Index
Image Processing IV
Visualizational aspects
A-Z Image Processing V
Classification - unsupervised

Image Processing IV (cont.)
Concentrating on essentials

The techniques brought up here fit in more functional categories, so do not strictly rely upon this outline. More thorough discussions of the following topics would require lots more space, so I strongly recommend reading (geo)statistical textbooks (also check the recently published sample exercise from the Remote Sensing Core Curriculum about Principal Components Analysis).

Data reduction - PCA (Principal Components Analysis)

Link back to the previous page and take a look at the 7 spectral bands from the Landsat TM data. You may notice more or less strong correspondence in the data - these are redundancies, 'digital ballast' not always necessary to carry along with the data. With the statistical method of principal components analysis we 'extract the essentials' by producing so-called 'principal component images' and statistical information output in IDRISI:

Table: PCA tabular output in IDRISI
7 spectral bands were feed into the PCA-function to produce 7 components

VAR/COVARtaurtm1taurtm2 taurtm3taurtm4taurtm5 taurtm6taurtm7
this is the variance/co-variance matrix between all cellvalues in each band combination
taurtm13087.332070.672530.54 974.94-517.63-449.01-34.25
taurtm22070.671546.111842.13 844.92-359.73-300.25-54.99
taurtm32530.541842.132218.84 960.56-411.23-362.97-38.62
taurtm4974.94844.92960.56 1063.28106.61-16.30-22.00
taurtm5-517.63-359.73-411.23 106.611021.40288.10482.89
taurtm6-449.01-300.25-362.97 -16.30288.10157.7497.38
taurtm7-34.25-54.99-38.62 -22.00482.8997.38290.16

COR-MATRXtaurtm1taurtm2 taurtm3taurtm4taurtm5 taurtm6taurtm7
this correlation matrix contains the PEARSON product-moment correlation coefficients (range: -1 to +1)
taurtm11.0000000.947761 0.9668470.538101-0.291492-0.643424 -0.036191
taurtm20.9477611.000000 0.9945710.658977-0.286261-0.607986 -0.082100
taurtm30.9668470.994571 1.0000000.625371-0.273162-0.613545 -0.048126
taurtm40.5381010.658977 0.6253711.0000000.102303-0.039810 -0.039612
taurtm5-0.291492-0.286261 -0.2731620.1023031.0000000.717772 0.887027
taurtm6-0.643424-0.607986 -0.613545-0.0398100.7177721.000000 0.455205
taurtm7-0.036191-0.082100 -0.048126-0.0396120.8870270.455205 1.000000

COMPONENTC1C2 C3C4C5C6 C7
these numbers show, how much variance is being explained by the components. Note that 98.38% of the overall variance is explained through the first three components!
%var.77.5114.42 6.451.080.37 0.100.06
eigenval.7274.641353.37 605.34101.6234.36 9.615.92

eigvec.10.637395-0.024205 -0.332783-0.6649080.078396-0.128519 -0.132783
Summary of eigenvectors per eigenvalue
eigvec.20.4536890.032000 0.0118310.5684480.0351830.001237 -0.684566
eigvec.30.5513950.026079 -0.0503500.4197400.048126-0.022627 0.716756
eigvec.40.2500480.409488 0.816091-0.233731-0.1018260.196958 0.000000
eigvec.5-0.1112670.798522 -0.2560410.062861-0.203832-0.488801 0.000000
eigvec.6-0.0925840.210568 0.0609130.0009700.968702-0.070669 0.000000
eigvec.7-0.0195340.384777 -0.3889840.0002050.0000000.836813 0.000000

LOADINGC1C2 C3C4C5C6 C7
the loadings are coefficients of the eigenvectors linear equation; they aid in analysing the relative importance of a band within a principal component (range: -1 to +1)
taurtm10.978414-0.016026 -0.147356-0.1206300.008270-0.007172 -0.005814
taurtm20.9841100.029939 0.0074030.1457320.0052450.000098 -0.042358
taurtm30.9984010.020367 -0.0262990.0898260.005989-0.001489 0.037021
taurtm40.6540420.461983 0.615764-0.072257-0.0183050.018729 0.000000
taurtm5-0.2969440.919171 -0.1971100.019828-0.037385-0.047423 0.000000
taurtm6-0.6287460.616786 0.1193280.0007780.452112-0.017447 0.000000
taurtm7-0.0978100.830999 -0.5618430.0001210.0000000.152325 0.000000

If you are not familiar with techies like 'eigenvalue', 'eigenvector', ... I recommend again diving deeper with statistical textbooks. To give you an expression of how that component images look like - find here 7 images from a subscene of the Hohe Tauern region (around the Matreier Tauernhaus):

Component 1
PCA Component 1
Component 2
PCA Component 2
Component 3
PCA Component 3
Component 4
PCA Component 4
Component 5
PCA Component 5
Component 6
PCA Component 6
Component 7
PCA Component 7

Such images might be simply used for enhancement, data reduction or even later on in the classification process (be careful in using them in the latter). Helpful not only for interpreting the results of the PCA are additional XY-scatterplots of one band versus another. In the IDRISI for DOS version you could create simple scatterplots with the routine SCATTER. The following scatterplot matrix has been prepared with the help of SPSS 7.0:

scatterplot matrix between 7 Landsat TM 5 bands e.g., 'TM 1' means thematic mapper band 1;

compare these plots against the correlation matrix part of the IDRISI output. Clearly visible are strong positive linear correlations between bands 2 + 3, 2 + 4, 3 + 4.



among my personal favorites:
    John C. DAVIS, 19862nd ed.: Statistics and Data Analysis in Geology. John Wiley & Sons. New York
    A. R. H. SWAN & M. SANDILANDS,.1995: Introduction to Geological Data Analysis. Blackwell Sciene, Oxford
    Trevor C. BAILEY & Anthony C. GATRELL,.1995: Interactive Spatial Data Analysis. Longman, Essex.

Index
Image Processing IV
Visualizational aspects
A-Z Image Processing V
Classification - unsupervised
last modified: | Comments to Eric J. LORUP

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