1. Principal Component Analysis (PCA)
Designed to reduce redundancy in multispectral bands
Spectral correlation from band to band
Either enhancement prior to visual interpretation or pre-processing for classification or other analysis
Compress all info originally in many bands into fewer bands
AABU

2. descriptive image statistics
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 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
The mean is the arithmetic average and is defined as the sum of all brightness value (BV)oobservations divided by the number of observations.

3. Cont’ Min Max Variance Standard deviation
Coefficient of variation (CV)
Skewness
Kurtosis

5. Measures of Distribution (Histogram) Asymmetry and Peak Sharpness
Skewness is a measure of the asymmetry of a histogram and is computed using the formula:
A perfectly symmetric histogram has a skewness value of zero. If a distribution has a long right tail of large values, it is positively skewed, and if it has a long left tail of small values, it is negatively skewed.

6. Measures of Distribution (Histogram) Asymmetry and Peak Sharpness
A histogram may be symmetric but have a peak that is very sharp or one that is subdued when compared with a perfectly normal distribution. A perfectly normal distribution (histogram) has zero kurtosis. The greater the positive kurtosis value, the sharper the peak in the distribution when compared with a normal histogram. Conversely, a negative kurtosis value suggests that the peak in the histogram is less sharp than that of a normal distribution. Kurtosis is computed using the formula:

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

9. Covariance
The different remote-sensing-derived spectral measurements for each pixel often change together in some predictable fashion. If there is no relationship between the brightness value in one band and that of another for a given pixel, the values are mutually independent; that is, an increase or decrease in one band’s brightness value is not accompanied by a predictable change in another band’s brightness value. Because spectral measurements of individual pixels may not be independent, some measure of their mutual interaction is needed. This measure, called the covariance, is the joint variation of two variables about their common mean.

10. The Sample Covariance
A'kif RS

11. Correlation
To estimate the degree of interrelation between variables in a manner not influenced by measurement units, the correlation coefficient, is commonly used. The correlation between two bands of remotely sensed data, rkl, is the ratio of their covariance (covkl) to the product of their standard deviations (sksl); thus:

12. 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 1” that can be accounted for or explained by a linear relationship with the values of the random variable “band 2.” 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”.

13. BAND 1 VS BAND 2
Example: investigate relationship between band 1 and band 2
Data: sample group response data on DNs of band 1, and corresponding band 2 DNs
A'kif RS

14. BAND 1 VS BAND 2 N Band 1 DNs (X ) Band 2 DNs (Y ) 1 45 2 5 42 3 10 3345 2 5 42 3 10 33 4 15 31 20 29
A'kif RS

15. BAND 1 VS BAND 2

16. Band 1 vs band2 DNS
Observe that as band 1 DNs goes up, corresponding Band 2 DNS goes down
Variables covary inversely
Covariance and Correlation quantify relationship
A'kif RS

17. Covariance
Variables that covary inversely, like band 1 and band , tend to appear on opposite sides of the group means
Average product of deviation measures extent to which variables covary, the degree of linkage between them
A'kif RS

18. Calculating Covariance
Band 1 (X )
Band 2 (Y ) 45 5 42 10 33 15 31 20 29 36
A'kif RS

19. Calculating Covariance
Band 1
(X )
Band 2
(Y )
-10
-90 9 45 5 -5
-30 6 42 10 -3 33 15
-25 31 20
-70 -7 29
∑= -215
A'kif RS

20. Covariance Calculation
(2)
Evaluation yields,
A'kif RS

21. Table for Calculating rxy
Band 1
(X )
X 2 XY
Y 2
Band 2
(Y )
2025 45 5 25
210
1764 42 10
100
330
1089 33 15
225
465
961 31 20
400
580
841 29 ∑= 50
750
1585
6680
180
A'kif RS

22. Computing rxy from Table
A'kif RS

23. Computing Correlation
A'kif RS

24. example Band 1 (Band 1 x Band 2) Band 2 130 7,410 57 165 5,775 35 100
Pixel
Band 1 (green)
Band 2 (red)
Band 3 (ni)
Band 4 (ni)
(1,1)
130 57
180
205
(1,2)
165 35
215
255
(1,3)
100 25
135
195
(1,4) 50
200
220
(1,5)
145 65
235
Band 1
(Band 1 x Band 2)
Band 2
130
7,410 57
165
5,775 35
100
2,500 25
135
6,750 50
145
9,425 65
675
31,860
232

25. Univariate statistics Band 1 Band 2 Band 3 Band 4
Mean (mk)
135
46.40
187
222
Variance (vark)
562.50
264.80
1007
570
(sk)
23.71
16.27
31.4
23.87
(mink)
100 25
195
(maxk)
165 65
215
255
Range (BVr) 40 80 60
Univariate statistics
Band 1
Band 2
Band 3
Band 4
562.25 -
135
264.80
718.75
275.25
537.50 64
663.75
570
Band 1
Band 2
Band 3
Band 4 -
0.35
0.95
0.53
0.94
0.16
0.87
covariance
Covariance
Correlation coefficient

26. Dimension rotation y y’ x’ 0.7x,0.7y -0.7x, 0.7y + x y 0.5x,0.87y

27. Principal Component Analysis (PCA)
Designed to reduce redundancy in multispectral bands
Spectral correlation from band to band
Either enhancement prior to visual interpretation or pre-processing for classification or other analysis
Compress all info originally in many bands into fewer bands
AABU

28. Principal Component Analysis (PCA) - The math behind the button
In the simple case of 45º axis rotation, Finding q
PC1
PC2’ [ ] = [ ] [ ] [ ] [ ]
DN1’
DN2’
cos q sin q
-sin q cos q
DN1
DN2
n11 n12
n21 n22 =
cov =
q = 45º
Cov’=RTcovR; cov’ is the matrix having eigenvalues as diagonal elements and
RT is the transpose of R. Eigenvalues can be found by diagonalizing cov. R has eigenvectors as column vectors
AABU

29. Covariance under Affine Transformation
A'kif RS

30. Covariance under Affine Transf
(2)
A'kif RS

31. Principal Component Analysis
In the simple case of 45º axis rotation,
The rotation in PCA depends on the data. In the top case, all the image data have similar DN2/DN1 ratios but different intensities, and PC1 passes through the elongated cluster.
In the bottom example, vegetation causes there to be 2 mixing lines (different DN4/DN3 ratios (and the “tasseled cap” distribution such that PC1 still passes through the centroid of the data, but is a different rotation that in the top case.
PC1
PC2

32. Tasseled Cap Transformation
Transforms (rotates) the data so that the majority of the information is contained in 3 bands that are directly related to physical scene characteristics
Brightness (weighted sum of all bands – principal variation in soil reflectance)
Greenness (contrast between NIR and VIS bands
Wetness (canopy and soil moisture)

33. Tasseled Cap Transformation (TCT)
TCT is a fixed rotation that is designed so that the mixing line connecting shadow and sunlit green vegetation parallels one axis and shadow-soil another. It is similar to the PCT.
Soil
Green

34. Next lecture – Spectral Mixture Analysis