Image Registration via Maximization of Mutual Information

1
Image Registration via Maximization of Mutual
Information

• Jianchun He
• Gary Christensen
• CEIG Seminar March 7, 2001

2
Introduction

• Image registration is a process of optimization,
  whose goal is to find the best spatial
  correspondence between images
• Different objective function results in different
  method
• Mutual information was proposed as an objective
  function in image registration in 1995
  concurrently by Viola and Wells, and by Collignon
  et al.

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Introduction

• Images can be viewed as deterministic or random
  process
• MI takes the 2nd approach and measures the
  statistical dependence between the intensity of
  corresponding voxels of the images to be
  registered
• The basis of MI method is that MI is maximized
  when the images are registered

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Method

• In information theory, the uncertainty of a
  discrete random variable A with probability mass
  function pA(a) is measured by its entropy
• H(A) -Sa pA(a) log pA(a)
• H(A) is zero when A is deterministic

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Method

• The joint entropy of two RVs A and B with joint
  pmf pAB(a,b) is
• H(A,B) -Sa,b pAB(a,b) log pAB(a,b)

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Method

• The conditional entropy of one variable given the
  other is
• H(AB) -Sa,b pAB(a,b) log pAB(ab)
• H(AB) -Sa,b pAB(a,b) log pBA(ba)
• It measures the uncertainty of one variable when
  the information about the other is given

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Method

• Mutual information measures how much information
  of one variable is contained in the other
• I(A,B) H(A) - H(AB)
• H(B) - H(BA)
• H(A) H(B) - H(A,B)

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Method

• Estimation of the joint probability is critical.
• Joint intensity histogram is always used.
• The histogram is usually binned for less
  computation (and better accuracy?).
• Marginal probabilities are obtained by summation
  over rows and columns.

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MI Study

• 1-D 16 gray level bars (image size 64x64)

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MI Study

• 2-D 5 gray level blocks (image size 50x50)

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MI Study

• 2-D MRI brain image (image size 256x256)

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MI Study

• Image registration of multi-modality is the same
  as that of one modality if 1-1 correspondence
  between the intensities exists.
• The MI function is not convex. Imperfect
  interpolation causes local extrema, especially in
  grid-aligned transformation and in low resolution
  images.
• Computation increases exponentially with the
  number of parameters in transformation.

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Using prior probability

• Likar and Pernus proposed two methods (2001) to
  smooth the MI function.
• Including the prior joint probability
• p(A, TB)l p(A,TB)(1- l)p(A,B)
• Random re-sampling--to break the grid-alignment.

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Likar and Pernuss results(1)
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Likar and Pernuss results(2)
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Optimization Step

• No analytic expression between the MI and the
  transformation parameters exists.
• Usually Powells method, which searches the
  parameter space around a given starting point, is
  used to find the optimal parameters.
• Also used is the Metropolis Algorithm.

17
Results

• Registrations of 2-D T1-weighted and Spin Density
  MRI images are studied.
• The images are randomly deformed by an affine
  transformation and then registered in both
  directions separately.

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Results -- MRI Images
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Results -- w/o Prior Probability
Image 3 to 1
Image 3 to 2
difference
Image 4 to 1
Image 4 to 2
difference
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Results