Tensor Field Regularization using Normalized Convolution and Markov Random Fields in a Bayesian Framework
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.
KeywordsProbability Density Function Fractional Anisotropy Minimum Mean Square Error Simulated Annealing Algorithm Minimum Mean Square Error Estimation
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- 5.S. M. Kay. Fundamentals of statistical signal processing: estimation theory. Prentice-Hall, 1993.Google Scholar
- 6.G Kindlmann. Superquadric tensor glyphs. In IEEE TVCG/EG Symposium on Visualization, pp. 147–154, 2004.Google Scholar
- 7.H. Knutsson and C.-F. Westin. Normalized and differential convolution: Methods for interpolation and filtering of incomplete and uncertain data. In Computer Vision and Pattern Recognition, pp. 515–523, 1993.Google Scholar
- 8.M. Martin-Fernandez, C. Alberola-Lopez, J. Ruiz-Alzola, and C. F. Westin. Regularization of diffusion tensor maps using a non-gaussian markov random field approach. In Medical Image Computing and Computer-Assisted Intervention, volume 2879 of Lecture Notes in Computer Science, pp. 92–100, 2003.CrossRefGoogle Scholar
- 10.M. Martin-Fernandez, C.-F. Westin, and C. Alberola-Lopez. 3d bayesian regularization of diffusion tensor mri using multivariate gaussian markov random fields. In Medical Image Computing and Computer-Assisted Intervention, volume 3216 of Lecture Notes in Computer Science, pp. 351–359, 2004.Google Scholar
- 12.C. Pierpaoli, P. Jezzard, P. J. Basser, A. Barnett, and G. Di Chiro. Diffusion tensor MR imaging of the human brain. Radiology, 201:637, 1996.Google Scholar
- 14.C. Tomasi and R. Manduchi. Bilateral filtering for gray and color images. In International Conference on Computer Vision, pp. 839–846, 1998.Google Scholar
- 16.C.-F. Westin. A Tensor Framework for Multidimensional Signal Processing. PhD thesis, Linköping University, Sweden, 1994.Google Scholar
- 19.G. Winkler. Image Analysis, Random Fields and Markov Chain Monte Carlo Methods, Applications of Mathematics, Stochastic Modelling and Applied Probability. Springer Verlag, 2003.Google Scholar