HMM-BASED HMM-BASED PATTERN DETECTIONPATTERN DETECTION

Image Processing and ReconstructionImage Processing and Reconstruction

Winter 2002Winter 2002

OutlineOutline

Markov Process Hidden Markov Models

• Elements• Basic Problems

Evaluation Optimization Training

• Implementation• 2-D HMM

Application Simulation and Results

Markov ProcessMarkov Process Can be described at any time to

be in one state among N distinct states

Its probabilistic description just requires a fixed specificationof current and previous states actual state at time t

state transition probability

Each state corresponds to a physical (observable) event

Too restrictive for sophisticated applications

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Extension to Hidden Markov Extension to Hidden Markov ModelsModels

A conditionally independent process on a Markov chain States correspond to clusters of context with similar

distribution

Elements of HMM:

• State transition probability

• The observation symbol probability in each state

• The initial state distribution

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Fundamental Problems for HMMFundamental Problems for HMM

Evaluation the probability of the observation O=O1O2…OT given the model , P(O| )

OptimizationChoosing optimal state sequence given the observation and the model .

Training

Estimating model parameters to maximize P(O| )

Evaluation the Model; Forward-Evaluation the Model; Forward-Backward AlgorithmBackward Algorithm

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Forward-Backward Procedure with order of Forward variable: Backward variable:

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Optimal States Sequence; Optimal States Sequence; Solution(s)Solution(s)

One solution: choose the states which are individually most likely.

This optimal solution has to be a valid state sequence!!

Vitterbi Algorithm: find the single best state sequence that maximizes P(Q|O,)

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Continuous Observation Continuous Observation

DistributionsDistributions In most of the applications (Speech, Image, …),

observations can not be characterized as discrete symbols from finite alphabet and should be considered by probability density function (PDF).

The most general representation of the PDF is a finite mixture of normal distributions with different means and variances for each state.

Estimating mean and variance instead of estimating bj(k)

Implementation ConsiderationsImplementation Considerations Scaling: Dynamic range of and will exceed the

precision range of any machine

Multiple observations for training

Initial Estimation of HMM Parametersfor convergence, good initial values of PDF are really helpful.

Choice of Model, Number of states, Choice of observation PDF

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Two-Dimensional HMMTwo-Dimensional HMM

Set of Markovian states within each super-state

Transition probability

Useful for segmentation

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Super-State

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Application: Pattern DetectionApplication: Pattern Detection

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SimulationsSimulations Feature Vector: DCT Coefficients or their averages over some of them

Block Size: 16*16

Both images in training set and test set have different rotation of “jinc”s, but the distance and center of them are fixed.

Running K-means Clustering Algorithm For initial estimation Comparing with template matching and Learning Vector Quantization

Distance measure for LVQ: is the computed variance of each coefficients in reference centroid

Average of Absolute value of the Coefficients

Results and Conclusion! Results and Conclusion! Detection Error