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Image Quality Assessment
Geography
& Statistical Evaluation
KHU
Jinmu Choi
1. Image Error and Quality
2. Sampling Theory
3. Univariate Descriptive Image Statistics
4. Multivariate Statistics
5. Geostatistics for RS
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Remote Sensing
1
1. Error in Remote Sensor

Cause of Error (or noise) into the remote
sensor data by:



the environment (e.g., atmospheric scattering),
random or systematic malfunction of the
remote sensing system (e.g., an uncalibrated
detector creates striping), or
improper airborne or ground processing of the
remote sensor data prior to actual data analysis
(e.g., inaccurate analog-to-digital conversion).
Remote Sensing
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Image Quality Assessment




Looking at the frequency of occurrence of
individual brightness values in a histogram
Viewing on individual pixel brightness values at
specific locations or within a geographic area,
Computing univariate descriptive statistics to
determine if unusual anomalies in the image,
Computing multivariate statistics to determine
the amount of between-band correlation (e.g.,
to identify redundancy).
Remote Sensing
3
RS Raster (Matrix) Data Format
(source: Jensen, 2011)
i = a row (or line) in the imagery
j = a column (or sample) in the imagery
n = total number of picture elements
(pixels) in an array
k = a band of imagery
BVijk = brightness value in a row i,
column j, of band k
l = another band of imagery
BVik = ith brightness value in band k
Image Processing Mathematical Notation
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
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
quantk = quantization level of band k (e.g., 28
= 0 to 255; 212 = 0 to 4095)
Mc = mean vector for class c
µk = mean of band k
µck = mean value of the data in
class c, band k
vark = variance of band k
sk = standard deviation of band k
skewnessk = skewness of a band k distribution
Md = mean vector for class d
sck = standard deviation of the data
in class c, band k
kurtosisk = kurtosis of a band k distribution
vckl = covariance matrix of class c
for bands k through l; shown as Vc
covkl = covariance between pixel values in two
bands, k and l
vdkl = covariance matrix of class d
for bands k through l; shown as Vd
2. RS Sampling Theory

Population: an infinite or finite set of
elements.

Sample: a subset of the elements taken from a
population

If selecting images obtained only in the
summer, it is a biased sample.

Sampling error: the difference between a
population and a sample
Remote Sensing
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RS Sampling Theory



Large samples produce
a symmetrical
frequency distribution.
Bell-shaped graph of
the distribution: a
normal distribution.
Statistical tests in the
analysis of RS data
assume that the
brightness values are
normally distributed.
Remote Sensing
(source: Jensen, 2011)
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RS Sampling Theory

Histogram: useful
graphic representation of
the information content
of a remotely sensed
image
(source: Jensen, 2011)
Remote Sensing
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3. Display of Brightness Values

Viewing individual pixel brightness values is for assessing
the quality and information content of the data.
(source: Jensen, 2011)
Remote Sensing
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Univariate Descriptive Statistics

Central Tendency

The mode : most
frequent value in
a distribution

Median: midway
value in the
frequency
distribution
n

Mean: arithmetic
average
k 
Remote Sensing
 BV
i 1
n
ik
10
(source: Jensen, 2011)
Hypothetical Dataset of
Brightness Values
Pixel
Band 1
(green)
Band 2
(red)
Band 3
Band 4
(near(nearinfrared) infrared)
(1,1)
130
57
180
205
(1,2)
165
35
215
255
(1,3)
100
25
135
195
(1,4)
135
50
200
220
(1,5)
145
65
205
235
Remote Sensing
(source: Jensen, 2011)
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Univariate Statistics
Band 1
(green)
Band 2
(red)
Band 3
(nearinfrared)
Band 4
(nearinfrared)
Mean (mk)
135
46.40
187
222
Variance
(vark)
562.50
264.80
1007
570
Standard
deviation
(sk)
23.71
16.27
31.4
23.87
Minimum
(mink)
100
25
135
195
Maximum
(maxk)
165
65
215
255
Range (BVr)
65
40
80
60
Remote Sensing
(source: Jensen, 2011)
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RS Univariate Statistics

Range: difference between the maximum (maxk)
and minimum (mink) values
rangek  max k  min k

Variance: average squared deviation (SS) of all
possible observations from the sample mean
n
vark 

 BV
i 1
 k 
2
ik
n
SS
vark 
n 1
Standard deviation: positive square root of the
variance.
sk   k  vark
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Standard Deviation
(source: Jensen, 2011)
Asymmetry and Peak Sharpness

Skewness : a measure of the asymmetry of a
histogram:
 BVik   k


sk
i 1 
skewnessk 
n

n



3
Kurtosis: very sharp peak or not compared with
a perfectly normal distribution
 1 n  BV  
k
kurtosisk     ik
sk
 n i 1 
Remote Sensing



4

 3

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4. RS Multivariate Statistics

Measurement of radiant flux reflected or emitted
from an object in more than one band

Multivariate statistical measures (covariance,
correlation) among the several bands to
determine how the measurements covary.

Usage

Principal components analysis (PCA)

Feature selection

Classification and accuracy assessment.
Remote Sensing
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Covariance

Covariance : the joint variation of two variables
about their common mean

For covariance, first compute the corrected sum
of products (SP) defined by the equation:
n
SPkl   BVik   k BVil  l 
i 1
n
n
SPkl   BVik  BVil  
n
 BV  BV
ik
i 1
n
i 1

il
i 1
Covariance between brightness values in bands
k and l, covkl, is equal to:
SP
cov kl 
Remote Sensing
kl
n 1
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Variance-Covariance Matrix
BV ik  k BV il  l 

i
n
COV
kl

1
n 1
Band 1
(green)
Band 2
(red)
Band 3
Band 4
(near(nearinfrared) infrared)
Band 1
SS1
cov1,2
cov1,3
cov1,4
Band 2
cov2,1
SS2
cov2,3
cov2,4
Band 3
cov3,1
cov3,2
SS3
cov3,4
Band 4
cov4,1
cov4,2
cov4,3
SS4
Remote Sensing
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Computation of Var.-Cov.
Band 1
(Band 1 x Band 2)
Band 2
130
7,410
57
165
5,775
35
100
2,500
25
135
6,750
50
145
9,425
65
675
31,860
232
n
n
SPkl   BVik  BVil  
i 1
cov kl 
SPkl
n 1
n
 BV  BV
i 1
ik
i 1
n
il
SP12  (31,860) 
675232
540
 135
Remote Sensing 4
5
cov12 
19
Result Var.-Cov. Matrix
Band 1
(green)
Band 2
(red)
Band 3
(nearinfrared)
Band 4
(nearinfrared)
Band 1
562.25
-
-
-
Band 2
135
264.80
-
-
Band 3
718.75
275.25
1007.50
-
Band 4
537.50
64
663.75
570
Remote Sensing
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Correlation

Correlation coefficient, (r) : the degree of
interrelation between two bands of RS data, rkl,

Ratio of their covariance (covkl) to the product of
their standard deviations (sk and sl)
cov kl
rkl 
s k sl

Coefficient of determination (r2) : proportion of
total variation in the values of “band l” that can
be explained by a linear relationship with the
values of “band k.”
Remote Sensing
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Using Univariate Statistics
Band 1
(green)
Band 2
(red)
Band 3
(nearinfrared)
Band 4
(nearinfrared)
Mean (mk)
135
46.40
187
222
Variance
(vark)
562.50
264.80
1007
570
Standard
deviation
(sk)
23.71
16.27
31.4
23.87
Minimum
(mink)
100
25
135
195
Maximum
(maxk)
165
65
215
255
Range (BVr)
65
40
80
60
Remote Sensing
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Resulting Correlation Matrix
cov kl
rkl 
s k sl
Band 1
(green)
Band 2
(red)
Band 3
(nearinfrared)
Band 4
(nearinfrared)
Band 1
-
-
-
-
Band 2
0.35
-
-
-
Band 3
0.95
0.53
-
-
Band 4
0.94
0.16
0.87
-
Remote Sensing
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Band
1
2
3
4
5
6
7
Min
51
17
14
5
0
0
102
Max
242
115
131
105
193
128
124
Mean
Standard Deviation
65.163137
10.231356
25.797593
5.956048
23.958016
8.469890
26.550666
15.690054
32.014001
24.296417
15.103553
12.738188
110.734372
4.305065
Covariance Matrix
Band Band 1
Band 2
1 104.680654 58.797907
2 58.797907 35.474507
3 82.602381 48.644220
4 69.603136 45.539546
5 142.947000 90.661412
6 94.488082 57.877406
7 24.464596 14.812886
Correlation Matrix
Band Band 1 Band 2
1 1.000000 0.964874
2 0.964874 1.000000
3 0.953195 0.964263
4 0.433582 0.487311
5 0.575042 0.626501
6 0.724997 0.762857
7 0.555425 0.577699
Band 3
82.602381
48.644220
71.739034
76.954037
149.566052
91.234270
23.827418
Band 3
0.953195
0.964263
1.000000
0.579068
0.726797
0.845615
0.653461
Band 4
69.603136
45.539546
76.954037
246.177785
342.523400
157.655947
46.815767
Band 4
0.433582
0.487311
0.579068
1.000000
0.898511
0.788821
0.693087
Univariate and
Multivariate Statistics of
Landsat TM Data of,
Charleston SC
Band 5
142.947000
90.661412
149.566052
342.523400
590.315858
294.019002
82.994241
Band 6
94.488082
57.877406
91.234270
157.655947
294.019002
162.261439
44.674247
Band 5 Band 6
Band 7
0.575042 0.724997 0.555425
0.626501 0.762857 0.577699
0.726797 0.845615 0.653461
0.898511 0.788821 0.693087
1.000000 0.950004 0.793462
0.950004 1.000000 0.814648
0.793462 0.814648 1.000000
Band 7
24.464596
14.812886
23.827418
46.815767
82.994241
44.674247
18.533586
(source: Jensen, 2011)
Feature Space Plots



Visual inspection
How two bands features in a remote
sensing dataset
The greater the frequency of
occurrence of unique pairs of values,
the brighter the feature space pixel.
Remote Sensing
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Geostatistical Analysis of RS Data




Image values record spatial properties of the
Earth’s surface
Autocorrelation: Things that are close to one
another are more alike than those farther away
How measure
autocorrelation in images?
Geostatistical analysis
incorporates spatial
autocorrelation information
in the kriging interpolation
process
Remote Sensing
(source: Jensen, 2011)
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5. Geostatistical Analysis - Kriging


Kriging : a family of least-squares linear
regression algorithms for interpolation
Weights in Kriging based not only on the
distance between the measured points and the
point to be predicted , but also on the overall
spatial arrangement among the measured points
(i.e., their autocorrelation)

Methods of kriging : simple, ordinary, universal,
probability, indicator, disjunctive, and multiple
variable co-kriging
Remote Sensing
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Variogram

Two tasks of kriging process:



quantifying the spatial structure of the surrounding
data points, (Variography)
producing a prediction at a new location
Semivariogram: measurements of the spatial
structure (the amount of spatial separation
m
between samples)
2






z
x

z
x

h

i
i
 (h)  i 1
m

Here, (BV) z of pixels x, h intervals (lag distance), m
possible pairs, semivariogram g(h).
Remote Sensing
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Semivariance
Semivariance calculation
with lag distance (h)
along a transect of pixels
extracted from an image.
m
 ( h) 
2






z
x

z
x

h
 i
i
i 1
m
(source: Jensen, 2011)
Characteristics of Semivariogram

Important characteristics of
the semivariogram include:

lag distance (h) on the x-axis,

sill (s),

range (a),

nugget variance (Co), and

spatially dependent structural
variance partial sill (C).
(source: Jensen, 2011)
Remote Sensing
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Summary

Image Error and Quality

Sampling Theory

Univariate Descriptive Image Statistics
Multivariate Statistics

Geostatistics for RS

Remote Sensing
31
Next



Exercise: Image Assessment with RS
software
Lecture: Initial Display Alternatives and
Scientific Visualization
Source:

Jensen and Jensen, 2011, Introductory Digital
Image Processing, 4th ed, Prentice Hall.