[PPT] - Nonparametric Regression Splines for Nonparametric Regression PowerPoint Presentation

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Nonparametric Regression Splines for Nonparametric Regression Splines for Regional Atmospheric Correction Regional Atmospheric Correction

Semih Kuter a Gerhard-Wilhelm Weber b Zuhal Akyürek c Aye Özmen b

a Çankırı Karatekin University, Faculty of Forestry, Department of Forest Engineering, 18200, Çankırı, Turkey b Middle East Technical University, Institute of Applied Mathematics, 06800, Ankara, Turkey c Middle East Technical University, Faculty of Engineering, Department of Civil Engineering, 06800, Ankara, Turkey

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Outline Outline Outline Outline

Introduction

Atmospheric correction, Why?
Atmospheric correction, How?
MARS & CMARS

Study Area, Data Set & Methodology

MARS & CMARS on MODIS Images

Results & Concluding Remarks

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Atmospheric correction, Why? Atmospheric correction, Why?

RS of Earth by space RS of Earth by space-based sensors based sensors

Measurement of Earth’s surface reflectance from a

space-based Remote Sensing Remote Sensing (RS) platform.

RS of Earth by space RS of Earth by space-based sensors based sensors

space-based Remote Sensing Remote Sensing (RS) platform.

Perturbations are introduced by the passage of

radiation through the Earth’s atmosphere.

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Basic geometry of the problem Basic geometry of the problem

Atmospheric correction, How? Atmospheric correction, How?

Basic geometry of the problem Basic geometry of the problem

Sun Sun Satellite

θs : View zenith angle
θv : Solar zenith angle
φ : Relative azimuth
φ : Relative azimuth

Target pixel

4

pixel

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Atmospheric correction, Why? Atmospheric correction, Why?

Factors affecting EM radiation in the atmosphere Factors affecting EM radiation in the atmosphere

Molecular absorption and scattering

Factors affecting EM radiation in the atmosphere

Factors affecting EM radiation in the atmosphere

N2, O2, O3, CO2, water wapor (RH) etc.

Emission

Aerosols

Emission

Absorption Scattering

Aerosols

Found in the atmospheric boundary layer and transported

by wind turbulence and atmospheric convection.

Larger particles

Fog, cloud, rain and snow.

Ionosphere Ionosphere

Ionization by extreme ultraviolet and X-radiation from the

Sun.

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Atmospheric correction, Why? Atmospheric correction, Why?

What is measured by the satellite? What is measured by the satellite?

RS satellite measures the top

top-of

f-atmospheric

atmospheric (TOA)

What is measured by the satellite? What is measured by the satellite?

reflectance value of the target pixel.

TOA is not the real surface reflectance value of the target

pixel, but a perturbed (or distorted) one by the atmosphere.

Dream

Dream situation situation: What would be the reflectance value measured at the satellite level, if there were no atmosphere between the satellite and the target pixel? Not possible to completely remove the effect

f

the

Not

possible to completely remove the effect

f

the atmosphere.

However, it may be reduced at certain level by using However, it may be reduced at certain level by using

Radiative Radiative Transfer Transfer (RT) models.

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Atmospheric correction, How? Atmospheric correction, How?

Main approaches in atmospheric correction Main approaches in atmospheric correction

Absolute atmospheric correction by RT models

Main approaches in atmospheric correction

Main approaches in atmospheric correction

The whole process of atmospheric attenuation is

numerically modeled.

Top Of Atmospheric
Temperature
Pressure
RH
Top Of Atmospheric

(TOA) Reflectance Radiative Transfer (RT) Model Atmospherically corrected reflectance

f the target pixel

Numerous RT models during the last three decades:

RH
Visibility
Angles etc.
5S

5S – Simulation of a Satellite Signal in the Solar Spectrum

6S

6S – Second Simulation of a Satellite Signal in the Solar Spectrum

MODTRAN

MODTRAN – MOD MODerate Resolution Atmospheric TRAN TRANsmission

FLAASH

FLAASH – Fast Line-of-sight Atmospheric Analysis of Spectral Hypercubes

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Basic RT equation Basic RT equation

Atmospheric correction, How? Atmospheric correction, How?

Basic RT equation Basic RT equation

2

3 2 2

( , , ) ( , , ) ( ) ( ) 2

H O TOA s v g R R A R g

U T O O CO T ρ θ θ ϕ ρ ρ ρ

+

 ∆ = + −   2 

2

( ) 1

s R A g H O s R A

T T U S ρ ρ

+ +

 +  − 

where

ρs: surface reflectance of the target,
ρ

: TOA reflectance at the satellite level,

1

s R A

S ρ

+

− 

ρTOA: TOA reflectance at the satellite level,
θs and θv: view zenith angle (VZA) and solar zenith angle (SZA),
φ: relative azimuth between the Sun and the satellite direction,
Tg: total gaseous transmission,
ρR: molecular scattering reflectance,
ρR+A: reflectance of the molecules and aerosols,

R+A

TR+A: transmission of the molecules and aerosols,
SR+A: spherical albedo.

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Drawbacks of RT models Drawbacks of RT models

Atmospheric correction, How? Atmospheric correction, How?

Drawbacks of RT models Drawbacks of RT models

When working with long time series data acquired by

LFOV LFOV (Large Field Of View) sensors: LFOV LFOV (Large Field Of View) sensors:

Different observational geometry for each pixel,
Atmospheric parameters do not remain constant (or

unknown).

Detailed Detailed RT Codes Codes Operationally Operationally too expensive and too expensive and Processing of a single MSG

MSG-SEVIRI SEVIRI image subset:

Detailed Detailed RT Codes Codes too expensive and too expensive and time consuming time consuming

Spinning Enhanced Visible and InfraRed Imager
Temporal resolution = 15 min.
4.4 million pixels for each of the three visible and NIR
4.4 million pixels for each of the three visible and NIR

channels = 20 min. for 6S

Backlog develops! Backlog develops!

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Atmospheric correction, How? Atmospheric correction, How?

Alternative to RT codes: Alternative to RT codes: SMAC SMAC

In the early 90s by H. Rahman & G. Dedieu,

Alternative to RT codes: Alternative to RT codes: SMAC SMAC

“Simplified Model for Atmospheric Correction” “Simplified Model for Atmospheric Correction”

Several hundred times faster than detailed RT codes, Based on the parametrization of RT equations:

ρ′

( ) ( ) ( )

, ,

s s s v G s v s

T T T S ρ ρ θ θ ϕ θ θ ρ ′ ∆ = ′ +

( ) ( )

ρ ρ θ θ ϕ ρ θ θ ϕ ′ = ∆ − ∆ ,

where

ρ : surface reflectance

( ) ( )

, , , ,

s TOA s v G a s v

T ρ ρ θ θ ϕ ρ θ θ ϕ ′ = ∆ − ∆

ρs: surface reflectance
TG: total gaseous transmission,
T(θs), T(θv): total downward and upward transmittences,
S: albedo of the land surface,
ρa: component of measured reflectance produced by the atmosphere.

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Alternative to RT codes: Alternative to RT codes: SMAC SMAC

Atmospheric correction, How? Atmospheric correction, How?

Alternative to RT codes: Alternative to RT codes: SMAC SMAC

Input variables for SMAC:

TOA reflectance (ρTOA)
Solar zenith angle (SZA)
Viewing zenith angle (VZA)
Solar azimuth angle (SAA)

Computing Computing surface reflectance surface reflectance pixel pixel-by by-pixel pixel

Solar azimuth angle (SAA)
Viewing azimuth angle (VAA)
Atmospheric optical depth at

550 nm (τ550)

Water vapor (uWV)
Ozone content (uO )

pixel pixel-by by-pixel pixel Process time for Process time for MSG MSG-SEVIRI subset: SEVIRI subset: Why popular despite its age?

Ozone content (uO3)
Sensor-specific predefined

coefficients

nly 25 sec.
nly 25 sec.

Why popular despite its age?

Atmospheric correction in a short time,
New sensors? No problem… Only the band specific
New sensors? No problem… Only the band specific

coefficients need to be updated,

Free and open-source!!!

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MARS & CMARS MARS & CMARS

Multivariate Adaptive Regression Splines Multivariate Adaptive Regression Splines

Nonparametric adaptive regression procedure

introduced by J. Friedman in 1991.

Multivariate Adaptive Regression Splines Multivariate Adaptive Regression Splines

introduced by J. Friedman in 1991.

Expansions of the truncated piecewise linear functions:  Reflected pair Reflected pair

( )

, , 0, , x if x x

therwise

τ τ τ

+

− >  − =   , , x if x τ τ − <  General model:

( ) Y f ε = + x

( )

, , 0, . x if x x

therwise

τ τ τ

+

− <  − =  

, x τ ∈ℝ

General model:

Y: response variable,
x = (x1, x2,…,xp)T, vector of predictors,

( ) Y f ε = + x

= (x1, x2,…,xp) , vector of predictors,
ε: observation error with zero mean and finite variance.

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MARS & CMARS MARS & CMARS

Multivariate Adaptive Regression Splines Multivariate Adaptive Regression Splines Multivariate Adaptive Regression Splines Multivariate Adaptive Regression Splines

τ

[ ( )] max( ( ),0) x x τ τ − − = − −

Reflected or “mirrored” pair Reflected or “mirrored” pair Reflected or “mirrored” pair Reflected or “mirrored” pair

[ ( )] max(( ),0) x x τ τ

+

+ − = − [ ( )] max( ( ),0) x x τ τ

+

− − = − −

τ A linear model The MARS model τ

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MARS & CMARS MARS & CMARS

Multivariate Adaptive Regression Splines Multivariate Adaptive Regression Splines Multivariate Adaptive Regression Splines Multivariate Adaptive Regression Splines

Can only piecewise linear functions be formed? Can only piecewise linear functions be formed? Reflected pairs can be multiplied together to form non-

linear functions:

1 2

( 1) (1 ) x x

+ +

− −

1 2

( 1) ( 1) x x

+ +

− −

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Interaction of (x1-1)+ with (1-x2)+ and (x2 -1)+ forming higher order terms.

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MARS & CMARS MARS & CMARS

Multivariate Adaptive Regression Splines Multivariate Adaptive Regression Splines Multivariate Adaptive Regression Splines Multivariate Adaptive Regression Splines

Spline fitting in higher Spline fitting in higher Tensor products of univariate Tensor products of univariate Multivariate spline BFs: dimensions dimensions spline functions spline functions Multivariate spline BFs:

( )

: . x

m m m m j j j

K m

B s x

κ κ κ

τ   = −    

∏

where

: total no. of truncated linear functions multiplied in the mth BF,

( )

1

j j j

m j κ κ κ = +

   

∏

K

: total no. of truncated linear functions multiplied in the mth BF,
: input var. of the kth truncated lin. function in the mth BF,
: knot value for

m

K

m j

xκ

m

κ

τ

m

K

x

: knot value for
m

j

κ

τ

m j

K

x { }

1

m j

sκ ∈ ±

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MARS & CMARS MARS & CMARS

Multivariate Adaptive Regression Splines Multivariate Adaptive Regression Splines Multivariate Adaptive Regression Splines Multivariate Adaptive Regression Splines

Estimation Estimation Estimation Estimation

1st

st stage: Forward Pass

stage: Forward Pass

Estimation Estimation

f model
f model

function function f( f(x) Estimation Estimation

f model
f model

function function f( f(x)

1 stage: Forward Pass stage: Forward Pass 2nd

nd stage: Backward Pass

stage: Backward Pass

f( f(x) f( f(x)

2nd

nd stage: Backward Pass

stage: Backward Pass Forward Pass:

Initial model:
( )

( )

1

, x x

M m m m

f B β β

=

= +∑

Analogous to forward stepwise linear regression,
The product resulting in the largest decrease in RSS

RSS added into the current model. into the current model.

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MARS & CMARS MARS & CMARS

Multivariate Adaptive Regression Splines Multivariate Adaptive Regression Splines Multivariate Adaptive Regression Splines Multivariate Adaptive Regression Splines

In In In In

Large Model Large Model

In In Forward Forward Pass Pass In In Forward Forward Pass Pass

Prune it by Backward Pass!.. Prune it by Backward Pass!.. Backward Pass: Prune it by Backward Pass!.. Prune it by Backward Pass!..

Decrease the complexity of the model without degrading

the fit to the data,

Removing BFs giving the smallest increase in RSS

RSS,

Produce an estimated best model with the optimal α,

ˆ fα

To find the optimal α

GCV (“lack

lack-of

f-fit”

fit” criteria).

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MARS & CMARS MARS & CMARS

Multivariate Adaptive Regression Splines Multivariate Adaptive Regression Splines

GCV

GCV

Multivariate Adaptive Regression Splines Multivariate Adaptive Regression Splines

( )

2 1 2

ˆ ˆ ( ) ( ) : x

N i i i

y f LOF f GCV

α α

α

=

− = = ∑

Q(α)=u + dK
K: no. of knots in FP
u: lin. independent func.
d: cost for each BF

( )

2

( ) ( ) : 1 LOF f GCV Q N

α

α α = = −

d: cost for each BF
ptimization
N: no. of observations
The numerator: Conventional RSS

RSS,

The denominator: Penalizes the numerator,
i.e., larger Q(α)

smaller model with less BFs, and smaller Q(α) larger model with more BFs,

The best model according to Backward Pass

Backward Pass that

The best model according to Backward Pass

Backward Pass that minimizes GCV.

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MARS & CMARS MARS & CMARS

Conic Multivariate Adaptive Regression Splines Conic Multivariate Adaptive Regression Splines

Alternative method to MARS by utilizing statistical

learning, inverse problems and multiobjective

Conic Multivariate Adaptive Regression Splines Conic Multivariate Adaptive Regression Splines

learning, inverse problems and multiobjective

ptimization theories.

Modern techniques of continuous optimization and

Conic Quadratic Programming Conic Quadratic Programming CQP CQP Conic Quadratic Programming Conic Quadratic Programming (CQP CQP).

Instead of applying backward step of MARS: PRSS PRSS with the with the BFs BFs built up built up during forward step of during forward step of MARS MARS Penalty terms added to the Penalty terms added to the least squares estimation least squares estimation

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least squares estimation least squares estimation to control the to control the LOF LOF

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MARS & CMARS MARS & CMARS

Conic Multivariate Adaptive Regression Splines Conic Multivariate Adaptive Regression Splines

PRSS

PRSS summed up during the forward pass of MARS:

Conic Multivariate Adaptive Regression Splines Conic Multivariate Adaptive Regression Splines

( )

max

2 2 2 2 , 1 1 1

:

M N m m i i m m r s m r s i m

PRSS y f G B d

δ δ

ϕ λ

< = = =

  = − +  

∑ ∑ ∑ ∑ ∫

x h h

where

( )

1 2

1 1 1 , ,

m T m

r s i m Q r s V δ δ δ δ < = = = ∈ =

 

∑ ∑ ∑ ∑ ∫

where

: denotes the variable set related with the mth BF,
: vector of variables contributing to the mth BF,

m

V

m

h

: penalty parameters (trade-off between accuracy & complexity),
: integration domains,
m

ϕ

m

Q

( ) ( )( )

m m m m δ δ δ δ

= ∂ ∂ ∂

( )

T

Finally, for ,

, where .

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( ) ( )( )

1 2

,

:

m m m m r s m m r s

G B B h h

δ δ δ δ

= ∂ ∂ ∂ h h

( )

1 2

,

T

δ δ = δ δ δ δ

1 2

: δ δ = + δ δ δ δ

{ }

1 2

, 0,1 δ δ ∈

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MARS & CMARS MARS & CMARS

Conic Multivariate Adaptive Regression Splines Conic Multivariate Adaptive Regression Splines

Trade-off between accuracy and complexity established

Conic Multivariate Adaptive Regression Splines Conic Multivariate Adaptive Regression Splines

ϕ

through the penalties ,

m

ϕ

PRSS PRSS approximated approximated as as PRSS PRSS approximated approximated as as

( )

max

1 2 2 2 2 1 1

Km

N M m im m m i

PRSS L ϕ λ

+ = =

≈ − + ∑

∑

ɶ y B g λ λ λ λ

where

as as as as

1 1 m i = =

where

: the vector of responses,
: an (N (Mmax+1))-matrix,

y ( )

ɶ B g

×

: an (N (Mmax+1))-matrix,
: Euclidian norm.

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( )

ɶ B g

2

i

×

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MARS & CMARS MARS & CMARS

Conic Multivariate Adaptive Regression Splines Conic Multivariate Adaptive Regression Splines

One penalty parameter employed rather than using

Conic Multivariate Adaptive Regression Splines Conic Multivariate Adaptive Regression Splines

2

: ϕ ϕ φ = =

distinct penalty parameters: . Then,

2

:

m

ϕ ϕ φ = =

( )

2 2

PRSS ϕ ≈ − + ɶ y B g L λ λ λ λ λ λ λ λ

where

( )

2 2

PRSS ϕ ≈ − + ɶ y B g L λ λ λ λ λ λ λ λ

: a diagonal ((Mmax+1)

(Mmax+1))-matrix,

: an ((Mmax+1)

1)-parameter vector.

L

λ

× ×

The PRSS PRSS problem A classical Tikhonov Regularization Tikhonov Regularization (TR TR)

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The PRSS PRSS problem Tikhonov Regularization Tikhonov Regularization (TR TR) problem with φ>0, φ=ϕ2

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MARS & CMARS MARS & CMARS

Conic Multivariate Adaptive Regression Splines Conic Multivariate Adaptive Regression Splines

TR

TR problem treated by CQP CQP with a convenient choice of

Conic Multivariate Adaptive Regression Splines Conic Multivariate Adaptive Regression Splines

ɶ

bound :

Z ∈ ɶ

minimize to

, h

( )

, 2 2

minimize to , subject to ,

h

h h Z − ≤ ≤ ɶ ɶ y B g L

λ λ λ λ

λ λ λ λ λ λ λ λ

By applying methods of continuous optimization, CQP

CQP re-written: re-written:

minimize ,

T x

c x

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( )

2

subject to 1,2, ,

T i i i i

q k − ≤ − …

x

G x g p x

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MARS & CMARS on MODIS Images MARS & CMARS on MODIS Images

MODIS MODIS

MOD

MODerate –resolution Imaging Spectroradiometer:

MODIS MODIS

Two scientific instruments operated by NASA

NASA,

On board the Terra

Terra satellite since 1999 & the Aqua Aqua since

On board the Terra

Terra satellite since 1999 & the Aqua Aqua since 2002,

Capturing data in 36 spectral bands (0.4 μm – 14.4 μm ),
Spatial resolutions: 2 bands at 250 m, 5 bands at 500 m
Spatial resolutions: 2 bands at 250 m, 5 bands at 500 m

and 29 bands at 1000 m,

Temporal resolution: 1 to 2 days,
Large-scale global dynamics including Earth’s cloud cover,
Large-scale global dynamics including Earth’s cloud cover,

radiation budget, processes in the oceans, on land and the lower atmosphere etc.,

6S

6S is the basic code underlying MODIS atmospheric

6S

6S is the basic code underlying MODIS atmospheric correction algorithm.

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Image data set taken by the Terra satellite Image data set taken by the Terra satellite

MARS & CMARS on MODIS Images MARS & CMARS on MODIS Images

5 images over the Alps in Europe:

Image data set taken by the Terra satellite Image data set taken by the Terra satellite

Europe

Data Set Date Used in EU1 11.10.2001 Tst. EU2 10.03.2002 Tst. EU2 10.03.2002 Tst. EU3 06.12.2003 Tst. EU4 22.04.2005

Trng. & Tst.

EU5 13.01.2006 Tst.

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Model training & testing Model training & testing

MARS & CMARS on MODIS Images MARS & CMARS on MODIS Images

Model training & testing Model training & testing

RGB color composite MODIS image of the study area (22.04.2005-EU4)

Training area size 483000 km2 Training area size 483000 km2 Test area size 53700 km2

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Model training & testing Model training & testing

MARS & CMARS on MODIS Images MARS & CMARS on MODIS Images

CSRef(x): model function

Model training & testing Model training & testing

estimated by MARS giving the atmospherically corrected surface reflectance,

1 1 1

( ) y CSRef ε             x

x=(λ, ϕ, κ)T: λ and ϕ refer to the

geographic longitude and latitude respectively, and κ is the TOA

2 2 2

( ) ( ) y CSRef y CSRef ε ε                   = +             x ⋮ ⋮ ⋮

respectively, and κ is the TOA reflectance value,

N: total no. of observations,

( ) ( )

k k k

y CSRef y CSRef ε ε                                     x x ⋮ ⋮ ⋮

yk: atmospherically corrected

surface reflectance at location

xk=(λk, ϕk, κk)T,

( )

N N N

y CSRef ε             x

xk=(λk, ϕk, κk) ,

εk: observation error.

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Model training & testing Model training & testing

MARS & CMARS on MODIS Images MARS & CMARS on MODIS Images

Model training & testing Model training & testing

After the backward pass, the final MARS model: 15. 0.0495191 14 0.00966538 13 16.8514 11 0.14844 10 0.0109712 8 0.00163948 7 5.85391 3 2.85519 2 1.56786 1 1.39677 0.681299 = BF BF BF BF BF BF BF BF BF BF Y × − × − × + × − × + × − × − × − × − × +

GCV = 0.004951 adj R2 = 0.8913

For CMARS, all BFs obtained in the forward pass are

used for the formulation of PRSS

PRSS in the form of a CQP

CQP problem:

4 0.0776 3 2.7299 2 0.4603 1 0.6372 0.5043 = BF BF BF BF Y × − × − × − × + 15. 0.0257 14 0.0093 13 14.4283 12 0.6916 11 0.4940 10 0.104 9 0.0176 8 0.0015 7 5.5764 6 0.0053 5 0.0038 4 0.0776 3 2.7299 2 0.4603 1 0.6372 0.5043 = BF BF BF BF BF BF BF BF BF BF BF BF BF BF BF Y × − × − × + × − × + × − × − × − × − × + × + × − × − × − × +

28

15. 0.0257 14 0.0093 13 14.4283 BF BF BF × − × − × +

SLIDE 29

Results Results

Comparing the results Comparing the results

Performances of MARS, CMARS

MARS, CMARS and SMAC SMAC on the predefined test areas compared against surface

Comparing the results Comparing the results

predefined test areas compared against surface reflectance values of 6S 6S product in terms of RMSE:

Data Set SMAC MARS CMARS EU1 0.08629 0.08296 0.08168 EU2 0.07150 0.04618 0.04569 EU3 EU3 0.11834 0.06505 0.06465 EU4 0.27556 0.24085 0.24011 EU5 0.18680 0.17004 0.16844 EU5 0.18680 0.17004 0.16844

29

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Results Results

Comparing the results Comparing the results

Test area for EU1 (11.10.2001)

Comparing the results Comparing the results

30 a) MARS, b) CMARS, c) SMAC, d) 6S

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Concluding remarks Concluding remarks

As a result… As a result…

According to the test results:

As a result…

As a result…

MARS & CMARS outperform SMAC,
Eventhough model training on one image & testing on all

image sets, MARS & CMARS still give better results than SMAC.

Within the light of preliminary results:

MARS & CMARS as an alternative tool for atmospheric
MARS & CMARS as an alternative tool for atmospheric

correction and for other problems related with different research areas in RS & GIS.

Future work: Future work:

Focus on the application of MARS & CMARS, as well as

Robust CMARS (RCMARS RCMARS) on larger data sets with different wavelength bands. different wavelength bands.

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References References

Anderson, G.P., Pukall, B., Allred, C.L., Jeong, L.S., Hoke, M., Chetwynd, J.H., Adler-

Golden, S.M., Berk, A., Bernstein, L.S., Richtsmeier, S.C., Acharya, P.K., and Matthew, M.W., “FLAASH and MODTRAN4: state-of-the-art atmospheric correction for hyperspectral data”, IEEE Aerospace Conference Proceedings 4, 1999, pp. 177-181.

Durmaz, M., Karslıoglu, M.O., and Nohutcu, M., “Regional VTEC modeling with

multivariate adaptive regression splines”, Advances in Space Research 46, 2010, pp. 180-189.

Friedman, J.H., “Multivariate adaptive regression splines”, The Annals of Statistics, 19,

1991, pp. 1-141. 1991, pp. 1-141.

Gao, B., Montes, M.J., Davis, C.O., and Goetz, A.F.H., “Atmospheric correction

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f Environment 113, 2009, pp. S17-S24.
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Mining, Inference, and Prediction, 2nd ed.”, Springer, 2009.

Kotchenova, S.Y., Vermote, E.F., Matarrese, R., and Klemm, F.M., “Validation of a vector

version of the 6S radiative transfer code for atmospheric correction of satellite data. Part I: Path radiance”, Applied Optics 45, 2006, pp. 6762-6774. I: Path radiance”, Applied Optics 45, 2006, pp. 6762-6774.

Rahman, H., and Dedieu, G., “SMAC: A simplified method for the atmospheric correction
f satellite measurements in the solar spectrum”, International Journal of Remote

Sensing 15, 1994, pp. 123-143.

Richards, J.A., and Jia, X., “Remote sensing digital image analysis: an introduction, 4th

ed.”, Springer, 2006.

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Thank You… Thank You… Thank You… Thank You…

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