Image Processing \u0026 Analysis Topic 3: Extraction of Basic Statistical Information

$Page 1: Image Processing \u0026 Analysis Topic 3: Extraction of Basic Statistical Information$
Image Processing & Analysis

Prof. Dr. Shattri Mansor

Department of Civil Engineering

Faculty of Engineering

Universiti Putra Malaysia

[email protected]

Topic 3: Extraction of Basic Statistical Information

Outline

Brief Review

Histograms

Univariate Image Statistics

Multivariate Image Statistics

Brief Review

A population is an infinite or finite set of elements.

A sample is a subset of elements taken from a population.

Samples are used to make inferences about populations.

Brief Review

Biased samples are observations that exclude certain population characteristics.

Sampling error is the difference between the actual population and the inferences drawn from samples.

Brief Review

Large samples drawn randomly from natural populations usually produce a symmetrical (normal) frequency distribution.

Many statistical tests used to analyze remotely sensed data assume normal distribution.

Adapted from Jensen (1996)

Histograms

The histogram is a graphic representation of the information content of a remotely sensed image.

Histograms provide measures of the quality of the data (high contrast, low contrast, multi-modal, etc.).




Measures of central tendency:

Mode – most frequently occurring DN.

Median – DN midway in the frequency distribution.

Mean () is the mathematical average:

n

n

i

ik

k

1

DN


For normal distributions, sample mean closely estimates population mean.

Mean is a poor measure of central tendency when the data set is skewed.

Negatively skewed – mode is to the right of mean.

Positively skewed – mode is to the left of mean.



Measures of dispersion from mean:

Range – difference between highest (max) and lowest (min) DN in a band.

Range may be misleading if max or min are extreme values.

rangek = maxk - mink



Variance (vark) – average squared deviation of all observations from the sample mean.

1var

n

1

2) - (DN

ni

kik

k

Also called the sum of squares (SS)



Standard Deviation (sk) – positive square root of variance.

Small sk indicates observations are clustered closely around the mean.

Large sk indicates widely scattered data.

kks var


For normal distributions:

68% of observations fall within 1 sk



Standard Deviation and Variance are the basis for several image processing procedures.

Example Data Set

Univariate Statistics


Because remote sensing often deals with reflectance or emittance in more than one band, multivariate techniques are also important.

There are many relevant multivariate techniques.

We will begin by examining covariance and correlation.


Corresponding pixel DNs across bands often vary together (co-vary) in a predictable fashion.

If there is no relationship between DNs in one band and corresponding DNs in another, the data sets are mutually independent.

Covariance is a measure of mutual interaction between corresponding pixel DNs in different bands.


Two steps to calculating covariance:

Calculate corrected sum of products (SP):

Divide SP by n-1:

n

i

n

i

n

i

ilik

ilikkln1

1 1

DNDN

)DNDN(SP

1cov

n

SPklkl


SS (var) and SP can be computed for all possible band combinations and displayed in a variance-covariance matrix:

Matrix Format Example Data


To more clearly distinguish the degree of interrelation between variables, correlation coefficient (r) is often used.

The correlation between two bands (rkl) is the ratio of their covariance to the product of their standard deviations:

lk

klkl

ssr

cov


r ranges between –1 and +1.

+1 indicates perfect linear relationship.

-1 indicates perfect inverse relationship.

Correlation Matrix


Date post:	27-Nov-2023
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Image Processing \u0026 Analysis Topic 3: Extraction of Basic Statistical Information

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