1. Histogram Manipulation
CHAPTER 6
IMAGE ANALYSIS
Histogram Manipulation
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2. x = 0 preserved as “no data” code
The image histogram
(e.g. p = 8, 8-bit) x = 1, 2, …, 255
Pixel values for a p-bit digital image: x = 1, 2, …, 2p-1
x = 0 preserved as “no data” code nx N
fx = =
no of pixels having the value x
total number of image pixels
Frequency of value x :
Image histogram:
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3.  Nx = nz Nx N Fx = = no of pixels having the value x
The image histogram
Nx = nz
z =1 x 
Number of pixels having value  x : Nx N
Fx = =
no of pixels having the value x
total number of image pixels
Cumulative frequency of value x : x Fx
Image cumulative histogram: 1
128
255
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4. Histogram Equalization
Image with optimal contrast: all values of gray equally present
f (x) = 1
255
f (x) = constant =
2p-1
Corresponding histogram f (x) :
homogeneous histogram !
p = 8 (8-bit):
F (x) = x
255
2p-1
Corresponding cumulative histogram F (x) :
p = 8 (8-bit):
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5. Histogram Equalization
Contrast Enhancement: Transformation of histogram to homogeneous one
original
cumulative
histogram
homogeneous
Continuous case:
Each pixel value x is replaced
with a new value x
such that F(x) = F (x )
Corresponding realistic discrete case
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6. Histogram Equalization
Problems appearing in discrete histogram equalization:
no values are mapped into
some particular values
of the new equalized histogram
Different values are mapped
into the same value
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7. Histogram Equalization
Original image and histogram
Resulting image and histogram
Note departure
from ideal
homogeneous
histogram !
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8. Modifying an image so that its histogram F(x) is transformed
Histogram matching
Modifying an image so that its histogram F(x) is transformed
into a prescribed histogram F (x )
(usually that of another image – Result: images of similar contrast)
“target”
cumulative
histogram
Each pixel value x is replaced
with a new value x
such that F(x) = F (x )
original
cumulative
histogram
Same as histogram equalization
with homogeneous histogram replaced by a given histogram
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9. Histogram matching The original image and its histogram
The resulting image and its histogram
The target image and its histogram
Note that
histograms
are not exactly
identical
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10. A & B values, selected so that xmin  1 & xmax  L
Linear streching
Original image with pixel values
limited to an interval xmin  x  xmax
Resulting image with pixel values
covering all possible values 0  x  L
(xmax – x) + L (x – xmin)
x =
xmax – xmin
x  x = Ax + B
A & B values, selected so that
xmin  1 & xmax  L
Application of a linear transformation
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11. Linear streching Original 3 bands of a Landsat TM image
The same 3 bands after linear stretching of their histograms
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12. Saturated linear streching
Linear transformation with
(a > xmin)  1 and (b < xmax)  L
instead of
xmin  1 and xmax  L
Saturation:
(values 1  x < a)  1
(values b < x  L)  L
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13. Saturated linear streching
Use of saturated linear stretching for the enhancement of particular features:
Boat identification
Original
Resulting
Bathymetry determination
Original
Resulting
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