data models - university of arizonadial/ece531/data_models.pdf · 2005-09-13 · image algorithms...
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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)
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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