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ECE/OPTI 531 – Image Processing Lab for Remote Sensing Fall 2005

Data Models

Reading: Chapter 4

Fall 2005Data Models 2

Data Models

• Univariate Data Statistics• Multivariate Data Statistics• Scattergrams and Scatterplots• Scene and Sensor Effects

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Fall 2005Data Models 3

Introduction

• What Factors Affect Remote Sensing Imagery?– atmospheric conditions– characteristics of the earth’s surface– sensor characteristics

• These factors can be modeled for their affectson remote sensing data

• Many image processing algorithms, in turn, relyon these data models

radiation models

scenemodels

sensormodels

datamodels

image

algorithmsprocessing

solar,

thermal

models

Fall 2005Data Models 4

Univariate Data Statistics

• Band-by-band statistics of a multispectral image• Do not measure interrelationships among bands• Can, in rare cases, be applied to the

multivariate case• Definition:

– Digital Number (DN) – numerical value of a pixel, eithera scalar (univariate, single band) or vector(multivariate, multiband)

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Fall 2005Data Models 5

DN Histogram• Measures brightness distribution

– Define equally sized DN “bins”,– Count the number of pixels in each bin and divide by the

total number of pixels in the image

• Properties– Analogous to the continuous Probability Density Function

(PDF) of statistics

– Histogram contains no information about the spatialdistribution of pixels

– Usually skewed, with a “tail” towards higher DNs• Applications

– contrast enhancement– DN thresholding

Fall 2005Data Models 6

DN Moments

• DN mean

– µ is a measure of average brightness– first moment (centroid) of the histogram

• DN variance

– Standard deviation σ is a measure of average contrast– second central moment of the histogram

• Efficient to calculate if histogram is alreadyavailable

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Fall 2005Data Models 7

0

0.05

0.1

0.15

0 20 40 60 80 100

Gaussian

image

frac

tio

n o

f to

tal

pix

els

DN

µ µ + !µ " !

Example image histogram comparedto equivalent Gaussian distribution

Histogram Features

• Higher order moments– skewness measures

asymmetry of thehistogram

– kurtosis measuressharpness of thehistogram peak relative toa normal distribution

• Other statistics– mode has the maximum

count– median divides histogram

area in half

Fall 2005Data Models 8

Cumulative Histogram• Cumulative fraction of

pixels with value less thanor equal to DN

• Properties– Analogous to the

Cumulative DistributionFunction (CDF) of statistics

– Monotonic function of DN• Applications

– contrast enhancement(histogram equalization)

– relative radiometricnormalization ofmultitemporal images

– noise removal (destriping)

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Gaussian

image

frac

tio

n o

f to

tal

pix

els

DN

Example image CDF comparedto equivalent Gaussian CDF

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Fall 2005Data Models 9

Data Models

• Univariate Data Statistics• Multivariate Data Statistics• Scattergrams and Scatterplots• Scene and Sensor Effects

Fall 2005Data Models 10

Multivariate Data Statistics

• K-band multispectral image• Measures statistical relationships among bands

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Fall 2005Data Models 11

• Each multispectral pixel is a K-dimensional column vector– Components of the vector are the DN values in each of the K

bands– At pixel (i,j):

• Quantization– Q = number of bits/pixel/band– Data space is quantized into (2Q)K cells– With 3 bands and 8 bits/pixel/band —> 2563 = 16,777,216

possible data vectors

DNp3

DNp1 DNp2

DNp

DNp1

DNp2

DNp3

=

DN3

DN1DN2

Vector Representation

Fall 2005Data Models 12

Covariance

• DN mean vector

– centroid of the data in K-D space• DN covariance matrix

– where covariance between bands m and n:

– diagonal element ckk is the DN variance in band k– C is a measure of the “spread” of the distribution

about the mean vector, similar to the 1-D variance

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Fall 2005Data Models 13

Correlation

• DN correlation matrix

• Correlation coefficient between bands m and n:

– Covariances are normalized by standard deviation ofeach band

• Interpretation of correlation– Measures the amount of linear relationship between

pairs of bands

Fall 2005Data Models 14

C and R Matrix Properties

• Properties of C and R– Symmetric, i.e., cmn = cnm and ρmn = ρnm

– If C and R are diagonal, the pixel values in bands mand n are uncorrelated

– Multiband images can be transformed by the PrincipalComponents Transform (PCT) so that C and R arediagonal, i.e. the transformed data are uncorrelated

• Applications– Designing decision boundaries for pattern recognition– Removing redundancy among spectral bands (data

compression)– Designing color contrast enhancement methods

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Fall 2005Data Models 15

Joint Distributions

high correlation moderate correlation low correlation

Idealized distributions with different correlations

Fall 2005Data Models 16

Multivariate Histogram

• Count the number of pixels in each DN “bin” anddivide by the total number of pixels in the image

• Scalar function of a vector– Each value is the pixel count for a particular DN vector,

divided by the total number of pixels– Not easily visualized for K > 2

• K-D Histogram Models (for reference)– Gaussian (Normal) distribution usually assumed for

mathematical convenience

1-D:

K-D:

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Fall 2005Data Models 17

Data Models

• Univariate Data Statistics• Multivariate Data Statistics• Scattergrams and Scatterplots• Scene and Sensor Effects

Fall 2005Data Models 18

Scattergrams

• Projection from K-D to 2D• Scattergram retains pixel counts in 2-D

“clusters” from the Briones and San Pablo Reservoirs in Fig. 2–13

“z-axis” is thefraction of pixelswith a given(DNm,DNn) vector

DN2 DN2 DN3

DN3

DN4

DN4

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Fall 2005Data Models 19

Scatterplots

• Scatterplot is a binary plot– Loses pixel counts– Project 3-D scatterplot onto 2-D, or threshold the 2-D

scattergram

Example for K = 3, viewed from different directions

Fall 2005Data Models 20

2-D Scatterplots

band 3 vs. band 2 band 4 vs. band 2 band 4 vs. band 3

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Fall 2005Data Models 21

Scatterplot Combinations2 3 4 5 6 7

1

2

3

4

5

6All possible 2-D scatterplotsfor a 7-band TM image

Fall 2005Data Models 22

Data Models

• Univariate Data Statistics• Multivariate Data Statistics• Scattergrams and Scatterplots• Scene and Sensor Effects

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Fall 2005Data Models 23

Topography Effects Study

• Model: radiance of scene is proportional tocosine of solar-surface angle (Chapter 2)

Topography Effects

Digital Elevation Model (DEM)

Create shaded-relief image from DEM using cosine irradiance model

Create spatial maps for spectral classes soil and vegetation

Create randomized spectra for each class

Multiply spectral maps by shaded-relief image

Compare before/after spectral scattergrams

Fall 2005Data Models 24

Shaded Relief and Class Maps

• Create shaded-relief imagefrom DEM using cosineirradiance model

• Create class maps forspectral classes soil andvegetation– class maps are mutually

exclusive, i.e. a pixel iseither soil or vegetation

solar irradiance spatial variation

spatial distribution of classes

soil vegetation

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Fall 2005Data Models 25

Synthetic Scene With Relief

• Create randomized spectra foreach class (Gaussian)

• Set each pixel to a sample fromthe appropriate classdistribution– ρ(x,y) for each class

• Multiply spectral maps byshaded-relief image

• Combine classes in each band

soil

vegetation

Red Band NIR Band

for each class

Red Band NIR Band

Fall 2005Data Models 26

• Without topography (flat terrain): two circulardistributions, zero correlation

• With topography: two elliptical distributions, highcorrelation

• Conclusion: topography creates spectral correlation

Scattergram Analysis

withouttopography

withtopography

spectral scattergrams without and with topographyDNNIR

DNred

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Fall 2005Data Models 27

Sensor Spatial Response Study

• Model: Sensor spatial response causes imageblurring and mixing of spectral signatures(Chapter 3)

Sensor Spatial Effects

Create synthetic scene containing soil, vegetation, and water

Create randomized spectra for each class

Convolve with 5 x 5 pixel Gaussian spatial response

Compare before/after spectral scattergrams

Fall 2005Data Models 28

Synthetic Scene

• Create class maps forspectral classes soil,vegetation, water– class maps are mutually

exclusive, i.e. a pixel isonly one class

– no topography• Create randomized

spectra for each class(Gaussian)

• Set each pixel to a samplefrom the appropriate classdistribution– ρ(x,y) for each class

• Combine classes in eachband

red band NIR band

class maps, no texture

uncorrelated texture

spatially-correlated texture

for each band

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Fall 2005Data Models 29

Spectral Scattergrams

NIR

red

uncorrelated texture spatially-correlated texture

no texture

Fall 2005Data Models 30

Simulate Blurring

• Convolve with 5 x 5 pixel Gaussian spatial response– represents image from simulated sensor

red band NIR band

uncorrelated texture

spatially-correlated texture

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Fall 2005Data Models 31

Scattergram Analysis

• With sensor spatial response: new spectralvalues between the original signatures

• Conclusion: spatial integration creates “mixing”of spectral signatures

NIR

reduncorrelated texture spatially-correlated texture

“after” sensor spatial

response

mixed soil/vegetation

pixels

“before” sensor spatial

response