ch5 spectral transforms pt 2 - engineeringmanian/ch5_part2.pdf · standardized pc (spc) • the...

InelInel 60076007Introduction to Remote SensingIntroduction to Remote Sensing

Chapter 5Chapter 5Chapter 5Chapter 5Spectral Transforms Spectral Transforms ––PCT, PCT,

t t h tt t h tcontrast enhancementcontrast enhancementProf. Vidya Manian

D t f El t i l d C tDept. of Electrical and ComptuerEngineering

INEL6007(Spring 2010) Chapter 5 - 1ECE, UPRM

Affine TransformationsAffine Transformations

x bxAz += T

z

K-dimensional p-dimensionalp

INEL6007(Spring 2010) 2ECE, UPRM

AffineAffine Transformations

deal with Rotation and Scaling of the

S t lSpectral Signatures


Linear Projectionj

Band j P*

P

VVTPVV

Band iINEL6007(Spring 2010) 4ECE, UPRM

Projection to a Lower Dimension

PCA

SVDSS


Projection to a Lower Dimension

OIDPP

IDSS


Principal Component TransformationPrincipal Component Transformation

PC Transformation is multivariate statistical analysisPC Transformation is multivariate statistical analysis technique used in remote sensing for:

• Dimensionality reductionDimensionality reduction.

• Denoising

• To visualize data with more than 3 bands.• PCA stretch (more later)

• Also known as the Karhunen-Loeve Transformation (KLT)and the Hotelling transformation


Some basic multivariable statisticsSome basic multivariable statistics

• The first principal component is selected by p p p yfinding a linear projection that maximizes variance T bz += xa

• The variance of z1 is given by111 bz += xa

• a is determined by solving the optimization

( ) 1XT11zvar aCa=

• a1 is determined by solving the optimization problem

1subject tomaxarg TT qqqCqa 1subject to ,max arg X1 == qqqCqaq


Digression: Eigenvalues & EiEigenvectors

λ= vAv[ ] 0λ

λ

i

i

=−=

i

ii

vIAvAv

[ ]i i

where λi is the i-th eigenvalue and vi is the corresponding

[ ] 0λdet =IAFi di th Ei l

i-th eigenvector

[ ] 0λdet i =− IAFinding the Eigenvalues

Finding the Eigenvectors [ ] 0λ =− vIAby solving [ ] 0λi =− ivIAINEL6007(Spring 2010) 9ECE, UPRM

Basic multivariable statistics (cont )Basic multivariable statistics (cont.)• It can be shown that the optimal solution for the first PC

isa1=v1

the first eigenvector of the covariance matrix CX and that var(z1)=λ1= λmax( 1) 1 max

its maximum eigenvalue.• The remaining principal components zi for i=2,3,...,K are

given byg yzi=vi

Tx+biwith var(zi)=λi where λi and vi are the i-th eigenvalue and eigenvector respectively.eigenvector respectively.

• Notice that eigenvalues are ordered in descending order: λ1 ≥ λ2≥…≥ λK

and the eigenvectors are orthonormal i eand the eigenvectors are orthonormal, i.e. vi

Tvi=1 and viTvj=0INEL6007(Spring 2010) 10ECE, UPRM

Principal Components TranformPrincipal Components Tranform

z=VTx+b

where V=[v v v ] is the matrix ofwhere V=[v1, v2, … , vn] is the matrix of covariance eigenvectors as defined previouslypreviously

• The bias term can be selected as b=0 or b=-VTμx


Pictorial RepresentationpBand 2

v 1v 1v 2

μ2

Band 1μ1INEL6007(Spring 2010) 12ECE, UPRM

F.L.Scarpace, Spring 2005

PCA Transform Summary

• Data are transformed along orthogonal g gaxes which are dependent on the variance within the image.g

• The first Principal Component is along the axis of maximum variance.the axis of maximum variance.

• Each succeeding axis has less variancevariance.


P i i l C t P tiPrincipal Components Properties

z=VTx+b

C =VTC V=diag{λ λ }n,1,2,i L== xvT

iizCZ=VTCXV=diag{λ1,…, λn}

• PCs are uncorrelated• PCs are uncorrelatedcov(zi,zj)=[CZ]ij=0, for i≠j


Simple MATLAB AlgorithmSimple MATLAB Algorithm

• X = image matrix with one column per bandX image matrix with one column per band• C=cov(X);• [U S V]=svd(C);[U,S,V]=svd(C);

– For covariance matrices SVD = Eigenvalue Eigenvector decomposition

– SVD produces singular values in descending order– V is the transformation matrix

• PCA = X*U =X*VT


PC TransformationPC Transformation

• For most remote sensing data sets theFor most remote sensing data sets, the first few PC components contain > 98% of the variancethe variance.

∑=×= n

p

1iiλ

100ty(p)variabili%∑

=

n

1iiλ

y(p)

• Most of the time the higher PCs (in number) contain only noise.) y


PCT Example (cont )PCT Example (cont.)• Step 1. Find eigenvaluesp g

– Solve the characteristic equation (two solutions)

– Therefore

– Note that PC1 contains 89% of the total data variance, and PC2 contains 11% i econtains 11%, i.e.


PCT Example (cont )PCT Example (cont.)

• Step 2. Find eigenvectorsStep 2. Find eigenvectors– Substitute eigenvalues into [λI-CX]v=0

• First eigenvalue yields dependent equationsFirst eigenvalue yields dependent equations

• Solving either equation e11=1.43v21

• PCT requires orthonormality of the eigenvectors• PCT requires orthonormality of the eigenvectors ei

Tei=1


Principal Component 2-band Example

[CX = 5 33 4 ]μX= 0

0[ ]4

5

2

3

0

1

-2

-1

-5 -4 -3 -2 -1 0 1 2 3 4 5-4

-3


Principal Component ExampleEigenvector of the largest eigenvalue

CX = 5 3]μX= 0[ ] [5

Eigenvalues =

3 4]μX0[ ] [

2

3

4Eigenvalues

5.8498 00 1 2879[ ]

The first PC contains

% var = 5 8498/(5 8498+1 2879)*100

0

1

20 1.2879[ ] % var = 5.8498/(5.8498+1.2879)*100= 82% of the total variability

-2

-1Eigenvectors =

0 7612 0 6486[ ]-5 -4 -3 -2 -1 0 1 2 3 4 5

-4

-3-0.7612 0.6486-0.6486 -0.7612[ ]INEL6007(Spring 2010) 21ECE, UPRM

Linear Projectionjx2

x*

x

vixvTv ixviiv

x2INEL6007(Spring 2010) 22ECE, UPRM

Histogram of the data projected in the principal component

18

20

12

14

16

z = v T x

8

10

12z1 v1 x

2

4

6

-6 -4 -2 0 2 4 60

2


2nd Example2nd Example

10

15

5

0

-5

-4 -2 0 2 4 6 8-10


2nd Example: First Principal C t

First Principal component

Component

10

15p p

5

0

-5

-4 -2 0 2 4 6 8-10


Histogram of the data projected in the principal componentin the principal component

50

60Projection on the 1st principal component

40

z = v T x

20

30

z1 v1 x

10

20

-10 -5 0 5 10 150


2nd Example: Second C t

2 d i i l t

Component

10

152nd principal component

5

10

0

-5

-4 -2 0 2 4 6 8 10-10


Histogram of the data projected in the second component

Projection on the 2nd principal component

in the second component

35

40Projection on the 2nd principal component

25

30

z2 = v2T x

15

20

z2 v2 x

5

10

-4 -2 0 2 4 6 80


LANDSAT TM Simulator PCA Example

Añasco, PR


1 2

Band 1 Band 23 4

Band 3 Band 4INEL6007(Spring 2010) 30ECE, UPRM

5 6

7

Band 5 Band 6

Band 7INEL6007(Spring 2010) 31ECE, UPRM

Covariance MatrixCovariance Matrix


PCA Variances and Transformation Matrix


Cummulative VariancesCummulative Variances

P C V a r i a n c e s f o r L a n d s a t i m a g e .S i n g u l a r

V a l u eM a g n i t u d e % V a r . C u m u l a t i v e

V a r i a b i l i t yy

λ 1 6 . 1 7 3 1 × 1 0 3 8 1 . 2 9 4 9 8 1 . 2 9 4 9

λ 2 1 . 2 5 1 0 × 1 0 3 1 6 . 4 7 4 0 9 7 . 7 6 8 9

λ 3 0 . 1 3 2 5 × 1 0 3 1 . 7 4 4 9 9 9 . 5 1 3 8

λ 4 0 . 0 3 0 0 × 1 0 3 0 . 3 9 5 4 9 9 . 9 0 9 3

λ 5 0 . 0 0 4 3 × 1 0 3 0 . 0 5 7 3 9 9 . 9 6 6 5

λ 6 0 . 0 0 1 5 × 1 0 3 0 . 0 1 9 5 9 9 . 9 8 6 0

λ 7 0 . 0 0 1 1 × 1 0 3 0 . 0 1 4 0 1 0 0 . 0 0 0 0INEL6007(Spring 2010) 34ECE, UPRM

PC1 PC2 PC3

PC4 PC5 PC6

PC7

7 principal components for the LANDSAT TM Simulator Imageover Añasco


PCT BenefitsPCT Benefits• Why use the PCT?

– Decorrelates spectral data• Multispectral bands are often

highly-correlated because oft i l t l l ti– material spectral correlation

– topography– sensor band overlap

• Decorrelation separates• Decorrelation separates “independent” components into separate “bands”– Compresses the variancep

λ1 ≥ λ2≥…≥ λK– Compression can potentially

reduce data computation b dburden


PCT DecorrelationPCT Decorrelation

• Non-vegetated sceneNon vegetated scene– PCT removes spectral redundancy


PCT Contrast ExtractionPCT Contrast Extraction• Vegetated sceneg

– PCT extracts contrast between bands 3 and 4 (red and NIR) due to the vegetation “red edge”


PCT Noise Detection• PCT can isolate spectrally-uncorrelated noise


PC Dimensionality ReductionPC Dimensionality Reduction• Choose p by selecting the number of PCs that

l i th i bilit b th h ld Texplain the variability above a threshold T.λ

100( )i bili%

p

1ii∑

Tλ

100ty(p) variabili% n

1ii

1i ≥×=

∑=

=

• T is usually selected above 95%

• Most of the time the higher PCs – p+1, p+2, p+3, …, n

contain mostly “noise.”INEL6007(Spring 2010) 40ECE, UPRM

PC Dimensionality Reduction (cont.)PC Dimensionality Reduction (cont.)

• Set V =[v1 v2 v ]=V(: 1:p) the matrixSet Vp=[v1, v2, …, vp]=V(:,1:p) the matrix

vT11 xz

⎟⎟⎞

⎜⎜⎛

⎟⎞

⎜⎛

xVv

z Tp

T

T22

p

xz=

⎟⎟⎟⎟⎟

⎜⎜⎜⎜⎜

=

⎟⎟⎟⎟⎟

⎜⎜⎜⎜⎜

=MM

• In MATLAB notation Zp=X*V(: 1:p)T

vTpp xz ⎟

⎠⎜⎝

⎟⎠

⎜⎝

In MATLAB notation Zp X V(:,1:p)• Need to use reshape to get back the

imagesimages.INEL6007(Spring 2010) 41ECE, UPRM

PC Dimensionality ReductionPC Dimensionality Reduction

• Many applications use the extracted PCAMany applications use the extracted PCA, z∈Rp, features instead of the original spectral signature x∈Rn with n>>pspectral signature, x∈R with n>>p


PCA for signal denoisingPCA for signal denoising• PCA Summaryy

– Original PCT

VTz=VTx,– Inverse PCT

x=Vzx=Vz– “Signal Components”– “Noise Components”– Noise Components

• Zp+1, zp+2, … , zn


PCA for signal denoising (cont )PCA for signal denoising (cont.)• Typical denoising involves computation of the PC and yp g p

neglection of the higher order PCs (associated with noise) and reconstruction of the image.– Dimensionality reductiony

zp=VpTx

– Filtered reconstruction

xp=Vpzpwhere xp is the filtered version of x, Vp=[v1,v2, …,vp] is a matrix formed by the first p eigenvectors of the covariance matrix andformed by the first p eigenvectors of the covariance matrix, and zp=[z1, z2, …,zp]T are the first p PCs.

• Optimal from a reconstruction point of viewMean squared error optimality– Mean squared error optimality


Denoising example (T=98%)Denoising example (T 98%)

Measured Hyperion Band 1 Filtered Hyperion Band 1


Standardized PC (SPC)• The standardized principal components ri for i=1,2,3,...,n are given

by

μx uqTii

i

iii r

σμxu =

−=

where qi is the i-th eigenvector of the correlation matrix RX and ui are the normalized random variables with unit variance and zero mean.

• Notice that correlation eigenvalues are ordered in descending order: ρ1 ≥ ρ2≥…≥ ρn

and eigenvectors are also orthonormal a d e ge ecto s a e a so o t o o aqi

Tqi=1 and qiTqj=0

• Notice that var(rk) = ρk

• Particularly useful with data of differing scales• Particularly useful with data of differing scales.


Why not usePCT?Why not usePCT?• It is data-dependentp• V coefficients change from scene-to-scene. V is

computed from the covariance of the data CX. • Makes consistent interpretation of PC images difficult• Makes consistent interpretation of PC images difficult

– Spectral details, particularly in small areas, may be lost if higher-order PCs are ignored

• Computationally expensive for large images or for many• Computationally expensive for large images or for many spectral bands– Calculation of covariance matrix is the culprit

Alt t th d i SVD b t till VERY t ti ll• Alternate method using SVD but still VERY computationally expensive

• Not necessarily optimal from a classification point of viewCl t il t d– Classes are not necessarily separated


Review questionsReview questions

• What is a principal componentWhat is a principal component transformation. What does it achieve with respect to the RS datarespect to the RS data.

• What are the advantages and disadvantages of the methoddisadvantages of the method.

• How is the PCT calculated. • Ex 5-1, 5-2, 5-3, 5-4.

INEL6007(Spring 2010) ECE, UPRM 50

Tasseled-Cap ComponentsTasseled Cap Components


Different ProjectionsDifferent Projections


Tasseled-Cap ComponentsTasseled Cap Components

• Linear spectralLinear spectral transform like the PCT

w=ATx• The TCT matrix is• The TCT matrix is

fixed for a given sensor

TC transformation Matrix for Different SensorsINEL6007(Spring 2010) 53ECE, UPRM

TCT BenefitsTCT Benefits• Why use the TCT?y

– It is a fixed reference• Same reference for every

image from a givenimage from a given sensor permits consistent interpretation– Components are related to– Components are related to

geophysical properties of the scene

• First component is “soilFirst component is soil brightness”

• Second component is “greeness”greeness


ExampleExample


TCT DrawbacksTCT Drawbacks• Why not use the TCT?

– Nonoptimal compression of datap p– Derivation of WTC requires multitemporal

data for each sensor


Contrast EnhancementContrast Enhancement


Problems in Visual Analysis of M l i l IMultispectral Imagery

• Most images do not fill the dynamic range of the sensorg y g• Most images also do not fill the dynamic range of the

display system


Contrast EnhancementContrast Enhancement• Contrast enhancement means “stretching” the g

data range to fill the display system range• Contrast enhancement is a mapping from the

i i l DN d l l (GL)original DN data space to a gray level (GL) display space

GL = T(DN)GL = T(DN)– Examples:

• Linear Stretch • Histogram Equalization Stretch

• Parameters of transformation T based on global or local image statistics (histogram)or local image statistics (histogram)


Contrast StretchesContrast StretchesExample of a Linear Stretch FunctionExample of a Linear Stretch Function

255255tp

ut D

N

114

Out

00

25550 15095Input DN


Histogram for Linear StretchHistogram for Linear StretchFirst band (Green) for sample SPOT imageFirst band (Green) for sample SPOT image


Raw Unstretched DataRaw Unstretched DataFirst band (Green) for sample SPOT imageFirst band (Green) for sample SPOT image


Linearly Stretched DataLinearly Stretched DataFirst band (Green) for sample SPOT imageFirst band (Green) for sample SPOT image


Different Mappings can be Defined for Contrast Enhancement


Histogram EqualizationHistogram Equalization

• One attempts to change the histogramOne attempts to change the histogram through the use of a function GL = T(DN) into a histogram that is constant for allinto a histogram that is constant for all brightness values.

• This would correspond to a brightness• This would correspond to a brightness distribution where all values are equally likelylikely.


Histogram Equalization StretchHistogram Equalization StretchHistogram Equalization StretchHistogram Equalization Stretch• Histogram D

N

255g

equalization is achieved by using the Cumulative O

utpu

t D

LUT

the Cumulative Density function of the image as a

fO

0transformation function, after appropriate scaling

0255

pp p gof the ordinate axis.

0

Input DN0 255


Histogram Equalization StretchHistogram Equalization StretchThe most numerous pixels have the greatest contrast


Histogram Equalization StretchHistogram Equalization StretchggFirst band (Green) for sample SPOT imageFirst band (Green) for sample SPOT image


Comparizon Between Histogram Comparizon Between Histogram Equalization MethodsEqualization Methods

Linear Histogram EqualizationStretch

g qStretch


Contrast Enhancement MapsContrast Enhancement Maps


Contrast Stretch ExamplesApplication to GOES Data


Normalization StretchNormalization Stretch• Linear scaling of DN mean and sigma to specified values, followed by

saturation (clipped at the ends)GL = aDN+b,

μGL=aμDN+b, σ2GL= a2σ2

DN

• Consistent behavior (robust) over wide range of images


Reference Stretch• Generalization of the Normalization Stretch

– Not only the second order statistics but the whole pdf (or CDF)M t h th CDF f th i b i d t f CDF• Match the CDF of the image being processed to a reference CDF, for example from another image, and then backward mapping– useful for

ltit l lti di t hi• multitemporal or multisensor radiance matching• matching image to reference contrast

• Gaussian Normalization is a common case


Multitemporal NormalizationMultitemporal Normalization


ThresholdingThresholding

• Binary “clipping” of DNs to low and high valuesBinary clipping of DNs to low and high values– Useful for

• segmentation of certain images, e.g. clouds/water, land/water


Color ImagesColor Images• Techniques used for single-band imagery can q g g y

be extended to color, but . . .– Sensitivity of the human vision system to shifts in

color and saturation require special attentionq p• Min-max stretch

– Stretch the DNs in each band over their respective minmax rangeminmax range

– Good news:• Easy to calculate and implement• No data lost by saturation• No data lost by saturation

– Bad news:• Sensitive to outlier DNs• Color balance can change unpredictably• Color balance can change unpredictably


Linearly Stretched DataLinearly Stretched DatayyThree band CIR combination: Three band CIR combination: Band 1 (spectral green) displayed as blueBand 1 (spectral green) displayed as blueB d 2 ( t l d) di l dB d 2 ( t l d) di l dBand 2 (spectral red) displayed as greenBand 2 (spectral red) displayed as greenBand 3 (spectral NIR) displayed as redBand 3 (spectral NIR) displayed as red

OriginalOriginal Linear StretchLinear StretchINEL6007(Spring 2010) 77ECE, UPRM

Histogram Equalization StretchHistogram Equalization StretchggThree band combination: Three band combination: Band 1 (spectral green) displayed as blueBand 2 (spectral red) displayed as greenBand 3 (spectral NIR) displayed as red

UnstretchedUnstretched Linear StretchLinear StretchINEL6007(Spring 2010) 78ECE, UPRM

Comparizon Between Histogram Comparizon Between Histogram Equalization MethodsEqualization Methods

Linear Stretch Histogram EqualizationLinear Stretch Histogram Equalization Stretch


Color NormalizationColor Normalization

• Normalization stretchNormalization stretch– “Standardized” stretch– Good news:

• Average color is grey• Contrast controlled by

single parameter, thesingle parameter, the desired output standard deviation

– Bad news:– Bad news:• Some data are lost in

saturation


Color DecorrelationColor Decorrelation• Decorrelation stretch

– Enhance small spectral deviations in highly correlated spectral bands

– Commonly used inCommonly used in geology

– Good news:• Decorrelates bands

E h i– Emphasizes differences among bands

– Can be applied to any number of bands

– Bad news:• Produces highly

saturated colors


Color Enhanced TM I fTM Images of Cuprite Mining District in NV


RGB Color Coordinate SystemRGB Color Coordinate System


Intensity Hue Saturation Color Coordinate System


IHS Color Coordinate SystemIHS Color Coordinate System• The vertical axis represents intensity (I) which variesp y ( )

from black (0) to white (255)• The circumference of the sphere represents hue (H),

hi h i th d i t l th f lwhich is the dominant wavelength of color.– 0 at the midpoint of red tones and increase counterclockwise

around the circumference of the sphere to conclude with 255adjacent to 0.

• Saturation (S) represents the purity of the color andranges from 0 at the center of the color sphere to 255 atranges from 0 at the center of the color sphere to 255 atthe circumference.– 0 represents a completely impure color in which all wavelengths

ll t d d hi h th ill iare equally represented and which the eye will perceive as ashade of gray that ranges from white to black depending onintensity.


Transformation EquationsTransformation Equations

GBG

BGRI ++=

3BIBGH −

=

3BI3BI

−−

I3BIS =


Color SpacesColor Spaces• HSI color coordinate system• Hexcone model

– similar to a cylindrical coordinate system, but based on RGB color cube

– value = max(R,G,B) used instead of intensity– efficient CST from RGB to Hue-Saturation-

Value (HSV)


Example Ramp Spectrum CSTsExample Ramp Spectrum CSTs


Color-Space TransformsColor Space Transforms

C l• Color-space transforms

Human vision system– Human vision system perceives hue (H), saturation (S) and intensity (I), not RGB

– Therefore, control over colorover color appearance is best done in HSI space


CST for Contrast EnhancementCST for Contrast Enhancement

• Intensity stretchIntensity stretch– Good news:

• Improves contrast without changing hue or saturation

• Based on human vision system model

– Bad news:• Can be applied only toCan be applied only to

color (3-band) images• Based on human vision

system modelsystem model


Color ContrastColor Contrast Enhancement


Examples (cont )Examples (cont.)original


3-D Scatterplots


Watch OutWatch Out


Watch OutWatch Out

New colors appearNew colors appear


Note the Change inChange in

Colors due to P iProcessing


Spatial Domain BlendingSpatial Domain Blending


Final RemarksFinal Remarks

• Several linear and nonlinear spectral transformsSeveral linear and nonlinear spectral transforms were studied– Qualitative and quantitative analysis– Band ratios Vegetation studies– PCA Data compression, data dependent, no-

if li tiuniform application– Tasseled-Cap Sensor specific

Color composites can be enhanced for visual analysis– Color-composites can be enhanced for visual analysis not for quantitative analysis

• Local transforms are also possiblep


Review questionsWh t i th f t t t t hi• What is the purpose of contrast stretching

• How does contrast stretch applied to PC work?• What is the difference between global and local transforms• What is the difference between global and local transforms• Explain linear stretch, nonlinear stretch, normalization and

reference stretch• Explain histogram equalization• Explain thresholding and for what purpose it is used.• What is the difference between stretching single band and

color images • What are color perceptual spaces• What are color perceptual spaces.• What is the best way to assign HSI for visual interpretation.• Ex 5-6 and 5-7

INEL6007(Spring 2010) ECE, UPRM 105

ch5 spectral transforms pt 2 - engineeringmanian/ch5_part2.pdf · standardized pc (spc) • the...

Documents