Correcting Nonuniformities in MRI Intensities Using Entropy Minimization Based on an Elastic Model

Nonuniformity in the pixel intensity in homogeneous regions of an observed image is modeled as a multiplicative smooth bias field. The multiplicative bias field tends to increase the entropy of the original image. Thus, the entropy of the observed image is minimized to estimate the original image. The entropy minimization should be constrained such that the estimated image is close to the observed image and the estimated bias field is smooth. To enforce these constraints, the bias field is modeled as a thin–plate deforming elastically. Mathematically, the elastic deformation is described using the partial differential equation (PDE) with the body force evaluated at each pixel. In our formulation, the body force is evaluated such that the overall entropy of the image decreases. In addition, modeling the bias field as an elastic deformation ensures that the estimated image is close to the observed image and that the bias field is smooth. This provides a mathematical formulation which is simple and devoid of weighting parameters for various constraints of interest. The performance of our proposed algorithm is evaluated using both 2D and 3D simulated and real subject brain MR images.