Transcript AAAS - University of the Western Cape

Image Quality Assessment and
Statistical Evaluation
Dr. John R. Jensen
Department of Geography
University of South Carolina
Columbia, SC 29208
Jensen, 2004
Image Quality Assessment and Statistical Evaluation
Many remote sensing datasets contain high-quality, accurate
data. Unfortunately, sometimes error (or noise) is introduced
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-todigital conversion).
Jensen, 2004
Image Quality Assessment and Statistical Evaluation
Therefore, the person responsible for analyzing the digital
remote sensor data should first assess its quality and statistical
characteristics. This is normally accomplished by:
• looking at the frequency of occurrence of individual
brightness values in the image displayed in a histogram
• viewing on a computer monitor individual pixel brightness
values at specific locations or within a geographic area,
• computing univariate descriptive statistics to determine if
there are unusual anomalies in the image data, and
• computing multivariate statistics to determine the amount of
between-band correlation (e.g., to identify redundancy).
Jensen, 2004
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
Jensen, 2004
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
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
Jensen, 2004
Image Processing Mathematical Notation
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
Jensen, 2004
Image Processing Mathematical Notation
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
Jensen, 2004
Remote Sensing Sampling Theory
A population is an infinite or finite set of elements. An
infinite population could be all possible images that might be
acquired of the Earth in 2004. All Landsat 7 ETM+ images of
Charleston, S.C. in 2004 is a finite population.
A sample is a subset of the elements taken from a population
used to make inferences about certain characteristics of the
population. For example, we might decide to analyze a June
1, 2004, Landsat image of Charleston. If observations with
certain characteristics are systematically excluded from the
sample either deliberately or inadvertently (such as selecting
images obtained only in the spring of the year), it is a biased
sample. Sampling error is the difference between the true
value of a population characteristic and the value of that
characteristic inferred from a sample.
Remote Sensing Sampling Theory
• Large samples drawn randomly from natural populations
usually produce a symmetrical frequency distribution. Most
values are clustered around some central value, and the
frequency of occurrence declines away from this central
point. A graph of the distribution appears bell shaped and is
called a normal distribution.
• Many statistical tests used in the analysis of remotely
sensed data assume that the brightness values recorded in a
scene are normally distributed. Unfortunately, remotely
sensed data may not be normally distributed and the analyst
must be careful to identify such conditions. In such instances,
nonparametric statistical theory may be preferred.
Jensen, 2004
Common
Symmetric and
Skewed
Distributions in
Remotely Sensed
Data
Jensen, 2004
Remote Sensing Sampling Theory
• The histogram is a useful graphic representation of the
information content of a remotely sensed image.
•It is instructive to review how a histogram of a single band
of imagery, k, composed of i rows and j columns with a
brightness value BVijk at each pixel location is constructed.
Jensen, 2004
Histogram of A
Single Band of
Landsat Thematic
Mapper Data of
Charleston, SC
Jensen, 2004
Histogram of
Thermal Infrared
Imagery of a
Thermal Plume
in the Savannah
River
Jensen, 2004
Remote Sensing Metadata
Metadata is “data or information about data”. Most quality
digital image processing systems read, collect, and store
metadata about a particular image or sub-image. It is
important that the image analyst have access to this metadata
information. In the most fundamental instance, metadata
might include:
the file name, date of last modification, level of quantization
(e.g, 8-bit), number of rows and columns, number of bands,
univariate statistics (minimum, maximum, mean, median,
mode, standard deviation), perhaps some multivariate
statistics, geo-referencing performed (if any), and pixel size.
Jensen, 2004
Viewing Individual Pixels
Viewing individual pixel brightness values in a remotely
sensed image is one of the most useful methods for
assessing the quality and information content of the data.
Virtually all digital image processing systems allow the
analyst to:
•
use a mouse-controlled cursor (cross-hair) to
identify a geographic location in the image (at a
particular row and column or geographic x,y
coordinate) and display its brightness value in n
bands,
•
display the individual brightness values of an
individual band in a matrix (raster) format.
Jensen, 2004
Cursor and Raster Display of Brightness Values
Jensen, 2004
Individual Pixel Display of
Brightness Values
Jensen, 2004
Raster Display of Brightness Values
Jensen, 2004
Two- and ThreeDimensional
Evaluation of
Pixel Brightness
Values within a
Geographic Area
Jensen, 2004
Univariate Descriptive Image Statistics
Measures of Central Tendency in Remote Sensor Data
• The mode is the value that occurs most frequently in a
distribution and is usually the highest point on the curve
(histogram). It is common, however, to encounter more than
one mode in a remote sensing dataset. The histograms of the
Landsat TM image of Charleston, SC and the predawn thermal
infrared image of the Savannah River have multiple modes.
They are nonsymmetrical (skewed) distributions.
•The median is the value midway in the frequency
distribution. One-half of the area below the distribution curve
is to the right of the median, and one-half is to the left.
Jensen, 2004
Common
Symmetric and
Skewed
Distributions in
Remotely Sensed
Data
Jensen, 2004
Univariate Descriptive Image Statistics
The mean is the arithmetic average and is defined as the sum of all
brightness value observations divided by the number of observations. It is
the most commonly used measure of central tendency. The mean (mk) of a
single band of imagery composed of n brightness values (BVik) is computed
using the formula:
n
mk 
 BV
ik
i 1
n
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 outlier.
Jensen, 2004
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
Jensen, 2004
Univariate Statistics for the Hypothetical Sample Dataset
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
Jensen, 2004
Remote Sensing Univariate Statistics - Variance
Measures of Dispersion
Measures of the dispersion about the mean of a distribution provide
valuable information about the image. For example, the range of a band of
imagery (rangek) is computed as the difference between the maximum
(maxk) and minimum (mink) values; that is,
rangek  maxk  mink
Unfortunately, when the minimum or maximum values are extreme or
unusual observations (i.e., possibly data blunders), the range could be a
misleading measure of dispersion. Such extreme values are not uncommon
because the remote sensor data are often collected by detector systems with
delicate electronics that can experience spikes in voltage and other
unfortunate malfunctions. When unusual values are not encountered, the
range is a very important statistic often used in image enhancement
functions such as min–max contrast stretching.
Jensen, 2004
Remote Sensing Univariate Statistics - Variance
Measures of Dispersion
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:
n
vark 
 BV
i 1
ik
 mk 
2
n
The numerator of the expression is the corrected sum of squares (SS). If
the sample mean (mk) were actually the population mean, this would be an
accurate measurement of the variance.
Jensen, 2004
Remote Sensing Univariate Statistics
Unfortunately, there is some underestimation because the sample mean was
calculated in a manner that minimized the squared deviations about it.
Therefore, the denominator of the variance equation is reduced to n – 1,
producing a larger, unbiased estimate of the sample variance:
SS
vark 
n 1
Jensen, 2004
Univariate Statistics for the Hypothetical Example Dataset
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
Jensen, 2004
Remote Sensing Univariate Statistics
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
sk   k  vark
Jensen, 2004
Jensen, 2004
Univariate Statistics for the Hypothetical Example Dataset
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
Jensen, 2004
Measures of Distribution (Histogram)
Asymmetry and Peak Sharpness
Skewness is a measure of the asymmetry of a histogram and is
computed using the formula:
 BVik  m k


sk
i 1 
skewnessk 
n
n



3
A perfectly symmetric histogram has a skewness value of zero.
Jensen, 2004
Measures of Distribution (Histogram)
Asymmetry and Peak Sharpness
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:
 1 n  BV  m
k
kurtosisk     ik
sk
 n i 1 



4

 3

Jensen, 2004
Remote Sensing Multivariate Statistics
Remote sensing research is often concerned with the
measurement of how much radiant flux is reflected or emitted
from an object in more than one band (e.g., in red and nearinfrared bands). It is useful to compute multivariate statistical
measures such as covariance and correlation among the several
bands to determine how the measurements covary. Later it will
be shown that variance–covariance and correlation matrices are
used in remote sensing principal components analysis (PCA),
feature selection, classification and accuracy assessment.
Jensen, 2004
Remote Sensing Multivariate Statistics
The different remote-sensing-derived spectral measurements for
each pixel often change together in some predictable fashion. If
there is no relationship between the brightness value in one
band and that of another for a given pixel, the values are
mutually independent; that is, an increase or decrease in one
band’s brightness value is not accompanied by a predictable
change in another band’s brightness value. Because spectral
measurements of individual pixels may not be independent,
some measure of their mutual interaction is needed. This
measure, called the covariance, is the joint variation of two
variables about their common mean.
Jensen, 2004
Remote Sensing Multivariate Statistics
To calculate covariance, we first compute the corrected sum of
products (SP) defined by the equation:
n
SPkl   BVik  m k BVil  ml 
i 1
Jensen, 2004
Remote
RemoteSensing
SensingUnivariate
Multivariate
Statistics
Statistics
It is computationally more efficient to use the following
formula to arrive at the same result:
n
n
SPkl   BVik BVil  
i 1
n
 BV  BV
i 1
ik
i 1
il
n
This quantity is called the uncorrected sum of products.
Jensen, 2004
Remote Sensing Multivariate Statistics
Just as simple variance was calculated by dividing the corrected
sums of squares (SS) by (n – 1), covariance is calculated by
dividing SP by (n – 1). Therefore, the covariance between
brightness values in bands k and l, covkl, is equal to:
SPkl
cov kl 
n 1
Jensen, 2004
Format of a Variance-Covariance Matrix
Band 1
(green)
Band 2
(red)
Band 3
(nearinfrared)
Band 4
(nearinfrared)
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
Jensen, 2004
Computation of Variance-Covariance Between
Bands 1 and 2 of the Sample Data
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
SP12  (31,860) 
cov12 
675232
540
 135
4
5
Jensen, 2004
Variance-Covariance Matrix of the Sample Data
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
Jensen, 2004
Correlation between Multiple Bands of
Remotely Sensed Data
To estimate the degree of interrelation between variables in a
manner not influenced by measurement units, the correlation
coefficient, r, is commonly used. The correlation between two
bands of remotely sensed data, rkl, is the ratio of their
covariance (covkl) to the product of their standard deviations
(sksl); thus:
covkl
rkl 
sk sl
Jensen, 2004
Correlation between Multiple Bands of
Remotely Sensed Data
If we square the correlation coefficient (rkl), we obtain the
sample coefficient of determination (r2), which expresses the
proportion of the total variation in the values of “band l” that
can be accounted for or explained by a linear relationship with
the values of the random variable “band k.” Thus a correlation
coefficient (rkl) of 0.70 results in an r2 value of 0.49, meaning
that 49% of the total variation of the values of “band l” in the
sample is accounted for by a linear relationship with values of
“band k”.
Jensen, 2004
Correlation Matrix of the Sample Data
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
Jensen, 2004
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 5
0.575042
0.626501
0.726797
0.898511
1.000000
0.950004
0.793462
Band 6
0.724997
0.762857
0.845615
0.788821
0.950004
1.000000
0.814648
Band 6
94.488082
57.877406
91.234270
157.655947
294.019002
162.261439
44.674247
Band 7
24.464596
14.812886
23.827418
46.815767
82.994241
44.674247
18.533586
Band 7
0.555425
0.577699
0.653461
0.693087
0.793462
0.814648
1.000000
Jensen, 2004
Feature Space Plots
The univariate and multivariate statistics discussed provide
accurate, fundamental information about the individual band
statistics including how the bands covary and correlate.
Sometimes, however, it is useful to examine statistical
relationships graphically.
Individual bands of remotely sensed data are often referred to
as features in the pattern recognition literature. To truly
appreciate how two bands (features) in a remote sensing dataset
covary and if they are correlated or not, it is often useful to
produce a two-band feature space plot.
Jensen, 2004
Feature Space Plots
A two-dimensional feature space plot extracts the brightness
value for every pixel in the scene in two bands and plots the
frequency of occurrence in a 255 by 255 feature space
(assuming 8-bit data). The greater the frequency of occurrence
of unique pairs of values, the brighter the feature space pixel.
Jensen, 2004
Two-dimensional
Feature Space
Plot of Landsat
Thematic Mapper
Band 3
and 4 Data of
Charleston, SC
obtained on
November 11,
1982
Jensen, 2004
Geostatistical Analysis of Remote Sensor Data
The Earth’s surface has distinct spatial properties. The
brightness values in imagery constitute a record of these spatial
properties. The spatial characteristics may take the form of
texture or pattern. Image analysts often try to quantify the
spatial texture or pattern. This requires looking at a pixel and its
neighbors and trying to quantify the spatial autocorrelation
relationships in the imagery. But how do we measure
autocorrelation characteristics in images?
Jensen, 2004
Geostatistical Analysis of Remote Sensor Data
A random variable distributed in space (e.g., spectral
reflectance) is said to be regionalized. We can use geostatistical
measures to extract the spatial properties of regionalized
variables. Once quantified, the regionalized variable properties
can be used in many remote sensing applications such as image
classification and the allocation of spatially unbiased sampling
sites during classification map accuracy assessment. Another
application of geostatistics is the prediction of values at
unsampled locations. Geostatistical interpolation techniques
could be used to evaluate the spatial relationships associated
with the existing data to create a new, improved systematic grid
of elevation values.
Jensen, 2004
Geostatistical Analysis of Remote Sensor Data
Geostatistics are now widely used in many fields and comprise
a branch of spatial statistics. Originally, geostatistics was
synonymous with kriging—a statistical version of interpolation.
Kriging is a generic name for a family of least-squares linear
regression algorithms that are used to estimate the value of a
continuous attribute (e.g., terrain elevation or percent
reflectance) at any unsampled location using only attribute data
available over the study area. However, geostatistical analysis
now includes not only kriging but also the traditional
deterministic spatial interpolation methods. One of the essential
features of geostatistics is that the phenomenon being studied
(e.g., elevation, reflectance, temperature, precipitation, a landcover class) must be continuous across the landscape or at least
capable of existing throughout the landscape.
Geostatistical Analysis of Remote Sensor Data
Autocorrelation is the statistical relationship among measured
points, where the correlation depends on the distance and
direction that separates the locations. We know from real-world
observation that spatial autocorrelation exists because we have
observed generally that things that are close to one another are
more alike than those farther away. As distance increases,
spatial autocorrelation decreases.
Geostatistical Analysis of Remote Sensor Data
Kriging makes use of the spatial autocorrelation information. Kriging is
similar to ‘distance weighted interpolation’ in that it weights the surrounding
measured values to derive a prediction for each new location. However, the
weights are based not only on the distance between the measured points and
the point to be predicted (used in inverse distance weighting), but also on the
overall spatial arrangement among the measured points (i.e., their
autocorrelation). Kriging uses weights that are defined statistically from the
observed data rather than a priori. This is the most significant difference
between deterministic (traditional) and geostatistical analysis. Traditional
statistical analysis assumes the samples derived for a particular attribute are
independent and not correlated in any way. Conversely, geostatistical
analysis allows a scientist to compute distances between observations and to
model autocorrelation as a function of distance and direction. This
information is then used to refine the kriging interpolation process,
hopefully, making predictions at new locations that are more accurate than
those derived using traditional methods. There are numerous methods of
kriging, including simple, ordinary, universal, probability, indicator,
disjunctive, and multiple variable co-kriging.
Geostatistical Analysis of Remote Sensor Data
The kriging process generally involves two distinct tasks:
• quantifying the spatial structure of the surrounding data points,
and
• producing a prediction at a new location.
Variography is the process whereby a spatially dependent model is fit to the
data and the spatial structure is quantified. To make a prediction for an
unknown value at a specific location, kriging uses the fitted model from
variography, the spatial data configuration, and the values of the measured
sample points around the prediction location.
Geostatistical Analysis of Remote Sensor Data
One of the most important measurements used to understand the spatial
structure of regionalized variables is the semivariogram, which can be used
to relate the semivariance to the amount of spatial separation (and
autocorrelation) between samples. The semivariance provides an unbiased
description of the scale and pattern of spatial variability throughout a region.
For example, if an image of a water body is examined, there may be little
spatial variability (variance), which will result in a semivariogram with
predictable characteristics. Conversely, a heterogeneous urban area may
exhibit significant spatial variability resulting in an entirely different
semivariogram.
Geostatistical Analysis of Remote Sensor Data
Phenomena in the real world that are close to one another (e.g., two nearby
elevation points) have a much greater likelihood of having similar values.
The greater the distance between two points, the greater the likelihood that
they have significantly different values. This is the underlying concept of
autocorrelation. The calculation of the semivariogram makes use of this
spatial separation condition, which can be measured in the field or using
remotely sensed data. This brief discussion focuses on the computation of
the semivariogram using remotely sensed data although it can be computed
just as easily using in situ field measurements.
Geostatistical Analysis of Remote Sensor Data
Consider a typical remotely sensed image over a study area. Now identify the
endpoints of a transect running through the scene. Twelve hypothetical individual
brightness values (BV) found along the transect are shown in the illustration. The
(BV) z of pixels x have been extracted at regular intervals z(x), where x = 1, 2, 3,..., n.
The relationship between a pair of pixels h intervals apart (h is referred to as the lag
distance) can be given by the average variance of the differences between all such
pairs along the transect. There will be m possible pairs of observations along the
transect separated by the same lag distance, h. The semivariogram gh), which is a
function relating one-half the squared differences between points to the directional
distance between two samples, can be expressed through the relationship:
m
g ( h) 
 z x   z x  h 
i 1
2
i
i
m
where gh) is an unbiased estimate of the average semivariance of the population.
Geostatistical Analysis of Remote Sensor Data
The total number of possible pairs m along the transect is computed by
subtracting the lag distance h from the total number of pixels present in the
dataset n, that is, m = n – h. In practice, semivariance is computed for pairs
of observations in all directions. Thus, directional semivariograms are
derived and directional influences can be examined.
The average semivariance is a good measure of the amount of dissimilarity
between spatially separate pixels. Generally, the larger the average
semivariance gh), the less similar are the pixels in an image (or the
polygons if the analysis was based on ground measurement).
Geostatistical Analysis of Remote Sensor Data
The semivariogram is a plot of the average semivariance value on the y-axis
(e.g., gh) is expressed in brightness value units if uncalibrated remote
sensor data are used) with the various lags (h) investigated on the x-axis.
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).
Geostatistical analysis incorporates spatial
autocorrelation information in the kriging
interpolation process. Phenomena that are
geographically closer together are generally
more highly correlated than things that are
farther apart.
Jensen, 2004
a) A hypothetical remote sensing
dataset used to demonstrate the
characteristics of lag distance (h)
along a transect of pixels
extracted from an image.
b) A semivariogram of the
semivariance g(h) characteristics
found in the hypothetical dataset
at various lag distances (h).
Jensen, 2004
The z-values of points (e.g., pixels in an image or locations
(or polygons) on the ground if collecting in situ data)
separated by various lag distances (h) may be compared and
their semivariance g(h) computed. The semivariance g(h) at
each lag distance may be displayed as a semivariogram with
the range, sill, and nugget variance characteristics.
Jensen, 2004
Original Images,
Semivariograms, and
Predicted Images.
a) The green band of a
0.61  0.61 m image
of cargo containers in
the port of Hamburg,
Germany, obtained on
May 10, 2002
(courtesy of
DigitalGlobe).
b) A subset containing
just cargo containers.
c) A subset continuing
just tarmac (asphalt).
Jensen, 2004
3-Dimensional
View of the
Thermal Infrared
Matrix of Data
Jensen, 2004