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Likelihood Models for Template MatchingUsing the PDF Projection Theorem
Arasanathan Thayananthan
Ramanan Navaratnam
Dr. Phil Torr
Prof. Roberto Cipolla
ProblemProblem
The correct template
The minimum chamfer score
Chamfer score 3.49 3.07
OverviewOverview
1. Problem Motivation
2. PDF Projection Theorem
3. Likelihood Modelling for Chamfer Matching
4. Experiments
5. Conclusion
MotivationMotivation
Template matching widely used in computer vision
Similarity measures are obtained from matching a template to a new image e.g. chamfer score, cross-correlation, etc.
A likelihood value need to be calculated from the similarity measures.
Chamfer score 3.58
Likelihood ?
MotivationMotivation
Is the similarity measure alone enough to calculate the likelihood ?
What are the probabilities of matching to a correct image and an incorrect image at this specific matching measure ?
Feature LikelihoodFeature Likelihood
Feature likelihood distributions, obtained by matching the templates to the real images they represent
They differ according to the shape and scale of the templates.
Feature LikelihoodsFeature Likelihoods
chamfer 6.0
likelihood 0.14 likelihood 0.03
Clutter LikelihoodsClutter Likelihoods
Clutter likelihood distributions are obtained by matching the template to the background clutter
Likelihood RatiosLikelihood Ratios
The ratio of the feature and clutter likelihood provides a robust likelihood measure.
Likelihood Ratio Tests (LRT) are often used in many classification problems
Jones & Ray [99], skin-colour classification
Sidenbladh & Black [01], limb-detector
Modelling the likelihoodModelling the likelihood
Need a principled framework for modelling the likelihood for template matching
Probability Distribution Function Projection Theorem ( Baggenstoss [99]) provides such a framework
OverviewOverview
1. Problem Motivation
2. PDF Projection Theorem
3. Likelihood Modelling for Chamfer Matching
4. Experiments
5. Conclusion
PDF Projection TheoremPDF Projection Theorem
Provides a mechanism to work in raw data space, I, instead of extracted feature space, z.
This is done by projecting the PDF estimates from the feature space back to the raw data space
PDF Projection TheoremPDF Projection Theorem
Neyman-Fisher factorisation states that if is a sufficient statistic for H, p(I|H) can be factored as
Applying Eq(1) for a hypothesis, H, and a reference Hypothesis, H0,
PDF Projection TheoremPDF Projection Theorem
Image space, I
I
PDF Projection TheoremPDF Projection Theorem
Image space, I Feature space, z
I z
PDF Projection TheoremPDF Projection Theorem
Image space, I Feature space, z
I z
Class-specific featuresClass-specific features
PDF Projection Theorem extends to class-specific features
Each hypothesis or class can have its feature set
Yet, we get consistent and comparable raw image likelihoods
Reference hypothesis H0 remains the same for all hypothesis
Class-specific featuresClass-specific features
I
OverviewOverview
1. Problem Motivation
2. PDF Projection Theorem
3. Likelihood Modelling for Chamfer Matching
4. Experiments
5. Conclusion
Chamfer MatchingChamfer Matching
Input image Canny edges
Distance transform Template
Chamfer MatchingChamfer Matching
We apply PDF projection Theorem to model likelihood in a chamfer matching scheme
Each template chooses its own subset of edge features, zj
Chamfer MatchingChamfer Matching
A common reference hypothesis is chosen for all templates
p(zj|H0) provides the probability of template matching to any image.
Difficulty is in learning p(zj|Hj) and p(zj|H0) for each template Tj
Learning the PDFsLearning the PDFs
Time-consuming to obtain real images for learning the PDFs
Software like “Poser” can create “near” real images
Becoming popular for learning image statistics e.g. Shakhnarovich [03]
For each template Tj, we learn p(zj|Hj) and p(zj|H0) from synthetic images.
Learning the PDFsLearning the PDFs
Example learning images for the template
For learning the feature likelihood p(zj|Hj)
For learning the reference likelihood p(zj|H0)
OverviewOverview
1. Problem Motivation
2. PDF Projection Theorem
3. Likelihood Modelling for Chamfer Matching
4. Experiments
5. Conclusion
ExperimentsExperiments
35 hand templates from a 3D hand model with 5 gestures at 7 different scales
Hypothesis, Hj, is that the image contains a hand pose similar to Template Tj, (in scale and gesture).
The distributions p(zj|Hj) and p(zj|H0) were learned off-line for each template.
ExperimentsExperiments
Aim of the experiment is to compare the matching performances of1. Zj, the chamfer score obtained by matching
the template Tj to the image
2. P(zj|Hj), the feature likelihood of Template Tj
3. P(I|Hj), the data likelihood value using the PDF projection theorem.
ExperimentsExperiments
Template matching on 1000 randomly created synthetic images.
Each synthetic image contains a hand pose similar in scale and pose to a randomly chosen template.
Three ROC curves were obtained for each matching measure.
ResultsResults
ResultsResults
PDF ProjectionTheorem
Chamfer
Chamfer score 4.96 4.06
feature likelihood 14.59 x 10-2 8.62 x 10-2
reference likelihood 88.69 x 10-5 383.32 x 10-5
data likelihood 0.164 x 103 0.022 x 103
ResultsResults
PDF ProjectionTheorem
Chamfer
Chamfer score 4.96 4.06
feature likelihood 14.59 x 10-2 8.62 x 10-2
reference likelihood 88.69 x 10-5 383.32 x 10-5
data likelihood 0.164 x 103 0.022 x 103
ResultsResults
PDF ProjectionTheorem
Chamfer
Chamfer score 4.96 4.06
feature likelihood 14.59 x 10-2 8.62 x 10-2
reference likelihood 88.69 x 10-5 383.32 x 10-5
data likelihood 0.164 x 103 0.022 x 103
ResultsResults
PDF ProjectionTheorem
Chamfer
Chamfer score 4.96 4.06
feature likelihood 14.59 x 10-2 8.62 x 10-2
reference likelihood 88.69 x 10-5 383.32 x 10-5
data likelihood 0.164 x 103 0.022 x 103
ResultsResults
PDF ProjectionTheorem
Chamfer
Chamfer score 3.49 3.07
feature likelihood 24.94x 10-2 27.88 x 10-2
reference likelihood 4.73 x 10-5 24.7 x 10-5
data likelihood 5.27 x 103 1.126 x 103
ResultsResults
PDF ProjectionTheorem
Chamfer
Chamfer score 3.49 3.07
feature likelihood 24.94x 10-2 27.88 x 10-2
reference likelihood 4.73 x 10-5 24.7 x 10-5
data likelihood 5.27 x 103 1.126 x 103
ResultsResults
PDF ProjectionTheorem
Chamfer
Chamfer score 3.49 3.07
feature likelihood 24.94x 10-2 27.88 x 10-2
reference likelihood 4.73 x 10-5 24.7 x 10-5
data likelihood 5.27 x 103 1.126 x 103
ResultsResults
PDF ProjectionTheorem
Chamfer
Chamfer score 3.49 3.07
feature likelihood 24.94x 10-2 27.88 x 10-2
reference likelihood 4.73 x 10-5 24.7 x 10-5
data likelihood 5.27 x 103 1.126 x 103
ResultsResults
PDF ProjectionTheorem
Chamfer
Chamfer score 3.72 3.54
feature likelihood 13.15 x 10-2 20.5 x 10-2
reference likelihood 8.5 x 10-5 108.0 x 10-5
data likelihood 1.547 x 103 0.191 x 103
ConclusionConclusion
Depending on raw matching score is less reliable in template matching
PDF Projection theorem provides a principled framework for modelling the likelihood in raw image data space.
Consistent and comparable likelihoods obtained through PDF projection theorem improves the efficiency of template matching scheme