Likelihood Models for Template Matching Using the PDF Projection Theorem

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Likelihood Models for Template Matching Using the PDF Projection Theorem

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Likelihood Models for Template Matching Using the PDF Projection Theorem

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Likelihood Models for Template MatchingUsing the PDF Projection Theorem

Arasanathan Thayananthan

Ramanan Navaratnam

Dr. Phil Torr

Prof. Roberto Cipolla

- Problem Motivation
- PDF Projection Theorem
- Likelihood Modelling for Chamfer Matching
- Experiments
- Conclusion

- 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 ?

- 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 likelihood distributions, obtained by matching the templates to the real images they represent
- They differ according to the shape and scale of the templates.

chamfer 6.0

likelihood 0.14

likelihood 0.03

- Clutter likelihood distributions are obtained by matching the template to the background clutter

- 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

- Need a principled framework for modelling the likelihood for template matching
- Probability Distribution Function Projection Theorem ( Baggenstoss [99]) provides such a framework

- Problem Motivation
- PDF Projection Theorem
- Likelihood Modelling for Chamfer Matching
- Experiments
- Conclusion

- 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

- Neyman-Fisher factorisation states that if is a sufficient statisticfor H, p(I|H) can be factored as
- Applying Eq(1) for a hypothesis, H, and a reference Hypothesis, H0,

I

Image space, I

z

I

Image space, I

Feature space, z

z

I

Image space, I

Feature space, z

- 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

I

- Problem Motivation
- PDF Projection Theorem
- Likelihood Modelling for Chamfer Matching
- Experiments
- Conclusion

Input image

Canny edges

Template

Distance transform

- We apply PDF projection Theorem to model likelihood in a chamfer matching scheme
- Each template chooses its own subset of edge features, zj

- 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

- 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.

Example learning images for the template

For learning the feature likelihood p(zj|Hj)

For learning the reference likelihood p(zj|H0)

- Problem Motivation
- PDF Projection Theorem
- Likelihood Modelling for Chamfer Matching
- Experiments
- Conclusion

- 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.

Aim of the experiment is to compare the matching performances of

- Zj, the chamfer score obtained by matching the template Tj to the image
- P(zj|Hj), the feature likelihood of Template Tj
- P(I|Hj), the data likelihood value using the PDF projection theorem.

- 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.

- 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