This process estimates a multiplicative bias field modeled by cubic B-splines. To compute splines coefficients, it tries to minimize an energy with 4 terms:
the entropy in the corrected MRI ;
the entropy in the white matter in the corrected MRI ;
the spatial regularity of the bias field ;
the mean intensity difference between the original and corrected images.
Description
The bias field is modeled by cubic B-splines. This means that it is parameterized by a few values regularely spaced in the image referential and is computed in any point by cubic interpolation (Figure 1).
Figure 1. Regularly spaced coefficients (left) and interpolated bias field (right). (Illustration from Wikipedia)
The four terms of the energy function are modulated by four coefficients (Figure 2):
Ke (coeff_entropy) modulates the entropy in the entire corrected image.
Kc (coeff_wm_entropy) modulates the entropy in a mask of points believed to belong to the white matter. These points are those of high curvature in the intensity space.
Kr (coeff_rigidity) modulates the regularity of the bias field. It pushes neighbouring coefficients to have similar values.
Ko (coeff_power_regul) modulates the global "power" of the bias field. The energy function is defined as the sum of square differences between intensities in the original and corrected images. Another way to see it is that it forces the mean bias value to be close to 1.
Figure 2. Coefficients modulating the minimized energy.
Troubleshooting
When failing, this step can cause havoc in the rest of the pipeline. A common issue is to find unusually high or low bias values in one part of the image when white matter points are badly selected. This would cause the apparition of outlier values in the bias-corrected image which would hamper the skull-stripping step (the k-means algorithm is particularly sensitive to outliers). If this problem arises, don't hesitate to give a bigger value to the energy regulating terms, coeff_rigidity and, especially, coeff_power_regul. Values as high as 100 can be used.
References
The princeps article was published by Mangin (2000), tough without the white matter entropy term.
J.-F. Mangin, "Entropy minimization for automatic correction of intensity nonuniformity," in Proceedings of the IEEE Workshop on Mathematical Methods in Biomedical Image Analysis, 2000, pp. 162-169. DOI: 10.1109/MMBIA.2000.852374
Commissure points. If provided, they are used to mask voxels too far below the brain (jaw, neck...), so that they don't impact the bias field estimation.
specie: OpenChoice ( optional, input )
Some parameters are automatically set at an appropriate value if the specie is provided.
contrast: OpenChoice ( optional, input )
The contrast (T1 or T2) must be provided. Indeed, white matter points are selected as points of high curvature, but we will select positive or negative curvature depending on the contrast.
grid_resolution_mm: Float ( optional, input )
Spacing between field coefficients in mm.
n_grid_pyramid: Integer ( optional, input )
Number of pyramid levels used for bias estimation.
coeff_entropy: Float ( optional, input )
Coefficient modulating the entropy term (Ke).
coeff_wm_entropy: Float ( optional, input )
Coefficient modulating the white matter entropy term (Kc).
coeff_rigidity: Float ( optional, input )
Coefficient modulating the field rigidity term (Kr).
coeff_power_regul: Float ( optional, input )
Coefficient modulating the field power term (i.e., it keeps the field centered around 1) (Ko).
VIP Bias Correction
