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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

David Bolin - [email protected] Spatial Statistics 1/35


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


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!



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.



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.



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



Some examples

◮ Classification (K-means and the EM algorithm)◮ LANDSAT

◮ Dependence structures; Markov random fields◮ Image reconstruction◮ Reconstruction of fields — Global ozone and temperature


Spatial Statistics Classification Spatial More K-means Bayesian EM-algorithm Example

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)



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.



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?



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



Discriminant analysis

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π1=0.5

π1=0.25

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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?



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.



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)



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.



LANDSAT image of Rio de Janeiro



Blue, Green, Red, and Infrared



Principal components extracted from 7 wavelengths



1D and 2D histograms



Bayesian pixel classification



Classified image pixels


Spatial Statistics Classification Spatial More Reconstruction Ozone Temperature

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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◮ 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.



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.



Image reconstruction — Corrupted pixels

image reconstruction

bad pixels − estimatebad pixels



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



Global ozone — Reconstruction

200

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Global ozone — Standard errors

1.522.533.544.555.56

2468101214161820



Global temperature — Data

January 2003 July 2003



Global temperature — Reconstruction

Global mean: 15◦C.



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/


Download - Spatial Statistics with Image Analysis · Spatial Statistics Classiﬁcation Spatial More Hierarchical Estimation Hierarchical models We often have some prior knowledge of the reality

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