Another Bias Correction Method John Ashburner

P(yi|k,k,k) = (2k2)-1/2 exp(-(yi-k)2/(2k2))

To incorporate bias correction, extra parameters are used to model smooth intensity variations. A field modelling the variation at element i is denoted i(), where  is a vector of unknown parameters. Intensities from the ith cluster are assumed normally distributed, with mean k/i() and variance (k/i())2:

P(yi|k,k,k,)

• = (2(k /i())2)-1/2 exp {-(yi-k /i())2/(2(k/i())2)}
• =i() (2k2)-1/2 exp {-(yi-k /i())2/(2k2)}

Mixture of Gaussians Based Derivation - II

• Probability of a voxel, irrespective of intensity, belonging to kth Gaussian, given the proportion of voxels belonging to that Gaussian is P(k|k)=k .

• Therefore, the joint probability of cluster k and intensity yi is:

• P(yi,k|k,k,k,) = P(yi|k,k,k,) P(k|k)

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