Chapter 4: Image Statistics and Histograms

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Chapter 4 Image Statistics and Histograms
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Univariate Statistics - used to describe and
assess image quality of one band of
imagery Multivariate Statistics - used to
describe and assess the relationship between two
or more bands of data
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Image Statistics Brightness
Contrast Image Energy Information Content
Mean grey level
Variance, Standard deviation
Root Mean Squared Energy (RMSE)
Entropy
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Image Statistics skewness a
measure of the asymmetry of the histogram of an
image kurtosis measure of the sharpness of the
peak of an image histogram as compared to a
normal distribution covariance a measure of
the joint variation of two bands of
data correlation measure of the interrelation
between pixel values
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Image Processing Mathematical Notation
The following notation is used to describe the
mathematical operations applied to the digital
remote sensor data i a row (or line) in the
imagery j a column (or sample) in the imagery
k a band of imagery l another band of
imagery n total number of picture elements
(pixels) in an array BVijk brightness
value in a row i, column j, of band k BVik ith
brightness value in band k
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Image Processing Mathematical Notation
(continued)
BVil ith brightness value in band l mink
minimum value of band k maxk maximum value of
band k rangek range of actual brightness
values in band k quantk quantization level of
band k (e.g., 28 0 to 255 212
0 to 4095) µk mean of band k vark variance
of band k sk standard deviation of band k
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Image Processing Mathematical Notation
(continued)
skewnessk skewness of a band k
distribution kurtosisk kurtosis of a band k
distribution covkl covariance between pixel
values in two bands, k and l rkl correlation
between pixel values in two bands, k and l Xc
measurement vector for class c composed of
brightness values (BVijk) from row i, column j,
and band k
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Image Processing Mathematical Notation
(continued)
Mc mean vector for class c Md mean vector for
class d µck mean value of the data in class c,
band k sck standard deviation of the data in
class c, band k vckl covariance matrix of
class c for bands k through l shown as
Vc vdkl covariance matrix of class d for bands
k through l shown as Vd
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Image Brightness
The mean of a single band of imagery composed of
n brightness values (BVik) is computed using the
formula The sample mean, mk, is an unbiased
estimate of the population mean. For symmetrical
distributions, the sample mean tends to be closer
to the population mean than any other unbiased
estimate (such as the median or mode).
Unfortunately, the sample mean is a poor measure
of central tendency when the set of observations
is skewed or contains an extreme value.
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Image Contrast
- relates to the variation of image intensity
about the mean - image contrast is
low when the image is uniformly grey -
image contrast is high when the image lacks
intermediate shades of grey - can be measured
using equations for variance or standard
deviation
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Image Contrast
The variance of a sample is the average squared
deviation of all possible observations from the
sample mean. The variance of a band of imagery,
vark, is computed using the equation The
numerator of the expression is the corrected sum
of squares (SS).
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Image Contrast
The standard deviation is the positive square
root of the variance. The standard deviation of
the pixel brightness values in a band of imagery,
sk, is computed as
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Image Histograms
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Skewness is a measure of the asymmetry of a
histogram and is computed using the formula
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A histogram may be symmetric but have a peak that
is very sharp or one that is subdued when
compared with a perfectly normal distribution. A
perfectly normal distribution (histogram) has
zero kurtosis. The greater the positive kurtosis
value, the sharper the peak in the distribution
when compared with a normal histogram.
Conversely, a negative kurtosis value suggests
that the peak in the histogram is less sharp than
that of a normal distribution. Kurtosis is
computed using the formula
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Image Histogram or Grey level distribution For
an image with G possible intensity values (grey
levels) the grey level distribution is defined
as cg the number of pixels with grey level
g (for
each g0,1,..G-1) pg the relative frequency
of grey level g pg
cg
MN
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Grey level Distribution plotted as a Histogram
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Grey level Distribution plotted as a Histogram
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Grey level Distribution plotted as a Histogram
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Grey level Distribution plotted as a Histogram
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Alternative Equations for Image Statistics
G-1
g pg
average std dev RMSE
g0
G-1
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(g - average) pg
g0
G-1
g pg
2
g0
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Image Statistics
G-1
pg log (pg)
entropy
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g0
(for all pg gt 0)
Entropy is sometimes is used to characterize
information content in an image. For example
an image which is uniformly grey has entropy 0.
The maximum possible entropy occurs when all
grey levels are present with equal probability.
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Cumulative Grey Level Distribution Pg
p0 p1 . . . pg Pg is the
proportion of pixels with grey level value less
than or equal to g.
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camera image
histogram cumulative
histogram
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Image Energy rmse (root mean square image
energy) rmse contrast average
M-1 N-1
1
2
i m,n
MN
m0 n0
2
2
2
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Average 118.7 Contrast 62.3 RMS Energy 134.08
Average 119.7 Contrast 53.2 RMS Energy 131.1