Diapositive 1

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Lionel Gueguen, Mihai Datcu
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Satellite Image Time Series (SITS)

• The series is compound of SPOT images acquired
  over two years. Images have been coregistered and
  intercalibrated.
• Characteristics - 20m/pel
• - 3 spectral bands (green, red, infrared)
• - non-uniform time sampling
• ADAM data set
• http//kalideos.cnes.fr/

Acquisition time
SPOT 1
SPOT 1
SPOT 2
SPOT 4
SPOT 4
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Rate-Distortion

• For a random variable X iid, the rate-distortion
  curve is

where
The conditional probabilities can be seen as a
soft assignment to clusters. So the optimisation
gives a clustering of the realizations of X.
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Informational Distortion

• Let consider the Kullback-Leibler divergence as
  our distortion. We consider a new random variable
  Y.
• The distortion is

• The mean distortion is

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Rate Distortion the inference
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Information Bottleneck

• We take the Lagrangian formulation of the
  rate-distortion curve and replace the mean
  distance.

• As the mutual information between X and Y does
  not vary, the criterion becomes

Statistics dependances
X
X
Y
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Multi Information Bottleneck

• We consider n random variables Yi which are
  independant. We extend the previous criterion to
  the Multi Information Bottleneck.

Y1
Y2

Statistics dependances
X
X
Yn
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Information Bottleneck the inference
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Example and experiment

• We take two random variables Y1 and Y2
• Y1 represents the spectral information
• Y2 represents the textural information
• The realizations of X are given by a rectangular
  partition of the SITS.

time
A realization of the random variable X
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Spectral information

• For each realizations xi of X, we estimate the
  mean and variance of xi.
• So the estiamted mean and variance,yi, are a
  realization of Y1.
• Thus, we can estimate the conditional probability
  p(XY1) by considering all realizations of X and
  Y1 .
• p(XY1)p(xiyj)i,j

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Textural Information

• The same concepts as previous with the random
  variable Y2.
• We use Gauss Markov Random Field to estimate the
  parameters of the probabilistic model p(xy2).
• When all the realizations of Y2, are computed, it
  is easy to compute the conditional probability
  p(XY2)

Where e is a white Gaussian noise.
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Experiments

• Size of blocks which are the realizations
  10x10x5
• Size of the subsequence 100x100x6
• There are 600 realizations.
• Number of cluster found is 7. We highlight tree
  clusters in the data space, by marking the blocks
  in red.

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Results on SITS
Cluster 1
Cluster 2
Cluster 3