OBJECTIVES 1. Define “point operation” 2. Overview of applications ...

Point Operations
OBJECTIVES
1. Define “point operation”
2. Overview of applications
3. Example (linear point op)
4. Equation for predicting output
histogram given input
histogram & point op
1

Definition of Point Operation
A(x, y) - Input B(x, y) - Output
Point operations modify way in which
image fills available gray-scale range.
Also called contrast enhancement,
contrast stretching and gray-scale
transformations (GST).
2

Mathematical Definition of Point Operation
A(x, y) - Input B(x, y) - Output
f
3

Overview of Applications
• Contrast Enhancement
• Photometric Calibration
• Thresholding (discussed)
• Clipping (discussed)
4

Overview of Applications
Contrast Enhancement
Features of interest may occupy a
narrow range of the gray scale.
May wish to expand contrast so
features of interest occupy a larger
range of displayed gray levels.
Before
After histogram
equalization
We will look at histogram equalization
and histogram matching later.
5

Overview of Applications
Photometric Calibration
Digitizer may have a nonlinear
response to light intensity, but we
may want gray levels in image to be
proportional to intensity.
Can design point operation to get rid
of nonlinearity.
6

Overview of Applications
• Display Calibration
Relationship between brightness of
cathode-ray tube (CRT) phosphors to
applied voltage is nonlinear - gamma
response characteristic.
Can design a point operation to
compensate for nonlinearity.
• Clipping
An operation may produce an image
having gray levels outside the
allowed range (0 - D max).
Set negative values to 0 and those
greater than D max to D max.
What is D max for an 8-bit image?.
7

Linear Point Operation
255
b
0
D B
a
( )
f D = aD +
b
A A
255
D A
8

Linear Point Operation
b affects brightness of image
by shifting gray levels.
E.g., a = 1 (no gain) and b = -74
Before After
Q: What is effect of positive b?
9

Linear Point Operation
a affects contrast as well as
brightness.
E.g., a = 1.14 = 255/224, b = 0
Before After
Q: What is effect of 0 < |a| < 1?
Q: What is effect of a < 0?
10

Equation for Output Histogram
GIVEN:
(1) Point operation f(D A) that maps
input image A(x, y) into output B(x, y)
(2) Histogram of input, H A(D A)
DERIVE: Expression for histogram of
output, H B(D B), in terms of f(D A) and
H A(D A).
WHY?
So we can design point operation to
achieve desired histogram.
11

D B = f(D A)
D B
Equation for Output Histogram
255
D B + ∆∆∆∆D B
∆∆∆∆D B
D B
D A
0
H A(D A)
H B(D B)
∆∆∆∆D A
12
D A
255
D A + ∆∆∆∆D A
D A
0

Summary
• Point operations transform the gray
scale of an image
• Mathematically specified by the grayscale
transformation (GST) f (D) that
expresses the mapping between
input and output gray levels
• Useful for contrast enhancement,
photometric & display calibration,
thresholding, clipping
• Given the histogram of the input,
HA(DA), and the GST f (DA), can
predict output histogram using the
equation:
( )
H D
f
B B
where
′ =
df
dD
=
A
( −1(
) )
− 1(
)
H f D
A B
( ) B
f ′ f D
13