Remote Sensing Image Statistics
Supplement to Lecture 2
prepared by R. Lathrop 9/99
updated 8/01
Readings:
ERDAS Field Guide 5th Ed App A:Math Topics
Digital Images
• Digital Number (DN) or Brightness Value
(BV) - the tonal gray scale expressed as a
number, typically 8-bit number (0-255)
• Dimensionality - determined by the number
of data layers (bands)
• Measurement Vector of a pixel - is the set of
data file values for one pixel in all n bands
Digital Image Storage Formats
• Band sequential (BSQ) - each band
contained in a separate file
• Band interleaved by line (BIL) - each record
in the file contains a scan line (row) of data
for one band, with successive bands
recorded as successive lines
• Band Interleaved by Pixel (BIP)
Calculating disk space
[ ( (x * y * b) * n) ] x 1.4 = output file size
where:
y = rows
x = columns
b = number of bytes per pixel
n = number of bands
1.4 adds 30% for pyramid layers and 10%
for other info
Summarizing data distributions
• Frequency distributions - method of
describing or summarizing large volumes of
data by grouping them into a limited
number of classes or categories
• Histograms - graphical representation of a
frequency distribution in the form of a bar
chart
Measures of Central Location
• Mean - simple arithmetic average, the sum
of all observations divided by the number of
observations
• Median - the middle number in a data set,
midway in the frequency distribution
• Mode - the value that occurs with the
greatest frequency, the peak in a histogram
Measures of Central Location example
BV band 1
130
Mean = 675 / 5 = 135
165
100
135
145
sum = 675
Median = 135
Statistical Notation
•
•
•
•
•
uk = mean of band k
sk = standard deviation of band k
Vark = variance of band k
COVkl = covariance of bands k & l
rkl = correlation between bands k & l
Measures of Dispersion
• Range - the difference between the largest
and smallest value
• Variance - the average of the squared
deviations between the data values and the
mean
• Standard Deviation - the square root of the
variance, in the units of data measurement
Measures of Dispersion example
BVik Bvik- uk ( Bvik- uk)2
Var = SUM(( Bvik- uk)2
n-1
130
-5
25
165
30
900
100
-35
1225
= 562.5 BV2
135
0
0
Sk = SQRT(562.5)
145
10
100
675
0
2250
Vark = 2250 / 4
= 23.71 BV
Feature Space Image
• Visualization of 2 bands of image data
simultaneously through a 2 band scatterplot
- the graph of the data file values of one
band of data against the values of another
band
• Feature space - abstract space that is defined
by spectral units
Spectral distance
• Spectral distance - the Euclidean distance in
n-dimensional spectral space
• D = SQRT[(sum (dk - ek)2]
where dk = BV of pixel d in band k
where ek = BV of pixel e in band k
• the equation is summed across k = 1 to n
bands
Spectral Distance example
Y
92, 153
180, 85
X
Spectral Distance example
Distance between [x1,y1] & [x2, y2]
[180, 85] & [92, 153]
D = SQRT[(sum (dk - ek)2]
D = SQRT[(180-92)2 + (85-153)2]
= SQRT[(88)2 + (-68)2]
= SQRT[7744 + 4624]
= SQRT[12,368] = 111.2
Spectral Distance example
Y
92, 153
Yd = 85-153
180, 85
Xd = 180 -92
X
Measures of inter-relation
between bands
• Independence - no relationship
• Dependence - inter-relationship
• Statistical measures of covariance and
correlation
Covariance - measure of the joint
variation of 2 variables about their
common mean
• COVk,l = Sum [(BVik - uk)(BVil - ul)]
(n-1)
• COVk,l = 0, statistical independence
• COVk,l > 0, positive relationship,
as k increases, l increases
• COVk,l < 0, negative relationship
as k increases, l decreases
Correlation - the degree of inter-relation
in a manner not influenced by the
measurement units (unitless)
• rkl = COVkl
Sk * Sl
• -1 < rkl < 1
• rkl = 0, statistically independent
• rkl = -1, perfect negative relationship
• rkl = 1, perfect positive relationship
Covariance & Correlation example
BV1
BV1- u1 BV2
BV2- u2 (D1)(D2)
130
-5
57
10.6
-53
165
30
35
-11.4
-342
100
-35
25
-21.4
749
135
0
50
3.6
0
(23.7)(16.3)
145
10
65
18.6
186
= 135 / 386.3
675
0
232
0
540
= 0.35
uk = 46.4
COV12 = 540 / 4
= 135
r12 = _
135
Covariance & Correlation Matrices
• Provide a useful summary of data
relationships
• High variance suggests a higher information
content for that band
• High correlation suggests a substantial
amount of redundancy
• Low correlation suggests that each band
provides information not found in the other
Covariance Matrix
Diagonals represent band variances. Example, variance for Band 3 =
273.1
Off-diagonals represent co-variances. Example, covariance of Band 1 and
4 = -35.3; same as covariance of Band 4 and 1. Negative covariance: as
one band increases, the other decreases.
Covariance matrix
1
2
1
232.3 139.1
2
139.1 89.9
3
237.2 153.4
4
-35.3 -4.6
5
191.9 142.1
6
42.0 24.1
7
182.6 122.0
3
237.2
153.4
273.1
-26.9
249.1
46.4
219.4
4
-35.3
-4.6
-26.9
341.1
216.0
-38.1
25.3
5
191.9
142.1
249.1
216.1
555.2
33.5
305.3
6
42.0
24.1
46.4
-38.1
33.5
31.22
40.6
7
182.6
122.0
219.4
25.3
305.3
40.6
227.6
Homework 2(Optional): Image Statistics
NBTM88.img is a 7 band TM image with 1000 rows by 1500 columns.
Image statistics for NBTM88.img:
B1
B2
B3
B4
Mean 102.8 41.7
47.0
74.9
Covariance matrix
1
2
1
232.3 139.1
2
139.1 89.9
3
237.2 153.4
4
-35.3 -4.6
5
191.9 142.1
6
42.0 24.1
7
182.6 122.0
3
237.2
153.4
273.1
-26.9
249.1
46.4
219.4
4
-35.3
-4.6
-26.9
341.1
216.0
-38.1
25.3
5
191.9
142.1
249.1
216.1
555.2
33.5
305.3
B5
77.4
6
42.0
24.1
46.4
-38.1
33.5
31.22
40.6
B6
155.8
7
182.6
122.0
219.4
25.3
305.3
40.6
227.6
B7
32.8
Homework 2: Image Statistics
1. What is the computer disk space storage size for the
NBTM88.img in ERDAS Imagine format? ______________
2. What is the variance for Band 4? ____________________
3. Do Band 3 and Band 4 positively covary? ______________
4. List the bands in order of decreasing "information" content.
High
a. ______
b. ______
c. ______
d. ______
e. ______
f. ______
Low
g. ______