Index of Contents Anatomic Structures › com3371 › week7 › ruiz.pdfAugust 1, 2000 Juan Ruiz-Alzola [email protected] 1 Statistical Classification of Anatomic Structures Juan

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August 1, 2000 Juan [email protected]

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Statistical Classification ofAnatomic Structures

Juan Ruiz-Alzola, PhD

ULPGC University, Spain, andSPL, Harvard Medical School &Brigham and Women’s Hospital


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Index of Contents

• General Issues and Motivation.

• Decision Theory.• Background Removal.• Towards Automatic Segmentation.

• Conclusions.• References.


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General Issues and MotivationProcedure:

•Sequence of slices

•Mental reconstruction

•Fixed viewpoint

•Difficult fussion

•Subjetive assesment

•Requires high skillsAugust 1, 2000 Juan Ruiz-Alzola

[email protected]

General Issues and MotivationSegmentation is needed for:

Tissular/funct. analysis Models construction

These processes can be implicit or explicit.

The later can be human or automatic.


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General Issues and Motivation

Analysis:

•Pathology detection

•Therapy evaluation

•Basic research

Model construction:

•Training

•Surgical planning

•Surgical guidance

Segmentation


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General Issues and MotivationAutomatization => Explicit segmentation

Human:

•Structure outline

•Very slow. High cost

•Impossible in real-time

•High variability (15 %)

Automatic:

•Intensity-based(statistical)

•Template-based

•Combined

Mixed: operator controlled

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General Issues and MotivationAtlas


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General Issues and MotivationVirtual Colonoscopy


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General Issues and MotivationSurgical planning

T1w with contrast T1w with contrast T2w


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General Issues and MotivationSurgical planning


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General Issues and MotivationSurgical guidance

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Magnetic Resonance Therapy at BWH


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Decision Theory

Nature choses

θ ∈Θ (states)z

Observation space: Z

δ5

δ4

δ3

δ2

δ1pz(z/θ)

Generate an observation

Make a decisionminimizing Loss L(θ,δ)

ϕ(z) ∈ϑ

δi ∈∆

(decisions)State of the

economyStocks market

•Buy

•Sell

•...

Example:


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Decision TheoryTaxonomy of problems in Decision Theory

Decision space ∆

Discrete Continous

Binary

H0: null hypothesis

H1: alternative hypothesis

Example: Detection ofSignal in Noise

Hypothesis testing

Multiple

H1

Hn

Example: Classificationand Pattern Recognition

Examples:•PDF learning•Noise filtering•Signal restauration

Estimation theory


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Decision Theory

An agent is confronted to the following problem: one coin out of two possiblenon-fair ones is tossed. After getting the result (head or tail) the agent is tomake a decision about which one was actually tossed. The agent has prioraccess to both coins in order to analyze them before the experiment is done.

A first example: Tossing two non-fair coins

Problem model in terms of Decision Theory

State of nature => coin (a or b) Is there any preference to pick a coin?

Probability pz(z/θ)pz(z/a)

pa

1-pa

H T z

pz(z/b)pb

1-pb

H T z


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Decision TheoryLet´s admit that we know the probabilty function of both coins:

HEAD TAIL

Coin: a pz(H/a) = 0.2 pz(T/a) = 0.8

Coin: b pz(H/b) = 0.9 pz(T/b) = 0.1

Likelihoods

Probalilities

A common sense decision rule would be:

H => Choose b

T => Choose aMaximum Likelihood rule


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Decision TheoryEstimation of the probabilities: model learning or training

Parametric estimation: the functional form of the PDF is known. In our case(H=0, T=1):

Benoulli Trial: pz(z/θ) = pθδ[z] + (1-pθ)δ[z-1]

Learning: repeat the experiment in order to estimate the parameters.

Supervised learning: it´s known which class generates the observation.

Several estimation approaches are possible. The most intuitive is the

Law of Large Numbers (direct averaging):

p(H/θ) = NHEADS / NTOTAL p(T/ θ) = NTAILS / NTOTAL

Exercise: Apply the Maximum Likelihood principle to obtain the same estimator

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Decision TheoryBinary Classification or Detection

States: Θ = {θ0 , θ1}

Observation pdf: pz(z/θ)

H0: θo generates the observation

H1: θ1 generates the observation

A hypothesis is selected after getting an observation using a decisionrule that optimizes some criterium.

Decision rule orDiscriminant function

d(z) =0 if z ∈ Z0 (select H0)

1 if z ∈ Z1 = Z0C(select H1){

Detection:H0 : only noise

H1 : signal (and noise){August 1, 2000 Juan Ruiz-Alzola

[email protected]

Decision TheoryThe Maximum Likelihood Detector

Choose the hypothesis that makes the obeservation more probable

pz(z/θ0) pz(z/θ1)><

H0

H1

The Likelihood Ratio Test

pz(z/θ0)

pz(z/θ1)l(z) =

H0><H1

Tpz(z/θ0)

pz(z/θ1)log l(z) = log

H0><H1

log (T)( )T = 1 <=> Maximum Likelihood Detector


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Decision TheoryThe Maximum a Posteriori Detector

The states of Nature are controlled by an underlying a priori probability lawthat is incorporated into the decision rule.

Chooses the hypothesis that maximizes the posterior PDF

pθ0 (θ0/z) p θ1(θ1/z)><

H0

H1

Applying Bayes Theorem it´s easy to formulate a Likelihood Ratio Test withT = P(θ1) / P(θ1) . It can be shown that this test minimizes the prob. of error


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Decision TheoryExtension to Multicategory Classifiers

Both ML and MAP rules are easily extended to the multicategory case justselecting the hypothesis (out of N>2) that leads the ML or the MAP.

The discriminant function partitions the observation space (usuallyvector based). The borders are easily found making equalities in the MLand MAP rules.

z1

z2

C1

C2

C3


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Decision TheoryA very brief introduction to non-parametric classsifiers

Very often we don´t know the functional form of the PDF´s. In thesecases we cannot apply a parameter estimation approach to learn themodel and we cannot apply a likelihood ratio test either.

In these cases the k-NN (k nearest neighbors) is commonly used:

z1

z2

Green: class 1

Red: class 2

Blue: Unknown

Training set (supervised)}Alternative way of finding discriminant boundaries


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Background RemovalAny dataset obtained from a physical sensor is perturbed by noise.

Two fundamental operationsFiltering: estimate the value of the signal

Detection: decide if there is signal present{Signal in noise model : I(x) = S(x) + n(x)

Detection of signal: hypothesis test applied to each voxel

H0 : voxel only has noise

H1 : voxel has signal and noise

A binary mask is constructed, setting to 1 every voxel where H1 is accepted.

Morphological operations are applied afterwards to obtain spatial coherence.

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Background RemovalDataset from Brigham & Women´s open magnet

Slice #15 Intensity histogram


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Background RemovalComparison of histograms

Slice #15

Slice #20

Slice #25

Whole volume


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Background RemovalHistogram equalization to visualize noise

Pointwise transformation Slice #15


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Background RemovalEstimation of the probabilities: model learning or training

Simple model: the conditional PDF´s are gaussian

We have to estimate the mean and the variance of each PDF

Background from image frame Foreground from image center

Sample mean:

Sample variance:

{ } ∑=

==N

i izN

zE1

1/ˆˆ θη

( ) ∑=

−−==

N

i izN

zarV1

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1/ˆ2ˆ ηθσ )


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Background RemovalConstructing the likelihood ratio test

p(z / background) = G(z; ηb,σb)

p(z / foreground) = G(z; ηf,σf)ln

G(z;ηb,σb)

G(z; ηf,σf)( )

B

>

<

F

lnP(F)

P(B)

The boundaries are easily found solving a second order equation.

Usually, only one of the solutions is physically feasible.

FBz

0


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Background RemovalBinary mask post-processing

The noise makes unavoidable the presence of False Detections.

It´s necessary a postprocessing to eliminate those detections.

1. Binary image formed from the output of the detector

2. Morphological operations enforcing consistency

2.1 Conected components analysis

2.2 Largest island selection

2.3 Holes filling


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Background RemovalSlice #15

Detector output Postprocessing


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Background Removal

Detector outputSlice #20

Postprocessing

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Background Removal

Detector outputSlice #25

Postprocessing


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Towards Automatic SegmentationExample: gray matter, white matter and lesions. Two channels.

PDw T2w

Images provided by Dr. S. Warfield, SPL, B&WH and Harvard Univ.


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Towards Automatic Segmentation

All classes White matter + lesion White matter Lesion


Overlapping of class distributions (joint PWd and T2w distribution)

There are no clear boundaries => New features must be added

Incorporation of a priori anatomic knowledge (atlas)

Feature space analysis


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Towards Automatic Segmentation


Atlas-moderated statistical classification paradigm


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Towards Automatic SegmentationSegmentation Results


kNN Atlas moderated


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Conclusions•Importance of image segmentation in medicalapplications

•Direct use of well-nown statistical techniques insimple problems as foreground segmentation.

•Need of more advanced techniques for generaltissue segmentation

•Importance of embedding anatomic knowledge inthe classifiers

•Automatic tissue segmentation: Hot research topic

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References•Pattern Classification and Scene Analysis, Duda &Hart, John Wiley & Sons, 1973

•Surgical Planning Lab web page:http:\\www.spl.harvard.edu

•Proceedings of the MICCAI conference (1998,1999,2000), Lecture Notes on Computer Science,Springer-Verlag.

Documents

Index of Contents Anatomic Structures › com3371 › week7 › ruiz.pdfAugust 1, 2000 Juan Ruiz-Alzola [email protected] 1 Statistical Classification of Anatomic Structures Juan