1 tree crown extraction using marked point processes guillaume perrin xavier descombes – josiane...

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Tree Crown Extraction Using Marked Point

Processes

Guillaume Perrin

Xavier Descombes – Josiane Zerubia

ARIANA, joint research group CNRS/INRIA/UNSAINRIA Sophia Antipolis, FRANCEhttp://www-sop.inria.fr/ariana

MAS, Applied Mathematics LaboratoryEcole Centrale Paris, FRANCE

http://www.mas.ecp.fr

EUSIPCO 2004 – 10th, September 2004

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Contents

Motivations

Notations and Definitions

Our Model for Tree Crown Extraction

Results

Conclusion

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3


Motivations

Remote sensing in forestry management- Near infrared images

- Could avoid human investigations : economic considerations

- More control on forest stands evolution

900nm

520nm

Forestry statistics to estimate- Stem number

- Diameter distribution

- Forestry cover area

Automatic Extraction- [Gougeon 95] ; valley following

- [Larsen 97] : template based model

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French Inventory (IFN)- Aerial images - 50 cm/pixel

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Contents

Motivations



Results

Conclusion

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Object Space U

An Object (position/marks)

A Configuration

A Marked Point Process X with- Probability Distribution PX

- Unnormalized Density h(.)

- Reference Poisson measure (uniform point process)

Example : Strauss Process

X

Ykpy

px

u

)x().x(x)P(Xx)(PX dhdd

ninteractioin pairsnb)x(

0parameter )x(.exp)x(

s

tsth

rs=5 with t1

P

+

+

++ +

+

+

++ +

+

nn Uxx ,...,x 1

4

P

+

+

+

++++

+ +

+ +

s=1 with t2<t1

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Contents

Motivations



Results

Conclusion

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Objects of the process- Disk process : position of the center and

radius

Density of the marked point process (1)

- Prior density (knowledge)

1. Penalizes intersections of disks

2. Favours alignments

3. Hard Core (stability reasons)

)x(exp)x( pp Uh )x()x()x()x( hcp

ap

ipp UUUU

0

)(),(min)x(

~

jIi xx ji

jiI

Ip xAxA

xxAU

0,)x(~

jAi xx

jiAAp xxU

0

,dminif)x(

jiHC

p

xxU

],[],0[],0[ MmMM RRYXKPU

pU

>0 repulsive

<0 attractive

)x().x()x(exp)x()x( ILhUIhh p

Proposed model for Tree Crown Extraction

Z

hf

)x()x(

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Density of the marked point process (2)- Likelihood = Gaussian Mixture

- Each pixel belongs to one of these 2 classes :- Tree Class, with normal distribution- Background Class, with normal distribution

Stability condition of the density

22

-)(exp

π2

1)x()x(

i

i

p ip

pIpLIL

11 , 22 ,

x.x hMuh

)x().x()x(exp)x()x( ILhUIhh p

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MCMC Simulation of point processes [Geyer 98] - Markov Chain (X) with equilibrium distribution PX (ergodic

convergence)

- Algorithm : Metropolis Hastings with Reversible Jumps [Green 95]

- Application to feature extraction :

Maximum A Posteriori Estimator

)x().x(x)P(Xx)(PX dhdd

0,)x()x(1

iT Thh i )x(argmaxx XMAP h

1. Simulate a point process defined by a density h(.)

2. Explore the whole state space

3. Find one of the global maxima of h(.)

Simulated Annealing

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Reversible Jump MCMC Algorithm

Configuration of objects Xi = x

Simulate y ~ Q(x,.) (proposal kernel)

Evaluate Green ratio R=F[Q(.,.),h(.),x,y]

Accept y with probability min(1,R)

Xi Y

Proposal Kernel :

- Birth / Death

- Translation

- Dilation

- Split / Merge

- BD alignment, …

Goal :

find the MAP as fast as possible and avoid local maxima of h(.)

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Contents

Motivations



Results

Conclusion

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Results

AI ,

99995.0,.1 aTaT ii

Results depend on- Simulated annealing scheme

- In theory : logarithmic decrease to get the MAP estimator

- In practice : geometric decrease.every N iterations

- Parameters of density h(.)- Which parameters for the priori ?

- Experimental / Parameter Estimation

- Which parameters for the likelihood ?- KMeans / Parameter Estimation

2211 ,,,

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Results

SLOW : 50M iterations

~ 17 minutes

304 objects - U=138431

FAST : 1,5M iterations

~ 30 seconds

299 objects - U=140348

Original image

80 105

TT

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Results

Extraction evolution / Green ratio

- High Temperature : Green ratio dominated by Poisson measure ratio

- Low Temperature : Green ratio dominated by density ratio

)y,x(Q

)x,y(Q

)(

)(

)x(

)y()y,x(

1

d

d

dx

dy

h

hR

iT

Critical Temperature

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Density ratio

Poisson measure

ratio

Kernel ratio

Diaporama

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Contents

Motivations



Results

Conclusion

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Conclusion

Advantages of the modeling- Geometrical information of stands taken into account

- Can be adapted to multi-species extraction

Drawbacks- Computational time

- Trees have to be separable(pb on too dense areas)

Future work- Parameter estimation on the global model (in progress)

- Texture information in the density (distinguish btw different species)

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References

[Gougeon 95] A crown-following approach to the automatic delineation of individual tree crowns in high spatial resolution aerial images – Canadian Journal of Remote Sensing – 21(3), 274-284, 1995.

[Larsen 97] Using ray-traced templates to find individual trees in aerial photographs – Proc. 10th Scandinavian Conference on Image Analysis, vol.2, 1007-1014, 1997.

[Green 95] Reversible Jump Markov Chain Monte Carlo computation and Bayesian model determination – Biometrika 82, 711-732, 1995.

[Geyer 98] Stochastic geometry, likelihood and computation : “Likelihood inference for spatial point processes”, Chapman et Hall, London, 1998.

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Results

Extraction evolution / Green ratio

- High Temperature : Green ratio dominated by Poisson measure ratio

- Low Temperature : Green ratio dominated by density ratio

)y,x(Q

)x,y(Q

)(

)(

)x(

)y()y,x(

1

d

d

dx

dy

h

hR

iT

Critical Temperature

13

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Density ratio

Poisson measure

ratio

Kernel ratio

1 tree crown extraction using marked point processes guillaume perrin xavier descombes – josiane...

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