1. Image Quality Assessment andImage Quality Assessment and Statistical EvaluationStatistical Evaluation Image Quality Assessment andImage Quality Assessment and Statistical EvaluationStatistical Evaluation Mahesh Kumar JatMahesh Kumar Jat Department of Civil EngineeringDepartment of Civil Engineering Malaviya National Institute of Technology,Malaviya National Institute of Technology, JaipurJaipur Mahesh Kumar JatMahesh Kumar Jat Department of Civil EngineeringDepartment of Civil Engineering Malaviya National Institute of Technology,Malaviya National Institute of Technology, JaipurJaipur

2. Many remote sensing datasets contain high-quality, accurate data. Unfortunately, sometimes error (or noise) is introduced into the remote sensing 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 sensing 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 3. 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 4. Image Processing Mathematical NotationImage Processing Mathematical NotationImage Processing Mathematical NotationImage Processing Mathematical Notation The following notation is used to describe the mathematicalThe 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 ll = 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 bandth brightness value in band kk The following notation is used to describe the mathematicalThe 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 ll = 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 bandth brightness value in band kk 5. Image Processing Mathematical NotationImage Processing Mathematical NotationImage Processing Mathematical NotationImage Processing Mathematical Notation BVBVilil == iith brightness value in bandth brightness value in band ll minminkk = minimum value of brightness in= minimum value of brightness in bandband kk maxmaxkk = maximum value of brightness in= maximum value of brightness in bandband kk rangerangekk = range of actual brightness values in band= range of actual brightness values in band kk quantquantkk = quantization level of band= quantization level of band kk (e.g., 2(e.g., 288 = 0 to 255;= 0 to 255; 221212 = 0 to 4095)= 0 to 4095) kk = 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 bandth brightness value in band ll minminkk = minimum value of brightness in= minimum value of brightness in bandband kk maxmaxkk = maximum value of brightness in= maximum value of brightness in bandband kk rangerangekk = range of actual brightness values in band= range of actual brightness values in band kk quantquantkk = quantization level of band= quantization level of band kk (e.g., 2(e.g., 288 = 0 to 255;= 0 to 255; 221212 = 0 to 4095)= 0 to 4095) kk = 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 6. Image Processing Mathematical NotationImage Processing Mathematical NotationImage Processing Mathematical NotationImage Processing Mathematical Notation skewnessskewnesskk = skewness of a band= skewness of a band kk distributiondistribution kurtosiskurtosiskk = kurtosis of a band= kurtosis of a band kk distributiondistribution covcovklkl = covariance between pixel values in two bands,= covariance between pixel values in two bands, kk andand ll rrklkl = correlation between pixel values in two bands,= correlation between pixel values in two bands, kk andand ll XXcc = measurement vector for class= measurement vector for class cc composed ofcomposed of brightness values (brightness values (BVBVijkijk) from row) from row ii, column, column jj, and, and bandband kk skewnessskewnesskk = skewness of a band= skewness of a band kk distributiondistribution kurtosiskurtosiskk = kurtosis of a band= kurtosis of a band kk distributiondistribution covcovklkl = covariance between pixel values in two bands,= covariance between pixel values in two bands, kk andand ll rrklkl = correlation between pixel values in two bands,= correlation between pixel values in two bands, kk andand ll XXcc = measurement vector for class= measurement vector for class cc composed ofcomposed of brightness values (brightness values (BVBVijkijk) from row) from row ii, column, column jj, and, and bandband kk 7. Image Processing Mathematical NotationImage Processing Mathematical NotationImage Processing Mathematical NotationImage Processing Mathematical Notation MMcc = mean vector for class= mean vector for class cc MMdd = mean vector for class= mean vector for class dd ckck = 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 throughthrough l;l; shown asshown as VVcc vvdkldkl = covariance matrix of class= covariance matrix of class dd for bandsfor bands kk throughthrough ll;; shown asshown as VVdd MMcc = mean vector for class= mean vector for class cc MMdd = mean vector for class= mean vector for class dd ckck = 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 throughthrough l;l; shown asshown as VVcc vvdkldkl = covariance matrix of class= covariance matrix of class dd for bandsfor bands kk throughthrough ll;; shown asshown as VVdd 8. Remote Sensing Sampling TheoryRemote Sensing Sampling TheoryRemote Sensing Sampling TheoryRemote Sensing Sampling Theory AA populationpopulation is an infinite or finite set of elements. Anis an infinite or finite set of elements. An infinite population could be all possible images that might beinfinite population could be all possible images that might be acquired of the Earth in 2004.acquired of the Earth in 2004. AA samplesample is a subset of the elements taken from a populationis a subset of the elements taken from a population used to make inferences about certain characteristics of theused to make inferences about certain characteristics of the population. For example, we might decide to analyze a Seppopulation. For example, we might decide to analyze a Sep 16, 2007, IRS P6 image of Jaipur. If observations with16, 2007, IRS P6 image of Jaipur. If observations with certain characteristics are systematically excluded from thecertain characteristics are systematically excluded from the sample either deliberately or inadvertently (such as selectingsample either deliberately or inadvertently (such as selecting images obtained only in the spring of the year), it is aimages obtained only in the spring of the year), it is a biasedbiased sample.sample. Sampling errorSampling error is the difference between the trueis the difference between the true value of a population characteristic and the value of thatvalue of a population characteristic and the value of that characteristic inferred from a sample.characteristic inferred from a sample. AA populationpopulation is an infinite or finite set of elements. Anis an infinite or finite set of elements. An infinite population could be all possible images that might beinfinite population could be all possible images that might be acquired of the Earth in 2004.acquired of the Earth in 2004. AA samplesample is a subset of the elements taken from a populationis a subset of the elements taken from a population used to make inferences about certain characteristics of theused to make inferences about certain characteristics of the population. For example, we might decide to analyze a Seppopulation. For example, we might decide to analyze a Sep 16, 2007, IRS P6 image of Jaipur. If observations with16, 2007, IRS P6 image of Jaipur. If observations with certain characteristics are systematically excluded from thecertain characteristics are systematically excluded from the sample either deliberately or inadvertently (such as selectingsample either deliberately or inadvertently (such as selecting images obtained only in the spring of the year), it is aimages obtained only in the spring of the year), it is a biasedbiased sample.sample. Sampling errorSampling error is the difference between the trueis the difference between the true value of a population characteristic and the value of thatvalue of a population characteristic and the value of that characteristic inferred from a sample.characteristic inferred from a sample. 9. Remote Sensing Sampling TheoryRemote Sensing Sampling TheoryRemote Sensing Sampling TheoryRemote Sensing Sampling Theory Large samples drawn randomly from natural populationsLarge samples drawn randomly from natural populations usually produce ausually produce a symmetrical frequency distributionsymmetrical frequency distribution. Most. Most values are clustered around some central value, and thevalues are clustered around some central value, and the frequency of occurrence declines away from this centralfrequency of occurrence declines away from this central point. A graph of the distribution appears bell shaped and ispoint. A graph of the distribution appears bell shaped and is called acalled a normal distributionnormal distribution.. Many statistical tests used in the analysis of remotelyMany statistical tests used in the analysis of remotely sensed data assume that the brightness values recorded in asensed data assume that the brightness values recorded in a scene are normally distributed. Unfortunately, remotelyscene are normally distributed. Unfortunately, remotely sensed data maysensed data may notnot be normally distributed and the analystbe normally distributed and the analyst must be careful to identify such conditions. In suchmust 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 populationsLarge samples drawn randomly from natural populations usually produce ausually produce a symmetrical frequency distributionsymmetrical frequency distribution. Most. Most values are clustered around some central value, and thevalues are clustered around some central value, and the frequency of occurrence declines away from this centralfrequency of occurrence declines away from this central point. A graph of the distribution appears bell shaped and ispoint. A graph of the distribution appears bell shaped and is called acalled a normal distributionnormal distribution.. Many statistical tests used in the analysis of remotelyMany statistical tests used in the analysis of remotely sensed data assume that the brightness values recorded in asensed data assume that the brightness values recorded in a scene are normally distributed. Unfortunately, remotelyscene are normally distributed. Unfortunately, remotely sensed data maysensed data may notnot be normally distributed and the analystbe normally distributed and the analyst must be careful to identify such conditions. In suchmust be careful to identify such conditions. In such instances,instances, nonparametricnonparametric statistical theory may be preferred.statistical theory may be preferred. 10. CommonCommon Symmetric andSymmetric and SkewedSkewed Distributions inDistributions in Remotely SensedRemotely Sensed DataData CommonCommon Symmetric andSymmetric and SkewedSkewed Distributions inDistributions in Remotely SensedRemotely Sensed DataData 11. Remote Sensing Sampling TheoryRemote Sensing Sampling TheoryRemote Sensing Sampling TheoryRemote Sensing Sampling Theory TheThe histogramhistogram is a useful graphic representation of theis 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 singleIt is instructive to review how a histogram of a single bandband of imageryof imagery,, kk, composed of, composed of ii rowsrows andand jj columnscolumns with awith a brightness valuebrightness value BVBVijkijk at each pixel location is constructed.at each pixel location is constructed. TheThe histogramhistogram is a useful graphic representation of theis 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 singleIt is instructive to review how a histogram of a single bandband of imageryof imagery,, kk, composed of, composed of ii rowsrows andand jj columnscolumns with awith a brightness valuebrightness value BVBVijkijk at each pixel location is constructed.at each pixel location is constructed. 12. Histogram of AHistogram of A Single Band ofSingle Band of Landsat ThematicLandsat Thematic Mapper Data ofMapper Data of Charleston, SCCharleston, SC Histogram of AHistogram of A Single Band ofSingle Band of Landsat ThematicLandsat Thematic Mapper Data ofMapper Data of Charleston, SCCharleston, SC 13. Histogram ofHistogram of Thermal InfraredThermal Infrared Imagery of aImagery of a Thermal Plume inThermal Plume in the Savannahthe Savannah RiverRiver Histogram ofHistogram of Thermal InfraredThermal Infrared Imagery of aImagery of a Thermal Plume inThermal Plume in the Savannahthe Savannah RiverRiver 14. Remote Sensing MetadataRemote Sensing MetadataRemote Sensing MetadataRemote Sensing Metadata MetadataMetadata is data or information about data. Most qualityis data or information about data. Most quality digital image processing systems read, collect, and storedigital image processing systems read, collect, and store metadata about a particular image or sub-image. It ismetadata about a particular image or sub-image. It is important that the image analyst have access to this metadataimportant that the image analyst have access to this metadata information. In the most fundamental instance, metadatainformation. In the most fundamental instance, metadata might include:might include: the file name, date of last modification, level of quantizationthe 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 multivariatemode, 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 qualityis data or information about data. Most quality digital image processing systems read, collect, and storedigital image processing systems read, collect, and store metadata about a particular image or sub-image. It ismetadata about a particular image or sub-image. It is important that the image analyst have access to this metadataimportant that the image analyst have access to this metadata information. In the most fundamental instance, metadatainformation. In the most fundamental instance, metadata might include:might include: the file name, date of last modification, level of quantizationthe 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 multivariatemode, standard deviation), perhaps some multivariate statistics, geo-referencing performed (if any), and pixel size.statistics, geo-referencing performed (if any), and pixel size. 15. Viewing Individual PixelsViewing Individual Pixels Viewing individual pixel brightness valuesViewing individual pixel brightness values in a remotelyin a remotely sensed image is one of the most useful methods forsensed 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 theVirtually 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 remotelyin a remotely sensed image is one of the most useful methods forsensed 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 theVirtually 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. 16. Cursor and Raster Display of Brightness ValuesCursor and Raster Display of Brightness ValuesCursor and Raster Display of Brightness ValuesCursor and Raster Display of Brightness Values 17. Individual Pixel Display ofIndividual Pixel Display of Brightness ValuesBrightness Values Individual Pixel Display ofIndividual Pixel Display of Brightness ValuesBrightness Values 18. Raster Display of Brightness ValuesRaster Display of Brightness ValuesRaster Display of Brightness ValuesRaster Display of Brightness Values 19. Two- and Three-Two- and Three- DimensionalDimensional Evaluation ofEvaluation of Pixel BrightnessPixel Brightness Values within aValues within a Geographic AreaGeographic Area Two- and Three-Two- and Three- DimensionalDimensional Evaluation ofEvaluation of Pixel BrightnessPixel Brightness Values within aValues within a Geographic AreaGeographic Area 20. Univariate Descriptive Image StatisticsUnivariate Descriptive Image StatisticsUnivariate Descriptive Image StatisticsUnivariate Descriptive Image Statistics Measures of Central Tendency in Remote Sensor DataMeasures 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 DataMeasures 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. 21. CommonCommon Symmetric andSymmetric and SkewedSkewed Distributions inDistributions in Remotely SensedRemotely Sensed DataData CommonCommon Symmetric andSymmetric and SkewedSkewed Distributions inDistributions in Remotely SensedRemotely Sensed DataData 22. Univariate Descriptive Image StatisticsUnivariate Descriptive Image StatisticsUnivariate Descriptive Image StatisticsUnivariate Descriptive Image Statistics TheThe 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 ofsingle band of imagery composed of nn brightness valuesbrightness 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. Foris an unbiased estimate of the population mean. For symmetrical distributions, the sample mean tends to be closer to thesymmetrical distributions, the sample mean tends to be closer to the population mean than any other unbiased estimate (such as the median orpopulation mean than any other unbiased estimate (such as the median or mode). Unfortunately, the sample mean is a poor measure of centralmode). 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. TheThe 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 ofsingle band of imagery composed of nn brightness valuesbrightness 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. Foris an unbiased estimate of the population mean. For symmetrical distributions, the sample mean tends to be closer to thesymmetrical distributions, the sample mean tends to be closer to the population mean than any other unbiased estimate (such as the median orpopulation mean than any other unbiased estimate (such as the median or mode). Unfortunately, the sample mean is a poor measure of centralmode). 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 BV n i ik k = = 1 23. PixelPixel Band 1Band 1 (green)(green) Band 2Band 2 (red)(red) Band 3Band 3 (near-(near- infrared)infrared) Band 4Band 4 (near-(near- 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 24. Band 1Band 1 (green)(green) Band 2Band 2 (red)(red) Band 3Band 3 (near-(near- infrared)infrared) Band 4Band 4 (near-(near- infrared)infrared) Mean (Mean (kk)) 135135 46.4046.40 187187 222222 VarianceVariance ((varvarkk)) 562.50562.50 264.80264.80 10071007 570570 StandardStandard deviationdeviation ((sskk)) 23.7123.71 16.2716.27 31.431.4 23.8723.87 MinimumMinimum ((minminkk)) 100100 2525 135135 195195 MaximumMaximum ((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 25. 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 minmax contrast stretching. 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 minmax contrast stretching. kkkrange minmax = 26. Remote Sensing Univariate Statistics - VarianceRemote Sensing Univariate Statistics - VarianceRemote Sensing Univariate Statistics - VarianceRemote Sensing Univariate Statistics - Variance Measures of DispersionMeasures of Dispersion TheThe variancevariance of a sample is the average squared deviation of all possibleof 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. Measures of DispersionMeasures of Dispersion TheThe variancevariance of a sample is the average squared deviation of all possibleof 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 BV n i kik k = = 1 2 var 27. Remote Sensing Univariate StatisticsRemote Sensing Univariate StatisticsRemote Sensing Univariate StatisticsRemote Sensing Univariate Statistics Unfortunately, there is some underestimation because the sample mean wasUnfortunately, 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 toTherefore, 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 wasUnfortunately, 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 toTherefore, 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: 1 var = n SS k 28. Band 1Band 1 (green)(green) Band 2Band 2 (red)(red) Band 3Band 3 (near-(near- infrared)infrared) Band 4Band 4 (near-(near- infrared)infrared) Mean (Mean (kk)) 135135 46.4046.40 187187 222222 VarianceVariance ((varvarkk)) 562.50562.50 264.80264.80 10071007 570570 StandardStandard deviationdeviation ((sskk)) 23.7123.71 16.2716.27 31.431.4 23.8723.87 MinimumMinimum ((minminkk)) 100100 2525 135135 195195 MaximumMaximum ((maxmaxkk)) 165165 6565 215215 255255 Range (Range (BVBVrr)) 6565 4040 8080 6060 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 29. Remote Sensing Univariate StatisticsRemote Sensing Univariate StatisticsRemote Sensing Univariate StatisticsRemote Sensing Univariate Statistics TheThe standard deviationstandard deviation is the positive square root of theis the positive square root of the variance. The standard deviation of the pixel brightness valuesvariance. 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 theis the positive square root of the variance. The standard deviation of the pixel brightness valuesvariance. 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== 30. Band 1Band 1 (green)(green) Band 2Band 2 (red)(red) Band 3Band 3 (near-(near- infrared)infrared) Band 4Band 4 (near-(near- infrared)infrared) Mean (Mean (kk)) 135135 46.4046.40 187187 222222 VarianceVariance ((varvarkk)) 562.50562.50 264.80264.80 10071007 570570 StandardStandard deviationdeviation ((sskk)) 23.7123.71 16.2716.27 31.431.4 23.8723.87 MinimumMinimum ((minminkk)) 100100 2525 135135 195195 MaximumMaximum ((maxmaxkk)) 165165 6565 215215 255255 Range (Range (BVBVrr)) 6565 4040 8080 6060 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 31. Measures of Distribution (Histogram)Measures of Distribution (Histogram) Asymmetry and Peak SharpnessAsymmetry and Peak Sharpness 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 isis a measure of the asymmetry of a histogram and is computed using the formula:computed using the formula: A perfectly symmetric histogram has aA perfectly symmetric histogram has a skewnessskewness value of zero.value of zero. SkewnessSkewness is a measure of the asymmetry of a histogram and isis a measure of the asymmetry of a histogram and is computed using the formula:computed using the formula: A perfectly symmetric histogram has aA perfectly symmetric histogram has a skewnessskewness value of zero.value of zero. n s BV skewness n i k kik k = = 1 3 32. A histogram may be symmetric but have a peak that is veryA histogram may be symmetric but have a peak that is very sharp or one that is subdued when compared with a perfectlysharp 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 zerohas 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 asharper the peak in the distribution when compared with a normal histogram. Conversely, a negative kurtosis valuenormal histogram. Conversely, a negative kurtosis value suggests that the peak in the histogram is less sharp than that ofsuggests 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 veryA histogram may be symmetric but have a peak that is very sharp or one that is subdued when compared with a perfectlysharp 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 zerohas 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 asharper the peak in the distribution when compared with a normal histogram. Conversely, a negative kurtosis valuenormal histogram. Conversely, a negative kurtosis value suggests that the peak in the histogram is less sharp than that ofsuggests 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: 3 1 1 4 = = n i k kik k s BV n kurtosis Measures of Distribution (Histogram)Measures of Distribution (Histogram) Asymmetry and Peak SharpnessAsymmetry and Peak Sharpness Measures of Distribution (Histogram)Measures of Distribution (Histogram) Asymmetry and Peak SharpnessAsymmetry and Peak Sharpness 33. Remote Sensing Multivariate StatisticsRemote Sensing Multivariate StatisticsRemote Sensing Multivariate StatisticsRemote Sensing Multivariate Statistics Remote sensing research is often concerned with theRemote sensing research is often concerned with the measurement of how much radiant flux is reflected or emittedmeasurement 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 computeinfrared bands). It is useful to compute multivariatemultivariate statisticalstatistical measures such asmeasures such as covariancecovariance andand correlationcorrelation among the severalamong the several bands to determine how the measurements covary. Later it willbands to determine how the measurements covary. Later it will be shown that variancecovariance and correlation matrices arebe shown that variancecovariance 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 theRemote sensing research is often concerned with the measurement of how much radiant flux is reflected or emittedmeasurement 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 computeinfrared bands). It is useful to compute multivariatemultivariate statisticalstatistical measures such asmeasures such as covariancecovariance andand correlationcorrelation among the severalamong the several bands to determine how the measurements covary. Later it willbands to determine how the measurements covary. Later it will be shown that variancecovariance and correlation matrices arebe shown that variancecovariance 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. 34. Remote Sensing Multivariate StatisticsRemote Sensing Multivariate StatisticsRemote Sensing Multivariate StatisticsRemote Sensing Multivariate Statistics The different remote-sensing-derived spectral measurementsThe different remote-sensing-derived spectral measurements for each pixel often change together in some predictablefor each pixel often change together in some predictable fashion. If there is no relationship between the brightness valuefashion. If there is no relationship between the brightness value in one band and that of another for a given pixel, the values arein one band and that of another for a given pixel, the values are mutually independent; that is, an increase or decrease in onemutually independent; that is, an increase or decrease in one bands brightness value is not accompanied by a predictablebands brightness value is not accompanied by a predictable change in another bands brightness value. Because spectralchange in another bands 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. Thissome measure of their mutual interaction is needed. This measure, called themeasure, 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 measurementsThe different remote-sensing-derived spectral measurements for each pixel often change together in some predictablefor each pixel often change together in some predictable fashion. If there is no relationship between the brightness valuefashion. If there is no relationship between the brightness value in one band and that of another for a given pixel, the values arein one band and that of another for a given pixel, the values are mutually independent; that is, an increase or decrease in onemutually independent; that is, an increase or decrease in one bands brightness value is not accompanied by a predictablebands brightness value is not accompanied by a predictable change in another bands brightness value. Because spectralchange in another bands 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. Thissome measure of their mutual interaction is needed. This measure, called themeasure, 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. 35. Remote Sensing Multivariate StatisticsRemote Sensing Multivariate StatisticsRemote Sensing Multivariate StatisticsRemote Sensing Multivariate Statistics To calculate covariance, we first compute theTo calculate covariance, we first compute the corrected sum ofcorrected sum of productsproducts ((SPSP)) defined by the equation:defined by the equation: To calculate covariance, we first compute theTo calculate covariance, we first compute the corrected sum ofcorrected sum of productsproducts ((SPSP)) defined by the equation:defined by the equation: ( )( )lil n i kikkl BVBVSP = =1 36. Remote Sensing Univariate StatisticsRemote Sensing Univariate Statistics It is computationally more efficient to use the followingIt 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 theThis quantity is called the uncorrected sum of productsuncorrected sum of products.. It is computationally more efficient to use the followingIt 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 theThis quantity is called the uncorrected sum of productsuncorrected sum of products.. ( ) n BVBV BVBVSP n i n i ilikn i ilikkl = = = = 1 1 1 Remote Sensing Multivariate StatisticsRemote Sensing Multivariate StatisticsRemote Sensing Multivariate StatisticsRemote Sensing Multivariate Statistics 37. Just as simple variance was calculated by dividing the correctedJust as simple variance was calculated by dividing the corrected sums of squares (sums of squares (SSSS) by) by ((n 1n 1)),, covariancecovariance is calculated byis calculated by dividingdividing SPSP by (by (n 1n 1). Therefore, the covariance between). Therefore, the covariance between brightness values in bandsbrightness values in bands kk andand ll,, covcovklkl, is equal to:, is equal to: Just as simple variance was calculated by dividing the correctedJust as simple variance was calculated by dividing the corrected sums of squares (sums of squares (SSSS) by) by ((n 1n 1)),, covariancecovariance is calculated byis calculated by dividingdividing SPSP by (by (n 1n 1). Therefore, the covariance between). Therefore, the covariance between brightness values in bandsbrightness values in bands kk andand ll,, covcovklkl, is equal to:, is equal to: 1 cov = n SPkl kl Remote Sensing Multivariate StatisticsRemote Sensing Multivariate StatisticsRemote Sensing Multivariate StatisticsRemote Sensing Multivariate Statistics 38. Band 1Band 1 (green)(green) Band 2Band 2 (red)(red) Band 3Band 3 (near-(near- infrared)infrared) Band 4Band 4 (near-(near- infrared)infrared) Band 1Band 1 SSSS11 covcov1,21,2 covcov1,31,3 covcov1,41,4 Band 2Band 2 covcov2,12,1 SSSS22 covcov2,32,3 covcov2,42,4 Band 3Band 3 covcov3,13,1 covcov3,23,2 SSSS33 covcov3,43,4 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 39. Band 1Band 1 (Band 1 x Band 2)(Band 1 x Band 2) Band 2Band 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 BetweenComputation of Variance-Covariance Between Bands 1 and 2 of the Sample DataBands 1 and 2 of the Sample Data Computation of Variance-Covariance BetweenComputation of Variance-Covariance Between Bands 1 and 2 of the Sample DataBands 1 and 2 of the Sample Data ( )( ) 135 4 540 cov 5 232675 )860,31( 12 12 == =SP 40. Band 1Band 1 (green)(green) Band 2Band 2 (red)(red) Band 3Band 3 (near-(near- infrared)infrared) Band 4Band 4 (near-(near- 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 41. Correlation between Multiple Bands ofCorrelation between Multiple Bands of Remotely Sensed DataRemotely Sensed Data Correlation between Multiple Bands ofCorrelation between Multiple Bands of Remotely Sensed DataRemotely Sensed Data To estimate the degree of interrelation between variables in aTo estimate the degree of interrelation between variables in a manner not influenced by measurement units, themanner not influenced by measurement units, the correlationcorrelation coefficient, r,coefficient, r, is commonly used. The correlation between twois 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 aTo estimate the degree of interrelation between variables in a manner not influenced by measurement units, themanner not influenced by measurement units, the correlationcorrelation coefficient, r,coefficient, r, is commonly used. The correlation between twois 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 kl kl ss r cov = 42. Correlation between Multiple Bands ofCorrelation between Multiple Bands of Remotely Sensed DataRemotely Sensed Data Correlation between Multiple Bands ofCorrelation between Multiple Bands of Remotely Sensed DataRemotely Sensed Data If we square the correlation coefficient (rkl), we obtain the sample coefficient of determination (r2 ), which expresses the proportion of the total variation in the values of band l that can be accounted for or explained by a linear relationship with the values of the random variable band k. Thus a correlation coefficient (rkl) of 0.70 results in an r2 value of 0.49, meaning that 49% of the total variation of the values of band l in the sample is accounted for by a linear relationship with values of band k. 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. 43. 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 2Band 2 (red)(red) Band 3Band 3 (near-(near- infrared)infrared) Band 4Band 4 (near-(near- 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 -- 44. 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 Matrix Band 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 Matrix Band 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 Matrix Band 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 Matrix Band 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 andUnivariate and MultivariateMultivariate Statistics of LandsatStatistics of Landsat TM Data ofTM Data of Charleston, SCCharleston, SC Univariate andUnivariate and MultivariateMultivariate Statistics of LandsatStatistics of Landsat TM Data ofTM Data of Charleston, SCCharleston, SC 45. Feature Space PlotsFeature Space PlotsFeature Space PlotsFeature Space Plots The univariate and multivariate statistics discussed provideThe univariate and multivariate statistics discussed provide accurate, fundamental information about the individual bandaccurate, fundamental information about the individual band statistics including how the bandsstatistics including how the bands covarycovary andand correlatecorrelate.. Sometimes, however, it is useful to examine statisticalSometimes, however, it is useful to examine statistical relationshipsrelationships graphicallygraphically.. Individual bands of remotely sensed data are often referred toIndividual bands of remotely sensed data are often referred to asas featuresfeatures in the pattern recognition literature. To trulyin 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 tocovary and if they are correlated or not, it is often useful to produce a two-bandproduce a two-band feature space plotfeature space plot.. The univariate and multivariate statistics discussed provideThe univariate and multivariate statistics discussed provide accurate, fundamental information about the individual bandaccurate, fundamental information about the individual band statistics including how the bandsstatistics including how the bands covarycovary andand correlatecorrelate.. Sometimes, however, it is useful to examine statisticalSometimes, however, it is useful to examine statistical relationshipsrelationships graphicallygraphically.. Individual bands of remotely sensed data are often referred toIndividual bands of remotely sensed data are often referred to asas featuresfeatures in the pattern recognition literature. To trulyin 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 tocovary and if they are correlated or not, it is often useful to produce a two-bandproduce a two-band feature space plotfeature space plot.. 46. Feature Space PlotsFeature Space PlotsFeature Space PlotsFeature Space Plots AA two-dimensional feature space plottwo-dimensional feature space plot extracts the brightnessextracts the brightness value for every pixel in the scene in two bands and plots thevalue for every pixel in the scene in two bands and plots the frequency of occurrence in a 255 by 255 feature spacefrequency 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. AA two-dimensional feature space plottwo-dimensional feature space plot extracts the brightnessextracts the brightness value for every pixel in the scene in two bands and plots thevalue for every pixel in the scene in two bands and plots the frequency of occurrence in a 255 by 255 feature spacefrequency 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. 47. Two-dimensionalTwo-dimensional Feature SpaceFeature Space Plot of LandsatPlot of Landsat Thematic MapperThematic Mapper Band 3Band 3 and 4 Data ofand 4 Data of Charleston, SCCharleston, SC obtained onobtained on November 11,November 11, 19821982 Two-dimensionalTwo-dimensional Feature SpaceFeature Space Plot of LandsatPlot of Landsat Thematic MapperThematic Mapper Band 3Band 3 and 4 Data ofand 4 Data of Charleston, SCCharleston, SC obtained onobtained on November 11,November 11, 19821982 48. Geostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor Data The Earths surface has distinct spatial properties. TheThe Earths surface has distinct spatial properties. The brightness values in imagery constitute a record of these spatialbrightness values in imagery constitute a record of these spatial properties. The spatial characteristics may take the form ofproperties. The spatial characteristics may take the form of texture or pattern. Image analysts often try to quantify thetexture or pattern. Image analysts often try to quantify the spatial texture or pattern. This requires looking at a pixel andspatial texture or pattern. This requires looking at a pixel and its neighbors and trying to quantify theits neighbors and trying to quantify the spatial autocorrelationspatial 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 Earths surface has distinct spatial properties. TheThe Earths surface has distinct spatial properties. The brightness values in imagery constitute a record of these spatialbrightness values in imagery constitute a record of these spatial properties. The spatial characteristics may take the form ofproperties. The spatial characteristics may take the form of texture or pattern. Image analysts often try to quantify thetexture or pattern. Image analysts often try to quantify the spatial texture or pattern. This requires looking at a pixel andspatial texture or pattern. This requires looking at a pixel and its neighbors and trying to quantify theits neighbors and trying to quantify the spatial autocorrelationspatial 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? 49. Geostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor Data A random variable distributed in space (e.g., spectralA random variable distributed in space (e.g., spectral reflectance) is said to bereflectance) is said to be regionalizedregionalized. We can use. We can use geostatisticalgeostatistical measures to extract the spatial properties ofmeasures to extract the spatial properties of regionalized variables.regionalized variables. Once quantified, the regionalizedOnce quantified, the regionalized variable properties can be used in many remote sensingvariable properties can be used in many remote sensing applications such as image classification and the allocation ofapplications such as image classification and the allocation of spatially unbiased sampling sites during classification mapspatially unbiased sampling sites during classification map accuracy assessment. Another application ofaccuracy assessment. Another application of geostatisticsgeostatistics is theis the prediction of values at unsampled locations. Geostatisticalprediction of values at unsampled locations. Geostatistical interpolation techniques could be used to evaluate the spatialinterpolation 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., spectralA random variable distributed in space (e.g., spectral reflectance) is said to bereflectance) is said to be regionalizedregionalized. We can use. We can use geostatisticalgeostatistical measures to extract the spatial properties ofmeasures to extract the spatial properties of regionalized variables.regionalized variables. Once quantified, the regionalizedOnce quantified, the regionalized variable properties can be used in many remote sensingvariable properties can be used in many remote sensing applications such as image classification and the allocation ofapplications such as image classification and the allocation of spatially unbiased sampling sites during classification mapspatially unbiased sampling sites during classification map accuracy assessment. Another application ofaccuracy assessment. Another application of geostatisticsgeostatistics is theis the prediction of values at unsampled locations. Geostatisticalprediction of values at unsampled locations. Geostatistical interpolation techniques could be used to evaluate the spatialinterpolation 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. 50. Geostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor Data GeostatisticsGeostatistics are now widely used in many fields and compriseare now widely used in many fields and comprise a branch ofa branch of spatial statisticsspatial statistics. Originally, geostatistics was. Originally, geostatistics was synonymous withsynonymous with krigingkriginga statistical version of interpolation.a statistical version of interpolation. Kriging is a generic name for a family of least-squares linearKriging is a generic name for a family of least-squares linear regression algorithms that are used to estimate the value of aregression algorithms that are used to estimate the value of a continuous attribute (e.g., terrain elevation or percentcontinuous attribute (e.g., terrain elevation or percent reflectance) at anyreflectance) at any unsampledunsampled location using only attribute datalocation using only attribute data available over the study area. However, geostatistical analysisavailable over the study area. However, geostatistical analysis now includes not only kriging but also the traditionalnow includes not only kriging but also the traditional deterministic spatial interpolation methods. One of the essentialdeterministic spatial interpolation methods. One of the essential features of geostatistics is that the phenomenon being studiedfeatures 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 becover class) must be continuouscontinuous across the landscape or at leastacross 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 compriseare now widely used in many fields and comprise a branch ofa branch of spatial statisticsspatial statistics. Originally, geostatistics was. Originally, geostatistics was synonymous withsynonymous with krigingkriginga statistical version of interpolation.a statistical version of interpolation. Kriging is a generic name for a family of least-squares linearKriging is a generic name for a family of least-squares linear regression algorithms that are used to estimate the value of aregression algorithms that are used to estimate the value of a continuous attribute (e.g., terrain elevation or percentcontinuous attribute (e.g., terrain elevation or percent reflectance) at anyreflectance) at any unsampledunsampled location using only attribute datalocation using only attribute data available over the study area. However, geostatistical analysisavailable over the study area. However, geostatistical analysis now includes not only kriging but also the traditionalnow includes not only kriging but also the traditional deterministic spatial interpolation methods. One of the essentialdeterministic spatial interpolation methods. One of the essential features of geostatistics is that the phenomenon being studiedfeatures 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 becover class) must be continuouscontinuous across the landscape or at leastacross the landscape or at least capable of existing throughout the landscape.capable of existing throughout the landscape. 51. Geostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor Data Autocorrelation is the statistical relationship among measured points, where the correlation depends on the distance and direction that separates the locations. We know from real-world observation that spatial autocorrelation exists because we have observed generally that things that are close to one another are more alike than those farther away. As distance increases, spatial autocorrelation decreases. 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. 52. Geostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor Data KrigingKriging makes use of the spatial autocorrelation information. Kriging ismakes use of the spatial autocorrelation information. Kriging is similar to distance weighted interpolationsimilar to distance weighted interpolation in that it weights thein 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 theHowever, the weights are based not only on the distance between the measured points and the point to be predicted (used in inverse distancemeasured points and the point to be predicted (used in inverse distance weighting), but also on the overallweighting), but also on the overall spatialspatial arrangementarrangement among the measuredamong the measured points (i.e., their autocorrelation). Kriging uses weights that are definedpoints (i.e., their autocorrelation). Kriging uses weights that are defined statistically from the observed data rather thanstatistically from the observed data rather than a prioria priori. This is the most. This is the most significant difference between deterministic (traditional) and geostatisticalsignificant difference between deterministic (traditional) and geostatistical analysis. Traditional statistical analysis assumes the samples derived for aanalysis. 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 distancesConversely, geostatistical analysis allows a scientist to compute distances between observations and to model autocorrelation as a function of distancebetween observations and to model autocorrelation as a function of distance and direction. This information is then used to refine the krigingand direction. This information is then used to refine the kriging interpolation process, hopefully, making predictions at new locations thatinterpolation process, hopefully, making predictions at new locations that are more accurate than those derived using traditional methods. There areare 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 ismakes use of the spatial autocorrelation information. Kriging is similar to distance weighted interpolationsimilar to distance weighted interpolation in that it weights thein 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 theHowever, the weights are based not only on the distance between the measured points and the point to be predicted (used in inverse distancemeasured points and the point to be predicted (used in inverse distance weighting), but also on the overallweighting), but also on the overall spatialspatial arrangementarrangement among the measuredamong the measured points (i.e., their autocorrelation). Kriging uses weights that are definedpoints (i.e., their autocorrelation). Kriging uses weights that are defined statistically from the observed data rather thanstatistically from the observed data rather than a prioria priori. This is the most. This is the most significant difference between deterministic (traditional) and geostatisticalsignificant difference between deterministic (traditional) and geostatistical analysis. Traditional statistical analysis assumes the samples derived for aanalysis. 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 distancesConversely, geostatistical analysis allows a scientist to compute distances between observations and to model autocorrelation as a function of distancebetween observations and to model autocorrelation as a function of distance and direction. This information is then used to refine the krigingand direction. This information is then used to refine the kriging interpolation process, hopefully, making predictions at new locations thatinterpolation process, hopefully, making predictions at new locations that are more accurate than those derived using traditional methods. There areare 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. 53. Geostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor Data The kriging process generally involves two distinct tasks: quantifying the spatial structure of the surrounding data points, and producing a prediction at a new location. Variography is the process whereby a spatially dependent model is fit to the data and the spatial structure is quantified. To make a prediction for an unknown value at a specific location, kriging uses the fitted model from variography, the spatial data configuration, and the values of the measured sample points around the prediction location. 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. 54. Geostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor Data One of the most important measurements used to understand the spatial structure of regionalized variables is the semivariogram, which can be used to relate the semivariance to the amount of spatial separation (and autocorrelation) between samples. The semivariance provides an unbiased description of the scale and pattern of spatial variability throughout a region. For example, if an image of a water body is examined, there may be little spatial variability (variance), which will result in a semivariogram with predictable characteristics. Conversely, a heterogeneous urban area may exhibit significant spatial variability resulting in an entirely different semivariogram. 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. 55. Geostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor Data Phenomena in the real world that are close to one another (e.g., two nearby elevation points) have a much greater likelihood of having similar values. The greater the distance between two points, the greater the likelihood that they have significantly different values. This is the underlying concept of autocorrelation. The calculation of the semivariogram makes use of this spatial separation condition, which can be measured in the field or using remotely sensed data. This brief discussion focuses on the computation of the semivariogram using remotely sensed data although it can be computed just as easily using in situ field measurements. 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. 56. Geostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor Data Consider a typical remotely sensed image over a study area. Now identify the endpoints of a transect running through the scene. Twelve hypothetical individual brightness values (BV) found along the transect are shown in the illustration. The (BV) z of pixels x have been extracted at regular intervals z(x), where x = 1, 2, 3,..., n. The relationship between a pair of pixels h intervals apart (h is referred to as the lag distance) can be given by the average variance of the differences between all such pairs along the transect. There will be m possible pairs of observations along the transect separated by the same lag distance, h. The semivariogram (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 hxzxz h m i ii= + = 1 2 )( 57. Geostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor Data The total number of possible pairsThe total number of possible pairs mm along the transect is computed byalong the transect is computed by subtracting the lag distancesubtracting the lag distance hh from the total number of pixels present in thefrom the total number of pixels present in the datasetdataset nn, that is,, that is, m = n h.m = n h. In practice, semivariance is computed for pairsIn practice, semivariance is computed for pairs of observations in all directions. Thus, directional semivariograms areof 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 ofThe average semivariance is a good measure of the amount of dissimilaritydissimilarity between spatially separate pixels. Generally, the larger the averagebetween spatially separate pixels. Generally, the larger the average semivariancesemivariance ((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 pairsThe total number of possible pairs mm along the transect is computed byalong the transect is computed by subtracting the lag distancesubtracting the lag distance hh from the total number of pixels present in thefrom the total number of pixels present in the datasetdataset nn, that is,, that is, m = n h.m = n h. In practice, semivariance is computed for pairsIn practice, semivariance is computed for pairs of observations in all directions. Thus, directional semivariograms areof 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 ofThe average semivariance is a good measure of the amount of dissimilaritydissimilarity between spatially separate pixels. Generally, the larger the averagebetween spatially separate pixels. Generally, the larger the average semivariancesemivariance ((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). 58. Geostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor DataGeostatistical Analysis of Remote Sensor Data 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). 59. Geostatistical analysisGeostatistical analysis incorporates spatialincorporates spatial autocorrelation information in the krigingautocorrelation information in the kriging interpolation process. Phenomena that areinterpolation process. Phenomena that are geographically closer together are generallygeographically closer together are generally more highly correlated than things that aremore highly correlated than things that are farther apart.farther apart. Geostatistical analysisGeostatistical analysis incorporates spatialincorporates spatial autocorrelation information in the krigingautocorrelation information in the kriging interpolation process. Phenomena that areinterpolation process. Phenomena that are geographically closer together are generallygeographically closer together are generally more highly correlated than things that aremore highly correlated than things that are farther apart.farther apart. 60. a) A hypothetical remote sensinga) A hypothetical remote sensing dataset used to demonstrate thedataset used to demonstrate the characteristics of lag distance (characteristics of lag distance (hh)) along a transect of pixelsalong a transect of pixels extracted from an image.extracted from an image. a) A hypothetical remote sensinga) A hypothetical remote sensing dataset used to demonstrate thedataset used to demonstrate the characteristics of lag distance (characteristics of lag distance (hh)) along a transect of pixelsalong a transect of pixels extracted from an image.extracted from an image. b) A semivariogram of theb) A semivariogram of the semivariancesemivariance (h)(h) characteristicscharacteristics found in the hypothetical datasetfound in the hypothetical dataset at various lag distances (at various lag distances (hh).). b) A semivariogram of theb) A semivariogram of the semivariancesemivariance (h)(h) characteristicscharacteristics found in the hypothetical datasetfound in the hypothetical dataset at various lag distances (at various lag distances (hh).). 61. 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. 62. Original Images,Original Images, Semivariograms, andSemivariograms, and Predicted Images.Predicted Images. a) The green band of aa) The green band of a 0.610.61 0.61 m image0.61 m image of cargo containers inof cargo containers in the port of Hamburg,the port of Hamburg, Germany, obtained onGermany, obtained on May 10, 2002May 10, 2002 (courtesy of(courtesy of DigitalGlobe).DigitalGlobe). b) A subset containingb) A subset containing just cargo containers.just cargo containers. c) A subset continuingc) A subset continuing just tarmac (asphalt).just tarmac (asphalt). Original Images,Original Images, Semivariograms, andSemivariograms, and Predicted Images.Predicted Images. a) The green band of aa) The green band of a 0.610.61 0.61 m image0.61 m image of cargo containers inof cargo containers in the port of Hamburg,the port of Hamburg, Germany, obtained onGermany, obtained on May 10, 2002May 10, 2002 (courtesy of(courtesy of DigitalGlobe).DigitalGlobe). b) A subset containingb) A subset containing just cargo containers.just cargo containers. c) A subset continuingc) A subset continuing just tarmac (asphalt).just tarmac (asphalt).