This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).
Radiometric precision is difficult to maintain in orbital images due to several factors (atmospheric conditions, Earthsun distance, detector calibration, illumination, and viewing angles). These unwanted effects must be removed for radiometric consistency among temporal images, leaving only landleaving radiances, for optimum change detection. A variety of relative radiometric correction techniques were developed for the correction or rectification of images, of the same area, through use of reference targets whose reflectance do not change significantly with time,
Spectral images acquired from the same area at different times contain valuable information for regular monitoring of the Earth’s surface, allowing us to describe the landcover evolution, vegetation phenology, and natural hazard events. However, it is difficult to maintain radiometric accuracy in orbital images due to changes in atmospheric conditions, EarthSun distance, detector calibration, illumination angles, viewing angles, and sensor oscillation, as are required in order to highlight the spectral changes of interest. Another common problem in changedetection analysis from different sensors, such as Landsat Thematic Mapper (TM) and Landsat Enhanced Thematic Mapper plus (ETM+) images is that they, additionally, require the evaluation of radiometric consistency between sensors to ensure comparability between the temporal images [
In previous studies, two levels of radiometric correction were developed: absolute and relative [
The linear regression method is the most widely used approach for relative correction. Initially, the technique was based on simple regression that considered all the pixels of temporal images [
PIF selection through visual inspection establishes regions of interest for the dark and bright (DB) areas [
Computational methods for PIF selection are more accurate, considering an automatic extraction. There are two main strategies, which consider the different dimensions of the image (X, Y, and Z components): (a) bandbyband methods that compare a pair of bands at different times (X and Y dimensions of the image), and (b) pixelbypixel methods that compare a pair of spectra at different times for a pixel (Z dimension of the image).
PIF detection methods that perform comparisons between pairs of bands are the most commonly used. These include nochange buffer zone around a fitted line from linear regression [
Elvidge
Limitation in the use of the buffer zone indicates the need for a different approach, already described in statistical science in the 1970s [
The density scatter plot method (ridge method) compares two images made at different times in a scatter plot and calculates the point density. The PIFs describe a highdensity ridge along a straight line near the 1:1 line (the ridge), while areas with pronounced change (clouds and land cover changes) are characterized as having low density [
However, PIF detection methods using band comparisons (simple linear regression, robust regression, density scatter plot, and linear transformations) present errors when there are few PIFs in the image. If the preponderance of pixels contains systemic changes, the identification of PIF may contain biased distortions. Thus, the use of these methods must adopt a “PIF assumption” for their implementation [
This problem in PIF detection can be avoided by ignoring the relationship between bands (X and Y dimension) and focusing instead on the particular spectral relationship between each pair of temporal pixel (Z dimension). PIF detection methods, using spectral comparison, consider the distance measures (Euclidean and Mahalanobis distance) and similarity measures (cosine correlation and Pearson’s correlation) [
Canty
In this study, a new algorithm is formulated for radiometric normalization that combines three methods for the detection of PIFs: similarity and distance measures, a density scatter plot method, and robust linear regression. The methods presented are complementary and can be used together in consecutive steps that improve the selection of PIFs. The method was implemented in software developed in C++.
The proposed methods have been applied in an extensive irrigation project located in the Gorutuba region, Central Brazil. The irrigation project has established an important economic growth vector to agricultural activities and agroindustries, keeping the production under a high climate risk. The São Francisco River provides a steady water flow for use in irrigated farming. The topography is generally very flat, which favors the development of agriculture irrigation, involving the establishment of dikes and regulations. The agriculture irrigated produces mainly fruits; species include banana, pineapple, and mango. Thus, this area is suitable for this study because it presents different targets such as: lake, city, cultivated area, and natural vegetation.
The automatic radiometric normalization consists of the following steps: coregistration of the temporal images, selection of the PIFs, and linear regression from the PIFs. In this paper, the algorithm for the automatic detection of the PIFs combines methods that provide a gradual selection (
The method considers, as inputs, two coregistered temporal images with their three dimensions: X, Y, and Z (spectral profile) acquired in the same geographical area at two times (t1 and t2). In the present paper, Landsat TM (Thematic Mapper) images are used. These were acquired without clouds during the same month of sequential years (9 September 2001, and 21 September 2011) in order to minimize any differences in sun angle and vegetation growth stage (
The images were coregistered from twelve ground control points identified in each image using ENVI software. The warping method used was the firstorder polynomial (simplest method) combined with the nearestneighbor resampling method to avoid interference in the pixel spectrum. The root mean squared error (RMS) value reported by ENVI was <0.2 pixels. Images have to be superimposed within 0.2 pixels (RMS) to achieve an error of only 10% [
The PIFs can be determined by distance and similarity measures between the spectra of images with the same spatial position at different times [
The spectral measures provide different information about the target. This is because each measure has its own characteristics, which justifies the use of different measures in accordance with the spectral behavior and data type [
The spectral measures can be combined in PIF detection, increasing efficiency. Thus, the user can choose the number of spectral measures to be used in PIF detection. PIF selection is only performed if the values are consistent with the thresholds of each method.
CCA was developed by Hotelling [
This approach is a complement of the multiple regression method that establishes a relationship of a set of predictors (
Each pair of bands is orthogonal and independent of all other bands derived from the same data set. Successive pairs of canonical variates are based on residual variance. The first pair of canonical variates demonstrates the highest intercorrelation, the next pair, the secondhighest correlation, and so forth. Thus, canonical correlation is an appropriate and powerful multivariate technique for both multiple dependent and independent variables. In this paper, the algorithm for the calculation of canonical variates, developed by Borga [
Nielsen
Therefore, MAD analysis takes the difference between linear combinations of the original data that have maximal correlation; it consists in normalization for a change detection scheme.
Spectral distance metrics can be calculated from temporal CVs. Canty
However, this formulation can provide extremely high values that prevent the display of the distance image (ruleimage). In this paper, we consider the Normalized Euclidean Distance (NED) more appropriate to image comparison with the other methods. NED is a simplification of the Mahalanobis Distance.
NED application in the magnitude of MAD images creates an ellipsoidal distribution that best represents the probability distribution of the estimated set by standardization of canonical variates. However, spectrum standardization is not always desired. CCA provides an ordering of image quality (according to its eigenvalues), where the uncorrelated noises with equal variance (
NED is not independent of the scene because its formulation uses the standard deviation (dependent of the scene). This characteristic differs from other measures; which consider only operations restricted to a given pixel; regardless of the size and value of the other pixels [
Scatterplot analysis is a powerful approach to exploratory multivariate data analysis. Song
The mathematical model that best describes the normalization bitemporal images involves linear regression of the PIFs. The algorithm assumes that the PIFs at t_{2} are linearly related to the pixels at t_{1}. This implies that the spectral reflectance properties of the sampled pixels have not changed over time.
Several estimators have been developed in order to be insensitive to small deviations from an idealized assumption for straightline fit [
Optionally, targets with high susceptibility to change over time can be disregarded as being unreliable, such as vegetation. The removal of these targets should be made by use of a binary mask, which includes the value of 1 in the analysis and ignores the value of 0. Thus, undesirable targets may be disposed even with spectral and density measures compatible with a PIF.
Greater accuracy implies more similarity between the frequency distribution of reference and normalized data,
However the sample size has a significant impact on the statistical results. A smaller sampling of PIFs regularly gives better performance because it reduces noise and outliers, which can substantially degrade the radiometric normalization. Variations in the number of samples modify the statistical data for the same method. Therefore, the comparison of methods must use the same number of pixels in order to eliminate this effect. In the present study, we developed a program that considers three alternatives for the PIFs selection from distance or similarity images (pixelbypixel methods): threshold values of the spectral measures, number of pixels, and image percentage (all userdefined).
These statistical procedures, despite being widely used should be evaluated carefully, since they represent the linear regression fits, and not directly the quality of PIFs. The comparison of the spectral behavior of known targets is still the most straightforward way to judge the overall performance of the radiometric normalization methods.
Initially, the PIFs detection using different spectral measures from original images were evaluated.
Most of the invariant points are identified independently using the three different measures; however there are differences among the results and these can be visualized from the intersection PIFs among the methods (
The spatial location of PIFs from the different methods can be performed by color composite images (
Statistical data from the simple linear regression for the selected PIFs are compared in order to evaluate the different spectral measures (
Measures of dispersion differ among PIFs from ED and measures of similarities (SCM and SAM) (
MAD components have an ordering image that highlights the reduction of intercorrelation between the CV pairs and increased noise interference (
The scatterplot between the pair of the latest CVpair appears spherically distributed around the data. This is evidence of the predominance of the uncorrelated noises with equal variance in all bands (
The high correlation frequently exists in multitemporal data provides a redundancy reduction and increasing the signal magnitudes in the first CVs. Once data have been transformed into canonical variates with decreasing noise, the subsequent calculation of the distance measures between the temporal variables can disregard the noisiest components. This practice should be emphasized with the use of the NED or Mahalanobis distance that lead to data normalization,
In this work we calculated the NED measures considering two scenarios: with all MAD components and removing the noisy components. The NED^{2} image originally described by Canty [
The number of coincident PIFs from the spectral measures (SCM, SAM, and ED) using the original images and the NED using the MAD components are extremely low, indicating a significant difference between the two procedures (
PIFs from NEDMAD compared to PIFs from ED_{CD} exhibit higher correlation coefficients for linear regression in the first and second bands and the lower for the rest. An inverse behavior is found for the RMSE values, where PIFs from NEDMAD presents lower values for bands 1, 2, 3, and 4. PIFs from NEDMAD show best fit regression line compared to the two similarity measures (SCM_{CD} and SAM_{CD}) applied to the original images.
The NEDMAD, considering all the bands, shows the smallest differences for bands 1, 2, and 5. However, the differences of the means of PIFs from NEDMAD procedures are larger in all bands compared to PIFs from EDCD (
Spectral analysis between the reference and normalized images was made from the dark radiometric control targets, considering water body. Dimension of the water body may be important, and small sizes may not be representative [
Results show a different behavior of the fitting information of the linear regression model from PIFs, where for example the distance measures have lower RMSE values and higher R values (
The similarity measures are not consistently related to the accuracy of the prediction,
The worst results for the NEDMAD measures are probably derived from the susceptibility of CCA to be influenced by the correlation of current changes in the ground target. CCA generates eigenvectors that describes the linear relations from the effects of atmosphere, illumination, and sensor calibration (that are external to the ground target). Temporal images with the majority of consistently changed ground targets generate eigenvalues conditioned to it, which interferes and modify the true PIFs, damaging their detection.
These results demonstrate that statistical data from the fits of the linear regression (RMS, correlation coefficient, and measures of central tendency and dispersion restricted to selected points) does not evidence necessarily the PIFs quality.
The use of ridge method on the previously selected PIFs allows a supplementary pixels reduction.
One problem in density thresholds is the elimination of points having low density that are positioned along the line (ridge). These are usually points at each end of the ridge. Therefore, this processing step needs to be done carefully to avoid loss of relevant information. The pixels eliminated in a particular band can be eliminated in all five other bands.
The final step is to perform the robust regression for determining the line of best fit to the data from the reference and subject image for each image band. Robust regression procedure automatically omits eventual outliers in the calculations considering an absolute deviation value. The robust procedures fit the calibration line through the dark and bright targets.
The proposed combination of methods enables one to extract points from various conditions that can be applied from image to image. Thus, areas labeled as invariant points in a method can be described as change point in the next methods. The method allows one to select a number of targets to cover the range from bright to dark data values. This highlights the importance of using difference techniques.
Statistical analyses of these complementary methods (ridge and robust regression), following pixelbypixel, always provide the best fit straight line since it eliminates outliers, making the demonstration unnecessary.
The main functions of the program are organized in the main window interface, which contains the temporal image input boxes, mask image input box, spectral measures box, density scatter plot box, robust regression box, and image display (
Input files are required for the two temporal images (
Four options for the automatic detection of PIFs using spectral measures are provided: SCM, SAM, Euclidean distance, and NEDMAD. The user can select the measures and their threshold values. The program allows the user to select different spectral measures simultaneously. The PIFs are only selected if they meet all conditions set out in the spectral measures. The program allows a preview of PIF images, obtained by the spectral measures, facilitating the user to, if necessary, reset the threshold values.
The ridge method can be applied from preselected PIFs from the previous step. The density scatter plot can be visualized from the preview button, which opens a window interface for the ridge method (
In the scatterplot density window, the user can choose to display the pair of bands and a presaved color table. At the bottom of the window, the user can set the density threshold value in order to eliminate the outliers. The elimination of the targets can be made by considering a single value for each pair of bands (multiple thresholds) or a single value for all scatter plots (single threshold). Density threshold value can be set with a scrollbar ranging from 0 to 255, derived by linear conversion of density values.
Finally, the robust regression using PIFs can be applied. In the main window the user can choose between the simple linear regression and robust regression from a check box. Thus, both method results can be compared. In the robust regression, the user must set the absolute deviation value between the regression line and the points in order to select the outliers. When the absolute deviation is greater than the set value, the observation is removed and the procedure is repeated. The decrease in the absolute deviation value by the user increases the number of outliers.
The linear regression window can be visualized with the preview button (
After the generation of calibrated images, the program generates statistics on the following images: reference, uncalibrated, and calibrated, considering the following statistical values: mean, variance, range, and coefficient of variation.
The program generates the following output files: radiometrically normalized images and PIF images. All inputs and results are shown in the File List, so it is possible to visualize them by choosing “Gray Scale” or “RGB” composite. The display interface provides basic functions for image visualization such as zoom areas and pixel values. Moreover, the results (output files) can be read with other images viewers.
The spectral measures can be performed on the original images, or on CVs from CCA. The use of CCA aims to correct temporal images (gain and offset). Thus CCA generates matrices of eigenvectors for both temporal sets that are invariants to linear and affine scaling, provided that the linear transformations are homogeneous for the entire image (such as bandbyband method). CCA operates throughout the bands; consequently specific effects in different parts of the image are not corrected. In addition, the calculation of eigenvectors in CCA is influenced, not only by the gain and offset from sensorilluminationatmospheric effects, but also by systematic changes in the ground. Therefore, the contributions of other correlated attributes though the temporal bands (e.g., seasonal changes in vegetation, planting cycle) are difficult to determine due to mathematical formulation. The issue of the CCA application is to introduce a bias associated with correlations of surface changes to the true PIFs, hindering their detection. The canonical analysis (such as the principal components and factor analysis) should use images with a prevalence of unchanged targets. Moreover, CVs images have a spatial noise that should be considered in the calculation of the distance measures. The elimination of noisy components of CVs should be explored as an improvement to the method.
Spectral measures on the original temporal images consider both land cover changes as undesirable effects (e.g., different atmospheric conditions, variations in the solar illumination angles, and sensor calibration trends). Normally, the relative radiometric correction adopts the assumption that the linear effects are much greater than nonlinear effects [
Statistical data of the linear regression (RMSE, correlation coefficient, and measures of central tendency, and dispersion restricted to selected points) does not always describe the PIFs quality.
The present method integrates different radiometric normalization algorithms with distinct mathematical and statistical operations that provide a set of alternatives according to the characteristics of the data. The methods can be combined in different ways, not needing to use all available methods. Compared to some previous relative radiometric normalization methods, this new method offers several improvements and advantages because a single method cannot be used in all situations. The best combination of methods is to adopt a pixelbypixel method followed by the bandbyband method. Thus, PIFs selection accuracy is distributed in the different methods that provide a sequential selection, taking into account, not only, the statistical behavior of the samples but also their physical properties, such as surface moisture, shadow,
While this approach is straightforward and brings together different algorithms, it still shows problems, especially with the effects of heterogeneous aerosol scattering and water vapor content in a scene. All methods of PIFs detection presume a uniform behavior for atmospheric effects; being suitable for molecular scattering and absorption by ozone and oxygen because their concentrations are quite stable over both time and space. Contrary to this, most aerosols are heterogeneously distributed which makes this task more difficult and requires the development of new algorithms.
Furthermore, atmospheric effects on the two images can be different, not only spatially, but also in the spectral dimension damaging PIFs detection by pixelbypixel methods. Considering aerosol effects, a simple procedure is used with infrared bands that are less contaminated, in the specific case of TM/ETM+ imagery the bands 4, 5, and 7. However, this procedure is ineffective if there are thick aerosols and thin clouds. Therefore, a challenge in improving the methods of relative correction is the development of procedures to consider the heterogeneity of the image,
The automatic selection of the invariant targets eliminates the need of a detailed manual comparison between each pair of the temporal image. PIFs detection from only bandbyband method show great limitations, requiring assume that the majority of the targets are invariant between a given image and the reference image. The improved efficiency of the bandbyband method is obtained by the association with pixelbypixel methods. Thus this paper proposes a new alternative to radiometric normalization between temporal images considering the combination of different procedures for automatic PIF selection. There is no one method that is better than others in every situation. Thus, the pixelbypixel approach (similarity and distance measures considering original image and Canonical Variates (CVs)) and bandbyband approach (density scatter plot and robust regression) should be used in conjunction to PIF estimation. This is because different methods have particular properties, some of which may be desirable for certain applications but not for others.
The combination of these methods provides a sequence of statistical techniques with independent criteria that provides the error minimization and best set of invariant targets. The sequential PIF selection obtains a number of invariant targets, which consider the entire brightness range of the images, ensuring the linear relationship between the image pairs during the calibration. The procedures described in this paper, still present limitations for nonlinear effects, which pose a great challenge for future studies.
This study was funded through a project grant “Integrated model for monitoring and sustainable development of the São Francisco Basin” from the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq). The authors are grateful for financial support from CNPq fellowship (Osmar Abílio de Carvalho Júnior, Renato Fontes Guimarães, Roberto Arnaldo Trancoso Gomes) and CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior) fellowship (Nilton Correia da Silva). Special thanks are given to Fundação de Apoio a Pesquisa do Distrito Federal (FAPDF) and Financiadora de Estudos e Projetos (FINEP) for additional support and Instituto Nacional de Pesquisas Espaciais (INPE) by Landsat images. Finally, the authors acknowledge the contribution from anonymous reviewers.
The authors declare no conflict of interest.
Flowchart of the main methods developed for radiometric normalization.
Operational pseudoinvariant features (PIFs) identification and radiometric normalization processing.
Location map of the Gorutuba region, Central Brazil. Color composite images of 2001 and 2011 (RGB: TM345).
(
(
Differences in PIFs detection between similarity and distance measures: (
Multivariate Alteration Detection (MAD) components of study area obtained by subtraction between the canonical variates: (
Scatterplots between the sixth pair of canonical variable images.
Frequency histogram of the 1st MAD (black line) and 6th MAD images (red line).
(
Mean of the absolute deviations between reference and normalized spectra for a 55 pixels of water body (dark radiometric control target).
(
Distribution in two dimensions using robust regression of points fitted to a straight line, considering 106 PIFs from the intersection of SCM and ED methods: (
Main window interface contains image input boxes, PIF detection using spectral measures, ridge method, and robust regression.
Density scatter plot window has the following attributes: scrolling list of density lookup table ramp, scrolling list of the bands, scatter plot viewer, threshold check box, and threshold value scrollbar.
Linear regression window with the following attributes: scrolling list of density lookup table ramp, scrolling list of the bands, scatterplot viewer with the straight lines from the robust regression, and simple linear regression.
Spectral Measures in Temporal Analysis.
Euclidean Distance for change detection (ED_{CD}) 

Sensitive to offset and gain factor 
Spectral Angle Mapper for change detection (SAM_{CD}) 

Negative correlation is not detected 
Spectral Correlation Mapper for change detection (SCM_{CD}) 

Negative correlation is detected 
Intersection number of PIFs from different methods (SCM_{CD}, SAM_{CD}, and ED_{CD}) considering a threshold value of 20%.
200,000  160,398  106,981  
160,398  200,000  102,774  
106,981  102,774  200,000 
Least squares regression on training PIFs (200,000) from SCM_{CD}, SAM_{CD}, and ED_{CD} measures;
 

Band1  −7.33  1.114  5.437  0.865  −3.316  1.061  5.121  0.851  −0.869  0.99  2.805  0.93 
Band2  −3.679  1.162  4.220  0.905  −3.073  1.143  3.901  0.902  0.248  1.006  1.988  0.959 
Band3  −3.675  1.144  6.615  0.944  −2.547  1.122  6.307  0.93  −0.263  1.026  3.254  0.977 
Band4  −0.025  1.097  6.642  0.964  2.352  1.066  6.454  0.94  2.518  1.037  4.462  0.98 
Band5  −1.345  1.06  11.398  0.974  −1.315  1.068  11.688  0.954  2.539  0.991  5.086  0.993 
Band7  −0.149  1.076  6.642  0.968  0.128  1.076  6.846  0.948  2.026  0.989  3.582  0.986 
Comparison of mean intensities of holdout test PIFs (20%) for the three PIF methods using the 2001 scene before and after normalization to the 2011 scene with least squares regression.
 
 
76.461  38.203  51.968  64.288  126.890  59.212  
77.273  40.199  55.236  69.964  132.700  63.073  
77.830  40.713  55.783  70.484  133.170  63.540  
0.550  0.513  0.546  0.520  0.465  0.466  
 
 
 
77.189  38.927  54.111  66.257  132.930  62.097  
78.030  40.927  57.686  72.463  140.140  66.420  
78.546  41.431  58.182  72.987  140.630  66.921  
0.516  0.504  0.495  0.523  0.491  0.500  
 
 
 
72.809  34.229  44.099  56.736  108.240  48.797  
70.809  34.229  44.412  60.840  109.33  49.791  
71.224  34.680  44.977  61.367  109.840  50.271  
0.414  0.4505  0.564  0.527  0.503  0.480 
Comparison of dispersion measures of holdout test PIFs (20%) for the three measures (SCM_{CD}, SAM_{CD} and ED_{CD}) using the 2001 scene before and after normalization to the 2011 scene with least squares regression.
 
 
70.69438  59.44261  273.3577  480.6458  2131.235  562.5406  
88.19824  79.76944  358.481  581.9668  2387.68  646.6398  
117.2427  98.06528  401.5877  622.2835  2524.955  694.9333  
78  62  113  148  253  143  
87  72  129  162  255  153  
99  73  128  148  255  163  
0.109965  0.201812  0.318149  0.341022  0.363826  0.400559  
0.121536  0.22218  0.342774  0.344807  0.36822  0.403169  
0.139135  0.243237  0.359244  0.353919  0.377336  0.414881  
 
 
 
61.32971  50.99061  203.6081  275.8851  1223.483  361.7351  
69.58914  66.44938  256.8769  315.0233  1392.437  418.5648  
95.20167  81.86585  296.2384  355.2058  1531.65  465.3757  
133  84  127  145  252  140  
141  96  142  155  254  150  
99  72  117  138  255  162  
0.101456  0.183442  0.263701  0.250687  0.263134  0.306285  
0.106908  0.199176  0.277837  0.244937  0.266286  0.308022  
0.124221  0.218387  0.295823  0.258223  0.278294  0.32236  
 
 
 
51.63264  44.39962  208.6694  451.3844  1791.989  443.5186  
51.6276  44.39962  219.4595  488.523  1768.854  441.4567  
58.49063  48.87633  230.1959  505.5454  1786.699  446.3774  
91  65  113  148  253  140  
90  65  115  154  251  138  
99  69  119  141  254  135  
0.098691  0.194667  0.327568  0.374467  0.391084  0.431584  
0.101473  0.194667  0.333561  0.363291  0.384677  0.421984  
0.107378  0.201592  0.337334  0.36639  0.384839  0.420275 
Number of PIFs from the intersection of different methods considering a threshold value of 20%: NEDMAD using all bands, NEDMAD using first five components, NEDMAD using first three components, SCM_{CD}, SAM_{CD} and ED_{CD}.
200,000  160,923  139,654  52,188  59,408  26,246  
160,923  200,000  155,799  47,496  55,353  14,738  
139,654  155,799  200,000  52,204  59,901  17,874 
Least squares regression on training PIF pixels (200,000) from the NEDMAD using all bands, NEDMAD image using first five components, NEDMAD using first three components;
 

Band1  −3.513  1.072  2.314  0.945  −2.249  1.057  2.190  0.955  −0.792  1.039  1.856  0.968 
Band2  −0.469  1.128  1.721  0.967  0.563  1.104  1.607  0.973  1.439  1.079  1.578  0.974 
Band3  3.086  1.100  2.792  0.972  4.598  1.075  2.789  0.976  4.774  1.065  3.697  0.962 
Band4  13.533  0.983  3.646  0.962  14.757  0.967  3.594  0.961  18.918  0.888  5.929  0.913 
Band5  13.163  1.002  7.158  0.963  15.295  0.990  6.986  0.965  15.875  0.983  6.160  0.975 
Band7  5.482  1.038  4.053  0.966  8.028  1.004  5.218  0.945  8.203  1.002  5.066  0.952 
Comparison of mean intensities of holdout test PIFs (20%) for the NEDMAD method (all, 5, and 3 components), considering 2001 scene before and after normalization to the 2011 scene with least squares regression.
 
 
73.53389  34.98043  47.22892  59.63658  120.6294  54.58548  
74.80974  38.51939  54.49465  71.60842  133.6294  61.72854  
75.29658  38.97923  55.02094  72.13204  134.0009  62.14728  
0.48684  0.45984  0.526285  0.52361  0.37149  0.41874  
 
 
 
74.54509  35.99552  49.202  61.24536  125.102  57.13146  
75.96249  39.72751  56.94335  73.55464  138.619  65.13146  
76.55546  40.28415  57.46744  73.98557  139.0883  65.3966  
0.59298  0.55665  0.524095  0.430935  0.46925  0.265145  
 
 
 
74.39065  36.0114  49.06649  63.04542  124.0779  56.60791  
75.96621  39.70047  56.54078  74.37767  137.4494  64.60791  
76.50336  40.29939  57.02833  74.90395  137.8846  64.93313  
0.53715  0.59892  0.487555  0.526275  0.43518  0.32522 
Comparison of dispersion measures of holdout test PIFs (20%) for the three PIF methods using the 2001 scene, before and after normalization, to the 2011 scene with least squares regression.
 
 
39.29503  33.80525  126.7082  171.6435  651.4781  210.9658  
44.91777  43.05102  153.9639  169.1287  651.4781  225.548  
50.49108  45.95556  162.0483  179.015  704.9602  243.7734  
60  58  95  133  238  139  
64  66  105  131  238  144  
64  59  101  131  237  148  
0.085247  0.166214  0.238339  0.219685  0.211591  0.26609  
0.089588  0.170338  0.227696  0.181612  0.191006  0.243295  
0.09437  0.173915  0.231363  0.185488  0.198141  0.25123  
 
 
 
44.14236  37.52932  137.2826  167.8967  671.9878  223.6594  
50.42474  46.9415  158.6868  156.1622  650.1234  223.6594  
54.12543  48.28298  166.2875  169.9436  706.807  252.7447  
63  58  100  124  238  131  
66  64  108  120  235  131  
68  59  104  126  238  132  
0.089127  0.170191  0.238136  0.211567  0.207213  0.261769  
0.093481  0.17246  0.221222  0.169894  0.18394  0.229616  
0.0961  0.17249  0.224392  0.1762  0.191144  0.243101  
 
 
 
47.69037  39.2796  150.0455  222.4742  760.6277  249.8103  
53.1722  46.57776  169.1822  175.7566  735.9215  249.8103  
54.9329  48.23015  183.8415  210.5953  773.4186  276.5494  
68  60  126  143  244  133  
71  65  134  127  240  133  
72  59  110  139  251  135  
0.092832  0.174038  0.249647  0.236585  0.222276  0.279208  
0.095989  0.171907  0.230047  0.178243  0.197366  0.244636  
0.09688  0.17233  0.237756  0.19374  0.201693  0.256106 