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

Image Quality Assessment and Image Quality Assessment and Statistical EvaluationStatistical Evaluation

Image Quality Assessment and Image Quality Assessment and Statistical EvaluationStatistical Evaluation

Jensen, 2004Jensen, 2004Jensen, 2004Jensen, 2004

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-to- digital conversion).

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-to- digital conversion).

Image Quality Assessment and Statistical EvaluationImage Quality Assessment and Statistical EvaluationImage Quality Assessment and Statistical EvaluationImage Quality Assessment and Statistical Evaluation

Jensen, 2004Jensen, 2004

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).

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).

Image Quality Assessment and Statistical EvaluationImage Quality Assessment and Statistical EvaluationImage Quality Assessment and Statistical EvaluationImage Quality Assessment and Statistical Evaluation


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

Jensen, 2004Jensen, 2004Jensen, 2004Jensen, 2004

The following notation is used to describe the mathematical The following notation is used to describe the mathematical operations applied to the digital remote sensor data: operations applied to the digital remote sensor 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 is used to describe the mathematical The following notation is used to describe the mathematical operations applied to the digital remote sensor data: operations applied to the digital remote sensor 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 2004. All Landsat 7 ETM+ images acquired of the Earth in 2004. All Landsat 7 ETM+ images of Charleston, S.C. in 2004 is a finite population. of Charleston, S.C. in 2004 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, 2004, Landsat image of Charleston. If observations with 1, 2004, 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 2004. All Landsat 7 ETM+ images acquired of the Earth in 2004. All Landsat 7 ETM+ images of Charleston, S.C. in 2004 is a finite population. of Charleston, S.C. in 2004 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, 2004, Landsat image of Charleston. If observations with 1, 2004, 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






• 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:

• 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,

• 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:

• 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,

• 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


Two- and Three-Two- and Three-Dimensional Dimensional Evaluation of Evaluation of

Pixel Brightness Pixel Brightness Values within a Values within a

Geographic AreaGeographic Area

Two- and Three-Two- and Three-Dimensional Dimensional Evaluation of Evaluation of

Pixel Brightness Pixel Brightness Values within a Values within a

Geographic AreaGeographic Area


Univariate Descriptive Image StatisticsUnivariate Descriptive Image StatisticsUnivariate Descriptive Image StatisticsUnivariate Descriptive Image Statistics

Measures of Central Tendency in Remote Sensor Data 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.

Measures of Central Tendency in Remote Sensor Data 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.









Univariate Descriptive Image StatisticsUnivariate Descriptive Image StatisticsUnivariate Descriptive Image StatisticsUnivariate Descriptive Image Statistics

The The meanmean 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 ((kk) of a ) of a

single band of imagery composed of single band of imagery composed of nn brightness values brightness values ((BVBVikik) is ) is

computed using the formula:computed using the formula:

The sample mean, The sample mean, kk,, is an unbiased estimate of the population mean. For is an unbiased estimate of the population mean. For

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

The The meanmean 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 ((kk) of a ) of a

single band of imagery composed of single band of imagery composed of nn brightness values brightness values ((BVBVikik) is ) is

computed using the formula:computed using the formula:

The sample mean, The sample mean, kk,, is an unbiased estimate of the population mean. For is an unbiased estimate of the population mean. For

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


n

BVn

iik

k

1

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

Hypothetical Dataset of Brightness ValuesHypothetical Dataset of Brightness ValuesHypothetical Dataset of Brightness ValuesHypothetical 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

Measures of DispersionMeasures 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,

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.


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,

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.


kkkrange minmax kkkrange minmax

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

The numerator of the expression is the corrected sum of squares (The numerator of the expression is the corrected sum of squares (SSSS). If ). If the sample mean (the sample mean (kk) were actually the population mean, this would be an ) were actually the population mean, this would be an

accurate measurement of the variance. accurate measurement of the variance.


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

The numerator of the expression is the corrected sum of squares (The numerator of the expression is the corrected sum of squares (SSSS). If ). If the sample mean (the sample mean (kk) were actually the population mean, this would be an ) were actually the population mean, this would be an

accurate measurement of the variance. accurate measurement of the variance.


n

BVn

ikik

k

1

2

var

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

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

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


1var

n

SSk 1

var

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


Univariate Statistics for the Hypothetical Example DatasetUnivariate Statistics for the Hypothetical Example DatasetUnivariate Statistics for the Hypothetical Example DatasetUnivariate Statistics for the Hypothetical Example Dataset

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

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



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


Univariate Statistics for the Hypothetical Example DatasetUnivariate Statistics for the Hypothetical Example DatasetUnivariate Statistics for the Hypothetical Example DatasetUnivariate Statistics for the Hypothetical Example Dataset

Measures of Distribution (Histogram) Measures of Distribution (Histogram) Asymmetry and Peak SharpnessAsymmetry and Peak Sharpness


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:

A perfectly symmetric histogram has a A perfectly symmetric histogram has a skewnessskewness value of zero. value of zero.

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:

A perfectly symmetric histogram has a A perfectly symmetric histogram has a skewnessskewness value of zero. value of zero.


n

s

BV

skewness

n

i k

kik

k

1

3

n

s

BV

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. a normal distribution. KurtosisKurtosis is computed using the formula: 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. a normal distribution. KurtosisKurtosis is computed using the formula: is computed using the formula:


31

1

4

n

i k

kikk s

BV

nkurtosis

31

1

4

n

i k

kikk s

BV

nkurtosis



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

Remote sensing research is often concerned with the Remote sensing research is often concerned with the measurement of how much radiant flux is reflected or emitted measurement of how much radiant flux is reflected or emitted from an object in more than one band (e.g., in red and near-from an object in more than one band (e.g., in red and near-infrared bands). It is useful to compute infrared bands). It is useful to compute multivariatemultivariate statistical statistical measures such as measures such as covariancecovariance and and correlationcorrelation among the several among the several bands to determine how the measurements covary. Later it will bands to determine how the measurements covary. Later it will be shown that variance–covariance and correlation matrices are be shown that variance–covariance and correlation matrices are used in remote sensing principal components analysis (PCA), used in remote sensing principal components analysis (PCA), feature selection, classification and accuracy assessment. feature selection, classification and accuracy assessment.

Remote sensing research is often concerned with the Remote sensing research is often concerned with the measurement of how much radiant flux is reflected or emitted measurement of how much radiant flux is reflected or emitted from an object in more than one band (e.g., in red and near-from an object in more than one band (e.g., in red and near-infrared bands). It is useful to compute infrared bands). It is useful to compute multivariatemultivariate statistical statistical measures such as measures such as covariancecovariance and and correlationcorrelation among the several among the several bands to determine how the measurements covary. Later it will bands to determine how the measurements covary. Later it will be shown that variance–covariance and correlation matrices are be shown that variance–covariance and correlation matrices are used in remote sensing principal components analysis (PCA), used in remote sensing principal components analysis (PCA), feature selection, classification and accuracy assessment. feature selection, classification and accuracy assessment.



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

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


n

BVBVBVBVSP

n

i

n

iilikn

iilikkl

1 1

1

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 ll,, 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 ll,, covcovklkl, is equal to: , is equal to:


1cov

n

SPklkl 1

cov

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

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

Correlation between Multiple Bands of Correlation between Multiple Bands of Remotely Sensed DataRemotely Sensed 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

lk

klkl ss

rcov



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”.

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”.


Correlation Matrix of the Sample DataCorrelation Matrix of the Sample DataCorrelation Matrix of the Sample DataCorrelation Matrix of 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

Feature Space PlotsFeature Space PlotsFeature Space PlotsFeature Space Plots

The univariate and multivariate statistics discussed provide The univariate and multivariate statistics discussed provide accurate, fundamental information about the individual band accurate, fundamental information about the individual band statistics including how the bands statistics including how the bands covarycovary and and correlatecorrelate. . Sometimes, however, it is useful to examine statistical Sometimes, however, it is useful to examine statistical relationships relationships graphicallygraphically. .

Individual bands of remotely sensed data are often referred to Individual bands of remotely sensed data are often referred to as as featuresfeatures in the pattern recognition literature. To truly in the pattern recognition literature. To truly appreciate how two bands (appreciate how two bands (featuresfeatures) in a remote sensing dataset ) in a remote sensing dataset covary and if they are correlated or not, it is often useful to covary and if they are correlated or not, it is often useful to produce a two-band produce a two-band feature space plotfeature space plot. .

The univariate and multivariate statistics discussed provide The univariate and multivariate statistics discussed provide accurate, fundamental information about the individual band accurate, fundamental information about the individual band statistics including how the bands statistics including how the bands covarycovary and and correlatecorrelate. . Sometimes, however, it is useful to examine statistical Sometimes, however, it is useful to examine statistical relationships relationships graphicallygraphically. .

Individual bands of remotely sensed data are often referred to Individual bands of remotely sensed data are often referred to as as featuresfeatures in the pattern recognition literature. To truly in the pattern recognition literature. To truly appreciate how two bands (appreciate how two bands (featuresfeatures) in a remote sensing dataset ) in a remote sensing dataset covary and if they are correlated or not, it is often useful to covary and if they are correlated or not, it is often useful to produce a two-band produce a two-band feature space plotfeature space plot. .


Feature Space PlotsFeature Space PlotsFeature Space PlotsFeature Space Plots

A A two-dimensional feature space plottwo-dimensional feature space plot extracts the brightness extracts the brightness value for every pixel in the scene in two bands and plots the value for every pixel in the scene in two bands and plots the frequency of occurrence in a 255 by 255 feature space frequency of occurrence in a 255 by 255 feature space (assuming 8-bit data). The greater the frequency of occurrence (assuming 8-bit data). The greater the frequency of occurrence of unique pairs of values, the brighter the feature space pixel. of unique pairs of values, the brighter the feature space pixel.

A A two-dimensional feature space plottwo-dimensional feature space plot extracts the brightness extracts the brightness value for every pixel in the scene in two bands and plots the value for every pixel in the scene in two bands and plots the frequency of occurrence in a 255 by 255 feature space frequency of occurrence in a 255 by 255 feature space (assuming 8-bit data). The greater the frequency of occurrence (assuming 8-bit data). The greater the frequency of occurrence of unique pairs of values, the brighter the feature space pixel. of unique pairs of values, the brighter the feature space pixel.


Two-dimensional Two-dimensional Feature Space Feature Space Plot of Landsat Plot of Landsat

Thematic Mapper Thematic Mapper Band 3 Band 3

and 4 Data of and 4 Data of Charleston, SC Charleston, SC

obtained on obtained on November 11, November 11,

19821982

Two-dimensional Two-dimensional Feature Space Feature Space Plot of Landsat Plot of Landsat

Thematic Mapper Thematic Mapper Band 3 Band 3

and 4 Data of and 4 Data of Charleston, SC Charleston, SC

obtained on obtained on November 11, November 11,

19821982


Geostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor Data

The Earth’s surface has distinct spatial properties. The The Earth’s surface has distinct spatial properties. The brightness values in imagery constitute a record of these spatial brightness values in imagery constitute a record of these spatial properties. The spatial characteristics may take the form of properties. The spatial characteristics may take the form of texture or pattern. Image analysts often try to quantify the texture or pattern. Image analysts often try to quantify the spatial texture or pattern. This requires looking at a pixel and spatial texture or pattern. This requires looking at a pixel and its neighbors and trying to quantify the its neighbors and trying to quantify the spatial autocorrelation spatial autocorrelation relationships in the imageryrelationships in the imagery. But how do we measure . But how do we measure autocorrelation characteristics in images? autocorrelation characteristics in images?

The Earth’s surface has distinct spatial properties. The The Earth’s surface has distinct spatial properties. The brightness values in imagery constitute a record of these spatial brightness values in imagery constitute a record of these spatial properties. The spatial characteristics may take the form of properties. The spatial characteristics may take the form of texture or pattern. Image analysts often try to quantify the texture or pattern. Image analysts often try to quantify the spatial texture or pattern. This requires looking at a pixel and spatial texture or pattern. This requires looking at a pixel and its neighbors and trying to quantify the its neighbors and trying to quantify the spatial autocorrelation spatial autocorrelation relationships in the imageryrelationships in the imagery. But how do we measure . But how do we measure autocorrelation characteristics in images? autocorrelation characteristics in images?



A random variable distributed in space (e.g., spectral A random variable distributed in space (e.g., spectral reflectance) is said to be reflectance) is said to be regionalizedregionalized. We can use . We can use geostatisticalgeostatistical measures to extract the spatial properties of measures to extract the spatial properties of regionalized variables.regionalized variables. Once quantified, the regionalized Once quantified, the regionalized variable properties can be used in many remote sensing variable properties can be used in many remote sensing applications such as image classification and the allocation of applications such as image classification and the allocation of spatially unbiased sampling sites during classification map spatially unbiased sampling sites during classification map accuracy assessment. Another application of accuracy assessment. Another application of geostatisticsgeostatistics is the is the prediction of values at unsampled locations. Geostatistical prediction of values at unsampled locations. Geostatistical interpolation techniques could be used to evaluate the spatial interpolation techniques could be used to evaluate the spatial relationships associated with the existing data to create a new, relationships associated with the existing data to create a new, improved systematic grid of elevation values.improved systematic grid of elevation values.

A random variable distributed in space (e.g., spectral A random variable distributed in space (e.g., spectral reflectance) is said to be reflectance) is said to be regionalizedregionalized. We can use . We can use geostatisticalgeostatistical measures to extract the spatial properties of measures to extract the spatial properties of regionalized variables.regionalized variables. Once quantified, the regionalized Once quantified, the regionalized variable properties can be used in many remote sensing variable properties can be used in many remote sensing applications such as image classification and the allocation of applications such as image classification and the allocation of spatially unbiased sampling sites during classification map spatially unbiased sampling sites during classification map accuracy assessment. Another application of accuracy assessment. Another application of geostatisticsgeostatistics is the is the prediction of values at unsampled locations. Geostatistical prediction of values at unsampled locations. Geostatistical interpolation techniques could be used to evaluate the spatial interpolation techniques could be used to evaluate the spatial relationships associated with the existing data to create a new, relationships associated with the existing data to create a new, improved systematic grid of elevation values.improved systematic grid of elevation values.



GeostatisticsGeostatistics are now widely used in many fields and comprise are now widely used in many fields and comprise a branch of a branch of spatial statisticsspatial statistics. Originally, geostatistics was . Originally, geostatistics was synonymous with synonymous with krigingkriging——a statistical version of interpolation. a statistical version of interpolation. Kriging is a generic name for a family of least-squares linear Kriging is a generic name for a family of least-squares linear regression algorithms that are used to estimate the value of a regression algorithms that are used to estimate the value of a continuous attribute (e.g., terrain elevation or percent continuous attribute (e.g., terrain elevation or percent reflectance) at any reflectance) at any unsampledunsampled location using only attribute data location using only attribute data available over the study area. However, geostatistical analysis available over the study area. However, geostatistical analysis now includes not only kriging but also the traditional now includes not only kriging but also the traditional deterministic spatial interpolation methods. One of the essential deterministic spatial interpolation methods. One of the essential features of geostatistics is that the phenomenon being studied features of geostatistics is that the phenomenon being studied (e.g., elevation, reflectance, temperature, precipitation, a land-(e.g., elevation, reflectance, temperature, precipitation, a land-cover class) must be cover class) must be continuouscontinuous across the landscape or at least across the landscape or at least capable of existing throughout the landscape.capable of existing throughout the landscape.

GeostatisticsGeostatistics are now widely used in many fields and comprise are now widely used in many fields and comprise a branch of a branch of spatial statisticsspatial statistics. Originally, geostatistics was . Originally, geostatistics was synonymous with synonymous with krigingkriging——a statistical version of interpolation. a statistical version of interpolation. Kriging is a generic name for a family of least-squares linear Kriging is a generic name for a family of least-squares linear regression algorithms that are used to estimate the value of a regression algorithms that are used to estimate the value of a continuous attribute (e.g., terrain elevation or percent continuous attribute (e.g., terrain elevation or percent reflectance) at any reflectance) at any unsampledunsampled location using only attribute data location using only attribute data available over the study area. However, geostatistical analysis available over the study area. However, geostatistical analysis now includes not only kriging but also the traditional now includes not only kriging but also the traditional deterministic spatial interpolation methods. One of the essential deterministic spatial interpolation methods. One of the essential features of geostatistics is that the phenomenon being studied features of geostatistics is that the phenomenon being studied (e.g., elevation, reflectance, temperature, precipitation, a land-(e.g., elevation, reflectance, temperature, precipitation, a land-cover class) must be cover class) must be continuouscontinuous across the landscape or at least across the landscape or at least capable of existing throughout the landscape.capable of existing throughout the landscape.


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.

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.


KrigingKriging makes use of the spatial autocorrelation information. Kriging is makes use of the spatial autocorrelation information. Kriging is similar to ‘distance weighted interpolation’similar to ‘distance weighted interpolation’ in that it weights the in that it weights the surrounding measured values to derive a prediction for each new location. surrounding measured values to derive a prediction for each new location. However, the weights are based not only on the distance between the However, the weights are based not only on the distance between the measured points and the point to be predicted (used in inverse distance measured points and the point to be predicted (used in inverse distance weighting), but also on the overall weighting), but also on the overall spatialspatial arrangementarrangement among the measured among the measured points (i.e., their autocorrelation). Kriging uses weights that are defined points (i.e., their autocorrelation). Kriging uses weights that are defined statistically from the observed data rather than statistically from the observed data rather than a prioria priori. This is the most . This is the most significant difference between deterministic (traditional) and geostatistical significant difference between deterministic (traditional) and geostatistical analysis. Traditional statistical analysis assumes the samples derived for a analysis. Traditional statistical analysis assumes the samples derived for a particular attribute are independent and not correlated in any way. particular attribute are independent and not correlated in any way. Conversely, geostatistical analysis allows a scientist to compute distances Conversely, geostatistical analysis allows a scientist to compute distances between observations and to model autocorrelation as a function of distance between observations and to model autocorrelation as a function of distance and direction. This information is then used to refine the kriging and direction. This information is then used to refine the kriging interpolation process, hopefully, making predictions at new locations that interpolation process, hopefully, making predictions at new locations that are more accurate than those derived using traditional methods. There are are more accurate than those derived using traditional methods. There are numerous methods of kriging, including simple, ordinary, universal, numerous methods of kriging, including simple, ordinary, universal, probability, indicator, disjunctive, and multiple variable co-kriging.probability, indicator, disjunctive, and multiple variable co-kriging.

KrigingKriging makes use of the spatial autocorrelation information. Kriging is makes use of the spatial autocorrelation information. Kriging is similar to ‘distance weighted interpolation’similar to ‘distance weighted interpolation’ in that it weights the in that it weights the surrounding measured values to derive a prediction for each new location. surrounding measured values to derive a prediction for each new location. However, the weights are based not only on the distance between the However, the weights are based not only on the distance between the measured points and the point to be predicted (used in inverse distance measured points and the point to be predicted (used in inverse distance weighting), but also on the overall weighting), but also on the overall spatialspatial arrangementarrangement among the measured among the measured points (i.e., their autocorrelation). Kriging uses weights that are defined points (i.e., their autocorrelation). Kriging uses weights that are defined statistically from the observed data rather than statistically from the observed data rather than a prioria priori. This is the most . This is the most significant difference between deterministic (traditional) and geostatistical significant difference between deterministic (traditional) and geostatistical analysis. Traditional statistical analysis assumes the samples derived for a analysis. Traditional statistical analysis assumes the samples derived for a particular attribute are independent and not correlated in any way. particular attribute are independent and not correlated in any way. Conversely, geostatistical analysis allows a scientist to compute distances Conversely, geostatistical analysis allows a scientist to compute distances between observations and to model autocorrelation as a function of distance between observations and to model autocorrelation as a function of distance and direction. This information is then used to refine the kriging and direction. This information is then used to refine the kriging interpolation process, hopefully, making predictions at new locations that interpolation process, hopefully, making predictions at new locations that are more accurate than those derived using traditional methods. There are are more accurate than those derived using traditional methods. There are numerous methods of kriging, including simple, ordinary, universal, numerous methods of kriging, including simple, ordinary, universal, probability, indicator, disjunctive, and multiple variable co-kriging.probability, indicator, disjunctive, and multiple variable co-kriging.


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.

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.


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.

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.


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.

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.


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 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:

where h) is an unbiased estimate of the average semivariance of the population.

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 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:

where h) is an unbiased estimate of the average semivariance of the population.

m

hxzxzh

m

iii

1

2

)(


The total number of possible pairs The total number of possible pairs mm along the transect is computed by along the transect is computed by subtracting the lag distance subtracting the lag distance hh from the total number of pixels present in the from the total number of pixels present in the dataset dataset nn, that is, , that is, m = n – h.m = n – h. In practice, semivariance is computed for pairs In practice, semivariance is computed for pairs of observations in all directions. Thus, directional semivariograms are of observations in all directions. Thus, directional semivariograms are derived and directional influences can be examined.derived and directional influences can be examined.

The average semivariance is a good measure of the amount of The average semivariance is a good measure of the amount of dissimilaritydissimilarity between spatially separate pixels. Generally, the larger the average between spatially separate pixels. Generally, the larger the average semivariance semivariance h)h), the less similar are the pixels in an image (or the , the less similar are the pixels in an image (or the polygons if the analysis was based on ground measurement).polygons if the analysis was based on ground measurement).

The total number of possible pairs The total number of possible pairs mm along the transect is computed by along the transect is computed by subtracting the lag distance subtracting the lag distance hh from the total number of pixels present in the from the total number of pixels present in the dataset dataset nn, that is, , that is, m = n – h.m = n – h. In practice, semivariance is computed for pairs In practice, semivariance is computed for pairs of observations in all directions. Thus, directional semivariograms are of observations in all directions. Thus, directional semivariograms are derived and directional influences can be examined.derived and directional influences can be examined.

The average semivariance is a good measure of the amount of The average semivariance is a good measure of the amount of dissimilaritydissimilarity between spatially separate pixels. Generally, the larger the average between spatially separate pixels. Generally, the larger the average semivariance semivariance h)h), the less similar are the pixels in an image (or the , the less similar are the pixels in an image (or the polygons if the analysis was based on ground measurement).polygons if the analysis was based on ground measurement).


The semivariogram is a plot of the average semivariance value on the y-axis (e.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).

The semivariogram is a plot of the average semivariance value on the y-axis (e.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 analysisGeostatistical analysis incorporates spatial incorporates spatial autocorrelation information in the kriging autocorrelation information in the kriging interpolation process. Phenomena that are interpolation process. Phenomena that are geographically closer together are generally geographically closer together are generally more highly correlated than things that are more highly correlated than things that are farther apart.farther apart.

Geostatistical analysisGeostatistical analysis incorporates spatial incorporates spatial autocorrelation information in the kriging autocorrelation information in the kriging interpolation process. Phenomena that are interpolation process. Phenomena that are geographically closer together are generally geographically closer together are generally more highly correlated than things that are more highly correlated than things that are farther apart.farther apart. Jensen, 2004Jensen, 2004

a) A hypothetical remote sensing a) A hypothetical remote sensing dataset used to demonstrate the dataset used to demonstrate the characteristics of lag distance (characteristics of lag distance (hh) ) along a transect of pixels along a transect of pixels extracted from an image. extracted from an image.

a) A hypothetical remote sensing a) A hypothetical remote sensing dataset used to demonstrate the dataset used to demonstrate the characteristics of lag distance (characteristics of lag distance (hh) ) along a transect of pixels along a transect of pixels extracted from an image. extracted from an image.


b) A semivariogram of the b) A semivariogram of the semivariance semivariance (h) (h) characteristics characteristics found in the hypothetical dataset found in the hypothetical dataset at various lag distances (at various lag distances (hh). ).

b) A semivariogram of the b) A semivariogram of the semivariance semivariance (h) (h) characteristics characteristics found in the hypothetical dataset found in the hypothetical dataset at various lag distances (at various lag distances (hh). ).


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 (h) computed. The semivariance (h) at each lag distance may be displayed as a semivariogram with the range, sill, and nugget variance characteristics.

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 (h) computed. The semivariance (h) at each lag distance may be displayed as a semivariogram with the range, sill, and nugget variance characteristics.


Original Images, Original Images, Semivariograms, and Semivariograms, and Predicted Images. Predicted Images.

a) The green band of a a) The green band of a 0.61 0.61 0.61 m image 0.61 m image of cargo containers in of cargo containers in the port of Hamburg, the port of Hamburg, Germany, obtained on Germany, obtained on May 10, 2002 May 10, 2002 (courtesy of (courtesy of DigitalGlobe). DigitalGlobe).

b) A subset containing b) A subset containing just cargo containers. just cargo containers.

c) A subset continuing c) A subset continuing just tarmac (asphalt).just tarmac (asphalt).

Original Images, Original Images, Semivariograms, and Semivariograms, and Predicted Images. Predicted Images.

a) The green band of a a) The green band of a 0.61 0.61 0.61 m image 0.61 m image of cargo containers in of cargo containers in the port of Hamburg, the port of Hamburg, Germany, obtained on Germany, obtained on May 10, 2002 May 10, 2002 (courtesy of (courtesy of DigitalGlobe). DigitalGlobe).

b) A subset containing b) A subset containing just cargo containers. just cargo containers.

c) A subset continuing c) A subset continuing just tarmac (asphalt).just tarmac (asphalt).

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


Dr. John R. Jensen Department of Geography University of South Carolina Columbia, SC 29208

Documents

band correlation

band lmink

band kvckl

band kquantk

band ksck

band kbvik

band of imageryn

band of imageryl