CHAPTER 10CHAPTER 10

Principal ComponentsPrincipal Components

BAND TRANSFORMATIONSBAND TRANSFORMATIONS

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Principal Components Principal Components

2

1

12

21

2

1

cossin

sincos

sin cos

sin cos

x

x

xx

xx

x

x

( )x R x

A two-dimensional illustration of the principal component transformation

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1 2 3( , , ) x R x

The principal components transformation (a three-dimensional illustration)

Principal Components Principal Components

1

2

3

b

b

b

x

PC1

PC2

PC3

x

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The original bands of a Landsat Thematic Mapper Image before applying the principal components transformation

The original bands of a Landsat Thematic Mapper Image before applying the principal components transformation

TM1 TM2 TM3 TM4

TM5 TM7 TM6

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The new bands resulting from the principal components transformation The new bands resulting from the principal components transformation

PC1 PC2 PC3 PC4 PC5 PC6 PC7

σj2 2543.73 192.64 68.11 9.61 5.12 3.25 1.08

σj 50.44 13.88 8.26 3.10 2.26 1.80 1.04

PC1 PC2 PC3 PC4 PC5 PC6 PC7

σj2 2543.73 192.64 68.11 9.61 5.12 3.25 1.08

σj 50.44 13.88 8.26 3.10 2.26 1.80 1.04

PC1 PC2 PC3

PC1 – PC2 – PC3

PC4 PC5 PC6 PC7

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A color composite presentation of the three first principal component bands (R=1, G=2, B=3), containing a significant part of the original image informationIt presents a type of classification of the image

A color composite presentation of the three first principal component bands (R=1, G=2, B=3), containing a significant part of the original image informationIt presents a type of classification of the image

PC1 PC2 PC3

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The original bands of a SPOT 4 image

The original bands of a SPOT 4 image

Correlation matrix:

standard deviations and variances:

175.066.092.0

75.0197.081.0

66.097.0170.0

92.081.070.01

R

σi σi2

8583 7367523615 1306523513 1234442622 68765

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The resultingl bands from the principal

components transformation

The resultingl bands from the principal

components transformation

standard deviations and variances:

σi σi2

97.94 9592.7129.79 887.23 9.79 95.90 4.51 20.30

correlation matrix == identity matrix

(R = I)

PC1 PC2

PC3 PC4

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A color composite of the first three principal components (R=1, G=3, B=2) where colors correspond to land cover classes.

Blue and green represent water and vegetation, while red corresponds to dry soil.

In particular the part of the forest destroyed by a recent fire is clearly outlined in the dark red area north-east of the city

A color composite of the first three principal components (R=1, G=3, B=2) where colors correspond to land cover classes.

Blue and green represent water and vegetation, while red corresponds to dry soil.

In particular the part of the forest destroyed by a recent fire is clearly outlined in the dark red area north-east of the city

PC1 PC2 PC3

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