Inel Inel 6007 6007 Introduction to Remote Sensing Introduction to Remote Sensing Chapter 5 Chapter 5 Chapter 5 Chapter 5 Spectral Transforms Spectral Transforms –PCT, PCT, t t h t t t h t contrast enhancement contrast enhancement Prof. Vidya Manian D t f El ti l d C t Dept. of Electrical and Comptuer Engineering INEL6007(Spring 2010) Chapter 5 - 1 ECE, UPRM
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CH5 Spectral Transforms pt 2 - Engineeringmanian/CH5_part2.pdf · Standardized PC (SPC) • The standardized principal components r i for i=1,2,3,...,n are given by x μ qTu i i i
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InelInel 60076007Introduction to Remote SensingIntroduction to Remote Sensing
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
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PCA for signal denoisingPCA for signal denoising• PCA Summaryy
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
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Denoising example (T=98%)Denoising example (T 98%)
Measured Hyperion Band 1 Filtered Hyperion Band 1
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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.
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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
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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.
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Tasseled-Cap ComponentsTasseled Cap Components
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Different ProjectionsDifferent Projections
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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
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ExampleExample
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TCT DrawbacksTCT Drawbacks• Why not use the TCT?
– Nonoptimal compression of datap p– Derivation of WTC requires multitemporal
data for each sensor
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Contrast EnhancementContrast Enhancement
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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
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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)
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Contrast StretchesContrast StretchesExample of a Linear Stretch FunctionExample of a Linear Stretch Function
255255tp
ut D
N
114
Out
00
25550 15095Input DN
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Histogram for Linear StretchHistogram for Linear StretchFirst band (Green) for sample SPOT imageFirst band (Green) for sample SPOT image
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Raw Unstretched DataRaw Unstretched DataFirst band (Green) for sample SPOT imageFirst band (Green) for sample SPOT image
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Linearly Stretched DataLinearly Stretched DataFirst band (Green) for sample SPOT imageFirst band (Green) for sample SPOT image
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Different Mappings can be Defined for Contrast Enhancement
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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.
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
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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
• 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
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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
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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
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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
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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
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Color Enhanced TM I fTM Images of Cuprite Mining District in NV
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RGB Color Coordinate SystemRGB Color Coordinate System
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Intensity Hue Saturation Color Coordinate System
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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.
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Transformation EquationsTransformation Equations
GBG
BGRI ++=
3BIBGH −
=
3BI3BI
−−
I3BIS =
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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)
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Example Ramp Spectrum CSTsExample Ramp Spectrum CSTs
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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
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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
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Color ContrastColor Contrast Enhancement
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Examples (cont )Examples (cont.)original
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3-D Scatterplots
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Watch OutWatch Out
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Watch OutWatch Out
New colors appearNew colors appear
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Note the Change inChange in
Colors due to P iProcessing
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Spatial Domain BlendingSpatial Domain Blending
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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
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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