dr. john r. jensen department of geography university of south carolina columbia, sc 29208

Dr. John R. JensenDr. John R. JensenDepartment of GeographyDepartment of Geography

University of South CarolinaUniversity of South CarolinaColumbia, SC 29208Columbia, SC 29208

Dr. John R. JensenDr. John R. JensenDepartment of GeographyDepartment of Geography

University of South CarolinaUniversity of South CarolinaColumbia, SC 29208Columbia, SC 29208

Initial Statistical Extraction and Initial Statistical Extraction and Image Quality AssessmentImage Quality Assessment

Initial Statistical Extraction and Initial Statistical Extraction and Image Quality AssessmentImage Quality Assessment

Jensen, 2003Jensen, 2003Jensen, 2003Jensen, 2003

The analyst responsible for analyzing the digital remote sensor The analyst responsible for analyzing the digital remote sensor data must first assess its quality. This is normally data must first assess its quality. This is normally performed by:performed by:

1.1. Computing fundamental image statistics and evaluating Computing fundamental image statistics and evaluating them to see if there are any unusual anomalies in the image them to see if there are any unusual anomalies in the image data that might be of concern,data that might be of concern, andand

2.2. performing a subjective evaluation of the appearance of the performing a subjective evaluation of the appearance of the remote sensor dataremote sensor data..

The analyst responsible for analyzing the digital remote sensor The analyst responsible for analyzing the digital remote sensor data must first assess its quality. This is normally data must first assess its quality. This is normally performed by:performed by:

1.1. Computing fundamental image statistics and evaluating Computing fundamental image statistics and evaluating them to see if there are any unusual anomalies in the image them to see if there are any unusual anomalies in the image data that might be of concern,data that might be of concern, andand

2.2. performing a subjective evaluation of the appearance of the performing a subjective evaluation of the appearance of the remote sensor dataremote sensor data..

Image Processing System ConsiderationsImage Processing System ConsiderationsImage Processing System ConsiderationsImage Processing System Considerations

Jensen, 2003Jensen, 2003

Image Processing Mathematical Notation Image Processing Mathematical Notation Image Processing Mathematical Notation Image Processing Mathematical Notation

Jensen, 2003Jensen, 2003Jensen, 2003Jensen, 2003

The following notation will be used to describe the The following notation will be used to describe the mathematical operations applied to the digital remote sensor mathematical operations applied to the digital remote sensor data: data:

ii = a row (or line) in the imagery = a row (or line) in the imagery

jj = a column (or sample) in the imagery = a column (or sample) in the imagery

kk = a band of imagery = a band of imagery

l l = another band of imagery= another band of imagery

nn = total number of picture elements (pixels) in an array = total number of picture elements (pixels) in an array

BVBVijkijk = brightness value in a row = brightness value in a row ii, column , column jj, of band , of band kk

BVBVikik = = iith brightness value in band th brightness value in band kk

The following notation will be used to describe the The following notation will be used to describe the mathematical operations applied to the digital remote sensor mathematical operations applied to the digital remote sensor data: data:

ii = a row (or line) in the imagery = a row (or line) in the imagery

jj = a column (or sample) in the imagery = a column (or sample) in the imagery

kk = a band of imagery = a band of imagery

l l = another band of imagery= another band of imagery

nn = total number of picture elements (pixels) in an array = total number of picture elements (pixels) in an array

BVBVijkijk = brightness value in a row = brightness value in a row ii, column , column jj, of band , of band kk

BVBVikik = = iith brightness value in band th brightness value in band kk



BVBVilil = = iith brightness value in band th brightness value in band ll

minmink k = minimum value of band = minimum value of band kk

maxmaxkk = maximum value of band = maximum value of band k k

rangerangek k = range of actual brightness values in band = range of actual brightness values in band kk

quantquantk k = quantization level of band = quantization level of band kk (e.g., 2 (e.g., 28 8 = 0 to 255; = 0 to 255;

221212 = 0 to 4095) = 0 to 4095)

µµkk = mean of band = mean of band kk

varvarkk = variance of band = variance of band kk

sskk = standard deviation of band = standard deviation of band kk

BVBVilil = = iith brightness value in band th brightness value in band ll

minmink k = minimum value of band = minimum value of band kk

maxmaxkk = maximum value of band = maximum value of band k k

rangerangek k = range of actual brightness values in band = range of actual brightness values in band kk

quantquantk k = quantization level of band = quantization level of band kk (e.g., 2 (e.g., 28 8 = 0 to 255; = 0 to 255;

221212 = 0 to 4095) = 0 to 4095)

µµkk = mean of band = mean of band kk

varvarkk = variance of band = variance of band kk

sskk = standard deviation of band = standard deviation of band kk



skewnessskewnesskk = skewness of a band = skewness of a band k k distributiondistribution

kurtosiskurtosiskk = kurtosis of a band = kurtosis of a band k k distributiondistribution

covcovklkl = covariance between pixel values in two bands, = covariance between pixel values in two bands,

kk and and ll

rrklkl = correlation between pixel values in two bands, = correlation between pixel values in two bands,

kk and and ll

XXcc = measurement vector for class = measurement vector for class c c composed of composed of

brightness values (brightness values (BVBVijkijk) from row ) from row ii, column , column jj, and , and band band

kk

skewnessskewnesskk = skewness of a band = skewness of a band k k distributiondistribution

kurtosiskurtosiskk = kurtosis of a band = kurtosis of a band k k distributiondistribution

covcovklkl = covariance between pixel values in two bands, = covariance between pixel values in two bands,

kk and and ll

rrklkl = correlation between pixel values in two bands, = correlation between pixel values in two bands,

kk and and ll

XXcc = measurement vector for class = measurement vector for class c c composed of composed of

brightness values (brightness values (BVBVijkijk) from row ) from row ii, column , column jj, and , and band band

kk



MMcc = mean vector for class = mean vector for class cc

MMdd = mean vector for class = mean vector for class dd

µµckck = mean value of the data in class = mean value of the data in class cc, band , band kk

ssckck = standard deviation of the data in class = standard deviation of the data in class cc, band , band kk

vvcklckl = covariance matrix of class c for bands = covariance matrix of class c for bands kk through through l; l;

shown as shown as VVcc

vvdkldkl = covariance matrix of class = covariance matrix of class dd for bands for bands kk through through ll; ;

shown as shown as VVdd

MMcc = mean vector for class = mean vector for class cc

MMdd = mean vector for class = mean vector for class dd

µµckck = mean value of the data in class = mean value of the data in class cc, band , band kk

ssckck = standard deviation of the data in class = standard deviation of the data in class cc, band , band kk

vvcklckl = covariance matrix of class c for bands = covariance matrix of class c for bands kk through through l; l;

shown as shown as VVcc

vvdkldkl = covariance matrix of class = covariance matrix of class dd for bands for bands kk through through ll; ;

shown as shown as VVdd

Remote Sensing Sampling Theory Remote Sensing Sampling Theory Remote Sensing Sampling Theory Remote Sensing Sampling Theory

A A populationpopulation is an infinite or finite set of elements. An is an infinite or finite set of elements. An infinite population could be all possible images that might be infinite population could be all possible images that might be acquired of the Earth in 2001. All Landsat 7 ETM+ images acquired of the Earth in 2001. All Landsat 7 ETM+ images of Charleston, S.C. in 2001 is a finite population. of Charleston, S.C. in 2001 is a finite population.

A A samplesample is a subset of the elements taken from a population is a subset of the elements taken from a population used to make inferences about certain characteristics of the used to make inferences about certain characteristics of the population. For example, we might decide to analyze a June population. For example, we might decide to analyze a June 1, 2001, Landsat image of Charleston. If observations with 1, 2001, Landsat image of Charleston. If observations with certain characteristics are systematically excluded from the certain characteristics are systematically excluded from the sample either deliberately or inadvertently (such as selecting sample either deliberately or inadvertently (such as selecting images obtained only in the spring of the year), it is a images obtained only in the spring of the year), it is a biasedbiased sample. sample. Sampling errorSampling error is the difference between the true is the difference between the true value of a population characteristic and the value of that value of a population characteristic and the value of that characteristic inferred from a sample.characteristic inferred from a sample.

A A populationpopulation is an infinite or finite set of elements. An is an infinite or finite set of elements. An infinite population could be all possible images that might be infinite population could be all possible images that might be acquired of the Earth in 2001. All Landsat 7 ETM+ images acquired of the Earth in 2001. All Landsat 7 ETM+ images of Charleston, S.C. in 2001 is a finite population. of Charleston, S.C. in 2001 is a finite population.

A A samplesample is a subset of the elements taken from a population is a subset of the elements taken from a population used to make inferences about certain characteristics of the used to make inferences about certain characteristics of the population. For example, we might decide to analyze a June population. For example, we might decide to analyze a June 1, 2001, Landsat image of Charleston. If observations with 1, 2001, Landsat image of Charleston. If observations with certain characteristics are systematically excluded from the certain characteristics are systematically excluded from the sample either deliberately or inadvertently (such as selecting sample either deliberately or inadvertently (such as selecting images obtained only in the spring of the year), it is a images obtained only in the spring of the year), it is a biasedbiased sample. sample. Sampling errorSampling error is the difference between the true is the difference between the true value of a population characteristic and the value of that value of a population characteristic and the value of that characteristic inferred from a sample.characteristic inferred from a sample.


• Large samples drawn randomly from natural populations Large samples drawn randomly from natural populations usually produce a usually produce a symmetrical frequency distributionsymmetrical frequency distribution. Most . Most values are clustered around some central value, and the values are clustered around some central value, and the frequency of occurrence declines away from this central frequency of occurrence declines away from this central point. A graph of the distribution appears bell shaped and is point. A graph of the distribution appears bell shaped and is called a called a normal distributionnormal distribution. .

• Many statistical tests used in the analysis of remotely Many statistical tests used in the analysis of remotely sensed data assume that the brightness values recorded in a sensed data assume that the brightness values recorded in a scene are normally distributed. Unfortunately, remotely scene are normally distributed. Unfortunately, remotely sensed data may sensed data may notnot be normally distributed and the analyst be normally distributed and the analyst must be careful to identify such conditions. In such must be careful to identify such conditions. In such instances, instances, nonparametricnonparametric statistical theory may be preferred. statistical theory may be preferred.

• Large samples drawn randomly from natural populations Large samples drawn randomly from natural populations usually produce a usually produce a symmetrical frequency distributionsymmetrical frequency distribution. Most . Most values are clustered around some central value, and the values are clustered around some central value, and the frequency of occurrence declines away from this central frequency of occurrence declines away from this central point. A graph of the distribution appears bell shaped and is point. A graph of the distribution appears bell shaped and is called a called a normal distributionnormal distribution. .

• Many statistical tests used in the analysis of remotely Many statistical tests used in the analysis of remotely sensed data assume that the brightness values recorded in a sensed data assume that the brightness values recorded in a scene are normally distributed. Unfortunately, remotely scene are normally distributed. Unfortunately, remotely sensed data may sensed data may notnot be normally distributed and the analyst be normally distributed and the analyst must be careful to identify such conditions. In such must be careful to identify such conditions. In such instances, instances, nonparametricnonparametric statistical theory may be preferred. statistical theory may be preferred.


Common Common Symmetric and Symmetric and

Skewed Skewed Distributions in Distributions in

Remotely Sensed Remotely Sensed DataData

Common Common Symmetric and Symmetric and

Skewed Skewed Distributions in Distributions in

Remotely Sensed Remotely Sensed DataData



• The The histogramhistogram is a useful graphic representation of the is a useful graphic representation of the information content of a remotely sensed image. information content of a remotely sensed image.

•It is instructive to review how a histogram of a single It is instructive to review how a histogram of a single bandband of imageryof imagery, , kk, composed of , composed of i i rowsrows and and jj columns columns with a with a brightness value brightness value BVBVijkijk at each pixel location is constructed. at each pixel location is constructed.

• The The histogramhistogram is a useful graphic representation of the is a useful graphic representation of the information content of a remotely sensed image. information content of a remotely sensed image.

•It is instructive to review how a histogram of a single It is instructive to review how a histogram of a single bandband of imageryof imagery, , kk, composed of , composed of i i rowsrows and and jj columns columns with a with a brightness value brightness value BVBVijkijk at each pixel location is constructed. at each pixel location is constructed.


Histogram of A Histogram of A Single Band of Single Band of

Landsat Thematic Landsat Thematic Mapper Data of Mapper Data of Charleston, SC Charleston, SC

Histogram of A Histogram of A Single Band of Single Band of

Landsat Thematic Landsat Thematic Mapper Data of Mapper Data of Charleston, SC Charleston, SC


Histogram of Histogram of Thermal Infrared Thermal Infrared

Imagery of a Imagery of a Thermal Plume Thermal Plume in the Savannah in the Savannah

RiverRiver

Histogram of Histogram of Thermal Infrared Thermal Infrared

Imagery of a Imagery of a Thermal Plume Thermal Plume in the Savannah in the Savannah

RiverRiver


Remote Sensing MetadataRemote Sensing MetadataRemote Sensing MetadataRemote Sensing Metadata

MetadataMetadata is “data or information about data”. Most quality is “data or information about data”. Most quality digital image processing systems read, collect, and store digital image processing systems read, collect, and store metadata about a particular image or sub-image. It is metadata about a particular image or sub-image. It is important that the image analyst have access to this metadata important that the image analyst have access to this metadata information. In the most fundamental instance, metadata information. In the most fundamental instance, metadata might include: might include:

the file name, date of last modification, level of quantization the file name, date of last modification, level of quantization (e.g, 8-bit), number of rows and columns, number of bands, (e.g, 8-bit), number of rows and columns, number of bands, univariate statistics (minimum, maximum, mean, median, univariate statistics (minimum, maximum, mean, median, mode, standard deviation), perhaps some multivariate mode, standard deviation), perhaps some multivariate statistics, geo-referencing performed (if any), and pixel size. statistics, geo-referencing performed (if any), and pixel size.

MetadataMetadata is “data or information about data”. Most quality is “data or information about data”. Most quality digital image processing systems read, collect, and store digital image processing systems read, collect, and store metadata about a particular image or sub-image. It is metadata about a particular image or sub-image. It is important that the image analyst have access to this metadata important that the image analyst have access to this metadata information. In the most fundamental instance, metadata information. In the most fundamental instance, metadata might include: might include:

the file name, date of last modification, level of quantization the file name, date of last modification, level of quantization (e.g, 8-bit), number of rows and columns, number of bands, (e.g, 8-bit), number of rows and columns, number of bands, univariate statistics (minimum, maximum, mean, median, univariate statistics (minimum, maximum, mean, median, mode, standard deviation), perhaps some multivariate mode, standard deviation), perhaps some multivariate statistics, geo-referencing performed (if any), and pixel size. statistics, geo-referencing performed (if any), and pixel size.


Viewing Individual PixelsViewing Individual Pixels

Viewing individual pixel brightness valuesViewing individual pixel brightness values in a remotely in a remotely sensed image is one of the most useful methods for sensed image is one of the most useful methods for assessing the quality and information content of the data. assessing the quality and information content of the data. Virtually all digital image processing systems allow the Virtually all digital image processing systems allow the analyst to:analyst to:

1. use a mouse-controlled cursorcursor (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,

2. display the individual brightness values of an individual band in a matrix (raster) format.

Viewing individual pixel brightness valuesViewing individual pixel brightness values in a remotely in a remotely sensed image is one of the most useful methods for sensed image is one of the most useful methods for assessing the quality and information content of the data. assessing the quality and information content of the data. Virtually all digital image processing systems allow the Virtually all digital image processing systems allow the analyst to:analyst to:

1. use a mouse-controlled cursorcursor (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,

2. display the individual brightness values of an individual band in a matrix (raster) format.


Cursor and Raster Display of Brightness Values Cursor and Raster Display of Brightness Values Cursor and Raster Display of Brightness Values Cursor and Raster Display of Brightness Values


Individual Pixel Display of Individual Pixel Display of Brightness Values Brightness Values

Individual Pixel Display of Individual Pixel Display of Brightness Values Brightness Values


Raster Display of Brightness Values Raster Display of Brightness Values Raster Display of Brightness Values Raster Display of Brightness Values


Three-Three-Dimensional Dimensional Evaluation of Evaluation of

Pixel Brightness Pixel Brightness Values within a Values within a

Geographic AreaGeographic Area

Three-Three-Dimensional Dimensional Evaluation of Evaluation of

Pixel Brightness Pixel Brightness Values within a Values within a

Geographic AreaGeographic Area


Remote Sensing Univariate StatisticsRemote Sensing Univariate StatisticsRemote Sensing Univariate StatisticsRemote Sensing Univariate Statistics

The The meanmean of a single band of imagery composed of of a single band of imagery composed of nn brightness values brightness values (BV(BVikik) is computed using the formula:) is computed using the formula:

The sample mean, The sample mean, kk,, is an unbiased estimate of the is an unbiased estimate of the

population mean. For symmetrical distributions, the sample population mean. For symmetrical distributions, the sample mean tends to be closer to the population mean than any other mean tends to be closer to the population mean than any other unbiased estimate (such as the median or mode). unbiased estimate (such as the median or mode). Unfortunately, the sample mean is a poor measure of central Unfortunately, the sample mean is a poor measure of central tendency when the set of observations is skewed or contains tendency when the set of observations is skewed or contains an extreme value.an extreme value.

The The meanmean of a single band of imagery composed of of a single band of imagery composed of nn brightness values brightness values (BV(BVikik) is computed using the formula:) is computed using the formula:

The sample mean, The sample mean, kk,, is an unbiased estimate of the is an unbiased estimate of the

population mean. For symmetrical distributions, the sample population mean. For symmetrical distributions, the sample mean tends to be closer to the population mean than any other mean tends to be closer to the population mean than any other unbiased estimate (such as the median or mode). unbiased estimate (such as the median or mode). Unfortunately, the sample mean is a poor measure of central Unfortunately, the sample mean is a poor measure of central tendency when the set of observations is skewed or contains tendency when the set of observations is skewed or contains an extreme value.an extreme value.


n

BVn

iik

k

1


PixelPixel Band 1Band 1

(green)(green)

Band 2 Band 2 (red)(red)

Band 3 Band 3 (near-(near-

infrared)infrared)


infrared)infrared)

(1,1)(1,1) 130130 5757 180180 205205

(1,2)(1,2) 165165 3535 215215 255255

(1,3)(1,3) 100100 2525 135135 195195

(1,4)(1,4) 135135 5050 200200 220220

(1,5)(1,5) 145145 6565 205205 235235

Sample Hypothetical Dataset of Brightness ValuesSample Hypothetical Dataset of Brightness ValuesSample Hypothetical Dataset of Brightness ValuesSample Hypothetical Dataset of Brightness Values


Band 1Band 1

(green)(green)

Band 2 Band 2

(red)(red)


infrared)infrared)


infrared)infrared)

Mean (Mean (kk)) 135135 46.4046.40 187187 222222

Variance Variance ((varvarkk))

562.50562.50 264.80264.80 10071007 570570

Standard Standard deviationdeviation

((sskk))

23.7123.71 16.2716.27 31.431.4 23.8723.87

MinimumMinimum

((minminkk))

100100 2525 135135 195195

Maximum Maximum ((maxmaxkk))

165165 6565 215215 255255

Range (Range (BVBVrr)) 6565 4040 8080 6060

Univariate Statistics for the Hypothetical Sample DatasetUnivariate Statistics for the Hypothetical Sample DatasetUnivariate Statistics for the Hypothetical Sample DatasetUnivariate Statistics for the Hypothetical Sample Dataset

Remote Sensing Univariate Statistics - VarianceRemote Sensing Univariate Statistics - VarianceRemote Sensing Univariate Statistics - VarianceRemote Sensing Univariate Statistics - Variance

The The variancevariance of a sample is the average squared deviation of of a sample is the average squared deviation of all possible observations from the sample mean. The variance all possible observations from the sample mean. The variance of a band of imagery,of a band of imagery, var varkk, is computed using the equation:, is computed using the equation:

The numerator of the expression is the corrected sum of The numerator of the expression is the corrected sum of squares (squares (SSSS). If the sample mean (). If the sample mean (kk) were actually the ) were actually the

population mean, this would be an accurate measurement of population mean, this would be an accurate measurement of the variance. the variance.

The The variancevariance of a sample is the average squared deviation of of a sample is the average squared deviation of all possible observations from the sample mean. The variance all possible observations from the sample mean. The variance of a band of imagery,of a band of imagery, var varkk, is computed using the equation:, is computed using the equation:

The numerator of the expression is the corrected sum of The numerator of the expression is the corrected sum of squares (squares (SSSS). If the sample mean (). If the sample mean (kk) were actually the ) were actually the

population mean, this would be an accurate measurement of population mean, this would be an accurate measurement of the variance. the variance.


n

BVn

ikik

k

1

2

var


Unfortunately, there is some underestimation because the Unfortunately, there is some underestimation because the sample mean was calculated in a manner that minimized the sample mean was calculated in a manner that minimized the squared deviations about it. Therefore, the denominator of the squared deviations about it. Therefore, the denominator of the variance equation is reduced to variance equation is reduced to n – 1n – 1, producing a larger, , producing a larger, unbiased estimate of the sample variance;unbiased estimate of the sample variance;

Unfortunately, there is some underestimation because the Unfortunately, there is some underestimation because the sample mean was calculated in a manner that minimized the sample mean was calculated in a manner that minimized the squared deviations about it. Therefore, the denominator of the squared deviations about it. Therefore, the denominator of the variance equation is reduced to variance equation is reduced to n – 1n – 1, producing a larger, , producing a larger, unbiased estimate of the sample variance;unbiased estimate of the sample variance;


1var

n

SSk


Band 1Band 1

(green)(green)

Band 2 Band 2

(red)(red)


infrared)infrared)


infrared)infrared)

Mean (Mean (kk)) 135135 46.4046.40 187187 222222


562.50562.50 264.80264.80 10071007 570570


((sskk))

23.7123.71 16.2716.27 31.431.4 23.8723.87

MinimumMinimum

((minminkk))

100100 2525 135135 195195


165165 6565 215215 255255




TheThe standard deviationstandard deviation is the positive square root of the is the positive square root of the variance. The standard deviation of the pixel brightness values variance. The standard deviation of the pixel brightness values in a band of imagery, in a band of imagery, sskk, is computed as , is computed as

TheThe standard deviationstandard deviation is the positive square root of the is the positive square root of the variance. The standard deviation of the pixel brightness values variance. The standard deviation of the pixel brightness values in a band of imagery, in a band of imagery, sskk, is computed as , is computed as


kkks var

Jensen, 2003VJensen, 2003V


Band 1Band 1

(green)(green)

Band 2 Band 2

(red)(red)


infrared)infrared)


infrared)infrared)

Mean (Mean (kk)) 135135 46.4046.40 187187 222222


562.50562.50 264.80264.80 10071007 570570


((sskk))

23.7123.71 16.2716.27 31.431.4 23.8723.87

MinimumMinimum

((minminkk))

100100 2525 135135 195195


165165 6565 215215 255255




SkewnessSkewness is a measure of the asymmetry of a histogram and is is a measure of the asymmetry of a histogram and is computed using the formula computed using the formula

SkewnessSkewness is a measure of the asymmetry of a histogram and is is a measure of the asymmetry of a histogram and is computed using the formula computed using the formula


n

sBV

skewness

n

i k

kik

k

1

3


A histogram may be symmetric but have a peak that is very A histogram may be symmetric but have a peak that is very sharp or one that is subdued when compared with a perfectly sharp or one that is subdued when compared with a perfectly normal distribution. A perfectly normal distribution (histogram) normal distribution. A perfectly normal distribution (histogram) has zero has zero kurtosiskurtosis. The greater the positive kurtosis value, the . The greater the positive kurtosis value, the sharper the peak in the distribution when compared with a sharper the peak in the distribution when compared with a normal histogram. Conversely, a negative kurtosis value normal histogram. Conversely, a negative kurtosis value suggests that the peak in the histogram is less sharp than that of suggests that the peak in the histogram is less sharp than that of a normal distribution. Kurtosis is computed using the formulaa normal distribution. Kurtosis is computed using the formula

A histogram may be symmetric but have a peak that is very A histogram may be symmetric but have a peak that is very sharp or one that is subdued when compared with a perfectly sharp or one that is subdued when compared with a perfectly normal distribution. A perfectly normal distribution (histogram) normal distribution. A perfectly normal distribution (histogram) has zero has zero kurtosiskurtosis. The greater the positive kurtosis value, the . The greater the positive kurtosis value, the sharper the peak in the distribution when compared with a sharper the peak in the distribution when compared with a normal histogram. Conversely, a negative kurtosis value normal histogram. Conversely, a negative kurtosis value suggests that the peak in the histogram is less sharp than that of suggests that the peak in the histogram is less sharp than that of a normal distribution. Kurtosis is computed using the formulaa normal distribution. Kurtosis is computed using the formula


31

1

4

n

i k

kikk s

BV

nkurtosis

Remote Sensing Multivariate StatisticsRemote Sensing Multivariate StatisticsRemote Sensing Multivariate StatisticsRemote Sensing Multivariate Statistics

The different remote-sensing-derived spectral measurements The different remote-sensing-derived spectral measurements for each pixel often change together in some predictable for each pixel often change together in some predictable fashion. If there is no relationship between the brightness value fashion. If there is no relationship between the brightness value in one band and that of another for a given pixel, the values are in one band and that of another for a given pixel, the values are mutually independent; that is, an increase or decrease in one mutually independent; that is, an increase or decrease in one band’s brightness value is not accompanied by a predictable band’s brightness value is not accompanied by a predictable change in another band’s brightness value. Because spectral change in another band’s brightness value. Because spectral measurements of individual pixels may not be independent, measurements of individual pixels may not be independent, some measure of their mutual interaction is needed. This some measure of their mutual interaction is needed. This measure, called the measure, called the covariancecovariance, is the joint variation of two , is the joint variation of two variables about their common mean. variables about their common mean.

The different remote-sensing-derived spectral measurements The different remote-sensing-derived spectral measurements for each pixel often change together in some predictable for each pixel often change together in some predictable fashion. If there is no relationship between the brightness value fashion. If there is no relationship between the brightness value in one band and that of another for a given pixel, the values are in one band and that of another for a given pixel, the values are mutually independent; that is, an increase or decrease in one mutually independent; that is, an increase or decrease in one band’s brightness value is not accompanied by a predictable band’s brightness value is not accompanied by a predictable change in another band’s brightness value. Because spectral change in another band’s brightness value. Because spectral measurements of individual pixels may not be independent, measurements of individual pixels may not be independent, some measure of their mutual interaction is needed. This some measure of their mutual interaction is needed. This measure, called the measure, called the covariancecovariance, is the joint variation of two , is the joint variation of two variables about their common mean. variables about their common mean.



To calculate covariance, we first compute the To calculate covariance, we first compute the corrected sum of corrected sum of productsproducts ( (SPSP) defined by the equation) defined by the equation

To calculate covariance, we first compute the To calculate covariance, we first compute the corrected sum of corrected sum of productsproducts ( (SPSP) defined by the equation) defined by the equation


lil

n

ikikkl BVBVSP

1

Remote Sensing Univariate StatisticsRemote Sensing Univariate Statistics

It is computationally more efficient to use the following It is computationally more efficient to use the following formula to arrive at the same result:formula to arrive at the same result:

This quantity is called the uncorrected sum of products. This quantity is called the uncorrected sum of products.

It is computationally more efficient to use the following It is computationally more efficient to use the following formula to arrive at the same result:formula to arrive at the same result:

This quantity is called the uncorrected sum of products. This quantity is called the uncorrected sum of products.


n

BVBVBVBVSP

n

i

n

iilikn

iilikkl

1 1

1


Just as simple variance was calculated by dividing the corrected Just as simple variance was calculated by dividing the corrected sums of squares (sums of squares (SSSS) by ) by ((n – 1n – 1)), , covariancecovariance is calculated by is calculated by dividing dividing SPSP by ( by (n – 1n – 1). Therefore, the covariance between ). Therefore, the covariance between brightness values in bands brightness values in bands kk and and l,l, covcovklkl, is equal to , is equal to

Just as simple variance was calculated by dividing the corrected Just as simple variance was calculated by dividing the corrected sums of squares (sums of squares (SSSS) by ) by ((n – 1n – 1)), , covariancecovariance is calculated by is calculated by dividing dividing SPSP by ( by (n – 1n – 1). Therefore, the covariance between ). Therefore, the covariance between brightness values in bands brightness values in bands kk and and l,l, covcovklkl, is equal to , is equal to


1cov

n

SPklkl



Band 1Band 1

(green)(green)

Band 2 Band 2

(red)(red)


infrared)infrared)


infrared)infrared)

Band 1Band 1 SSSS11covcov1,21,2 covcov1,31,3 covcov1,41,4

Band 2Band 2 covcov2,12,1 SSSS22covcov2,32,3 covcov2,42,4

Band 3Band 3 covcov3,13,1 covcov3,23,2 SSSS33covcov3,43,4

Band 4Band 4 covcov4,14,1 covcov4,24,2 covcov4,34,3 SSSS44

Format of a Variance-Covariance MatrixFormat of a Variance-Covariance MatrixFormat of a Variance-Covariance MatrixFormat of a Variance-Covariance Matrix

Band 1Band 1 (Band 1 x Band 2)(Band 1 x Band 2) Band 2 Band 2

130130 7,4107,410 5757

165165 5,7755,775 3535

100100 2,5002,500 2525

135135 6,7506,750 5050

145145 9,4259,425 6565

675675 31,86031,860 232232

Computation of Variance-Covariance Between Computation of Variance-Covariance Between Bands 1 and 2 of the Sample DataBands 1 and 2 of the Sample Data

Computation of Variance-Covariance Between Computation of Variance-Covariance Between Bands 1 and 2 of the Sample DataBands 1 and 2 of the Sample Data

1354

540cov

5

232675)860,31(

12

12

SP



Band 1Band 1

(green)(green)

Band 2 Band 2

(red)(red)


infrared)infrared)


infrared)infrared)

Band 1Band 1 562.25562.25 -- -- --

Band 2Band 2 135135 264.80264.80 -- --

Band 3Band 3 718.75718.75 275.25275.25 1007.501007.50 --

Band 4Band 4 537.50537.50 6464 663.75663.75 570570

Variance-Covariance Matrix of the Sample DataVariance-Covariance Matrix of the Sample DataVariance-Covariance Matrix of the Sample DataVariance-Covariance Matrix of the Sample Data


To estimate the degree of interrelation between variables in a To estimate the degree of interrelation between variables in a manner not influenced by measurement units, the manner not influenced by measurement units, the correlation correlation coefficient, r,coefficient, r, is commonly used. The correlation between two is commonly used. The correlation between two bands of remotely sensed data, bands of remotely sensed data, rrklkl, is the ratio of their , is the ratio of their

covariance (covariance (covcovklkl) to the product of their standard deviations ) to the product of their standard deviations

((sskkssll); thus:); thus:

To estimate the degree of interrelation between variables in a To estimate the degree of interrelation between variables in a manner not influenced by measurement units, the manner not influenced by measurement units, the correlation correlation coefficient, r,coefficient, r, is commonly used. The correlation between two is commonly used. The correlation between two bands of remotely sensed data, bands of remotely sensed data, rrklkl, is the ratio of their , is the ratio of their

covariance (covariance (covcovklkl) to the product of their standard deviations ) to the product of their standard deviations

((sskkssll); thus:); thus:


lk

klkl ss

rcov

Correlation Matrix for the Sample DataCorrelation Matrix for the Sample DataCorrelation Matrix for the Sample DataCorrelation Matrix for the Sample Data

Band 1Band 1

(green)(green)

Band 2 Band 2

(red)(red)


infrared)infrared)


infrared)infrared)

Band 1Band 1 -- -- -- --

Band 2Band 2 0.350.35 -- -- --

Band 3Band 3 0.950.95 0.530.53 -- --

Band 4Band 4 0.940.94 0.160.16 0.870.87 --



Band Min Max Mean Standard DeviationBand Min Max Mean Standard Deviation 1 51 242 65.163137 10.2313561 51 242 65.163137 10.231356 2 17 115 25.797593 5.9560482 17 115 25.797593 5.956048 3 14 131 23.958016 8.4698903 14 131 23.958016 8.469890 4 5 105 26.550666 15.6900544 5 105 26.550666 15.690054 5 0 193 32.014001 24.2964175 0 193 32.014001 24.296417 6 0 128 15.103553 12.7381886 0 128 15.103553 12.738188 7 102 124 110.734372 4.3050657 102 124 110.734372 4.305065

Covariance MatrixCovariance MatrixBand Band 1 Band 2 Band 3 Band 4 Band 5 Band 6 Band 7Band Band 1 Band 2 Band 3 Band 4 Band 5 Band 6 Band 7 1 104.680654 58.797907 82.602381 69.603136 142.947000 94.488082 24.4645961 104.680654 58.797907 82.602381 69.603136 142.947000 94.488082 24.464596 2 58.797907 35.474507 48.644220 45.539546 90.661412 57.877406 14.8128862 58.797907 35.474507 48.644220 45.539546 90.661412 57.877406 14.812886 3 82.602381 48.644220 71.739034 76.954037 149.566052 91.234270 23.8274183 82.602381 48.644220 71.739034 76.954037 149.566052 91.234270 23.827418 4 69.603136 45.539546 76.954037 246.177785 342.523400 157.655947 46.8157674 69.603136 45.539546 76.954037 246.177785 342.523400 157.655947 46.815767 5 142.947000 90.661412 149.566052 342.523400 590.315858 294.019002 82.9942415 142.947000 90.661412 149.566052 342.523400 590.315858 294.019002 82.994241 6 94.488082 57.877406 91.234270 157.655947 294.019002 162.261439 44.6742476 94.488082 57.877406 91.234270 157.655947 294.019002 162.261439 44.674247 7 24.464596 14.812886 23.827418 46.815767 82.994241 44.674247 18.5335867 24.464596 14.812886 23.827418 46.815767 82.994241 44.674247 18.533586

Correlation MatrixCorrelation MatrixBand Band 1 Band 2 Band 3 Band 4 Band 5 Band 6 Band 7Band Band 1 Band 2 Band 3 Band 4 Band 5 Band 6 Band 7 1 1.000000 0.964874 0.953195 0.433582 0.575042 0.724997 0.5554251 1.000000 0.964874 0.953195 0.433582 0.575042 0.724997 0.555425 2 0.964874 1.000000 0.964263 0.487311 0.626501 0.762857 0.5776992 0.964874 1.000000 0.964263 0.487311 0.626501 0.762857 0.577699 3 0.953195 0.964263 1.000000 0.579068 0.726797 0.845615 0.6534613 0.953195 0.964263 1.000000 0.579068 0.726797 0.845615 0.653461 4 0.433582 0.487311 0.579068 1.000000 0.898511 0.788821 0.6930874 0.433582 0.487311 0.579068 1.000000 0.898511 0.788821 0.693087 5 0.575042 0.626501 0.726797 0.898511 1.000000 0.950004 0.7934625 0.575042 0.626501 0.726797 0.898511 1.000000 0.950004 0.793462 6 0.724997 0.762857 0.845615 0.788821 0.950004 1.000000 0.8146486 0.724997 0.762857 0.845615 0.788821 0.950004 1.000000 0.814648 7 0.555425 0.577699 0.653461 0.693087 0.793462 0.814648 1.0000007 0.555425 0.577699 0.653461 0.693087 0.793462 0.814648 1.000000

Band Min Max Mean Standard DeviationBand Min Max Mean Standard Deviation 1 51 242 65.163137 10.2313561 51 242 65.163137 10.231356 2 17 115 25.797593 5.9560482 17 115 25.797593 5.956048 3 14 131 23.958016 8.4698903 14 131 23.958016 8.469890 4 5 105 26.550666 15.6900544 5 105 26.550666 15.690054 5 0 193 32.014001 24.2964175 0 193 32.014001 24.296417 6 0 128 15.103553 12.7381886 0 128 15.103553 12.738188 7 102 124 110.734372 4.3050657 102 124 110.734372 4.305065

Covariance MatrixCovariance MatrixBand Band 1 Band 2 Band 3 Band 4 Band 5 Band 6 Band 7Band Band 1 Band 2 Band 3 Band 4 Band 5 Band 6 Band 7 1 104.680654 58.797907 82.602381 69.603136 142.947000 94.488082 24.4645961 104.680654 58.797907 82.602381 69.603136 142.947000 94.488082 24.464596 2 58.797907 35.474507 48.644220 45.539546 90.661412 57.877406 14.8128862 58.797907 35.474507 48.644220 45.539546 90.661412 57.877406 14.812886 3 82.602381 48.644220 71.739034 76.954037 149.566052 91.234270 23.8274183 82.602381 48.644220 71.739034 76.954037 149.566052 91.234270 23.827418 4 69.603136 45.539546 76.954037 246.177785 342.523400 157.655947 46.8157674 69.603136 45.539546 76.954037 246.177785 342.523400 157.655947 46.815767 5 142.947000 90.661412 149.566052 342.523400 590.315858 294.019002 82.9942415 142.947000 90.661412 149.566052 342.523400 590.315858 294.019002 82.994241 6 94.488082 57.877406 91.234270 157.655947 294.019002 162.261439 44.6742476 94.488082 57.877406 91.234270 157.655947 294.019002 162.261439 44.674247 7 24.464596 14.812886 23.827418 46.815767 82.994241 44.674247 18.5335867 24.464596 14.812886 23.827418 46.815767 82.994241 44.674247 18.533586

Correlation MatrixCorrelation MatrixBand Band 1 Band 2 Band 3 Band 4 Band 5 Band 6 Band 7Band Band 1 Band 2 Band 3 Band 4 Band 5 Band 6 Band 7 1 1.000000 0.964874 0.953195 0.433582 0.575042 0.724997 0.5554251 1.000000 0.964874 0.953195 0.433582 0.575042 0.724997 0.555425 2 0.964874 1.000000 0.964263 0.487311 0.626501 0.762857 0.5776992 0.964874 1.000000 0.964263 0.487311 0.626501 0.762857 0.577699 3 0.953195 0.964263 1.000000 0.579068 0.726797 0.845615 0.6534613 0.953195 0.964263 1.000000 0.579068 0.726797 0.845615 0.653461 4 0.433582 0.487311 0.579068 1.000000 0.898511 0.788821 0.6930874 0.433582 0.487311 0.579068 1.000000 0.898511 0.788821 0.693087 5 0.575042 0.626501 0.726797 0.898511 1.000000 0.950004 0.7934625 0.575042 0.626501 0.726797 0.898511 1.000000 0.950004 0.793462 6 0.724997 0.762857 0.845615 0.788821 0.950004 1.000000 0.8146486 0.724997 0.762857 0.845615 0.788821 0.950004 1.000000 0.814648 7 0.555425 0.577699 0.653461 0.693087 0.793462 0.814648 1.0000007 0.555425 0.577699 0.653461 0.693087 0.793462 0.814648 1.000000

Univariate and Univariate and Multivariate Multivariate

Statistics of Landsat Statistics of Landsat TM Data of TM Data of

Charleston, SCCharleston, SC

Univariate and Univariate and Multivariate Multivariate

Statistics of Landsat Statistics of Landsat TM Data of TM Data of

Charleston, SCCharleston, SC

3-Dimensional 3-Dimensional View of the View of the

Thermal Infrared Thermal Infrared Matrix of Data Matrix of Data

3-Dimensional 3-Dimensional View of the View of the

Thermal Infrared Thermal Infrared Matrix of Data Matrix of Data


Two-dimensional Two-dimensional Feature Space Feature Space

Plot of TM Plot of TM Bands 3 and 4Bands 3 and 4

Two-dimensional Two-dimensional Feature Space Feature Space

Plot of TM Plot of TM Bands 3 and 4Bands 3 and 4


dr. john r. jensen department of geography university of south carolina columbia, sc 29208

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