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

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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

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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

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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.

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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)
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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.
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Model description

• Observation generated by set of unique
  M-dimensional vectors each with unit norm, Ee1,
  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.
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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
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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.

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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.

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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

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Expectation maximization algorithm for SCA
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Winner-take-all SCA
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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.
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Visual comparison

• 3 classes assumed

• Results demonstrate the feasibility of the
  proposed SCA concept.

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Quantitative comparison
Percent correctly classified

• Among the tested methods, the proposed algorithms
  have the best accuracy and lowest computational
  complexity.

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Sensitivity

• 4 classes assumed

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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.

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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).