SIMILARITY-BASED CLUSTERING USING THE EXPECTATION-MAXIMIZATION (EM) ALGORITHM

1. SIMILARITY-BASED CLUSTERING USING THE EXPECTATION-MAXIMIZATION (EM) ALGORITHM Jovan G. Brankov, Nikolas P. Galatsanos, Yongyi Yang, and Miles N. Wernick Illinois Institute of Technology Research supported by Whitaker Foundation and NIH/NHLBI HL65425 www.ipl.iit.edu

2. Motivation: Noise reduction in nuclear medicine • Frames of dynamic and gated imaging studies can be noisy; • Image sequences can benefit from special reconstruction techniques that utilize spatio-temporal correlations in the signal; • In practice temporal correlation is NOT spatially stationary. • The useful information is usually non-stationary. • Increase temporal correlation by: • Motion compensation; • gated myocardial perfusion study • Identifying spatial regions with similar temporal statistics to be processed similarly. • Hemodynamic response studies www.ipl.iit.edu

3. Ongoing 4D reconstruction project • Context within the project; • Temporal Karhunen-Loeve (KL) pre-smoothing (1995) (Method Ia); • Fully 4D reconstruction for dynamic PET using KL (1997); • 4D gated SPECT reconstruction by KL (1998); • Used unsupervised clustering + KL for fine-tuning (1999)(Method Ib); • 4D gated SPECT algorithm with motion compensated post smoothing (2001); • 4D gated SPECT algorithm with motion compensated reconstruction (2002). • Method Ia was designed for motionless objects with spatially stationary statistic; • In this paper, we propose an improved unsupervised clustering algorithm to be incorporated in Method Ib. www.ipl.iit.edu

4. Method Ib : Spatially adaptive temporal filtering • Identify spatial regions in projection domain having similar temporal characteristics; • k-means unsupervised clustering algorithm • Apply different temporal KLT to each spatial region, adapting the smoothing to the local temporal behavior; • Reconstruct images from smoothed projections. k-means algorithm is NOT well suited for this task (dependent on the signal amplitude) www.ipl.iit.edu

5. Motivation: Identifying region with similar temporal behavior Time activity curves (TAC) Realistic MRI voxel-based numerical brain phantom developed by Zubal et al. [11C] Carfentanil Study JJ Frost et al.1990 I. G. Zubal, C. R. Harrell, E. O. Smith, Z. Rattner, G. R. Ginde, and P. B. Hoffer, “Computerized three-dimensional segmented human anatomy,” Med. Phys, vol. 21, pp. 299-302, 1994. www.ipl.iit.edu

6. Model description • Observation generated by set of unique M-dimensional vectors each with unit norm,E=[e1, e2,... eK],; • Our objective is to estimate the parameters of the proposed model: the class label, the prior class probabilities, and the distinct directions . Model: Yn - nth observation Xn - class label; an - is the unknown amplitude of the nth observation. www.ipl.iit.edu

7. Probability density function: Basic Idea • For the same strength of additive noise, observed direction confidence increases with signal amplitude. Y1 ,Y2- observation Noise - additive noise eX1 =eX2- unique direction www.ipl.iit.edu

8. W1 W2 A2 A1 A2>A1 => W1<W2 Probability density function • Similarity measurement defined as the cosine of the angle between two vectors; • Similarity: • We approximate a angular distribution by the following truncated exponential distribution: where SNR is a concentration parameter and is a normalizing constant. • : www.ipl.iit.edu

9. Probability density function • Why truncated exponential distribution? • If M is 2 (2D case) this is a first order approximation of phase distribution for a signal corrupted with additive Gaussian random process (Rician pdf); • It can be shown that this is the distribution of spherically warped normal distribution (Madia, 1972); • Produces better results. www.ipl.iit.edu

10. Complete data • Now we can define a mixture model that can be solved by theexpectation maximization (EM) algorithm. • Complete data uniquely defines the model parameters; • Expected log-likelihood function of complete data: where with , and www.ipl.iit.edu

11. Expectation maximization algorithm for SCA www.ipl.iit.edu

12. Winner-take-all SCA www.ipl.iit.edu

13. Unsupervised clustering methods • Traditional clustering algorithms are dependent on the signal amplitude; • Gaussian mixture models (GMM)* (special case probabilistic PCA) • k-means* • winner-take-all variant of GMM • Principal component analysis (PCA); • basis functions are orthogonal • Independent component analysis (ICA); • components are independent • Clustered component analysis (CCA)1 (Bouman et al.) partially avoids the amplitude dependency*; ( also a special case probabilistic PCA) • Newly proposed method to determine distinct time activity curves existing in an image sequence (SCA).(want to neglect multiplicative scale factors) * compared with later 1C. A. Bouman, S. Chen, and M. J. Lowe, “Clustered Component Analysis for fMRI Signals estimation and Classification,” IEEE Tran. Image Proc., vol. 1, pp. 609-612, 2000. www.ipl.iit.edu

14. Visual comparison • 3 classes assumed • Results demonstrate the feasibility of the proposed SCA concept. www.ipl.iit.edu

15. Quantitative comparison Percent correctly classified • Among the tested methods, the proposed algorithms have the best accuracy and lowest computational complexity. www.ipl.iit.edu

16. Sensitivity • 4 classes assumed www.ipl.iit.edu

17. Conclusion • Results presented here demonstrate the feasibility of the proposed SCA concept. • Among the tested methods, the proposed algorithms have the best accuracy and lowest computational complexity. www.ipl.iit.edu

18. Future efforts • Aim: • Incorporating a minimum description length (MDL) criterion to automatically estimate number of classes. • Explorr possible applications: • Automated kinetic model parameter estimation; • Temporal pre/post smoothing; • Spatio-temporal reconstruction; • Image segmentation based on color (neglecting color intensity). www.ipl.iit.edu