# Another Bias Correction Method John Ashburner

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• ## Most current methods are either:

• Parametric: Bias correction is incorporated into a mixture of Gaussians type approach, possibly as a refinement of a tissue classification algorithm.
• Bias correction component is normally applied to log-transformed intensities (e.g. Wells III et al, 1996; Van Leemput et al, 1999).
• Non-parametric: Bias correction applied to histograms of intensities in order to maximise entropy.
• Most widely used approach is applied to histograms of log-transformed intensities (Sled et al, 1998).
• Another approach minimises entropy of the histogram of the original intensities, with a modification to preserve the average intensity in the image (Mangin, 2000).

• ## 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)}

• ## P(y|,,,) = i P(yi|,,,)

• = i{k i() (2k2)-1/2 exp {-(yi-k /i())2/(2k2)} }

• ## The algorithm is iterative, and involves alternating between:

• Re-estimating , while holding fixed.
• Re-estimating , while holding fixed.
• ## Re-estimating  involves building a probability density representation of the data, which have been corrected with the current bias estimate.

• Simplest case involves generating a simple histogram.
• An iterative method is needed for B-splines of degree greater than 1.
• Similar to algorithm for ML-EM reconstruction of PET images.

• ## Distinction between intensity variations due to:

• bias artefact due to MR physics
• different tissue properties.

• ## P(y,|) = P(y|,) P()

• where P() is assumed to be a d dimensional multi-normal distribution:

• ## Optimising the current objective function is equivalent to optimising the entropy of the probability distribution of log-transformed intensities.

• Entropy of log-transformed data is:
• H = -I-1i log{i() yi k k k(i()yi)} = I-1 E - I-1 i log{yi}
• where I is the number of voxels.
• ## Histograms are produced from intensities that are not log-transformed.

• Same method can be applied to images containing low or negative image intensities.
• ## Resolves a problem pointed out by Arnold et al (2001), with the SPM99 bias correction approach (Ashburner, 2000):

• The entropy of non-transformed intensity distribution is maximised when the estimated bias field is uniformly zero.
• Resulted in side effects because average bias (rather than average corrected image intensity) was constrained to one.

• ## W.M Wells III, W.E.L Grimson, R. Kikinis & F.A. Jolesz. Adaptive segmentation of MRI data. IEEE Transactions on Medical Imaging 15(4):429-442, 1996.

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