1. Gaussian Mixture Model classification of Multi-Color Fluorescence In Situ Hybridization (M-FISH) Images
Amin Fazel
2006
Department of Computer Science and Electrical Engineering University of Missouri – Kansas City

2. Motivation and Goals Chromosomes store genetic information
Chromosome images can indicate genetic disease, cancer, radiation damage, etc.
Research goals:
Locate and classify each chromosome in an image
Locate chromosome abnormalities
Thursday,
June, 2006
CS and EE Department
UMKC

3. Karyotyping 46 human chromosomes form 24 types
22 different pairs
2 sex chromosomes, X and Y
Grouped and ordered by length
Banding Patterns
Karyotype
Thursday,
June, 2006
CS and EE Department
UMKC

4. Multi-spectral Chromosome Imaging
Multiplex Fluorescence In-Situ Hybridization (M-FISH) [1996]
Five color dyes (fluorophores)
Each human chromosome type absorbs a unique combination of the dyes
32 (25) possible combinations of dyes distinguish 24 human chromosome types
Healthy Male
Thursday,
June, 2006
CS and EE Department
UMKC

5. M-FISH Images 6th dye (DAPI) binds to all chromosomes
M-FISH Image 5 Dyes
DAPI Channel 6th Dye
Thursday,
June, 2006
CS and EE Department
UMKC

6. M-FISH Images
Images of each dye obtained with appropriate optical filter
Each pixel a six dimensional vector
Each vector element gives contribution of a dye at pixel
Chromosomal origin distinguishable at single pixel (unless overlapping)
Unnecessary to estimate length, relative centromere position, or banding pattern
Thursday,
June, 2006
CS and EE Department
UMKC

7. Bayesian Classification
Based on probability theory
A feature vector is denoted as
x = [x1; x2; : : : ; xD]T
D is the dimension of a vector
The probability that a feature vector x belongs to class wk is p(wk|x) and this posteriori probability can be computed via
and
Probability density function of class wk
Prior probability
Thursday,
June, 2006
CS and EE Department
UMKC

8. Gaussian Probability Density Function
In the D-dimensional space
is the mean vector
is the covariance matrix
In the Gaussian distribution lies an assumption that the class model is truly a model of one basic class
Thursday,
June, 2006
CS and EE Department
UMKC

9. Gaussian mixture model GMM
GMM is a set of several Gaussians which try to represent groups / clusters of data
therefore represent different subclasses inside one class
The PDF is defined as a weighted sum of Gaussians
Thursday,
June, 2006
CS and EE Department
UMKC

10. Gaussian Mixture Models
Equations for GMMs:
multi-dimensional case:  becomes vector ,  becomes covariance matrix .
assume  is diagonal matrix:
211 1
222
233
-1 =
Thursday,
June, 2006
CS and EE Department
UMKC

11. GMM Gaussian Mixture Model (GMM) is characterized by
the number of components,
the means and covariance matrices of the Gaussian components
the weight (height) of each component
Thursday,
June, 2006
CS and EE Department
UMKC

12. GMM GMM is the same dimension as the feature space (6-dimensional GMM)
for visualization purposes, here are 2-dimensional GMMs:
likelihood
value1
value2
value2
Thursday,
June, 2006
CS and EE Department
UMKC

13. GMM
These parameters are tuned using a iterative procedure called the Expectation Maximization (EM)
EM algorithm: recursively updates distribution of each Gaussian model and conditional probability to increase the maximum likelihood.
Thursday,
June, 2006
CS and EE Department
UMKC

14. GMM Training Flow Chart (1)
Initialize the initial Gaussian means μi using the K-means clustering algorithm
Initialize the covariance matrices to the distance to the nearest cluster
Initialize the weights 1 / C so that all Gaussian are equally likely
K-means clustering
1. Initialization: random or max. distance.
2. Search:
for each training vector, find the closest code word, assign this training vector to that cell
3. Centroid Update:
for each cell, compute centroid of that cell. The
new code word is the centroid.
4. Repeat (2)-(3) until average distance falls below threshold
Thursday,
June, 2006
CS and EE Department
UMKC

15. GMM Training Flow Chart (2)
E step: Computes the conditional expectation of the complete log-likelihood, (Evaluate the posterior probabilities that relate each cluster to each data point in the conditional probability) assuming the current cluster parameters to be correct
M step: Find the cluster parameters that maximize the likelihood of the data assuming that the current data distribution is correct.
Thursday,
June, 2006
CS and EE Department
UMKC

16. GMM Training Flow Chart (3)
recompute wn,c using the new weights, means and covariances. Stop training if
wn+1,c - wn,c < threshold
Or the number of epochs reach the specified value. Otherwise, continue the iterative updates.
Thursday,
June, 2006
CS and EE Department
UMKC

17. GMM Test Flow Chart
Present each input pattern x and compute the confidence for each class k:
Where is the prior probability of class ck estimated by counting the number of training patterns
Classify pattern x as the class with the highest confidence.
Thursday,
June, 2006
CS and EE Department
UMKC

18. Results Training Input Data Thursday, June, 2006 CS and EE Department
UMKC

19. Results True label Correctness Two Gaussian Correctness One Gaussian
Thursday,
June, 2006
CS and EE Department
UMKC

20. Thanks for your patience !
Thursday,
June, 2006
CS and EE Department
UMKC