gnr401 principles of satellite image processingavikb/gnr401/dip/nr401-nov2012... · gnr401...

GNR401 Principles of Satellite Image

Processing

Instructor: Prof. B. Krishna MohanCSRE, IIT Bombay

[email protected] 5 Guest Lecture PCT and Band Arithmetic

November 07, 2012 9.30 AM – 10.55 AM

Contents of the urePrincipal Component Transform• Alternative Interpretations• Inverse PCTBand Arithmetic• Motivation for band arithmetic• Band ratio• Vegetation indices

IIT Bombay Slide 1

GNR401 B. Krishna Mohan

November 07, 2012 PCT and Band Arithmetic

Multiband Image Operations• Operations performed by combining gray levels

recorded in different bands for the same pixel• Applications

– Data reduction through decorrelation– Highlighting specific features with significant

difference in response in different bands– The transformed data may be viewed like

enhanced versions compared to originals

IIT Bombay Slide 2


Principal Component Transform• Highlights the redundancy in the data sets

due to similar response in some of the wavelengths

• Original bands variables represented along different coordinate axes, redundancy implies variables are correlated, not independent

• Gray level in a band at a pixel can be predicted from the knowledge of the pixel gray level in other bands

IIT Bombay Slide 3


Example of Redundancy in Data• Example: Highly

correlated data• Values along band b1

leads to knowledge along band b2 of the data element

• Linear variation (nearly) between b1 and b2

• Often true in case of visible bands

b1

b2

IIT Bombay Slide 4


Example of Redundancy in Data• Points projected

onto the line a small error in the position of the point.

• Points represented by only one coordinate b1’ half data reduced

• For highly correlated data, this error will be minimal b1

b2

b1’

b2’

IIT Bombay Slide 5


Decorrelating Multispectral Remotely Sensed Data

• How do we identify the optimum axes along which the remotely sensed data should be projected so that the transformed data would be uncorrelated?

• What should be the way to rank the new axesso that we can discard the least important dimensions of the transformed data?

• Invertibility of the transformation?

IIT Bombay Slide 6


Useful band statisticsIIT Bombay Slide 7


1 1

.

M Nki j

i jg

M N

2

1 1( )

.

M Nkij k

i jg

M N

1 1( )( )

.

M Nk lij k ij l

i jg g

M N

Mean

Variance

Covariance

Covariance Matrix• C = {Ckl | k = 1, …, K, l = 1, …, K}• K is the number of bands in which the

multispectral dataset was generated• C is a symmetric matrix• Ckl = Clk• Diagonal elements of C are the intra-band

variances• Off-diagonal elements are the inter-band

covariances

IIT Bombay Slide 8


Relation between correlation and covariance

• Correlation Rkl =

• It can be shown that Rkl = Ckl + mkml

• For data with zero-mean, correlation and co-variance will be equal

IIT Bombay Slide 9


1 1

.

M Nk l

i j i ji j

g g

M N

Principal Component Transformation

Problem to solve:• Find a transformation to be applied to the input

multispectral image such that the covariance matrix of the result is reduced to a diagonal matrix

• Further, we should find an axis vk such that the variance of the projected coordinates (zk = vk

t x) is maximum.

IIT Bombay Slide 10


SolutionGiven the transformed vector

zk = vkt x

The variance sz2 =

This simplifies to sz2 = vtCv (Dropping subscript k

for a moment!)

C, the covariance matrix is a positive, semi-definite, real symmetric matrix.

IIT Bombay Slide 11


1 1( )( )

.

M Nt t

ij k ij li j

v x x v

M N

Finding vector v• To maximize the projected variance sz

2, find a vsuch that vtCv is maximum, subject to the constraint vtv = 1. Combining the maximization function with the constraint, we can write

• vtCv – l(vtv – 1) = maximum• Differentiating w.r.t. v,

IIT Bombay Slide 12


( 1) 0t tCv

v v v v

Finding vThe derivative results in

Cv = lv (Verify!)Therefore, v is an eigenvector of CGiven that sz

2 = vtCvvtCv = vt(lv) = lvtv = l = sz

2

This implies that v is the eigenvector of C with the largest eigenvalue

Therefore all the eigenvectors with decreasing eigenvalues lead to axes with decreasing variance along them.

IIT Bombay Slide 13


Alternative Explanation to PCTLet the transformed pixel vector y = DtxCovariance matrix of y = Sy = DtSxD(Note that Sy = E{(y – my)(y-my)t}

= E{(Dtx – Dtmx)(Dtx – Dtmx)t}This simplifies to Sy = DtE(x – mx)(x – mx)tDD is a set of vectors independent of x)

IIT Bombay Slide 14


Alternative ExplanationCovariance matrix of y = Sy = DtSxDIt is desired that Sy be diagonal, i.e., the

data in the transformed domain is uncorrelated

Let Sy =

IIT Bombay Slide 15


1

2

0 ... 00 ... 0 ... 0 0 ... n

Alternative ExplanationLet Sy =

IIT Bombay Slide 16


1

2

0 ... 00 ... 0 ... 0 0 ... n

Then Sy = DtSxD is a similarity transformation with D containing eigenvectors of Sx

We can order li in such a way that they are in descending order. Given that y = Dtx, y1 corresponds to direction given by e1, that is the first row of Dt, …

Each transformed pixel vector y is obtained from scalar products of eigenvectors of Sx and x

Sample Eigenvectors and EigenvaluesIIT Bombay Slide 17


34.89 55.62 52.87 22.71 55.62 105.95 99.58 43.3352.87 99.58 104.02 45.8022.71 43.33 45.80 21.35

Covariance Matrix

0.34 −0.61 0.71 −0.060.64 −0.40 −0.65 −0.060.63 0.57 0.22 0.480.28 0.38 0.11 −0.88

Eigenvalues 253.44 7.91 3.96 0.89

Eigenvectors

Sample Eigenvectors and EigenvaluesIIT Bombay Slide 18


TransformationIIT Bombay Slide 19


New component value = dot product of eigenvector and pixel vector

(i,j) pixel position

n eigenvectors for n principal components

1st principal component dot product of pixel vector with eigenvector corresponding to largest eigenvalue

Principal ComponentsFor n input bands, n principal components are

computedThe utility of the principal components

gradually decreases from 1st towards the laste.g., For Landsat TM, last three PCs are

generally of very little value

IIT Bombay Slide 20


Visualization of PCTIIT Bombay Slide 21


From J.R. Jensen’s ure notes at Univ. South Carolina; used with permission

Comments on PCTIIT Bombay Slide 22


• For IRS / IKONOS images, out of four bands, 2-3 principal components capture most of the useful information. The last 1-2 bands are redundant.

• Advantages– Smaller data volume to handle– Principal components appear to be enhanced

versions of the originals, having contributions from all the four input bands

• Application scientists use composites of PC 1-2-3 for interpretation of various features such as geology

Band 1 (Blue)

IIT Bombay Slide 23


Band 2 (Green)

IIT Bombay Slide 24


Band 3 (Red)

IIT Bombay Slide 25


Band 4 (NIR)

IIT Bombay Slide 26


Band 5 (SWIR)

IIT Bombay Slide 27


Band 7 (SWIR)

IIT Bombay Slide 28


PC1

IIT Bombay Slide 29


PC2

IIT Bombay Slide 30


PC3IIT Bombay Slide 31


PC6

IIT Bombay Slide 32


Input Image FC

CIIT Bombay Slide 33


Decorrelation Stretch

IIT Bombay Slide 34


Inverse PCTIIT Bombay Slide 35


• Inverse PCT is used to generate the bands in the original domain

• If ALL PCTs are retained, inverse will give back the original bands

• If any PCTs are dropped, inverse will give new bands in the original domain that may be close to the original bands depending on how many PCTs are discarded



From the principle of PCT, we havey = Dtx

Dt contains eigenvectors of Sx, covariance matrix from the original image

Since Dt is an orthonormal matrix, (Dt)t = (Dt)-1

From each pixel vector in PC domain, x = (Dt)t y



For k band image, matrix D is square, of size k x kIf m principal components are dropped, we are left

with a matrix (D1) of size k x (k-m)The vector y is reduced to y1 of size k-m x 1Therefore the modified vector x1 is given by

x1 = D1y1The difference between x and x1 is a measure of

the loss of information due to removal of some of the PCs

Comments on PCTIIT Bombay Slide 38


• One of the other important applications of PCT is data fusion

• Replace first PC by the image from another sensor

• Apply inverse PC

Band Arithmetic

MotivationIIT Bombay Slide 39


Multiband ArithmeticIIT Bombay Slide 40


• In a given pair of bands the response of two objects is generally different.

• Pixel by pixel comparison between images can highlight pixels that have very high difference in ref ance in those bands

• Operations like band difference and band ratio or combinations of them are popularly used for this purpose

Band Ratio• Very common operation

Ratioi,j(m,n) = Bandi (m,n) / Bandj(m,n)

If Bandj(m,n) = 0, suitable adjustment has to be made (e.g., add +1 to the denom.)

Minimum ratio will be 0; Maximum ratio will be 255

IIT Bombay Slide 41


Input Image

IIT Bombay Slide 42


Input Image FC

CIIT Bombay Slide 43


IR/R

IIT Bombay Slide 44


Band Ratio• For fast computing, approximations can be

made such as:

0 ≤ Ratioi,j(m,n) ≤1, Ratioi,j(m,n)scaled =Round [Ratioi,j(m,n)x127]

1 < Ratioi,j(m,n) ≤ 255, Ratioi,j(m,n)scaled =Round [127 + Ratioi,j(m,n)/2]

• Advantage – in one pass image is generated in range 0-255

IIT Bombay Slide 45


Band Difference• Similar to band ratio, band difference can

also be used to account for difference in ref ance by objects in two wavelengths

• Band ratio - more popular in practical applications such as geological mapping

• Topographic effects on the images are reduced by ratioing.

IIT Bombay Slide 46


Band Multiplication• Pixel by pixel multiplication of two images• Not used to multiply gray levels in one

band with corresponding gray levels in another band

• Used in practice to mask some part of the image and retain the rest of it by preparing a mask image and performing image to image multiplication of pixels

IIT Bombay Slide 47


Band Addition• Similar to Band Multiplication, band addition has

no direct practical application in adding gray levels of two bands of an image

• This method too can be used to mask a portion of the image and retain the remaining part.

IIT Bombay Slide 48


Specialized Indices• Combination of band differences, ratios

and additions can result in useful outputs that can highlight features like green vegetation

• One such feature is Normalized Difference Vegetation Index (NDVI)

• NDVI(m,n) =

IIT Bombay Slide 49


( , ) ( , )( , ) ( , )

IR R

IR R

Band m n Band m nBand m n Band m n

NDVI• NDVI results in high values where IR dominates

red wavelength. This happens where vegetation is present

• Range of NDVI is [-1 +1]• NDVI has been widely used in a wide ranging of

agricultural, forestry and biomass estimation applications

• It is also used to measure the length of crop growth and dry-down periods by comparing NDVI computed from multidate images

IIT Bombay Slide 50


Input Image

IIT Bombay Slide 51


NIRIIT Bombay Slide 52


REDIIT Bombay Slide 53


NDVIIIT Bombay Slide 54


Other Vegetation Indices• Simple Ratio = Red/NIR• NDVI6 = (Band 6 – Band 5)/(Band 6 + Band 5)• NDVI7 = (Band 7 – Band 5)/(Band 7 + Band 5)• Standard NDVITM = (TM4 – TM3)/(TM4 + TM3)These are applicable when seven band data like Landsat

Thematic Mapper data are availableFor IRS LISS3 imagery, NDVIIRS =

IIT Bombay Slide 55


4 3

4 3

( , ) ( , )( , ) ( , )

Band m n Band m nBand m n Band m n

IRS L4-NDVI

IIT Bombay Slide 56


Fast Computation of NDVI• Range of NDVI [-1, +1]• Scale suitably to generate an NDVI image

• For example, NDVIscaled =127(1+NDVI)

• This ensures that the resultant NDVI has a range of [0 254]

IIT Bombay Slide 57


Se ed Ref ance CurvesIIT Bombay Slide 58


From J.R. Jensen’s ure notes at Univ. South CarolinaUsed with permission

Time Series of 1984 and 1988 NDVI Measurements Derived from AVHRR Global Area Coverage (GAC) Data Region around El Obeid, Sudan, in Sub-Saharan Africa

IIT Bombay Slide 59



Simple Ratio v/s NDVI

IIT Bombay Slide 60



Infrared Index• Traditional NDVI does not work very well when

the soil is moist, as in case of wetlands. The Infrared Index (II) can tackle this situation better

• Several bands needed in the infrared region, as in case of Landsat TM

IIT Bombay Slide 61


4 5

4 5

TM TM

TM TM

NIR MIRIINIR MIR

Soil LineIIT Bombay Slide 62



Perpendicular Vegetation IndexPVI is defined as

IIT Bombay Slide 63


2 2, , , ,S R V R S NIR V NIRPVI

A vegetation index that assumes that the ref ance in the NIR and red varies with increasing vegetation density (such as leaf area index) and that these variations are parallel to the soil baseline. Therefore, the perpendicular distance from the baseline in a NIR-red plot determines the vegetation density.See http://www.ccrs.nrcan.gc.ca/glossary/index_e.php?id=2179 for more definitions of various indices in remote sensing including PVI

Soil Adjusted Vegetation IndexThe soil adjusted vegetation index (SAVI) introduces a soil

calibration factor, L, to the NDVI equation to minimize soil background influences resulting from first order soil-plant spectral interactions:

IIT Bombay Slide 64


(1 )L NIR redSAVI

NIR red L

Ref:A. R. Huete, “A soil-adjusted vegetation index (SAVI),” Rem. Sens. Env., vol. 25, pp. 295-309, 1988.

Atmospherically Adjusted Vegetation Index (ARVI)

• The atmospheric effects are accounted for in ARVI

IIT Bombay Slide 65


* ** *

p nir p rbARVIp nir p rb

* * * *p rb p red p blue p red p* indicates the atmospherically corrected versions of NIR, Red and Blue bands for molecular scattering and

ozone absorption(Ref. J.R. Jensen’s notes)

Enhanced Vegetation IndexEVI is a mixture of SAVI and ARVI, in that both atmospheric effects and

soil effects are accounted for.

IIT Bombay Slide 66


1 2

* ** * *

p nir p redEVIp nir C p red C p blue L

C1C1 and and C2C2 describe the use of the blue band in correction of the red band describe the use of the blue band in correction of the red band

for atmospheric aerosol scattering. The coefficients, for atmospheric aerosol scattering. The coefficients, C1C1, , C2C2, and , and LL are are empirically determined as 6.0, 7.5, and 1.0, respectively for MODIS. This empirically determined as 6.0, 7.5, and 1.0, respectively for MODIS. This

algorithm has improved sensitivity to high biomass regions and improved algorithm has improved sensitivity to high biomass regions and improved vegetation monitoring through a devegetation monitoring through a de--coupling of the canopy background coupling of the canopy background

signal and a reduction in atmospheric influencessignal and a reduction in atmospheric influencesSource: Yoram J. Kaufman and Didier Tanre, Atmospherically Resistant Vegetation Index (ARVI) for EOS-MODIS, IEEE Trans. GERS, vol. 30, no. 2, pp. 261-270, 1992

IIT Bombay Slide 67


From J.R. Jensen’s ure notes at Univ. South Carolina; used with permission

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