HMM-BASED PATTERN DETECTION

1. HMM-BASED PATTERN DETECTION Image Processing and Reconstruction Winter 2002

2. Outline • Markov Process • Hidden Markov Models • Elements • Basic Problems • Evaluation • Optimization • Training • Implementation • 2-D HMM • Application • Simulation and Results

3. Markov Process Can be described at any time to be in one state among N distinct states Its probabilistic description just requires a fixed specification of 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 S2 S3 S1 a31

4. Extension to Hidden Markov Models • 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

5. Fundamental Problems for HMM • Evaluation the probability of the observation O=O1O2…OT given the model , P(O| ) • Optimization Choosing optimal state sequence given the observation and the model . • Training Estimating model parameters to maximize P(O| )

6. Evaluation the Model; Forward-Backward Algorithm This calculation is on order of Forward-Backward Procedure with order of • Forward variable: • Backward variable:

7. Optimal States Sequence; 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,)

8. Training the Model

9. Continuous Observation Distributions • 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)

10. Implementation Considerations • Scaling: Dynamic range of  and  will exceed the precision range of any machine • Multiple observations for training • Initial Estimation of HMM Parameters for convergence, good initial values of PDF are really helpful. • Choice of Model, Number of states, Choice of observation PDF

11. Two-Dimensional HMM • Set of Markovian states within each super-state • Transition probability • Useful for segmentation Sub-State Si-1,j Si,j-1 Si,j Super-State

12. Application: Pattern Detection SNR=-5 SNR=10

13. Simulations • 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

14. Results and Conclusion! Detection Error