Tensor Field Regularization using Normalized Convolution and Markov Random Fields in a Bayesian Framework

  • Carl-Fredrik Westin
  • Marcos Martin-Fernandez
  • Carlos Alberola-Lopez
  • Juan Ruiz-Alzola
  • Hans Knutsson
Part of the Mathematics and Visualization book series (MATHVISUAL)


This chapter presents two techniques for regularization of tensor fields. We first present a nonlinear filtering technique based on normalized convolution, a general method for filtering missing and uncertain data. We describe how the signal certainty function can be constructed to depend on locally derived certainty information and further combined with a spatially dependent certainty field. This results in reduced mixing between regions of different signal characteristics, and increased robustness to outliers, compared to the standard approach of normalized convolution using only a spatial certainty field. We contrast this deterministic approach with a stochastic technique based on a multivariate Gaussian signal model in a Bayesian framework. This method uses a Markov random field approach with a 3D neighborhood system for modeling spatial interactions between the tensors locally. Experiments both on synthetic and real data are presented. The driving tensor application for this work throughout the chapter is the filtering of diffusion tensor MRI data.


Probability Density Function Fractional Anisotropy Minimum Mean Square Error Simulated Annealing Algorithm Minimum Mean Square Error Estimation 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Carl-Fredrik Westin
    • 1
  • Marcos Martin-Fernandez
    • 1
  • Carlos Alberola-Lopez
    • 1
  • Juan Ruiz-Alzola
    • 1
  • Hans Knutsson
    • 1
  1. 1.Laboratory of Mathematics in Imaging, Brigham and Women’s HospitalHarvard Medical SchoolBostonUSA

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