Another Bias Correction Method John Ashburner

Re-estimating  makes use of the histogram and its gradient w.r.t. intensity - therefore zeroeth degree splines can not be used.

Regularisation + Optimisation

• Distinction between intensity variations due to:

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

• Don’t want to over-fit the data, so regularisation is used.

• Maximise the joint probability of y and , assuming independence of and :

• P(y,|) = P(y|,) P()

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

• P() = N (0,C0) = ((2)d|C0|)-1/2 exp{ -1/2 (-0)TC0 -1 (-0)}

• A Newton-Raphson optimisation scheme can be derived from this:

• First and second derivatives of E can be efficiently computed by parameterising the bias field in terms of separable basis functions.

A Quick Evaluation

Discussion

• 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.

• Accuracy of the results depends on form and magnitude of regularisation.

References

• J.B. Arnold, J.S. Liow, K.A. Schaper, J.J. Stern, J.G. Sled, D.W. Shattuck, A.J. Worth, M.S. Cohen, R.M. Leahy, J.C. Mazziotta & D.A. Rottenberg. Qualitative and quantitative evaluation of six algorithms for correcting intensity nonuniformity effect. NeuroImage 3(15):931-943, 2001.

• J. Ashburner & K.J. Friston. Voxel-based morphometry - the methods. NeuroImage 11:805-821, 2000.

• R.K.-S. Kwan, A.C. Evans & G.B. Pike. An Extensible MRI Simulator for Post-Processing Evaluation. Visualization in Biomedical Computing (VBC'96). Lecture Notes in Computer Science, vol. 1131. Springer-Verlag, 1996. 135-140

• J.-F. Mangin. Entropy minimisation for automatic correction of intensity nonuniformity. In Proc. IEEE Workshop on Mathematical Methods in Biomedical Image Analysis, 2000.

• J.G. Sled, A.P. Zijdenbos & A.C. Evans. A non-parametric method for automatic correction of intensity nonuniformity in MRI data. IEEE Transactions on Medical Imaging 17(1):87-97, 1998.

• K. Van Leemput, F. Maes, D. Vandermeulen & P. Seutens. Automated model-based bias field correction of MR images of the brain. IEEE Transactions on Medical Imaging 18(10):885-896, 1999.

• 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.