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Multiple Instance Hidden Markov Model: Application to Landmine Detection in GPR Data

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### Multiple Instance Hidden Markov Model: Application to Landmine Detection in GPR Data

Jeremy Bolton, SenihaYuksel, Paul Gader

CSI Laboratory

University of Florida

Highlights

- Hidden Markov Models (HMMs) are useful tools for landmine detection in GPR imagery
- Explicitly incorporating the Multiple Instance Learning (MIL) paradigm in HMM learning is intuitive and effective
- Classification performance is improved when using the MI-HMM over a standard HMM
- Results further support the idea that explicitly accounting for the MI scenario may lead to improved learning under class label uncertainty

Outline

- HMMs for Landmine detection in GPR
- Data
- Feature Extraction
- Training
- MIL Scenario
- MI-HMM
- Classification Results

GPR Data

- GPR data
- 3d image cube
- Dt, xt, depth
- Subsurface objects are observed as hyperbolas

GPR Data Feature Extraction

- Many features extracted from in GPR data measure the occurrence of an “edge”
- For the typical HMM algorithm (Gaderet al.),
- Preprocessing techniques are used to emphasize edges
- Image morphology and structuring elements can be used to extract edges

Image

Preprocessed

Edge Extraction

4-d Edge Features

Edge Extraction

Concept behind the HMM for GPR

- Using the extracted features (an observation sequence when scanning from left to right in an image) we will attempt to estimate some hidden states

Sampling HMM Summary

- Feature Calculation
- Dimensions (Not always relevant whether positive or negative diagonal is observed …. Just simply a diagonal is observed)
- HMMSamp: 2d
- Down sampling depth
- HMMSamp: 4
- HMM Models
- Number of States
- HMMSamp : 4
- Gaussian components per state (Fewer total components for probability calculation)
- HMMSamp : 1 (recent observation)

Training the HMM

- Xuping Zhang proposed a Gibbs Sampling algorithm for HMM learning
- But, given an image(s) how do we choose the training sequences?
- Which sequence(s) do we choose from each image?
- There is an inherent problem in many image analysis settings due to class label uncertainty per sequence
- That is, each image has a class label associated with it, but each image has multiple instances of samples or sequences. Which sample(s) is truly indicative of the target?
- Using standard training techniques this translates to identifying the optimal training set within a set of sequences
- If an image has N sequences this translates to a search of 2N possibilities

Training Sample Selection Heuristic

- Currently, an MRF approach (Collins et al.) is used to bound the search to a localized area within the image rather than search all sequences within the image.
- Reduces search space, but multiple instance problem still exists

Standard Learning vs. Multiple Instance Learning

- Standard supervised learning
- Optimize some model (or learn a target concept) given training samples and corresponding labels
- MIL
- Learn a target concept given multiplesets of samples and corresponding labels for the sets.
- Interpretation: Learning with uncertain labels / noisy teacher

Multiple Instance Learning (MIL)

- Given:
- Set of I bags
- Labeled + or -
- The ith bag is a set of Ji

samples in some feature space

- Interpretation of labels
- Goal: learn concept
- What characteristic is common to the positive bags that is not observed in the negative bags

Standard learning doesn’t always fit: GPR Example

- Standard Learning
- Each training sample (feature vector) must have a label
- But which ones and how many compose the optimal training set?
- Arduous task: many feature vectors per image and multiple images
- Difficult to label given GPR echoes, ground truthing errors, etc …
- Label of each vector may not be known

EHD: Feature Vector

Learning from Bags

- In MIL, a label is attached to a set of samples.
- A bag is a set of samples
- A sample within a bag is called an instance.
- A bag is labeled as positive if and only if at least one of its instances is positive.

POSITIVE BAGS

(Each bag is an image)

NEGATIVE BAGS

(Each bag is an image)

MI Learning: GPR Example

- Multiple Instance Learning
- Each training bag must have a label
- No need to label all feature vectors, just identify images (bags) where targets are present
- Implicitly accounts for class label uncertainty …

MI-HMM

- Assuming independence between the bags and assuming the Noisy-OR (Pearl) relationship between the sequences within each bag
- where

MI-HMM learning

- Due to the cumbersome nature of the noisy-OR, the parameters of the HMM are learned using Metropolis – Hastings sampling.

Sampling

- HMM parameters are sampled from Dirichlet
- A new state is accepted or rejected based on the ratio rat iteration t + 1
- where P is the noisy-or model.

Discrete Observations

- Note that since we have chosen a Metropolis Hastings sampling scheme using Dirichlets, our observations must be discretized.

MI-HMM Summary

- Feature Calculation
- Dimensions
- HMMSamp: 2d
- MI-HMM: 2d features are descretized into 16 symbols
- Down sampling depth
- HMMSamp: 4
- MI-HMM: 4
- HMM Models
- Number of States
- HMMSamp : 4
- MI-HMM: 4
- Components per state (Fewer total components for probability calculation)
- HMMSamp : 1 Gaussian
- MI-HMM: Discrete mixture over 16 symbols

Concluding Remarks

- Explicitly incorporating the Multiple Instance Learning (MIL) paradigm in HMM learning is intuitive and effective
- Classification performance is improved when using the MI-HMM over a standard HMM
- More effective and efficient
- Future Work
- Construct bags without using MRF heuristic
- Apply to EMI data: spatial uncertainty

Standard Learning vs. Multiple Instance Learning

- Standard supervised learning
- Optimize some model (or learn a target concept) given training samples and corresponding labels
- MIL
- Learn a target concept given multiplesets of samples and corresponding labels for the sets.
- Interpretation: Learning with uncertain labels / noisy teacher

Multiple Instance Learning (MIL)

- Given:
- Set of I bags
- Labeled + or -
- The ith bag is a set of Ji

samples in some feature space

- Interpretation of labels
- Goal: learn concept
- What characteristic is common to the positive bags that is not observed in the negative bags

MIL Application: Example GPR

- Collaboration: Frigui, Collins, Torrione
- Construction of bags
- Collect 15 EHD feature vectors from the 15 depth bins
- Mine images = + bags
- FA images = - bags

Standard vs. MI Learning: GPR Example

- Standard Learning
- Each training sample (feature vector) must have a label
- Arduous task
- many feature vectors per image and multiple images
- difficult to label given GPR echoes, ground truthing errors, etc …
- label of each vector may not be known

EHD: Feature Vector

Standard vs MI Learning: GPR Example

- Multiple Instance Learning
- Each training bag must have a label
- No need to label all feature vectors, just identify images (bags) where targets are present
- Implicitly accounts for class label uncertainty …

Random Set Brief

- Random Set

element is NOT the

target concept

How can we use Random Sets for MIL?- Random set for MIL: Bags are sets
- Idea of finding commonality of positive bags inherent in random set formulation
- Sets have an empty intersection or non-empty intersection relationship
- Find commonality using intersection operator
- Random sets governing functional is based on intersection operator
- Capacity functional : T

A.K.A. : Noisy-OR gate (Pearl 1988)

Random Set Functionals

- Capacity functionals for intersection calculation
- Use germ and grain model to model random set
- Multiple (J) Concepts
- Calculate probability of intersection given X and germ and grain pairs:
- Grains are governed by random radii with assumed cumulative:

Random Set model parameters

Germ

Grain

T

x

T

T

x

x

x

x

T

T

x

x

x

RSF-MIL: Germ and Grain Model- Positive Bags = blue
- Negative Bags = orange
- Distinct shapes = distinct bags

Multiple Concepts: Disjunction or Conjunction?

- Disjunction
- When you have multiple types of concepts
- When each instance can indicate the presence of a target
- Conjunction
- When you have a target type that is composed of multiple (necessary concepts)
- When each instance can indicate a concept, but not necessary the composite target type

Conjunctive RSF-MIL

- Previously Developed Disjunctive RSF-MIL (RSF-MIL-d)
- Conjunctive RSF-MIL (RSF-MIL-c)

Noisy-OR combination across concepts and samples

Standard noisy-OR for one concept j

Noisy-AND combination across concepts

Synthetic Data Experiments

- Extreme Conjunct data set requires that a target bag exhibits two distinct concepts rather than one or none

AUC (AUC when initialized near solution)

Disjunctive Target Concepts

- Using Large overlapping bins (GROSS Extraction) the target concept can be encapsulated within 1 instance: Therefore a disjunctive relationship exists

Target Concept

Type 1

NoisyOR

Target Concept

Type 2

NoisyOR

OR

…

Target Concept

Type n

NoisyOR

Target Concept Present?

What if we want features with finer granularity

- Fine Extraction
- More detail about image and more shape information, but may loose disjunctive nature between (multiple) instances

Constituent Concept 1

(top of hyperbola)

NoisyOR

Target Concept Present?

AND

…

Constituent Concept 2

(wings of hyperbola)

NoisyOR

Our features have more granularity, therefore our concepts may be constituents of a target, rather than encapsulating the target concept

GPR Experiments

- Extensive GPR Data set
- ~800 targets
- ~ 5,000 non-targets
- Experimental Design
- Run RSF-MIL-d (disjunctive) and RSF-MIL-c (conjunctive)
- Compare both feature extraction methods
- Gross extraction: large enough to encompass target concept
- Fine extraction: Non-overlapping bins
- Hypothesis
- RSF-MIL will perform well when using gross extraction whereas RSF-MIL-c will perform well using Fine extraction

Experimental Results

- Highlights
- RSF-MIL-d using gross extraction performed best
- RSF-MIL-c performed better than RSF-MIL-d when using fine extraction
- Other influencing factors: optimization methods for RSF-MIL-d and RSF-MIL-c are not the same

Gross Extraction

Fine Extraction

Future Work

- Implement a general form that can learn disjunction or conjunction relationship from the data
- Implement a general form that can learn the number of concepts
- Incorporate spatial information
- Develop an improved optimization scheme for RSF-MIL-C

HMM Model Visualization

Points =

Gaussian Component means

DTXTHMM

Falling

Diagonal

Color =

State Index

State index1

State index 2

State index 3

Rising Diagonal

Transition probabilitiesfrom state to state (red = high probability)

Initial probabilities

Pattern Characterized

MIL Example (AHI Imagery)

- Robust learning tool
- MIL tools can learn target signature with limited or incomplete ground truth

Which spectral signature(s) should we use to train a target model or classifier?

Spectral mixing

Background signal

Ground truth not exact

MI-RVM

- Addition of set observations and inference using noisy-OR to an RVM model
- Prior on the weight w

SVM review

- Classifier structure
- Optimization

MI-SVM Discussion

- RVM was altered to fit MIL problem by changing the form of the target variable’s posterior to model a noisy-OR gate.
- SVM can be altered to fit the MIL problem by changing how the margin is calculated
- Boost the margin between the bag (rather than samples) and decision surface
- Look for the MI separating linear discriminant
- There is at least one sample from each bag in the half space

mi-SVM

- Enforce MI scenario using extra constraints

At least one sample in each positive bag must have a label of 1.

Mixed integer program: Must find optimal hyperplane and optimal labeling set

All samples in each negative bag must have a label of -1.

Current Applications

- Multiple Instance Learning
- MI Problem
- MI Applications
- Multiple Instance Learning: Kernel Machines
- MI-RVM
- MI-SVM
- Current Applications
- GPR imagery
- HSI imagery

HSI: Target Spectra Learning

- Given labeled areas of interest: learn target signature
- Given test areas of interest: classify set of samples

Overview of MI-RVM Optimization

- Two step optimization
- Estimate optimal w, given posterior of w
- There is no closed form solution for the parameters of the posterior, so a gradient update method is used
- Iterate until convergence. Then proceed to step 2.
- Update parameter on prior of w
- The distribution on the target variable has no specific parameters.
- Until system convergence, continue at step 1.

1) Optimization of w

- Optimize posterior (Bayes’ Rule) of w
- Update weights using Newton-Raphsonmethod

2) Optimization of Prior

- Optimization of covariance of prior
- Making a large number of assumptions, diagonal elements of A can be estimated

Random Sets: Multiple Instance Learning

- Random set framework for multiple instance learning
- Bags are sets
- Idea of finding commonality of positive bags inherent in random set formulation
- Find commonality using intersection operator
- Random sets governing functional is based on intersection operator

MI issues

- MIL approaches
- Some approaches are biased to believe only one sample in each bag caused the target concept
- Some approaches can only label bags
- It is not clear whether anything is gained over supervised approaches

Side Note: Bayesian Networks

- Noisy-OR Assumption
- Bayesian Network representation of Noisy-OR
- Polytree: singly connected DAG

Side Note

- Full Bayesian network may be intractable
- Occurrence of causal factors are rare (sparse co-occurrence)
- So assume polytree
- So assume result has boolean relationship with causal factors
- Absorb I, X and A into one node, governed by randomness of I
- These assumptions greatly simplify inference calculation
- Calculate Z based on probabilities rather than constructing a distribution using X

Diverse Density (DD)

- Probabilistic Approach
- Goal:
- Standard statistics approaches identify areas in a feature space with high density of target samples and low density of non-target samples
- DD: identify areas in a feature space with a high “density” of samples from EACH of the postitive bags (“diverse”), and low density of samples from negative bags.
- Identify attributes or characteristics similar to positive bags, dissimilar with negative bags
- Assume t is a target characterization
- Goal:
- Assuming the bags are conditionally independent

Random Set Brief

- Random Set

Random Set Functionals

- Capacity and avoidance functionals
- Given a germ and grain model
- Assumed random radii

When disjunction makes sense

- Using Large overlapping bins the target concept can be encapsulated within 1 instance: Therefore a disjunctive relationship exists

Target Concept Present

OR

Theoretical and Developmental Progress

- Previous Optimization:
- Did not necessarily promote

diverse density

- Current optimization
- Better for context learning and MIL
- Previously no feature relevance or selection (hypersphere)
- Improvement: included learned weights on each feature dimension

- Previous TO DO list
- Improve Existing Code
- Develop joint optimization for context learning and MIL
- Apply MIL approaches (broad scale)
- Learn similarities between feature sets of mines
- Aid in training existing algos: find “best” EHD features for training / testing
- Construct set-based classifiers?

element is NOT the

target concept

How do we impose the MI scenario?: Diverse Density (Maronet al.)- Calculation (Noisy-OR Model):
- Inherent in Random Set formulation
- Optimization
- Combo of exhaustive search and gradient ascent

How can we use Random Sets for MIL?

- Random set for MIL: Bags are sets
- Idea of finding commonality of positive bags inherent in random set formulation
- Sets have an empty intersection or non-empty intersection relationship
- Find commonality using intersection operator
- Random sets governing functional is based on intersection operator
- Example:

Bags with target

{l,a,e,i,o,p,u,f}

{f,b,a,e,i,z,o,u}

{a,b,c,i,o,u,e,p,f}

{a,f,t,e,i,u,o,d,v}

Bags without target

{s,r,n,m,p,l}

{z,s,w,t,g,n,c}

{f,p,k,r}

{q,x,z,c,v}

{p,l,f}

intersection

union

Target concept =

\

{a,e,i,o,u,f}

{f,s,r,n,m,p,l,z,w,g,n,c,v,q,k}

= {a,e,i,o,u}

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