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Spatial Statistics Classification Spatial More
Spatial Statistics with Image Analysis
David Bolin1
1Mathematical StatisticsCentre for Mathematical Sciences
Lund University
Lund
October 5, 2012
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Outline
Spatial Statistics with Image Analysis
Hierarchical models
Estimation procedures
Pixel classification
K-means
Bayesian classification
EM-algorithm
Example
Spatial statistics
Image reconstruction
Ozone
Temperature
Learn more
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Spatial Statistics Classification Spatial More Hierarchical Estimation
A statistical approach
◮ All measurements contain random measurement errors/variation.
◮ Most natural phenomena have natural random variation.
◮ Often the uncertainty of an estimate is, at least, as important as the
estimate it self.
◮ We need to describe and model random variation and uncertainties!
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Hierarchical models
◮ We often have some prior knowledge of the reality.
◮ Given knowledge of the true reality, what can we say about images
and other data?
◮ Construct a model for observations given that we know the truth.
◮ Given data, what can we say about the unknown reality?
This is the inverse problem.
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Bayes’ Formula
Using a statistical formulation:
◮ A prior model for reality, π(x)
◮ A conditional model for data y given reality, p(y|x)
◮ We want the posterior distribution, p(x|y). The distribution for x
given y.
◮ Bayes’ Formula:
p(x|y) =p(y|x)π(x)
p(y)=
p(y|x)π(x)∑z
p(y|z)π(z),
where p(y) is the total density for data.
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Estimation procedures
Maximum A Posteriori (MAP): Maximise the posterior distribution
p(x|y) with respect to x.
◮ Standard optimisation methods◮ Specialised procedures, using the model structure
Simulation: Simulate samples from the posterior distribution p(x|y).Estimate statistical properties from these samples. The
samples can be seen as representative “possible realities”,
given the available data.
◮ Markov chain Monte Carlo (MCMC)◮ Gibbs sampling
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Some examples
◮ Classification (K-means and the EM algorithm)◮ LANDSAT
◮ Dependence structures; Markov random fields◮ Image reconstruction◮ Reconstruction of fields — Global ozone and temperature
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Object classification
We present object classification as an image segmentation problem, i.e.
we want to classify all pixels in an image.
◮ We assume that there is a set, {1, . . . ,K}, of K object categories.
◮ Let x̃i denote the class of each pixel, and assume that x̃i are
independent for all pixels.
◮ The prior probabilities πk = p(x̃i = k) describe the relative
abundance of each class.
◮ For each model class there is a data distribution.
◮ For each pixel we have the distribution p(yi|x̃i = k)
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The K-means algorithm
The K-means algorithm is a simple, popular algorithm for separating
data into different clusters.
1. Select K data-points at random, as initial cluster centres.
2. Assign all data points to their nearest cluster centre.
3. Compute the mean within each cluster, and let these be the new
cluster centres.
4. Repeat from 2.
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Drawbacks of K-means
◮ Handling of different πk?
◮ Different variation within clusters?
◮ Overlapping clusters?
If we assume that the data within each cluster is multivariate Normal,
how can we improve on K-means?
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Bayesian classification
◮ Assume that πk and the parameters of p(y|x̃ = k), μk andΣk, for
k = 1, . . . ,K, are known.
◮ Maximum A Posteriori classification chooses the object class for
each pixel that has the highest posterior probability
p(x̃ = k|y) =p(y|x̃ = k)π(k)
p(y)=
p(y|x̃ = k)πk∑k p(y|x̃ = k)πk
◮ If allΣk are equal, we obtain the method Linear discriminant
analysis. Otherwise, we obtain the method Quadratic
discriminant analysis
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Discriminant analysis
0 2 4 6 8 100
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10Linear Bayesian discrimination
π1=0.5
π1=0.25
π1=0.1
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10Quadratic discrimination
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Improving on K-means
1. Select K data-points at random, as initial cluster centres.
2. Assign all data points to their nearest cluster centre.
3. Compute the mean within each cluster, and let these be the newcluster centres.
3.1 Compute the amount of pixels in each class and use that asestimates of πk.
3.2 EstimateΣk, one for each cluster.
4. Use the estimated parameters to perform a MAP classification.
5. Repeat from 2 3.
Still one major issue: We are not sure of the classifications in each step.
How should the uncertainty be taken into account?
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Using the posterior probabilities
◮ Given estimates of πk, μk andΣk, compute the posterior class
probability for each data point.
◮ Let pi,k = P(x̃i = k|yi).
◮ Then,
π̂k =1
n
∑
i
pi,k
is the average probability for a data point to belong to class k.
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Using the posterior probabilities (cont)
We also use the probabilities pi,k as certainty weights when estimating
μk andΣk.
μ̂k =1
nπ̂k
∑
i
pi,kyi
Σ̂k =1
nπ̂k
∑
i
pi,k(yi − μ̂)T(yi − μ̂)
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The EM-algorithm for Normal mixtures
Given: Initial estimates π(0) andΘ(0) of π = {π1, . . . , πK} and
Θ = {μ1, . . . ,μK,Σ1, . . . ,ΣK}
◮ Let p(t)i,k
= P(xi = k|yi,π(t),Θ(t)) ∝ π
(t)k
p(yi|xi = k,Θ(t)).
◮ Let
π(t+1)k
=1
n
∑
i
p(t)i,k
μ(t+1)k
=1
nπ(t+1)k
∑
i
p(t)i,k
yi
Σ(t+1)k
=1
nπ(t+1)k
∑
i
p(t)i,k(yi − μ
(t+1))T(yi − μ(t+1))
This algorithm can be more rigorously derived using likelihood
principles.
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Spatial Statistics Classification Spatial More K-means Bayesian EM-algorithm Example
LANDSAT image of Rio de Janeiro
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Blue, Green, Red, and Infrared
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Principal components extracted from 7 wavelengths
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1D and 2D histograms
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Bayesian pixel classification
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Classified image pixels
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Stochastic fields
Many problems can be approached by modelling the spatial dependence
structure between pixels (or irregular locations).
◮ A random field is a collection of random variables x̃(u) with a
common density function.
◮ The random field can be used to model the dependence between
pixels, or other spatially varying data.
◮ For a random field x̃(u), the expectation function
mx̃(u) = E(x̃(u)) collects the point-wise expectations of the field.
◮ For the covariance (dependence) between different pixels, we write
r(u,v) = C(x̃(u), x̃(v)) = E((x̃(u)− mx̃(u))(x̃(v)− mx̃(v))).
◮ r(u,v) is called the covariance function.
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0 20 40 60
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nu=0.5
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nu=1.0
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nu=2.0
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nu=4.0
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nu=0.5
5 10 15 20 25 30
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nu=1.0
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nu=2.0
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nu=4.0
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Image reconstruction
◮ Given an image with missing pixels we want to reconstruct the
missing values.
◮ Assume that the image is a sample from an underlying field.
◮ Estimate parameters of the field using the known pixels.
◮ Given parameters and data, estimate values for the missing pixels.
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Image reconstruction
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Spatial Statistics Classification Spatial More Reconstruction Ozone Temperature
Image reconstruction — Corrupted pixels
◮ Typically we don’t know which pixels that are bad.
◮ A better model is then◮ Assume an underlying image, x.◮ Assume an indicator image for bad pixels, z.◮ Given the indicator we either observe the correct pixel value from x
or noise.
◮ Use Bayes’ Formula to compute the distribution for the unknown
image (and indicator) given observations and parameters.
◮ Estimate parameters and reconstruct.
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Spatial Statistics Classification Spatial More Reconstruction Ozone Temperature
Image reconstruction — Corrupted pixels
image reconstruction
bad pixels − estimatebad pixels
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Spatial statistics
◮ Why limit ourselfs to images?
◮ The principal works just as well for data that is not on a grid.◮ Weather stations◮ Environmental monitoring,◮ Depth measurements◮ etc.
◮ We can even handle data on the globe.◮ Example: Total Column Ozone
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Global ozone — Reconstruction
200
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500
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Global ozone — Standard errors
1.522.533.544.555.56
2468101214161820
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Global temperature — Data
January 2003 July 2003
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Global temperature — Reconstruction
Global mean: 15◦C.
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Learn more!
What?Spatial statistics with image analysis, FMSN20
When?HT2-2013, October–December
Where?Information and Matlab files will be available at
www.maths.lth.se/education/lth/courses/fmsn20masm25/
and
www.maths.lth.se/matstat/kurser/fmsn20masm25/
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