I All measurements contain random measurement errors ......Spatial Statistics Classication Spatial...

Spatial Statistics Classification Spatial More

Spatial Statistics with Image Analysis

Johan Lindström1

1Mathematical StatisticsCentre for Mathematical Sciences

Lund University

LundOctober 4, 2013

Johan Lindström - [email protected] Spatial Statistics 1/37


Outline

Spatial Statistics with Image AnalysisHierarchical ModelsEstimation Procedures

Pixel ClassificationK-meansBayesian ClassificationEM-algorithmExample

Spatial StatisticsSatellite DataImage ReconstructionTemperature

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 theestimate itself.

◮ 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 distributionp(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. Thesamples 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◮ Satellite data◮ Image reconstruction◮ Reconstruction of fields — Global mean 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 relativeabundance 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 separatingdata 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 newcluster 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, fork = 1, . . . ,K, are known.

◮ Maximum A Posteriori classification chooses the object class foreach 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 discriminantanalysis. Otherwise, we obtain the method Quadraticdiscriminant analysis



Discriminant Analysis

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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 new

cluster centres.3.1 Compute the amount of pixels in each class and use that as

estimates 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 classprobability for each data point.

◮ Let pi,k = P(x̃i = k|yi).◮ Then,

π̂k =1n

∑

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 estimatingμk andΣk.

μ̂k =1

nπ̂k

∑

i

pi,kxi

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} andΘ = {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 =1n

∑

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 likelihoodprinciples.



LANDSAT image of Rio de Janeiro



Blue, Green, Red, and Infrared



Why Infrared?

False colour composite (orIR-colour) images are used to pickout healthy vegetation(chlorophyll).

University of Wisconsin-MadisonCampus, artificial turf on thefootball field.

T. Lillesand, R. Kiefer, J. Chipman (2004),

Remote Sensing and Image Interpretation,

5ed:Plate 3.



Principal components extracted from 7 wavelengths



1D and 2D histograms



Bayesian Pixel Classification



Classified Image


Spatial Statistics Classification Spatial More Satellite Data Reconstruction Temperature

Stochastic Fields

Many problems can be approached by modelling the spatial dependencestructure between pixels (or irregular locations).

◮ A random field is a collection of random variables x̃(u) with acommon density function.

◮ The random field can be used to model the dependence betweenpixels, or other spatially varying data.

◮ For a random field x̃(u), the expectation functionmx̃(u) = E(x̃(u)) collects the point-wise expectations of the field.

◮ For the covariance (dependence) between different pixels, we writer(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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Satellite Data — Vegetation

January 1999

July 1999



Satellite Data — Trend in Vegetation

K2 Estimate

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K2 Estimate

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Image Reconstruction

Spatial Interpolation

Given observations at some locations (pixels), y(ui), i = 1 . . .nwe want to make statements about the value at unobserved location(s),x(u0).

The typical model consists of a latent Gaussian field

x ∈ N (μ,Σ) ,

observed at locations ui, i = 1, . . . ,n, with additive Gaussian noise(nugget or small scale variability)

yi = x(ui)+ εi εi ∈ N(

0,σ2ε).



Image Reconstruction (cont)

Assuming known parameters: μ,Σ, and σ2ε with

[Σ]kl = r(uk,ul)

the conditional expectation of the missing pixels is

E (x | y) = μ+ΣxyΣ−1yy (y − μ)

where

Σxy ∼ r(uj,ui){

j ∈ unobservedi ∈ observed

Σyy ∼ r(ui,ui)+ σ2ε i ∈ observed



Image Reconstruction



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 unknownimage (and indicator) given observations and parameters.



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.



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 atwww.maths.lth.se/matstat/kurser/fmsn20masm25/

Who?Lecturer: Jonas Wallin

MH:323


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