Fundamentals                      Chester F. Carlson Center for Imaging Science
The Image Histogram Has Many Uses

Use #2

A way to describe Imaging Devices.

Imaging Science Fundamentals            Chester F. Carlson Center for Imaging Science
The Image Histogram Has Many Uses
Use #2: A way to describe Imaging Devices.

For example,
a camera
Original                     Photo-copy

N                                N

R                                 R

Average R = 0.2                           Average R = 0.78

Imaging Science Fundamentals                  Chester F. Carlson Center for Imaging Science
Imaging Devices
Change Brightness and Contrast
The image
The image
you want
you have
to make

Input                              Output
An Imaging Device

N
Camera, scanner, transmission line,            N

computer, display monitor, printer, etc.
R
R
Anything in Group II of the Imaging Chain

Imaging Science Fundamentals                                     Chester F. Carlson Center for Imaging Science
The image
The image
you want
you have
to make

Input                       Output
An Imaging Device

N
N

R
R

Imaging Science Fundamentals                                Chester F. Carlson Center for Imaging Science
Think of the imaging device as something that
changes the histogram.

N
An Imaging Device                     N
Input                       Output
R
R

The image
you have                                                            The image
you want
to make

Imaging Science Fundamentals                               Chester F. Carlson Center for Imaging Science
The Imaging Device is described
by the way it transforms
gray levels in histograms.

N
An Imaging Device                     N
Input                        Output
Ro
Rc

Original                                                          Copy
gray levels,                                                     gray levels
Ro                     Rc                                        Rc.

Imaging Science Fundamentals                    Ro          Chester F. Carlson Center for Imaging Science
This graph is called the Tone Transfer Function (TTF).
The TTF is a characteristic of the Imaging Device.

Input                                                Output
Ro               An Imaging Device                  Rc

Rc

Ro
Different shape curves will produce output different histograms.

Imaging Science Fundamentals                            Chester F. Carlson Center for Imaging Science
How to measure a TTF

N
An Imaging Device                     N
Input                        Output
Ro
Rc

Original                                                          Copy
gray levels,                                                     gray levels
Ro                     Rc                                        Rc.

Imaging Science Fundamentals                    Ro          Chester F. Carlson Center for Imaging Science
How to measure a TTF
Use an image with a very simple histogram.
A gray scale has the same number
of pixels at each gray level.

Ro             An Imaging Device            Ro
Input
Original                                                                    Output

N                                                                        N

Ro                                                                        Ro

Compare the output values, Rc, versus the input values, Ro.

Imaging Science Fundamentals                               Chester F. Carlson Center for Imaging Science
How to measure a TTF
Pair up the input values, Rc, with the output values, Ro.

This Ro        becomes      this Rc

This Ro        becomes      this Rc

This Ro        becomes      this Rc

Original                                                                        Output
Plot the output values, Rc, versus the input values, Ro.
The graph is the
Rc                           TTF of the device.

Ro
Imaging Science Fundamentals                              Chester F. Carlson Center for Imaging Science
Device Property #1: Location of the TTF Curve

1

Rc      0.5

0
1
0
Imaging Science Fundamentals   Ro   Chester F. Carlson Center for Imaging Science
Property #1: Location of the TTF Curve

Device #1                     Device #2
Device #1                     Device #3
1

Device #2
Rc

0
Device #3       0                                    1
Ro

Imaging Science Fundamentals          Chester F. Carlson Center for Imaging Science
Location: Controls the brightness of the copy image.

Device #1                                   Device #2
Device #1                     Device #3
1

Device #2
Rc

0
Device #3                    0                                     1
Ro

Device #1 is most sensitive to brightness. It makes the copy brighter.

Device #3 is least sensitive to brightness. It makes the copy darker.

Imaging Science Fundamentals                        Chester F. Carlson Center for Imaging Science
A Useful Metric of Device Sensitivity

Device

Locate the middle gray for the output copy: Rc = 0.5
1       Devices #1        #2   #3

Rc      0.5

0
1
0
Imaging Science Fundamentals                Ro         Chester F. Carlson Center for Imaging Science
A Useful Metric of Device Sensitivity

Device

Next, locate the corresponding input values: Ros
1       Devices #1          #2          #3
Device         Ros

#1        0.30

Rc      0.5                                                             #2        0.50

#3        0.68

0
1
0      0.3        0.5        0.68
Imaging Science Fundamentals                Ro                  Chester F. Carlson Center for Imaging Science
A Useful Metric of Device Sensitivity

Device

It is useful to define an index of "sensitivity": S.                       1
1                                                                      S=
Devices #1         #2          #3                                    2Ros

S1=1.67                                       Device         Ros         S
S2=1.00
S3=0.74               #1        0.30        1.67
Rc       0.5
#2        0.50        1.00

#3        0.68        0.74
0
1
0               0.3   0.5        0.68
Imaging Science Fundamentals         Ro                          Chester F. Carlson Center for Imaging Science
Sensitivity           (location): Controls the brightness of the copy image.
1
Device #1
S1=1.67
#2            #3
Device #1                  Rc

S2=1.00
0
0                                     1
Device #2                                    Ro
Device       Ros          S
S3=0.74
#1       0.30         1.67
Device #3
#2       0.50         1.00

#3       0.68         0.74
S > 1 increases brightness.
S = 1 gives the same brightness.
S < 1 decreases brightness.
Imaging Science Fundamentals                        Chester F. Carlson Center for Imaging Science
Device Property #2: Slope of the TTF Curve

1

Rc      0.5

0
1
0
Imaging Science Fundamentals   Ro   Chester F. Carlson Center for Imaging Science
Property #2: Slope of the TTF Curve

Device #1
Device #2
1
Device #1

Device #2
Rc                        Device #3

0
Device #3       0                                    1
Ro

Imaging Science Fundamentals          Chester F. Carlson Center for Imaging Science
A Useful Metric of Device Slope

Device

Locate the middle gray for the output copy: Rc = 0.5
1       Devices TTF

Rc      0.5

0
1
0
Imaging Science Fundamentals                 Ro       Chester F. Carlson Center for Imaging Science
A Useful Metric of Device Slope

Device

Next, Draw a straight line through the point
so that it matches the slope at that point.
1     Devices TTF

Rc      0.5

0
1
0
Imaging Science Fundamentals             Ro             Chester F. Carlson Center for Imaging Science
A Useful Metric of Device Slope

Device

Locate the points where the straight line crosses Rc=0 and Rc=1.
Find the corresponding values of Ro. Call these Ro0 and Ro1.
1     Devices TTF

Ro0 = 0.40 Ro1 = 0.60

Rc      0.5                                                          1
Slope =
Ro1 - Ro0

1
Slope =
0.6 - 0.4
0
1 Slope = 5.0
0    0.4        0.6
Imaging Science Fundamentals          Ro           Chester F. Carlson Center for Imaging Science
A Useful Metric of Device Slope

Device

Call this slope the "Contrast" of the device,
symbolized by Greek letter "gamma", g.                               1
Slope =
1       Devices TTF
Ro1 - Ro0
Slope = 5.0
1
g=    Ro1 - Ro0                         Or,

Rc       0.5            g = 5.0                                  g=
1
Ro1 - Ro0

g = 5.0
0
1
0
Imaging Science Fundamentals                  Ro         Chester F. Carlson Center for Imaging Science
Device #2
1
Device #1

Device #1        Rc                           Device #3

0
0                                       1
Device #2                           Ro

Device        g
Device #3
#1       5.00

g > 1 increases brightness.                    #2       1.00
g = 1 gives the same brightness.               #3       0.20
g < 1 decreases brightness.
Imaging Science Fundamentals                 Chester F. Carlson Center for Imaging Science
A Summary of the Device TTF

Device

S=1 and g = 1      Output copy = Input Copy

S>1     Output copy is brighter
S<1     Output copy is darker

g >1    Output copy has higher contrast
g <1    Output copy has less contrast

Imaging Science Fundamentals                     Chester F. Carlson Center for Imaging Science
A Summary of the Device TTF

S<1 g>1             S>1 g>1
S=1 g>1

S>1 g=1
S<1 g=1

S=1 g<1
S<1 g<1              S>1 g<1

Imaging Science Fundamentals                    Chester F. Carlson Center for Imaging Science
A Summary of                                   Sensitivity & Gamma

S<1 g>1   S=1 g>1      S>1 g>1

S<1 g=1                     S>1 g=1

S=1 g<1
S<1 g<1               S>1 g<1

Imaging Science Fundamentals                           Chester F. Carlson Center for Imaging Science
A Cautious Note:
There are many ways to define indices of sensitivity and slope.
There are many other names used instead of sensitivity and slope.

Brightness -100% to 0 to +100% (instead of sensitivity < 1 to 1 to > 1)

Sensitivity

Gamma

Contrast -100% to 0 to +100% (instead of gamma < 1 to 1 to > 1)

In general, however, the TTF of most imaging devices are characterized
by some index of location (sensitivity, brightness, etc.)
and by some index of slope (gamma, contrast, etc.)
Imaging Science Fundamentals                             Chester F. Carlson Center for Imaging Science
Properties of the Image:

(I) The Histogram
A. Brightness (average R)
B. Contrast (Range of R)

(II) Others to be discussed later
(sharpness, granularity, etc.)

Note: Don't confuse the terms "brightness"
and "contrast" as used here for the terms
used by Adobe and others for location and
slope of the TTF. We are using the terms as
properties of the image, not properties of the
imaging device.

Imaging Science Fundamentals                                Chester F. Carlson Center for Imaging Science
Properties of the Imaging Device:

(I) The TTF (Transforms the histogram)
A. Sensitivity
B. Gamma
(II) Others to be discussed later
(sharpening/blurring, noise, etc.)
Note: All imaging devices are described by the TTF.
Most devices are given an index of location defined from the TTF.
Most devices are given an index of slope defined from the TTF.

However: Different organizations use different definitions for location and slope.
They also use different names for the TTF, the location, and the slope.
Also, terms like "sensitivity", "gamma", "brightness", and "contrast" are in
common use, but there is no universal agreement about how these terms
should be applied. Thus, all four are used by different groups to mean many
different things, both as device properties and as image properties. Be cautious
and ask for clarification if you are not sure what is meant.
Imaging Science Fundamentals                            Chester F. Carlson Center for Imaging Science

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