Nice, 17/18 December 2001 Autonomous mapping of natural fields using Random Closed Set Models Stefan Rolfes, Maria Joao Rendas rolfes,[email protected]

Nice, 17/18 December 2001

Autonomous mapping of natural fields using Random Closed Set Models

Stefan Rolfes, Maria Joao Rendas

rolfes,[email protected]

Laboratoire I3S, CNRS, UNSA

Sophia Antipolis, France


Outline

• Introduction

• Habitat mapping

• Representation using RCS models

• Navigation using RCS maps

• Simulation results

• Conclusion


Goals: • evaluate the total amount of living/dead maerl in Rousey Sound (Orkney

Islands, Scotland)• characterise the spatial distribution of maerl

Platform: PhantomSensor: vision

Maerl mapping


Individual delimitation of each maerl patch is impossible

Approach:(1) Learn the statistical characteristics of the field:

• the distribution of the patches sizes• the distribution of their shapes• how they are spatially scattered

(2) Relate the local distribution to the site’s characteristics (depth, currents, slope, bottom type) whenever this information is available.

Result:A “statistical map” of the area surveyed and a “statistical model” of its

properties.enables determination of the expected total amount of maerlprovides the basis for extrapolating the local observations to other (unobserved) areas.

Maerl mapping


image segmentation region classification statistical characterisation

raw images expert knowledge (model)

(labels)(homogenous regions)

model types

mappingrobot position

mappost-processing• total amount of maerl• relation of maerl distribution to

geophysical parameters

(shape, size and spatial distribution)

Data processing


Generalisation of K-means clustering algorithm in distribution space

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Image Segmentation: approach


Goal: find the “homogeneous” regions of the image

Approach: • model the set of pixels in each neighbourhood of the image as iid random

variables• discriminate distinct regions as realisations of distinct random variables

, two sequences

Hypotheses:

H0:

H1:

)(1

nx )(2nx

nnn pxx )(2

)(1 ,

nnnnnn pppxpx21,2

)(21

)(1 ,

Statistical Test


),()ˆ()ˆ(

0

1

21 Ln

H

H

DD

ln)( ED Kullback-Leibler divergence

Ljn

an

ijaixji ,,1,

1

1|

type of the sequence xi(n)

212

1ˆ mixture of 1 and 2.

Optimal test


K-means (Lloyd) algorithm


Iterate

2,1,)(#

1)1(

)(

n

kCkh

knCijij

nn1)

2) 2,1)1(minarg|)1(2,1

nkhDnijkC mij

mn

Geometrical view


Start with ji

jiwN

h,

,01

and randomly “split” h0 in two histograms h1(0) and h2(0) such that

[1,0]),0()1()0( 210 hhh

Randomly generate

L

iiihihh

1101 1)(),()(,

find that minimizes))()()1()(( 02211 hhNhND

where

)(1

1)( 102 hhh

)(()(#)( 211 hDhDN ijij

Replace Euclidean distance (between points)by Kullback divergence (between histograms)


Test “homogeneity” of the classes found by testing the distribution of the Kullback-Leibler distances of its members with respect to against the exponential distribution (theoretical dist. – type theory)

DeDp )(

Since is the average value of the exponential distribution, we use

ii D̂/1ˆ

and test the values of Diii eDpD

ˆˆ)(

)(Dpi )(ih

Determination of the number of classes


segmented image (1)original image 1 Examples


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all imageclass 1 class 2

classes histograms (3)

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original image (2)



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Describing natural scenes

Formal description of the geometry of the environment as a union of closed sets:

Observation : The ‘ Objects ’ tend to form random-like patterns

);pK( ii1i

KiK located at : ip

Assumption : Perceptual data (Images) have been segmented into

areas of distinct types (Preprocessing step).


Modelisation as Random Closed Set

)(1

iii

Each model is defined by a parameter vector ),(

Family of models : ,,, 21 MMM ),( iM

},,{ 21 l

},,{ 21 K

Doubly stochastic process :

1) Random point process (germ process)

describes spatial distribution of objects

2) Shape process (grain process)

determines the geometry of the objects


Examples of Random Closed Sets

Uniform distribution

Cluster process Regular structures

Non isotropic distribution


Map of the environment

4A

1A2A

3A

Segmentation of the workspace : ,1

ii

A

)(ii MA

Non isotropic

),(xx

isotropic

x Map of the environment


Perceptual observations : Hitting capacities

);()( KPKT

Knowledge of the hitting capacities for all compact sets is equivalent to knowledge of the RCS model determined by

Hit

Miss

)))((exp(1)( 02 KKT

E

Analytical expression for Boolean models :


Distribution of hitting capacities

Local observations in an observation window W induce a distribution on the hitting capacities for stationary RCS models

Its characterization is important for :

)W),X()K(T̂(p

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2

4

6

8

10

12

14

16

18

T

Boolean model : (0.002, r in (5,15))

Empirical distriution for K = square(0) Empirical distriution for K = square(18)

• Mapping (segmentation of the workspace)

• Localization (Bayesian approach)

Requires explicit detection of model change


Conclusions

• We proposed a novel environment description by RCS models

• Proposal of a new image segmentation algorithm (adaptively learns the number of classes)

– Methods for detection of model changes (region boundaries)

– Validation with real data

Future work

the workspace (boundary tracking)

Documents

Nice, 17/18 December 2001 Autonomous mapping of natural fields using Random Closed Set Models Stefan Rolfes, Maria Joao Rendas rolfes,[email protected]