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ECE/OPTI 531 – Image Processing Lab for Remote Sensing Fall 2005
Spectral Transforms
Reading: Chapter 5
Fall 2005Spectral Transforms 2
Spectral Transforms
• Feature Spaces• Spectral Band Ratios and VIs• Principal and Tasseled-Cap Components• Contrast Enhancement
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Multispectral Data Spaces
• Three “spaces” associated with multispectralimages:– image space – the DNb(x,y) space, i.e. an “image” in
band b– spectral (data) space – the DN = (DN1, DN2, . . ., DNK)
vector space– feature space – derived from the image or spectral
space
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Feature Spaces
• Good features reduce effects that hinder theextraction of information
• Nonlinear spectral transform
– multispectral ratios are one example• Linear spectral transform
– corresponds to a coordinate rotation of the DN space tothe DN´ space
– principal components transform is an example
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Spectral Transforms
• Feature Spaces• Spectral Band Ratios and VIs• Principal and Tasseled-Cap Components• Contrast Enhancement
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Simple Ratio:
Modulation Ratio:
Spectral Band Ratios
• Benefits– reduce topographic shading– emphasize spectral differences– correlate with geophysical variables
• Multispectral ratios
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Calibrated Spectral Ratios
• Simple physical model for radiometriccalibration
• Band-specific gain factor includes– sensor gain– solar spectral irradiance– atmospheric transmittance (2-way)
• is the solar irradiance projectionfactor due to surface topography (Chapter 2)
• Band-specific bias includes– atmospheric path radiance– sensor offset
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Calibrated Ratios (cont.)
• Partially-calibrated data– estimate DN offset and subtract
– topographic shading due to goes awaybecause it is the same in every band
• Fully-calibrated data– proportionality constant also goes away
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Partial Calibration
TM1 uncalibrated DN1 TM2 uncalibrated DN2
DN1 – b1 DN2 – b2
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Shading Reduction
DN band ratio TM2/TM1
DN offset bk removed before ratio
topographic
shading reduced
by bk removal
retainstopographic
shading
Suppression of topographic shading by partial calibration
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Geologic Discrimination
TM7/TM5 (uncalibrated)
TM5/TM4 (uncalibrated)partial
calibration
less important
in NIR and SWIR
Ratios for geologic discrimination in the SWIR
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Vegetation Indices
• All VIs defined in terms of reflectance, not DN• VIs therefore require scene calibrated data
• Soil-Adjusted Vegetation Index (Huete)
– L is an empirical constant, typically about 0.5– reduces to NDVI for L = 0– superior to NDVI for low vegetation cover
Ratio Vegetation Index:
Normalized Difference Vegetation Index:
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VI Interpretation
RVI NDVI
CIRcomposite
crops
wet soil
dry soil(Landsat TM) !red
!NIRRVI = 3 2 1
NDVI = 0.5 0.33 0
NDVI correlates better withvegetation density than does RVI
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Spectral Transforms
• Feature Spaces• Spectral Band Ratios and VIs• Principal and Tasseled-Cap Components• Contrast Enhancement
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Principal Components
• Principal Components Transform (PCT) is a linearmatrix transform
– Principal Component (PC) vector is K-dimensional, justlike DN vector
– Each PCk is a weighted sum of all spectral bands;weights are the rows of K×K WPC matrix
– Also known as the Karhonen-Loeve Transformation (KLT)and the Hotelling transformation
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PCT Properties
• Covariance matrix of PC bands CPC is related to originalcovariance matrix C by,
– By matrix properties, WPC diagonalizes C, i.e. CPC is a diagonalmatrix
– Since CPC is diagonal, the PC bands are uncorrelated• Diagonal elements of CPC are the data eigenvalues
– Each eigenvalue λk is equal to the variance of thecorresponding PCk
– The total data variance is preserved,
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PCT Calculation
• The rows of WPC are the eigenvectors of the data,
– Each eigenvector ek contains the weights applied to theoriginal bands to obtain PCk and is found by solving theequation,
– Eigenvalues are found by solving the characteristicequation
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• Original data in DN-space
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DN2
DN1
556455344433322221DN2DN1Pixel
PCT Example
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PCT Example (cont.)
• Step 1. Find eigenvalues– Solve the characteristic equation (two solutions)
– Therefore,
– Note that PC1 contains 89% of the total data variance,and PC2 contains 11%, i.e.
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PCT Example (cont.)
• Step 2. Find eigenvectors– Substitute eigenvalues into
• First eigenvalue yields dependent equations
• Solving either equation
• PCT requires orthonormality of the eigenvectors
– Solve simultaneous equations to yield eigenvectors, andthe PCT weight matrix
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PCT Example (cont.)
• Step 3. Transform DN data to PC-space– For Pixel 1:
– Final PC data and scatterplot
1.256.9560.436.3850.184.9941.574.7431.323.3520.502.781
PC2PC1Pixel
0
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2 3 4 5 6 7
PC2
PC1
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PCT Benefits
• Why use the PCT?– Decorrelates spectral data
• Multispectral bands areoften highly-correlatedbecause of
– material spectralcorrelation
– topography– sensor band overlap
• Decorrelation separatesindependent componentsinto separate “bands”
– Compresses the variance
• Compression canpotentially reduce datacomputation burden
DN1
PC1
PC2
DN2
DN3
DN4
PC1
PC2
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TM bands
PC bands
DN
vari
ance
band or PC index
Correlated
Uncorrelated
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TM1 TM2 TM3
TM4 TM5 TM7
PC1 PC2 PC3
PC4 PC5 PC6
PCT Decorrelation
• Non-vegetated scene– PCT removes spectral redundancy
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PCT Contrast Extraction
• Vegetated scene– PCT extracts contrast between bands 3 and 4 (red and
NIR) due to the vegetation “red edge”
TM1 TM2 TM3
TM4 TM5 TM7
PC1 PC2 PC3
PC4 PC5 PC6
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PCT Noise Detection
• PCT can isolate spectrally-uncorrelated noise
TM2 TM3 TM4
PC1 PC2 PC3
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PCT Drawbacks
• Why not use the PCT?– It is data-dependent
• W coefficients change from scene-to-scene• Makes consistent interpretation of PC images difficult
– Spectral details, particularly in small areas, may be lostif higher-order PCs are ignored
– Computationally expensive for large images or formany spectral bands
• Calculation of covariance matrix is the culprit
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Tasseled-Cap Components
• Linear spectraltransform like the PCT
• In this case, the WTCmatrix is fixed for agiven sensor
Tasseled-cap components for MSS and TM
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TCT Benefits
• Why use the TCT?– It is a fixed reference
• Same reference for everyimage from a given sensorpermits consistentinterpretation
– Components are related togeophysical properties ofthe scene
• First component is “soilbrightness”
• Second component is“greeness”
Brightness
Greeness
DN3
DN4PC2
PC1
TCT axes alignbetter with thesoil and vegetationdirections
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TCT Drawbacks
• Why not use the TCT?– Nonoptimal compression of data– Derivation of WTC requires multitemporal data for each
sensor
PC1 PC2 PC3
PC4 PC5 PC6
TC1 TC2 TC3
TC4 TC5 TC6
Comparison of PC and TC images
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Spectral Transforms
• Feature Spaces• Spectral Band Ratios and VIs• Principal and Tasseled-Cap Components• Contrast Enhancement
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Contrast Enhancement
• Two problems– Most images do not fill the dynamic range of the sensor
– Most images also do not fill the dynamic range of the displaysystem
• Contrast enhancement means “stretching” the data rangeto fill the display system range GL = T(DN)– Parameters of transformation T based on global or local
image statistics
signal rangerequired at
A/D input for fullrange DN output
ab
eb
anticipated range
optional high gainb
offsetb
all scenes
standard gainb
of detected signals low radiance scenes
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Global Single-Band Transforms
• Linear stretch– min-max
• scale range of image DNs to range of display GLs• sensitive to outliers
– saturation• scale a smaller range of DNs to range of display GLs• saturation of 1–3% pixels at each end usually acceptable
• Nonlinear stretch– piecewise-linear
• different contrast “gain” over different DN ranges– histogram equalization
• use scaled CDF of original image as the transformation
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DN-to-GL Transformations
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Contrast Stretch Examples
original min-max
3% saturation 8% saturation
piecewise linear histogram equalization
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original min-max
3% saturation 8% saturation
piecewise linear histogram equalization
Application to GOES image of North America
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Normalization Stretch
• Linear scale of DN mean and sigma to specifiedvalues, followed by saturation
• Consistent behavior (robust) over wide range ofimages
original µ = 128, σ = 32
µ = 128, σ = 64µ = 128, σ = 48
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Reference Stretch
• Match the CDF of the image being processed to areference CDF, for example from another image– useful for
• multitemporal or multisensor radiance matching• matching image to reference contrast
DN
0
1
CDF
DNref
0
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CDFref
Reference stretch is a double transformation
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Multitemporal Normalization
0 100
2 103
4 103
6 103
8 103
1 104
nu
mb
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of
pix
els
DN0 12763
0 100
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1 103
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2.5 103
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GL
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GL
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TM band 3, Dec 31, 1982 TM band 3, Aug 12, 1983
dark-light targetlinear stretch
CDF referencestretch
Reference
CDF reference stretch
dark-light linear stretch
Dec 31, 1982 Aug 12, 1983
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Thresholding
• Binary “clipping” of DNs to low and high values– Useful for
• segmentation of certain images, e.g. clouds/water,land/water
DNT = 50
DNT = 100 DNT = 150
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Formulas
Mathematical formulas for contrast enhancement techniques
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Color Images
• Techniques used for single-band imagery can beextended to color, but . . .– Sensitivity of the human vision system to shifts in color
and saturation require special attention• Min-max stretch
– Stretch the DNs in each band over their respective min-max range
– Good news:• Easy to calculate and implement• No data lost by saturation
– Bad news:• Sensitive to outlier DNs• Color balance can change unpredictably
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Color Normalization
• Normalization stretch– “Standardized” stretch– Good news:
• Average color is grey• Contrast controlled by
single parameter, thedesired outputstandard deviation
– Bad news:• Some data are lost in
saturation
Normalization Stretch
Linearly stretch each band to same DN mean ( typically 128) and same DN standard deviation (typically 32 – 48)
Clip to [0,255]
RGB RGB
linear stretchto µ, !
clip at [0,255]
linear stretchto µ, !
clip at [0,255]
linear stretchto µ, !
clip at [0,255]
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Color Decorrelation
• Decorrelation stretch– Enhance small spectral
deviations in highly-correlated spectralbands
– Commonly used ingeology
– Good news:• Decorrelates bands• Emphasizes differences
among bands• Can be applied to any
number of bands– Bad news:
• Produces highlysaturated colors
Decorrelation Stretch
PCT transform
Stretch each PC component to equalize variances
Inverse PCT transform
Clip to [0,255]
image data space PC space
PCT
PCT-1clip at [0,255]
equalizevariances
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Color Spaces
• HSI color coordinate system• Hexcone model
– similar to a cylindrical coordinatesystem, but based on RGB color cube
– value = max(R,G,B) used instead ofintensity
– efficient CST from RGB to Hue-Saturation-Value (HSV)
intensity
saturation
hue
I
S
H
whiteblack
P
red
yellow
green
cyan
blue
magenta
(255,0,0)
(0,255,0)
(0,0,255)
yellow
magenta
cyan
red
greenblue
black
white
GL3
GL2
GL1
project subcube faces onto orthogonal planeintersection at the vertex of subcube
cylindrical coordinates
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Color-Space Transforms
• Color-space transforms– Human vision system
perceives hue (H),saturation (S) andintensity (I), not RGB
– Therefore, control overcolor appearance is bestdone in HSI space
Color-Space Transform
Transform RGB to HSI
Modify HSI components as desired
Inverse transform modified HSI to RGB
Clip to [0,255]
RGB space HSI space
CST
CST-1clip at [0,255]
modify
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Example Ramp Spectrum CSTs
50% intensity
50% saturation
cycle hue
H S I
linear hue100% saturation100% intensity
100% intensity
linear hue
linear hue100% saturation
100% saturation100% intensity
RGB
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CST for Contrast Enhancement
• Intensity stretch– Good news:
• Improves contrastwithout changing hueor saturation
• Based on human visionsystem model
– Bad news:• Can be applied only to
color (3-band) images• Based on human vision
system model
Intensity Enhanced CST
Do CST
Stretch intensity component as desired
Inverse CST
Clip to [0,255]
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Color Contrast Enhancement
TM bands 3,2,1 TM bands 7,5,4
min-max stretch
normalization stretch
decorrelation stretch
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Examples (cont.)
original
histogram equalization stretch
min-max stretch
normalization stretch
HSI intensity stretch
decorrelation stretch
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3-D Scatterplots
original min-max
histogram equalized normalized
decorrelated HSI